Subgroups — C₆₇×#59429 #30738680

Order 17152.

1788 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2788	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2581	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2570	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2563	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2538	C₂	1	2	yes	maximal
7	C₆₇×#2497	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2497	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2465	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2440	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2367	C₂	1	2	yes	maximal
12	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#2345	C₂	1	2	yes	maximal, complemented
15	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#400×C₆₇	C₂²	1	4	yes	—
30	#400×C₆₇	C₂²	1	4	yes	—
31	#398×C₆₇	C₂²	1	4	yes	—
32	#396×C₆₇	C₂²	1	4	yes	—
33	#396×C₆₇	C₂²	1	4	yes	—
34	#395×C₆₇	C₂²	1	4	yes	—
35	#395×C₆₇	C₂²	1	4	yes	—
36	#393×C₆₇	C₂²	1	4	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	1	yes	characteristic
39	#391×C₆₇	C₂²	1	2	yes	—
40	#391×C₆₇	C₂²	1	2	yes	—
41	#390×C₆₇	C₂²	1	1	yes	characteristic
42	#390×C₆₇	C₂²	1	2	yes	—
43	#389×C₆₇	C₂²	1	1	yes	characteristic
44	#389×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	4	yes	—
48	#389×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	1	yes	characteristic
51	#389×C₆₇	C₂²	1	2	yes	—
52	#389×C₆₇	C₂²	1	2	yes	—
53	#388×C₆₇	C₂²	1	4	yes	—
54	#388×C₆₇	C₂²	1	4	yes	—
55	#388×C₆₇	C₂²	1	2	yes	—
56	#388×C₆₇	C₂²	1	2	yes	—
57	#387×C₆₇	C₂²	1	4	yes	—
58	#387×C₆₇	C₂²	1	4	yes	—
59	#387×C₆₇	C₂²	1	4	yes	—
60	#387×C₆₇	C₂²	1	4	yes	—
61	#385×C₆₇	C₂²	1	4	yes	—
62	#385×C₆₇	C₂²	1	4	yes	—
63	#384×C₆₇	C₂²	1	1	yes	characteristic
64	#384×C₆₇	C₂²	1	2	yes	—
65	#384×C₆₇	C₂²	1	2	yes	—
66	#384×C₆₇	C₂²	1	1	yes	characteristic
67	#382×C₆₇	C₂²	1	1	yes	characteristic
68	#382×C₆₇	C₂²	1	2	yes	—
69	#380×C₆₇	C₂²	1	2	yes	—
70	#380×C₆₇	C₂²	1	2	yes	—
71	#380×C₆₇	C₂²	1	2	yes	—
72	#380×C₆₇	C₂²	1	4	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
82	C₄³×C₆₇	C₂²	1	1	yes	characteristic
83	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
88	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
89	C₄⋊Q₈×C₆₇	—	2	4	no	—
90	C₄⋊Q₈×C₆₇	—	2	4	no	—
91	#124×C₆₇	Q₈	1	8	yes	complemented
92	C₂²⋊Q₈×C₆₇	—	4	8	no	—
93	C₄×Q₈×C₆₇	—	2	4	no	—
94	C₄×Q₈×C₆₇	—	4	4	no	—
95	C₄×Q₈×C₆₇	—	4	4	no	—
96	C₄×Q₈×C₆₇	—	4	4	no	—
97	C₄×D₄×C₆₇	—	2	4	no	—
98	C₄×D₄×C₆₇	—	4	4	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
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	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
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	4	yes	—
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	4	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	2	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	4	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	2	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
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	4	yes	—
153	#95×C₆₇	C₂³	1	2	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	#95×C₆₇	C₂³	1	4	yes	—
157	#95×C₆₇	C₂³	1	4	yes	—
158	#95×C₆₇	C₂³	1	4	yes	—
159	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	4	no	—
162	C₂×Q₈×C₆₇	—	2	4	no	—
163	C₂×Q₈×C₆₇	—	2	2	no	—
164	C₂×Q₈×C₆₇	—	2	4	no	—
165	C₂×Q₈×C₆₇	—	4	2	no	—
166	C₂×Q₈×C₆₇	—	4	2	no	—
167	C₂×Q₈×C₆₇	—	4	2	no	—
168	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
169	C₂×D₄×C₆₇	—	2	2	no	—
170	C₂×D₄×C₆₇	—	2	2	no	—
171	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	4	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	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₄⋊C₄×C₆₇	—	2	8	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	2	no	—
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	4	no	—
204	C₄⋊C₄×C₆₇	—	2	8	no	—
205	C₄⋊C₄×C₆₇	—	2	2	no	—
206	C₄⋊C₄×C₆₇	—	2	2	no	—
207	C₄⋊C₄×C₆₇	—	4	2	no	—
208	C₄⋊C₄×C₆₇	—	4	4	no	—
209	C₄⋊C₄×C₆₇	—	4	2	no	—
210	C₄⋊C₄×C₆₇	—	4	2	no	—
211	C₄⋊C₄×C₆₇	—	4	2	no	—
212	C₄⋊C₄×C₆₇	—	4	4	no	—
213	C₄⋊C₄×C₆₇	—	4	2	no	—
214	C₄⋊C₄×C₆₇	—	4	4	no	—
215	C₄⋊C₄×C₆₇	—	4	4	no	—
216	C₄⋊C₄×C₆₇	#41	1	4	yes	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	4	2	no	—
220	C₂²⋊C₄×C₆₇	—	4	4	no	—
221	C₂²⋊C₄×C₆₇	—	4	4	no	—
222	C₂²⋊C₄×C₆₇	—	4	4	no	—
223	C₂²⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
224	C₂²⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
225	C₄²×C₆₇	—	2	4	no	—
