Subgroups — C₆₇×#59457 #30738708

Order 17152.

1980 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2780	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2580	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2566	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2566	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2538	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2454	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2452	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2450	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2366	C₂	1	1	yes	maximal, characteristic
12	C₆₇×#2366	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2349	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2349	C₂	1	2	yes	maximal, complemented
15	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
17	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	#400×C₆₇	C₂²	1	4	yes	—
32	#399×C₆₇	C₂²	1	4	yes	—
33	#398×C₆₇	C₂²	1	4	yes	—
34	#396×C₆₇	C₂²	1	4	yes	—
35	#396×C₆₇	C₂²	1	4	yes	—
36	#396×C₆₇	C₂²	1	4	yes	—
37	#396×C₆₇	C₂²	1	4	yes	—
38	#393×C₆₇	C₂²	1	4	yes	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#390×C₆₇	C₂²	1	2	yes	—
42	#390×C₆₇	C₂²	1	2	yes	—
43	#390×C₆₇	C₂²	1	2	yes	—
44	#390×C₆₇	C₂²	1	2	yes	—
45	#390×C₆₇	C₂²	1	2	yes	—
46	#390×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	1	yes	characteristic
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	2	yes	—
51	#389×C₆₇	C₂²	1	2	yes	—
52	#389×C₆₇	C₂²	1	1	yes	characteristic
53	#389×C₆₇	C₂²	1	1	yes	characteristic
54	#389×C₆₇	C₂²	1	1	yes	characteristic
55	#389×C₆₇	C₂²	1	2	yes	—
56	#387×C₆₇	C₂²	1	2	yes	—
57	#387×C₆₇	C₂²	1	2	yes	—
58	#386×C₆₇	C₂²	1	4	yes	—
59	#386×C₆₇	C₂²	1	4	yes	—
60	#385×C₆₇	C₂²	1	4	yes	—
61	#385×C₆₇	C₂²	1	4	yes	—
62	#385×C₆₇	C₂²	1	4	yes	—
63	#385×C₆₇	C₂²	1	4	yes	—
64	#385×C₆₇	C₂²	1	4	yes	—
65	#385×C₆₇	C₂²	1	4	yes	—
66	#384×C₆₇	C₂²	1	2	yes	—
67	#384×C₆₇	C₂²	1	1	yes	characteristic
68	#384×C₆₇	C₂²	1	1	yes	characteristic
69	#382×C₆₇	C₂²	1	1	yes	characteristic
70	#382×C₆₇	C₂²	1	2	yes	—
71	#382×C₆₇	C₂²	1	2	yes	—
72	#380×C₆₇	C₂²	1	4	yes	—
73	#380×C₆₇	C₂²	1	4	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
81	C₂×#95×C₆₇	C₂²	1	4	yes	—
82	C₄³×C₆₇	C₂²	1	1	yes	characteristic
83	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
85	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
88	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
89	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
90	C₄⋊Q₈×C₆₇	—	2	4	no	—
91	#127×C₆₇	Q₈	1	8	yes	complemented
92	#125×C₆₇	—	2	4	no	—
93	#125×C₆₇	—	4	4	no	—
94	C₂²⋊Q₈×C₆₇	—	4	8	no	—
95	C₄×Q₈×C₆₇	—	4	4	no	—
96	C₄×D₄×C₆₇	—	2	8	no	—
97	C₄×D₄×C₆₇	—	2	8	no	—
98	C₄×D₄×C₆₇	—	4	4	no	—
99	C₄×D₄×C₆₇	—	4	4	no	—
100	C₄×D₄×C₆₇	—	4	4	no	—
101	C₄×D₄×C₆₇	—	4	4	no	—
102	C₄×D₄×C₆₇	—	4	4	no	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
131	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
133	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
134	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
135	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
140	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
141	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
143	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
147	#95×C₆₇	C₂³	1	2	yes	—
148	#95×C₆₇	C₂³	1	2	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	#95×C₆₇	C₂³	1	4	yes	—
