Subgroups — C₆₇×#59885 #30739136

Order 17152.

1986 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2809	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2808	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2733	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2591	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2572	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2543	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2532	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2459	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2436	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2372	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2353	C₂	1	4	yes	maximal, complemented
12	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₂²	1	2	yes	—
16	C₂×#124×C₆₇	C₂²	1	2	yes	—
17	C₂×#121×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#400×C₆₇	C₂²	1	2	yes	—
25	#400×C₆₇	C₂²	1	2	yes	—
26	#398×C₆₇	C₂²	1	4	yes	—
27	#397×C₆₇	C₂²	1	4	yes	—
28	#394×C₆₇	C₂²	1	4	yes	—
29	#394×C₆₇	C₂²	1	2	yes	—
30	#394×C₆₇	C₂²	1	2	yes	—
31	#393×C₆₇	C₂²	1	4	yes	—
32	#393×C₆₇	C₂²	1	4	yes	—
33	#392×C₆₇	C₂²	1	4	yes	—
34	#391×C₆₇	C₂²	1	2	yes	—
35	#391×C₆₇	C₂²	1	4	yes	—
36	#391×C₆₇	C₂²	1	1	yes	characteristic
37	#391×C₆₇	C₂²	1	1	yes	characteristic
38	#391×C₆₇	C₂²	1	1	yes	characteristic
39	#391×C₆₇	C₂²	1	1	yes	characteristic
40	#391×C₆₇	C₂²	1	2	yes	—
41	#390×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	2	yes	—
43	#388×C₆₇	C₂²	1	4	yes	—
44	#388×C₆₇	C₂²	1	4	yes	—
45	#388×C₆₇	C₂²	1	4	yes	—
46	#388×C₆₇	C₂²	1	4	yes	—
47	#388×C₆₇	C₂²	1	4	yes	—
48	#388×C₆₇	C₂²	1	4	yes	—
49	#388×C₆₇	C₂²	1	4	yes	—
50	#387×C₆₇	C₂²	1	4	yes	—
51	#387×C₆₇	C₂²	1	4	yes	—
52	#386×C₆₇	C₂²	1	4	yes	—
53	#386×C₆₇	C₂²	1	4	yes	—
54	#386×C₆₇	C₂²	1	4	yes	—
55	#386×C₆₇	C₂²	1	4	yes	—
56	#386×C₆₇	C₂²	1	4	yes	—
57	#383×C₆₇	C₂²	1	4	yes	—
58	#381×C₆₇	C₂²	1	4	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
61	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
65	#376×C₆₇	C₂²	1	4	yes	—
66	C₂×#95×C₆₇	C₂²	1	4	yes	—
67	C₂²×Q₈×C₆₇	—	2	4	no	—
68	C₂²×Q₈×C₆₇	—	4	2	no	—
69	C₂²×Q₈×C₆₇	—	4	2	no	—
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
82	#124×C₆₇	—	2	8	no	—
83	#124×C₆₇	—	4	4	no	—
84	#121×C₆₇	—	2	8	no	—
85	#121×C₆₇	—	4	4	no	—
86	#121×C₆₇	—	4	4	no	—
87	C₄×D₄×C₆₇	—	2	16	no	—
88	C₄×D₄×C₆₇	—	4	4	no	—
89	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄²×C₆₇	—	4	2	no	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
118	#95×C₆₇	C₂³	1	4	yes	—
119	#95×C₆₇	C₂³	1	4	yes	—
120	#95×C₆₇	C₂³	1	4	yes	—
121	#95×C₆₇	C₂³	1	4	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	4	yes	—
124	#95×C₆₇	C₂³	1	4	yes	—
125	#95×C₆₇	C₂³	1	2	yes	—
126	#95×C₆₇	C₂³	1	4	yes	—
127	#95×C₆₇	C₂³	1	2	yes	—
128	#95×C₆₇	C₂³	1	4	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	#95×C₆₇	C₂³	1	4	yes	—
131	#95×C₆₇	C₂³	1	4	yes	—
132	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
133	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂×Q₈×C₆₇	—	2	4	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	4	no	—
137	C₂×Q₈×C₆₇	—	4	1	no	—
138	C₂×Q₈×C₆₇	—	4	1	no	—
139	C₂×Q₈×C₆₇	—	4	1	no	—
140	C₂×Q₈×C₆₇	—	4	2	no	—
141	C₂×Q₈×C₆₇	—	4	1	no	—
142	C₂×Q₈×C₆₇	—	4	16	no	—
143	C₂×Q₈×C₆₇	#41	1	4	yes	—
144	C₂×D₄×C₆₇	—	2	8	no	—
145	C₂×D₄×C₆₇	—	2	8	no	—
146	C₂×D₄×C₆₇	—	2	8	no	—
147	C₂×D₄×C₆₇	—	2	4	no	—
148	C₂×D₄×C₆₇	—	2	4	no	—
149	C₂×D₄×C₆₇	—	4	2	no	—
150	C₂×D₄×C₆₇	—	4	2	no	—
151	C₂×D₄×C₆₇	—	4	2	no	—
152	C₂×D₄×C₆₇	—	4	2	no	—
153	C₂×D₄×C₆₇	—	4	2	no	—
154	C₂×D₄×C₆₇	—	4	4	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	8	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	4	no	—
167	C₂²×C₄×C₆₇	—	4	4	no	—
168	C₂²×C₄×C₆₇	#41	1	8	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₄⋊C₄×C₆₇	—	2	8	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	4	1	no	—