226	C₄²×C₆₇	—	2	2	no	—
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	—	2	2	no	—
229	C₄²×C₆₇	—	2	2	no	—
230	C₄²×C₆₇	—	2	4	no	—
231	C₄²×C₆₇	—	2	2	no	—
232	C₄²×C₆₇	—	4	1	no	—
233	C₄²×C₆₇	—	4	4	no	—
234	C₄²×C₆₇	—	4	4	no	—
235	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
236	C₄²×C₆₇	#41	1	4	yes	—
237	C₂³×C₆₇	—	2	2	no	—
238	C₂³×C₆₇	—	2	2	no	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	—	2	2	no	—
242	C₂³×C₆₇	—	2	2	no	—
243	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
244	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
245	Q₈×C₆₇	—	2	8	no	—
246	Q₈×C₆₇	—	4	8	no	—
247	Q₈×C₆₇	—	4	8	no	—
248	D₄×C₆₇	—	2	8	no	—
249	C₂×C₄×C₆₇	—	2	1	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	1	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	1	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	4	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	1	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	1	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	4	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	4	8	no	—
286	C₂×C₄×C₆₇	—	8	8	no	—
287	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
288	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
289	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
290	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
291	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
292	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
293	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
294	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
295	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
296	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
297	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
298	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
299	C₂²×C₆₇	—	2	4	no	—
300	C₂²×C₆₇	—	2	4	no	—
301	C₂²×C₆₇	—	2	4	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	2	no	—
304	C₂²×C₆₇	—	4	2	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
307	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
308	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
310	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
311	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
312	C₂²×C₆₇	#585	1	1	yes	characteristic
313	C₄×C₆₇	—	2	4	no	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	2	2	no	—
317	C₄×C₆₇	—	2	4	no	—
318	C₄×C₆₇	—	2	2	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	4	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	—	4	2	no	—
325	C₄×C₆₇	—	4	2	no	—
326	C₄×C₆₇	—	4	1	no	—
327	C₄×C₆₇	#554	1	4	yes	—
328	C₄×C₆₇	#556	1	4	yes	—
329	#59429	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
330	C₂×C₆₇	—	4	4	no	—
331	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
332	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
333	C₂×C₆₇	Q₈×#41	1	1	yes	characteristic
334	C₂×C₆₇	#3503	1	1	yes	characteristic
335	C₂×C₆₇	#3506	1	1	yes	characteristic
336	C₂×C₆₇	#3519	1	1	yes	characteristic
337	C₂×C₆₇	#3547	1	1	yes	characteristic
338	#2788	C₂×C₆₇	1	4	yes	complemented
339	#2581	C₂×C₆₇	1	2	yes	complemented
340	#2570	C₂×C₆₇	1	2	yes	complemented
341	#2567	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#2563	C₂×C₆₇	1	1	yes	characteristic
343	#2538	C₂×C₆₇	1	2	yes	—
344	#2497	C₂×C₆₇	1	2	yes	complemented
345	#2497	C₂×C₆₇	1	2	yes	complemented
346	#2465	C₂×C₆₇	1	4	yes	complemented
347	#2440	C₂×C₆₇	1	2	yes	complemented
348	#2367	C₂×C₆₇	1	2	yes	—
349	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
350	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
351	#2345	C₂×C₆₇	1	2	yes	complemented
352	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
353	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
354	C₄×#124	C₂×C₆₇	1	1	yes	characteristic
355	C₆₇	#59429	1	1	yes	characteristic, direct product factor, Sylow
356	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
357	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
360	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
362	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
363	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
364	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
366	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
367	#400	C₂²×C₆₇	1	4	yes	—
368	#400	C₂²×C₆₇	1	4	yes	—
369	#398	C₂²×C₆₇	1	4	yes	—
370	#396	C₂²×C₆₇	1	4	yes	—
371	#396	C₂²×C₆₇	1	4	yes	—
372	#395	C₂²×C₆₇	1	4	yes	—
373	#395	C₂²×C₆₇	1	4	yes	—