151	#95×C₆₇	C₂³	1	2	yes	—
152	#95×C₆₇	C₂³	1	2	yes	—
153	#95×C₆₇	C₂³	1	2	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	2	yes	—
156	#95×C₆₇	C₂³	1	2	yes	—
157	#95×C₆₇	C₂³	1	2	yes	—
158	#95×C₆₇	C₂³	1	2	yes	—
159	#95×C₆₇	C₂³	1	4	yes	—
160	#95×C₆₇	C₂³	1	4	yes	—
161	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
162	C₂×Q₈×C₆₇	—	2	4	no	—
163	C₂×Q₈×C₆₇	—	2	4	no	—
164	C₂×Q₈×C₆₇	—	4	2	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	4	no	—
167	C₂×D₄×C₆₇	—	2	4	no	—
168	C₂×D₄×C₆₇	—	2	4	no	—
169	C₂×D₄×C₆₇	—	4	2	no	—
170	C₂×D₄×C₆₇	—	4	2	no	—
171	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	—
172	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	—
173	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	—
174	C₂²×C₄×C₆₇	—	2	8	no	—
175	C₂²×C₄×C₆₇	—	2	8	no	—
176	C₂²×C₄×C₆₇	—	4	4	no	—
177	C₂²×C₄×C₆₇	—	4	4	no	—
178	C₂²×C₄×C₆₇	—	4	4	no	—
179	C₂²×C₄×C₆₇	—	4	4	no	—
180	C₂²×C₄×C₆₇	—	4	4	no	—
181	C₂²×C₄×C₆₇	—	4	4	no	—
182	C₂²×C₄×C₆₇	—	4	4	no	—
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
198	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	4	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	2	no	—
203	C₄⋊C₄×C₆₇	—	2	2	no	—
204	C₄⋊C₄×C₆₇	—	2	4	no	—
205	C₄⋊C₄×C₆₇	—	2	2	no	—
206	C₄⋊C₄×C₆₇	—	2	2	no	—
207	C₄⋊C₄×C₆₇	—	2	2	no	—
208	C₄⋊C₄×C₆₇	—	2	2	no	—
209	C₄⋊C₄×C₆₇	—	2	4	no	—
210	C₄⋊C₄×C₆₇	—	2	4	no	—
211	C₄⋊C₄×C₆₇	—	2	2	no	—
212	C₄⋊C₄×C₆₇	—	4	2	no	—
213	C₄⋊C₄×C₆₇	—	4	2	no	—
214	C₄⋊C₄×C₆₇	—	4	2	no	—
215	C₄⋊C₄×C₆₇	—	4	2	no	—
216	C₄⋊C₄×C₆₇	—	4	2	no	—
217	C₄⋊C₄×C₆₇	—	4	2	no	—
218	C₄⋊C₄×C₆₇	—	4	4	no	—
219	C₄⋊C₄×C₆₇	—	4	4	no	—
220	C₄⋊C₄×C₆₇	—	4	4	no	—
221	C₄⋊C₄×C₆₇	—	4	4	no	—
222	C₄⋊C₄×C₆₇	#41	1	4	yes	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₂²⋊C₄×C₆₇	—	2	8	no	—
225	C₂²⋊C₄×C₆₇	—	2	8	no	—
226	C₂²⋊C₄×C₆₇	—	4	4	no	—
227	C₂²⋊C₄×C₆₇	—	4	4	no	—
228	C₂²⋊C₄×C₆₇	—	4	4	no	—
229	C₂²⋊C₄×C₆₇	—	4	4	no	—
230	C₂²⋊C₄×C₆₇	—	4	4	no	—
231	C₂²⋊C₄×C₆₇	—	4	4	no	—
232	C₄²×C₆₇	—	2	4	no	—
233	C₄²×C₆₇	—	2	2	no	—
234	C₄²×C₆₇	—	2	2	no	—
235	C₄²×C₆₇	—	2	2	no	—
236	C₄²×C₆₇	—	2	4	no	—
237	C₄²×C₆₇	—	2	4	no	—
238	C₄²×C₆₇	—	2	4	no	—
239	C₄²×C₆₇	—	2	2	no	—
240	C₄²×C₆₇	—	4	1	no	—
241	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
242	C₄²×C₆₇	#41	1	4	yes	—
243	C₂³×C₆₇	—	2	4	no	—
244	C₂³×C₆₇	—	2	4	no	—
245	C₂³×C₆₇	—	2	4	no	—
246	C₂³×C₆₇	—	2	4	no	—
247	C₂³×C₆₇	—	2	4	no	—
248	C₂³×C₆₇	—	2	4	no	—
249	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
250	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
251	Q₈×C₆₇	—	4	8	no	—
252	D₄×C₆₇	—	2	16	no	—
253	D₄×C₆₇	—	2	16	no	—
254	D₄×C₆₇	—	4	8	no	—
255	C₂×C₄×C₆₇	—	2	1	no	—
256	C₂×C₄×C₆₇	—	2	1	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	1	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	1	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	4	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	4	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	2	1	no	—
288	C₂×C₄×C₆₇	—	2	2	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	8	8	no	—
293	C₂×C₄×C₆₇	—	8	8	no	—
294	C₂×C₄×C₆₇	—	8	8	no	—
295	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
296	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
297	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
298	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
299	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
