Subgroups — C₆₇×#59679 #30738930

Order 17152.

1828 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2800	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2566	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2545	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2513	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2512	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2512	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2466	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2447	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2367	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2357	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
14	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, 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×#126×C₆₇	C₂²	1	4	yes	—
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#124×C₆₇	C₂²	1	2	yes	—
23	C₂×#124×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×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	#397×C₆₇	C₂²	1	4	yes	—
31	#397×C₆₇	C₂²	1	4	yes	—
32	#397×C₆₇	C₂²	1	4	yes	—
33	#396×C₆₇	C₂²	1	4	yes	—
34	#396×C₆₇	C₂²	1	4	yes	—
35	#396×C₆₇	C₂²	1	4	yes	—
36	#394×C₆₇	C₂²	1	4	yes	—
37	#394×C₆₇	C₂²	1	4	yes	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	1	yes	characteristic
40	#391×C₆₇	C₂²	1	1	yes	characteristic
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	#390×C₆₇	C₂²	1	2	yes	—
48	#390×C₆₇	C₂²	1	2	yes	—
49	#390×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	1	yes	characteristic
53	#389×C₆₇	C₂²	1	2	yes	—
54	#389×C₆₇	C₂²	1	4	yes	—
55	#389×C₆₇	C₂²	1	2	yes	—
56	#388×C₆₇	C₂²	1	4	yes	—
57	#388×C₆₇	C₂²	1	4	yes	—
58	#388×C₆₇	C₂²	1	4	yes	—
59	#388×C₆₇	C₂²	1	4	yes	—
60	#388×C₆₇	C₂²	1	4	yes	—
61	#387×C₆₇	C₂²	1	2	yes	—
62	#387×C₆₇	C₂²	1	4	yes	—
63	#387×C₆₇	C₂²	1	2	yes	—
64	#385×C₆₇	C₂²	1	4	yes	—
65	#385×C₆₇	C₂²	1	4	yes	—
66	#384×C₆₇	C₂²	1	1	yes	characteristic
67	#384×C₆₇	C₂²	1	2	yes	—
68	#384×C₆₇	C₂²	1	1	yes	characteristic
69	#382×C₆₇	C₂²	1	2	yes	—
70	#382×C₆₇	C₂²	1	1	yes	characteristic
71	#381×C₆₇	C₂²	1	4	yes	—
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	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	2	yes	—
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	1	yes	characteristic
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₄⋊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	1	yes	characteristic
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
87	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
88	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
89	C₄⋊Q₈×C₆₇	—	2	4	no	—
90	#127×C₆₇	—	2	4	no	—
91	#126×C₆₇	—	4	8	no	—
92	#125×C₆₇	—	2	4	no	—
93	#124×C₆₇	—	2	8	no	—
94	#124×C₆₇	—	4	4	no	—
95	C₄×Q₈×C₆₇	—	2	4	no	—
96	C₄×Q₈×C₆₇	—	2	4	no	—
97	C₄×Q₈×C₆₇	—	4	4	no	—
98	C₄×D₄×C₆₇	—	2	8	no	—
99	C₄×D₄×C₆₇	—	4	4	no	—
100	C₄×D₄×C₆₇	—	4	4	no	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	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	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	1	yes	characteristic
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	2	yes	—
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	1	yes	characteristic
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	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	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	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×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
142	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
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	#95×C₆₇	C₂³	1	2	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	4	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	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	2	2	no	—