374	#393	C₂²×C₆₇	1	4	yes	—
375	#391	C₂²×C₆₇	1	2	yes	—
376	#391	C₂²×C₆₇	1	2	yes	—
377	#391	C₂²×C₆₇	1	2	yes	—
378	#391	C₂²×C₆₇	1	1	yes	characteristic
379	#390	C₂²×C₆₇	1	2	yes	—
380	#390	C₂²×C₆₇	1	1	yes	characteristic
381	#389	C₂²×C₆₇	1	2	yes	—
382	#389	C₂²×C₆₇	1	1	yes	characteristic
383	#389	C₂²×C₆₇	1	2	yes	—
384	#389	C₂²×C₆₇	1	1	yes	characteristic
385	#389	C₂²×C₆₇	1	2	yes	—
386	#389	C₂²×C₆₇	1	2	yes	—
387	#389	C₂²×C₆₇	1	2	yes	—
388	#389	C₂²×C₆₇	1	2	yes	—
389	#389	C₂²×C₆₇	1	2	yes	—
390	#389	C₂²×C₆₇	1	4	yes	—
391	#388	C₂²×C₆₇	1	4	yes	—
392	#388	C₂²×C₆₇	1	4	yes	—
393	#388	C₂²×C₆₇	1	2	yes	—
394	#388	C₂²×C₆₇	1	2	yes	—
395	#387	C₂²×C₆₇	1	4	yes	—
396	#387	C₂²×C₆₇	1	4	yes	—
397	#387	C₂²×C₆₇	1	4	yes	—
398	#387	C₂²×C₆₇	1	4	yes	—
399	#385	C₂²×C₆₇	1	4	yes	—
400	#385	C₂²×C₆₇	1	4	yes	—
401	#384	C₂²×C₆₇	1	1	yes	characteristic
402	#384	C₂²×C₆₇	1	1	yes	characteristic
403	#384	C₂²×C₆₇	1	2	yes	—
404	#384	C₂²×C₆₇	1	2	yes	—
405	#382	C₂²×C₆₇	1	2	yes	—
406	#382	C₂²×C₆₇	1	1	yes	characteristic
407	#380	C₂²×C₆₇	1	2	yes	—
408	#380	C₂²×C₆₇	1	2	yes	—
409	#380	C₂²×C₆₇	1	4	yes	—
410	#380	C₂²×C₆₇	1	2	yes	—
411	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
412	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
413	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
414	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
415	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
416	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
417	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
418	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
419	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
420	C₄³	C₂²×C₆₇	1	1	yes	characteristic
421	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
422	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
423	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
424	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
425	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
426	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
427	C₄⋊Q₈	—	2	4	no	—
428	C₄⋊Q₈	—	2	4	no	—
429	#124	Q₈×C₆₇	1	8	yes	complemented
430	C₂²⋊Q₈	—	4	8	no	—
431	C₄×Q₈	—	2	4	no	—
432	C₄×Q₈	—	4	4	no	—
433	C₄×Q₈	—	4	4	no	—
434	C₄×Q₈	—	4	4	no	—
435	C₄×D₄	—	2	4	no	—
436	C₄×D₄	—	4	4	no	—
437	C₂×C₄⋊C₄	—	2	8	no	—
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
440	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
441	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
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	4	yes	—
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
458	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
459	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
466	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
469	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
471	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
473	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
474	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
475	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
476	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
477	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
478	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
479	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
480	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
481	#95	C₂³×C₆₇	1	2	yes	—
482	#95	C₂³×C₆₇	1	2	yes	—
483	#95	C₂³×C₆₇	1	2	yes	—
484	#95	C₂³×C₆₇	1	4	yes	—
485	#95	C₂³×C₆₇	1	2	yes	—
486	#95	C₂³×C₆₇	1	2	yes	—
487	#95	C₂³×C₆₇	1	4	yes	—
488	#95	C₂³×C₆₇	1	4	yes	—
489	#95	C₂³×C₆₇	1	2	yes	—
490	#95	C₂³×C₆₇	1	4	yes	—
491	#95	C₂³×C₆₇	1	2	yes	—
492	#95	C₂³×C₆₇	1	2	yes	—
493	#95	C₂³×C₆₇	1	4	yes	—
494	#95	C₂³×C₆₇	1	4	yes	—
495	#95	C₂³×C₆₇	1	4	yes	—
496	#95	C₂³×C₆₇	1	4	yes	—
497	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	4	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	2	4	no	—
503	C₂×Q₈	—	4	2	no	—
504	C₂×Q₈	—	4	2	no	—
505	C₂×Q₈	—	4	2	no	—
506	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
507	C₂×D₄	—	2	2	no	—
508	C₂×D₄	—	2	2	no	—
509	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	8	no	—
514	C₂²×C₄	—	4	4	no	—
515	C₂²×C₄	—	4	4	no	—
516	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
517	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
522	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
527	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