300	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
301	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
302	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
303	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
304	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
305	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
306	C₂²×C₆₇	—	2	8	no	—
307	C₂²×C₆₇	—	2	8	no	—
308	C₂²×C₆₇	—	2	8	no	—
309	C₂²×C₆₇	—	4	4	no	—
310	C₂²×C₆₇	—	4	4	no	—
311	C₂²×C₆₇	—	4	4	no	—
312	C₂²×C₆₇	—	4	4	no	—
313	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
314	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
315	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
316	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
317	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
318	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
319	C₂²×C₆₇	#585	1	1	yes	characteristic
320	C₄×C₆₇	—	2	2	no	—
321	C₄×C₆₇	—	2	2	no	—
322	C₄×C₆₇	—	2	2	no	—
323	C₄×C₆₇	—	2	4	no	—
324	C₄×C₆₇	—	2	2	no	—
325	C₄×C₆₇	—	4	2	no	—
326	C₄×C₆₇	—	4	2	no	—
327	C₄×C₆₇	—	4	4	no	—
328	C₄×C₆₇	—	4	2	no	—
329	C₄×C₆₇	—	4	2	no	—
330	C₄×C₆₇	—	4	1	no	—
331	C₄×C₆₇	—	4	2	no	—
332	C₄×C₆₇	—	4	2	no	—
333	C₄×C₆₇	—	4	2	no	—
334	C₄×C₆₇	#554	1	4	yes	—
335	C₄×C₆₇	#554	1	4	yes	—
336	#59457	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
337	C₂×C₆₇	—	4	8	no	—
338	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
339	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
340	C₂×C₆₇	#3480	1	1	yes	characteristic
341	C₂×C₆₇	#3501	1	1	yes	characteristic
342	C₂×C₆₇	#3508	1	1	yes	characteristic
343	C₂×C₆₇	#3519	1	1	yes	characteristic
344	C₂×C₆₇	#3546	1	1	yes	characteristic
345	#2780	C₂×C₆₇	1	4	yes	complemented
346	#2580	C₂×C₆₇	1	2	yes	complemented
347	#2571	C₂×C₆₇	1	1	yes	characteristic, complemented
348	#2566	C₂×C₆₇	1	2	yes	complemented
349	#2566	C₂×C₆₇	1	2	yes	complemented
350	#2552	C₂×C₆₇	1	1	yes	characteristic, complemented
351	#2538	C₂×C₆₇	1	2	yes	complemented
352	#2454	C₂×C₆₇	1	2	yes	complemented
353	#2452	C₂×C₆₇	1	2	yes	complemented
354	#2450	C₂×C₆₇	1	4	yes	complemented
355	#2366	C₂×C₆₇	1	1	yes	characteristic
356	#2366	C₂×C₆₇	1	1	yes	characteristic
357	#2349	C₂×C₆₇	1	2	yes	complemented
358	#2349	C₂×C₆₇	1	2	yes	complemented
359	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
360	C₄×#127	C₂×C₆₇	1	1	yes	characteristic
361	C₄×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
362	C₆₇	#59457	1	1	yes	characteristic, direct product factor, Sylow
363	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
364	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
365	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
366	C₂×#125	C₂²×C₆₇	1	2	yes	—
367	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
368	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
369	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
370	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
371	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
372	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
373	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
374	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
375	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
376	#400	C₂²×C₆₇	1	4	yes	—
377	#399	C₂²×C₆₇	1	4	yes	—
378	#398	C₂²×C₆₇	1	4	yes	—
379	#396	C₂²×C₆₇	1	4	yes	—
380	#396	C₂²×C₆₇	1	4	yes	—