163	C₂×Q₈×C₆₇	—	2	2	no	—
164	C₂×Q₈×C₆₇	—	2	2	no	—
165	C₂×Q₈×C₆₇	—	2	2	no	—
166	C₂×Q₈×C₆₇	#41	1	4	yes	—
167	C₂×D₄×C₆₇	—	2	4	no	—
168	C₂×D₄×C₆₇	—	2	4	no	—
169	C₂×D₄×C₆₇	—	2	4	no	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	—	2	4	no	—
172	C₂×D₄×C₆₇	—	2	4	no	—
173	C₂×D₄×C₆₇	—	2	4	no	—
174	C₂×D₄×C₆₇	—	4	2	no	—
175	C₂×D₄×C₆₇	—	4	2	no	—
176	C₂²×C₄×C₆₇	—	2	8	no	—
177	C₂²×C₄×C₆₇	—	4	4	no	—
178	C₂²×C₄×C₆₇	—	4	4	no	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
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	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
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	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	2	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	2	no	—
204	C₄⋊C₄×C₆₇	—	2	4	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	4	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	2	2	no	—
211	C₄⋊C₄×C₆₇	—	4	2	no	—
212	C₄⋊C₄×C₆₇	—	4	2	no	—
213	C₄⋊C₄×C₆₇	—	4	2	no	—
214	C₄⋊C₄×C₆₇	—	4	4	no	—
215	C₄⋊C₄×C₆₇	—	4	4	no	—
216	C₂²⋊C₄×C₆₇	—	2	8	no	—
217	C₂²⋊C₄×C₆₇	—	2	8	no	—
218	C₂²⋊C₄×C₆₇	—	2	8	no	—
219	C₂²⋊C₄×C₆₇	—	2	8	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	—	4	4	no	—
224	C₂²⋊C₄×C₆₇	—	4	4	no	—
225	C₂²⋊C₄×C₆₇	—	4	4	no	—
226	C₂²⋊C₄×C₆₇	—	4	4	no	—
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	—	2	4	no	—
229	C₄²×C₆₇	—	2	4	no	—
230	C₄²×C₆₇	—	2	2	no	—
231	C₄²×C₆₇	—	2	2	no	—
232	C₄²×C₆₇	—	2	2	no	—
233	C₄²×C₆₇	—	2	4	no	—
234	C₄²×C₆₇	—	2	2	no	—
235	C₄²×C₆₇	—	4	1	no	—
236	C₄²×C₆₇	—	4	4	no	—
237	C₄²×C₆₇	—	4	4	no	—
238	C₄²×C₆₇	#41	1	4	yes	—
239	C₄²×C₆₇	#41	1	4	yes	—
240	C₂³×C₆₇	—	2	4	no	—
241	C₂³×C₆₇	—	2	4	no	—
242	C₂³×C₆₇	—	2	4	no	—
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	C₂⁵	1	1	yes	characteristic
248	Q₈×C₆₇	—	2	8	no	—
249	Q₈×C₆₇	—	4	4	no	—
250	D₄×C₆₇	—	4	8	no	—
251	D₄×C₆₇	—	4	8	no	—
252	D₄×C₆₇	—	4	8	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	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	1	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	1	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	4	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	1	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	2	1	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	2	2	no	—
288	C₂×C₄×C₆₇	—	2	2	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	8	8	no	—
291	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
292	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
293	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
294	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
295	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
296	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
297	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
298	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
299	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
300	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
301	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
302	C₂²×C₆₇	—	4	4	no	—
303	C₂²×C₆₇	—	4	4	no	—
304	C₂²×C₆₇	—	4	4	no	—
305	C₂²×C₆₇	—	4	4	no	—
306	C₂²×C₆₇	—	4	4	no	—
307	C₂²×C₆₇	—	4	4	no	—
308	C₂²×C₆₇	—	4	4	no	—
309	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
310	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
311	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
312	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
313	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
314	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
315	C₂²×C₆₇	#585	1	1	yes	characteristic
316	C₄×C₆₇	—	2	4	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	2	2	no	—
319	C₄×C₆₇	—	2	2	no	—
320	C₄×C₆₇	—	2	2	no	—
321	C₄×C₆₇	—	4	2	no	—