532	C₄⋊C₄	—	2	8	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	2	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	4	2	no	—
546	C₄⋊C₄	—	4	2	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₄⋊C₄	—	4	4	no	—
549	C₄⋊C₄	—	4	2	no	—
550	C₄⋊C₄	—	4	4	no	—
551	C₄⋊C₄	—	4	4	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	—	4	4	no	—
554	C₄⋊C₄	#41×C₆₇	1	4	yes	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	4	4	no	—
558	C₂²⋊C₄	—	4	4	no	—
559	C₂²⋊C₄	—	4	2	no	—
560	C₂²⋊C₄	—	4	4	no	—
561	C₂²⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
562	C₂²⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	2	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	2	2	no	—
567	C₄²	—	2	2	no	—
568	C₄²	—	2	4	no	—
569	C₄²	—	2	2	no	—
570	C₄²	—	4	4	no	—
571	C₄²	—	4	4	no	—
572	C₄²	—	4	1	no	—
573	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
574	C₄²	#41×C₆₇	1	4	yes	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	2	no	—
581	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
582	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
583	Q₈	—	2	8	no	—
584	Q₈	—	4	8	no	—
585	Q₈	—	4	8	no	—
586	D₄	—	2	8	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	4	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	4	8	no	—
624	C₂×C₄	—	8	8	no	—
625	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
626	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
627	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
628	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
629	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
630	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
631	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
632	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
633	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
634	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
635	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
636	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	4	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
645	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
646	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
647	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
648	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
649	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
650	C₂²	#585×C₆₇	1	1	yes	characteristic
651	C₄	—	2	4	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	4	no	—
656	C₄	—	2	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	1	no	—
660	C₄	—	4	2	no	—
661	C₄	—	4	4	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	4	no	—
664	C₄	—	4	2	no	—
665	C₄	#554×C₆₇	1	4	yes	—
666	C₄	#556×C₆₇	1	4	yes	—
667	C₂	—	4	4	no	—
668	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
669	C₂	C₂×#556×C₆₇	1	1	yes	characteristic
670	C₂	Q₈×#41×C₆₇	1	1	yes	characteristic
671	C₂	C₆₇×#3503	1	1	yes	characteristic
672	C₂	C₆₇×#3506	1	1	yes	characteristic
673	C₂	C₆₇×#3519	1	1	yes	characteristic
674	C₂	C₆₇×#3547	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
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×#2788	C₂	semidirect
2. C₆₇×#2581	C₂	semidirect
3. C₆₇×#2570	C₂	semidirect
4. C₆₇×#2567	C₂	semidirect
7. C₆₇×#2497	C₂	semidirect
8. C₆₇×#2497	C₂	semidirect
9. C₆₇×#2465	C₂	semidirect
10. C₆₇×#2440	C₂	semidirect
12. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
14. C₆₇×#2345	C₂	semidirect
15. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
16. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
91. #124×C₆₇	Q₈	semidirect
329. #59429	C₆₇	direct (N × H)
338. #2788	C₂×C₆₇	semidirect
339. #2581	C₂×C₆₇	semidirect
340. #2570	C₂×C₆₇	semidirect
341. #2567	C₂×C₆₇	semidirect
344. #2497	C₂×C₆₇	semidirect
345. #2497	C₂×C₆₇	semidirect
346. #2465	C₂×C₆₇	semidirect
347. #2440	C₂×C₆₇	semidirect
349. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
351. #2345	C₂×C₆₇	semidirect
352. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
353. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
355. C₆₇	#59429	direct (N × H)
429. #124	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×#2788	2
C₆₇×#2581	2
C₆₇×#2570	2
C₆₇×#2567	2
C₆₇×#2563	2
C₆₇×#2538	2
C₆₇×#2497	2
C₆₇×#2465	2
C₆₇×#2440	2
C₆₇×#2367	2
Q₈×C₄⋊C₄×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2345	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#124×C₆₇	2
#59429	67

Subgroups — C67×#59429 #30738680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59429 #30738680