381	#396	C₂²×C₆₇	1	4	yes	—
382	#396	C₂²×C₆₇	1	4	yes	—
383	#393	C₂²×C₆₇	1	4	yes	—
384	#391	C₂²×C₆₇	1	2	yes	—
385	#390	C₂²×C₆₇	1	2	yes	—
386	#390	C₂²×C₆₇	1	2	yes	—
387	#390	C₂²×C₆₇	1	2	yes	—
388	#390	C₂²×C₆₇	1	2	yes	—
389	#390	C₂²×C₆₇	1	2	yes	—
390	#390	C₂²×C₆₇	1	2	yes	—
391	#390	C₂²×C₆₇	1	2	yes	—
392	#389	C₂²×C₆₇	1	1	yes	characteristic
393	#389	C₂²×C₆₇	1	1	yes	characteristic
394	#389	C₂²×C₆₇	1	1	yes	characteristic
395	#389	C₂²×C₆₇	1	1	yes	characteristic
396	#389	C₂²×C₆₇	1	2	yes	—
397	#389	C₂²×C₆₇	1	2	yes	—
398	#389	C₂²×C₆₇	1	2	yes	—
399	#389	C₂²×C₆₇	1	2	yes	—
400	#389	C₂²×C₆₇	1	2	yes	—
401	#387	C₂²×C₆₇	1	2	yes	—
402	#387	C₂²×C₆₇	1	2	yes	—
403	#386	C₂²×C₆₇	1	4	yes	—
404	#386	C₂²×C₆₇	1	4	yes	—
405	#385	C₂²×C₆₇	1	4	yes	—
406	#385	C₂²×C₆₇	1	4	yes	—
407	#385	C₂²×C₆₇	1	4	yes	—
408	#385	C₂²×C₆₇	1	4	yes	—
409	#385	C₂²×C₆₇	1	4	yes	—
410	#385	C₂²×C₆₇	1	4	yes	—
411	#384	C₂²×C₆₇	1	1	yes	characteristic
412	#384	C₂²×C₆₇	1	1	yes	characteristic
413	#384	C₂²×C₆₇	1	2	yes	—
414	#382	C₂²×C₆₇	1	1	yes	characteristic
415	#382	C₂²×C₆₇	1	2	yes	—
416	#382	C₂²×C₆₇	1	2	yes	—
417	#380	C₂²×C₆₇	1	4	yes	—
418	#380	C₂²×C₆₇	1	4	yes	—
419	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
420	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
421	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
422	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
423	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
424	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
425	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
426	C₂×#95	C₂²×C₆₇	1	4	yes	—
427	C₄³	C₂²×C₆₇	1	1	yes	characteristic
428	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
429	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
430	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
431	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
432	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
433	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
434	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
435	C₄⋊Q₈	—	2	4	no	—
436	#127	Q₈×C₆₇	1	8	yes	complemented
437	#125	—	2	4	no	—
438	#125	—	4	4	no	—
439	C₂²⋊Q₈	—	4	8	no	—
440	C₄×Q₈	—	4	4	no	—
441	C₄×D₄	—	2	8	no	—
442	C₄×D₄	—	2	8	no	—
443	C₄×D₄	—	4	4	no	—
444	C₄×D₄	—	4	4	no	—
445	C₄×D₄	—	4	4	no	—
446	C₄×D₄	—	4	4	no	—
447	C₄×D₄	—	4	4	no	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
458	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
459	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
460	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
462	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
463	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
464	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
465	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
466	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
467	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
468	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
469	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
471	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
472	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
473	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
474	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
475	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
476	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