322	C₄×C₆₇	—	4	2	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	—	4	2	no	—
325	C₄×C₆₇	—	4	2	no	—
326	C₄×C₆₇	—	4	4	no	—
327	C₄×C₆₇	—	4	2	no	—
328	C₄×C₆₇	—	4	2	no	—
329	C₄×C₆₇	—	4	1	no	—
330	C₄×C₆₇	#553	1	4	yes	—
331	C₄×C₆₇	#556	1	4	yes	—
332	#59679	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
333	C₂×C₆₇	—	8	4	no	—
334	C₂×C₆₇	C₂×#553	1	1	yes	characteristic
335	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
336	C₂×C₆₇	#3480	1	1	yes	characteristic
337	C₂×C₆₇	#3506	1	1	yes	characteristic
338	C₂×C₆₇	#3508	1	1	yes	characteristic
339	C₂×C₆₇	#3522	1	1	yes	characteristic
340	C₂×C₆₇	#3554	1	1	yes	characteristic
341	#2800	C₂×C₆₇	1	4	yes	complemented
342	#2566	C₂×C₆₇	1	2	yes	complemented
343	#2548	C₂×C₆₇	1	1	yes	characteristic
344	#2548	C₂×C₆₇	1	1	yes	characteristic
345	#2545	C₂×C₆₇	1	2	yes	complemented
346	#2513	C₂×C₆₇	1	2	yes	complemented
347	#2512	C₂×C₆₇	1	2	yes	complemented
348	#2512	C₂×C₆₇	1	2	yes	complemented
349	#2466	C₂×C₆₇	1	4	yes	complemented
350	#2447	C₂×C₆₇	1	2	yes	complemented
351	#2367	C₂×C₆₇	1	2	yes	complemented
352	#2357	C₂×C₆₇	1	2	yes	complemented
353	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
354	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
355	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
356	C₄×#127	C₂×C₆₇	1	1	yes	characteristic
357	C₄×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
358	C₆₇	#59679	1	1	yes	characteristic, direct product factor, Sylow
359	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×#126	C₂²×C₆₇	1	4	yes	—
362	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
363	C₂×#124	C₂²×C₆₇	1	2	yes	—
364	C₂×#124	C₂²×C₆₇	1	2	yes	—
365	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
366	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
367	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
368	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
369	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
370	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
371	#397	C₂²×C₆₇	1	4	yes	—
372	#397	C₂²×C₆₇	1	4	yes	—
373	#397	C₂²×C₆₇	1	4	yes	—
374	#396	C₂²×C₆₇	1	4	yes	—
375	#396	C₂²×C₆₇	1	4	yes	—
376	#396	C₂²×C₆₇	1	4	yes	—
377	#394	C₂²×C₆₇	1	4	yes	—
378	#394	C₂²×C₆₇	1	4	yes	—
379	#391	C₂²×C₆₇	1	1	yes	characteristic
380	#391	C₂²×C₆₇	1	1	yes	characteristic
381	#391	C₂²×C₆₇	1	2	yes	—
382	#390	C₂²×C₆₇	1	2	yes	—
383	#390	C₂²×C₆₇	1	2	yes	—
384	#390	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	#389	C₂²×C₆₇	1	1	yes	characteristic
392	#389	C₂²×C₆₇	1	1	yes	characteristic
393	#389	C₂²×C₆₇	1	2	yes	—
394	#389	C₂²×C₆₇	1	2	yes	—
395	#389	C₂²×C₆₇	1	2	yes	—
396	#389	C₂²×C₆₇	1	4	yes	—
397	#388	C₂²×C₆₇	1	4	yes	—
398	#388	C₂²×C₆₇	1	4	yes	—
399	#388	C₂²×C₆₇	1	4	yes	—
400	#388	C₂²×C₆₇	1	4	yes	—
401	#388	C₂²×C₆₇	1	4	yes	—
402	#387	C₂²×C₆₇	1	2	yes	—
403	#387	C₂²×C₆₇	1	2	yes	—
404	#387	C₂²×C₆₇	1	4	yes	—
405	#385	C₂²×C₆₇	1	4	yes	—
406	#385	C₂²×C₆₇	1	4	yes	—
407	#384	C₂²×C₆₇	1	1	yes	characteristic
408	#384	C₂²×C₆₇	1	1	yes	characteristic
409	#384	C₂²×C₆₇	1	2	yes	—
410	#382	C₂²×C₆₇	1	1	yes	characteristic
411	#382	C₂²×C₆₇	1	2	yes	—
412	#381	C₂²×C₆₇	1	4	yes	—
413	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
414	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
415	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
416	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
417	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
418	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
419	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
420	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
421	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
422	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
423	C₄³	C₂²×C₆₇	1	1	yes	characteristic
424	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
425	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
426	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
427	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
428	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
429	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
430	C₄⋊Q₈	—	2	4	no	—
431	#127	—	2	4	no	—
432	#126	—	4	8	no	—
433	#125	—	2	4	no	—
434	#124	—	2	8	no	—
435	#124	—	4	4	no	—
436	C₄×Q₈	—	2	4	no	—
437	C₄×Q₈	—	2	4	no	—
438	C₄×Q₈	—	4	4	no	—
439	C₄×D₄	—	2	8	no	—
440	C₄×D₄	—	4	4	no	—
441	C₄×D₄	—	4	4	no	—
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
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	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
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	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
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	4	yes	—
461	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
462	C₂×C₄⋊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	4	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
469	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
470	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
471	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
472	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
473	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	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	2	yes	—
477	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
478	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
479	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
480	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
481	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
482	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
483	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
484	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
486	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
487	#95	C₂³×C₆₇	1	2	yes	—
488	#95	C₂³×C₆₇	1	2	yes	—
489	#95	C₂³×C₆₇	1	2	yes	—
490	#95	C₂³×C₆₇	1	4	yes	—
491	#95	C₂³×C₆₇	1	4	yes	—
492	#95	C₂³×C₆₇	1	4	yes	—
493	#95	C₂³×C₆₇	1	2	yes	—
494	#95	C₂³×C₆₇	1	2	yes	—
495	#95	C₂³×C₆₇	1	2	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	4	yes	—
501	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
502	C₂×Q₈	—	2	2	no	—
503	C₂×Q₈	—	2	2	no	—
504	C₂×Q₈	—	2	2	no	—
505	C₂×Q₈	—	2	2	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	#41×C₆₇	1	4	yes	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	4	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	—	2	4	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	4	2	no	—
517	C₂²×C₄	—	2	8	no	—
518	C₂²×C₄	—	4	4	no	—
519	C₂²×C₄	—	4	4	no	—
520	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
523	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
524	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
526	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
527	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
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	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	4	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	—	4	4	no	—
554	C₄⋊C₄	—	4	4	no	—
555	C₄⋊C₄	—	4	2	no	—
556	C₄⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	8	no	—
561	C₂²⋊C₄	—	4	4	no	—
562	C₂²⋊C₄	—	4	4	no	—
563	C₂²⋊C₄	—	4	4	no	—
564	C₂²⋊C₄	—	4	4	no	—
565	C₂²⋊C₄	—	4	4	no	—
566	C₂²⋊C₄	—	4	4	no	—
567	C₂²⋊C₄	—	4	4	no	—
568	C₄²	—	2	4	no	—
569	C₄²	—	2	4	no	—
570	C₄²	—	2	4	no	—
571	C₄²	—	2	2	no	—
572	C₄²	—	2	2	no	—
573	C₄²	—	2	2	no	—
574	C₄²	—	2	4	no	—
575	C₄²	—	2	2	no	—
576	C₄²	—	4	4	no	—
577	C₄²	—	4	4	no	—
578	C₄²	—	4	1	no	—