477	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
478	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
479	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
480	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
481	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
482	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
483	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
484	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
485	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
486	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
487	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
488	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
489	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
490	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
491	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
492	#95	C₂³×C₆₇	1	2	yes	—
493	#95	C₂³×C₆₇	1	2	yes	—
494	#95	C₂³×C₆₇	1	4	yes	—
495	#95	C₂³×C₆₇	1	4	yes	—
496	#95	C₂³×C₆₇	1	2	yes	—
497	#95	C₂³×C₆₇	1	2	yes	—
498	#95	C₂³×C₆₇	1	2	yes	—
499	#95	C₂³×C₆₇	1	2	yes	—
500	#95	C₂³×C₆₇	1	2	yes	—
501	#95	C₂³×C₆₇	1	2	yes	—
502	#95	C₂³×C₆₇	1	2	yes	—
503	#95	C₂³×C₆₇	1	2	yes	—
504	#95	C₂³×C₆₇	1	4	yes	—
505	#95	C₂³×C₆₇	1	4	yes	—
506	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
507	C₂×Q₈	—	2	4	no	—
508	C₂×Q₈	—	2	4	no	—
509	C₂×Q₈	—	4	2	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	4	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	4	2	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	—
517	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	—
518	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	—
519	C₂²×C₄	—	2	8	no	—
520	C₂²×C₄	—	2	8	no	—
521	C₂²×C₄	—	4	4	no	—
522	C₂²×C₄	—	4	4	no	—
523	C₂²×C₄	—	4	4	no	—
524	C₂²×C₄	—	4	4	no	—
525	C₂²×C₄	—	4	4	no	—
526	C₂²×C₄	—	4	4	no	—
527	C₂²×C₄	—	4	4	no	—
528	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
538	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
539	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
540	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
541	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
542	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
543	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	2	4	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	4	2	no	—
558	C₄⋊C₄	—	4	2	no	—
559	C₄⋊C₄	—	4	2	no	—
560	C₄⋊C₄	—	4	2	no	—
561	C₄⋊C₄	—	4	4	no	—
562	C₄⋊C₄	—	4	4	no	—
563	C₄⋊C₄	—	4	4	no	—
564	C₄⋊C₄	—	4	4	no	—
565	C₄⋊C₄	—	4	2	no	—
566	C₄⋊C₄	—	4	2	no	—
567	C₄⋊C₄	#41×C₆₇	1	4	yes	—
568	C₂²⋊C₄	—	2	8	no	—
569	C₂²⋊C₄	—	2	8	no	—
570	C₂²⋊C₄	—	2	8	no	—
571	C₂²⋊C₄	—	4	4	no	—
572	C₂²⋊C₄	—	4	4	no	—
573	C₂²⋊C₄	—	4	4	no	—
574	C₂²⋊C₄	—	4	4	no	—
575	C₂²⋊C₄	—	4	4	no	—
576	C₂²⋊C₄	—	4	4	no	—
577	C₄²	—	2	4	no	—
578	C₄²	—	2	2	no	—
579	C₄²	—	2	2	no	—
580	C₄²	—	2	2	no	—
581	C₄²	—	2	4	no	—
582	C₄²	—	2	4	no	—
583	C₄²	—	2	4	no	—
584	C₄²	—	2	2	no	—
585	C₄²	—	4	1	no	—
586	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
587	C₄²	#41×C₆₇	1	4	yes	—
588	C₂³	—	2	4	no	—
589	C₂³	—	2	4	no	—
590	C₂³	—	2	4	no	—
591	C₂³	—	2	4	no	—
592	C₂³	—	2	4	no	—
593	C₂³	—	2	4	no	—
594	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
595	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
596	Q₈	—	4	8	no	—
597	D₄	—	2	16	no	—
598	D₄	—	2	16	no	—
599	D₄	—	4	8	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	4	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	8	8	no	—
638	C₂×C₄	—	8	8	no	—
639	C₂×C₄	—	8	8	no	—
640	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
641	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
642	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
643	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
644	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
645	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
646	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
647	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
648	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
649	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
650	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
651	C₂²	—	2	8	no	—
652	C₂²	—	2	8	no	—
653	C₂²	—	2	8	no	—
654	C₂²	—	4	4	no	—
655	C₂²	—	4	4	no	—
656	C₂²	—	4	4	no	—
657	C₂²	—	4	4	no	—
658	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
659	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
660	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
661	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
662	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
663	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
664	C₂²	#585×C₆₇	1	1	yes	characteristic
665	C₄	—	2	2	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	4	no	—
669	C₄	—	2	2	no	—
670	C₄	—	4	2	no	—
671	C₄	—	4	2	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	2	no	—
675	C₄	—	4	4	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	2	no	—
678	C₄	—	4	2	no	—
679	C₄	#554×C₆₇	1	4	yes	—
680	C₄	#554×C₆₇	1	4	yes	—
681	C₂	—	4	8	no	—
682	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
683	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
684	C₂	C₆₇×#3480	1	1	yes	characteristic
685	C₂	C₆₇×#3501	1	1	yes	characteristic
686	C₂	C₆₇×#3508	1	1	yes	characteristic
687	C₂	C₆₇×#3519	1	1	yes	characteristic
688	C₂	C₆₇×#3546	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#2780	C₂	semidirect
2. C₆₇×#2580	C₂	semidirect
3. C₆₇×#2571	C₂	semidirect
4. C₆₇×#2566	C₂	semidirect
5. C₆₇×#2566	C₂	semidirect
6. C₆₇×#2552	C₂	semidirect
7. C₆₇×#2538	C₂	semidirect
8. C₆₇×#2454	C₂	semidirect
9. C₆₇×#2452	C₂	semidirect
10. C₆₇×#2450	C₂	semidirect
13. C₆₇×#2349	C₂	semidirect
14. C₆₇×#2349	C₂	semidirect
15. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
17. C₄×#125×C₆₇	C₂	semidirect
91. #127×C₆₇	Q₈	semidirect
336. #59457	C₆₇	direct (N × H)
345. #2780	C₂×C₆₇	semidirect
346. #2580	C₂×C₆₇	semidirect
347. #2571	C₂×C₆₇	semidirect
348. #2566	C₂×C₆₇	semidirect
349. #2566	C₂×C₆₇	semidirect
350. #2552	C₂×C₆₇	semidirect
351. #2538	C₂×C₆₇	semidirect
352. #2454	C₂×C₆₇	semidirect
353. #2452	C₂×C₆₇	semidirect
354. #2450	C₂×C₆₇	semidirect
357. #2349	C₂×C₆₇	semidirect
358. #2349	C₂×C₆₇	semidirect
359. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
361. C₄×#125	C₂×C₆₇	semidirect
362. C₆₇	#59457	direct (N × H)
436. #127	Q₈×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#2780	2
C₆₇×#2580	2
C₆₇×#2571	2
C₆₇×#2566	2
C₆₇×#2552	2
C₆₇×#2538	2
C₆₇×#2454	2
C₆₇×#2452	2
C₆₇×#2450	2
C₆₇×#2366	2
C₆₇×#2349	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#127×C₆₇	2
C₄×#125×C₆₇	2
#59457	67

Subgroups — C67×#59457 #30738708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59457 #30738708