579	C₄²	#41×C₆₇	1	4	yes	—
580	C₄²	#41×C₆₇	1	4	yes	—
581	C₂³	—	2	4	no	—
582	C₂³	—	2	4	no	—
583	C₂³	—	2	4	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	4	no	—
586	C₂³	—	2	4	no	—
587	C₂³	—	2	4	no	—
588	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
589	Q₈	—	2	8	no	—
590	Q₈	—	4	4	no	—
591	D₄	—	4	8	no	—
592	D₄	—	4	8	no	—
593	D₄	—	4	8	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	2	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	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	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	4	no	—
619	C₂×C₄	—	2	4	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	8	8	no	—
632	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
633	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
634	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
635	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
636	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
637	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
638	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
639	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
640	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
641	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
642	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
643	C₂²	—	4	4	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	4	no	—
646	C₂²	—	4	4	no	—
647	C₂²	—	4	4	no	—
648	C₂²	—	4	4	no	—
649	C₂²	—	4	4	no	—
650	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
651	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
652	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
653	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
654	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
655	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
656	C₂²	#585×C₆₇	1	1	yes	characteristic
657	C₄	—	2	2	no	—
658	C₄	—	2	4	no	—
659	C₄	—	2	2	no	—
660	C₄	—	2	2	no	—
661	C₄	—	2	2	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	2	no	—
669	C₄	—	4	4	no	—
670	C₄	—	4	2	no	—
671	C₄	#553×C₆₇	1	4	yes	—
672	C₄	#556×C₆₇	1	4	yes	—
673	C₂	—	8	4	no	—
674	C₂	C₂×#553×C₆₇	1	1	yes	characteristic
675	C₂	C₂×#556×C₆₇	1	1	yes	characteristic
676	C₂	C₆₇×#3480	1	1	yes	characteristic
677	C₂	C₆₇×#3506	1	1	yes	characteristic
678	C₂	C₆₇×#3508	1	1	yes	characteristic
679	C₂	C₆₇×#3522	1	1	yes	characteristic
680	C₂	C₆₇×#3554	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×#2800	C₂	semidirect
2. C₆₇×#2566	C₂	semidirect
5. C₆₇×#2545	C₂	semidirect
6. C₆₇×#2513	C₂	semidirect
7. C₆₇×#2512	C₂	semidirect
8. C₆₇×#2512	C₂	semidirect
9. C₆₇×#2466	C₂	semidirect
10. C₆₇×#2447	C₂	semidirect
11. C₆₇×#2367	C₂	semidirect
12. C₆₇×#2357	C₂	semidirect
13. C₆₇×#2355	C₂	semidirect
14. Q₈×C₄⋊C₄×C₆₇	C₂	semidirect
15. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
17. C₄×#125×C₆₇	C₂	semidirect
332. #59679	C₆₇	direct (N × H)
341. #2800	C₂×C₆₇	semidirect
342. #2566	C₂×C₆₇	semidirect
345. #2545	C₂×C₆₇	semidirect
346. #2513	C₂×C₆₇	semidirect
347. #2512	C₂×C₆₇	semidirect
348. #2512	C₂×C₆₇	semidirect
349. #2466	C₂×C₆₇	semidirect
350. #2447	C₂×C₆₇	semidirect
351. #2367	C₂×C₆₇	semidirect
352. #2357	C₂×C₆₇	semidirect
353. #2355	C₂×C₆₇	semidirect
354. Q₈×C₄⋊C₄	C₂×C₆₇	semidirect
355. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
357. C₄×#125	C₂×C₆₇	semidirect
358. C₆₇	#59679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2800	2
C₆₇×#2566	2
C₆₇×#2548	2
C₆₇×#2545	2
C₆₇×#2513	2
C₆₇×#2512	2
C₆₇×#2466	2
C₆₇×#2447	2
C₆₇×#2367	2
C₆₇×#2357	2
C₆₇×#2355	2
Q₈×C₄⋊C₄×C₆₇	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#127×C₆₇	2
C₄×#125×C₆₇	2
#59679	67

Subgroups — C67×#59679 #30738930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59679 #30738930