183	C₄⋊C₄×C₆₇	—	4	2	no	—
184	C₄⋊C₄×C₆₇	—	4	1	no	—
185	C₄⋊C₄×C₆₇	—	4	2	no	—
186	C₄⋊C₄×C₆₇	—	4	2	no	—
187	C₄⋊C₄×C₆₇	—	4	2	no	—
188	C₄⋊C₄×C₆₇	—	4	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	8	no	—
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	—	2	8	no	—
192	C₂²⋊C₄×C₆₇	—	2	8	no	—
193	C₂²⋊C₄×C₆₇	—	2	8	no	—
194	C₂²⋊C₄×C₆₇	—	2	8	no	—
195	C₂²⋊C₄×C₆₇	—	4	4	no	—
196	C₂²⋊C₄×C₆₇	—	4	4	no	—
197	C₂²⋊C₄×C₆₇	—	4	4	no	—
198	C₂²⋊C₄×C₆₇	—	4	4	no	—
199	C₂²⋊C₄×C₆₇	—	4	4	no	—
200	C₄²×C₆₇	—	2	8	no	—
201	C₄²×C₆₇	—	2	4	no	—
202	C₄²×C₆₇	—	2	4	no	—
203	C₄²×C₆₇	—	4	1	no	—
204	C₄²×C₆₇	—	4	1	no	—
205	C₄²×C₆₇	—	4	4	no	—
206	C₄²×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	2	4	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	2	4	no	—
214	C₂³×C₆₇	—	2	4	no	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	—	2	4	no	—
217	C₂³×C₆₇	C₂×#41	1	4	yes	—
218	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
219	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
220	Q₈×C₆₇	—	4	4	no	—
221	Q₈×C₆₇	—	4	8	no	—
222	Q₈×C₆₇	—	4	4	no	—
223	D₄×C₆₇	—	4	8	no	—
224	D₄×C₆₇	—	4	16	no	—
225	D₄×C₆₇	—	8	4	no	—
226	D₄×C₆₇	—	8	4	no	—
227	C₂×C₄×C₆₇	—	2	8	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	4	8	no	—
255	C₂×C₄×C₆₇	—	4	8	no	—
256	C₂×C₄×C₆₇	—	8	8	no	—
257	C₂×C₄×C₆₇	—	8	4	no	—
258	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
259	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
260	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
261	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
262	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	4	no	—
269	C₂²×C₆₇	—	4	4	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	4	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	4	4	no	—
274	C₂²×C₆₇	#555	1	4	yes	—
275	C₂²×C₆₇	C₂²×#41	1	2	yes	—
276	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂²×C₆₇	#585	1	2	yes	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	2	no	—
282	C₄×C₆₇	—	2	4	no	—
283	C₄×C₆₇	—	2	4	no	—
284	C₄×C₆₇	—	2	4	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	4	no	—
287	C₄×C₆₇	—	4	2	no	—
288	C₄×C₆₇	—	4	4	no	—
289	C₄×C₆₇	—	4	2	no	—
290	C₄×C₆₇	—	4	2	no	—
291	#59885	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₆₇	—	2	4	no	—
293	C₂×C₆₇	—	8	4	no	—
294	C₂×C₆₇	C₂×#555	1	1	yes	characteristic
295	C₂×C₆₇	#3480	1	2	yes	—
296	C₂×C₆₇	#3490	1	1	yes	characteristic
297	C₂×C₆₇	#3505	1	2	yes	—
298	C₂×C₆₇	#3545	1	1	yes	characteristic
299	#2809	C₂×C₆₇	1	1	yes	characteristic, complemented
300	#2808	C₂×C₆₇	1	2	yes	complemented
301	#2733	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#2591	C₂×C₆₇	1	2	yes	complemented
303	#2572	C₂×C₆₇	1	2	yes	complemented
304	#2543	C₂×C₆₇	1	4	yes	complemented
305	#2532	C₂×C₆₇	1	4	yes	complemented
306	#2459	C₂×C₆₇	1	4	yes	complemented
307	#2436	C₂×C₆₇	1	2	yes	complemented
308	#2372	C₂×C₆₇	1	2	yes	complemented
309	#2353	C₂×C₆₇	1	4	yes	complemented
310	C₄×#121	C₂×C₆₇	1	2	yes	complemented
311	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₆₇	#59885	1	1	yes	characteristic, direct product factor, Sylow
313	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#124	C₂²×C₆₇	1	2	yes	—
315	C₂×#124	C₂²×C₆₇	1	2	yes	—
316	C₂×#121	C₂²×C₆₇	1	2	yes	—
317	C₂×#121	C₂²×C₆₇	1	2	yes	—
318	C₂×#121	C₂²×C₆₇	1	2	yes	—
319	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
320	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
321	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
323	#400	C₂²×C₆₇	1	2	yes	—
324	#400	C₂²×C₆₇	1	2	yes	—
325	#398	C₂²×C₆₇	1	4	yes	—
326	#397	C₂²×C₆₇	1	4	yes	—
327	#394	C₂²×C₆₇	1	2	yes	—
328	#394	C₂²×C₆₇	1	2	yes	—
329	#394	C₂²×C₆₇	1	4	yes	—
330	#393	C₂²×C₆₇	1	4	yes	—
331	#393	C₂²×C₆₇	1	4	yes	—
332	#392	C₂²×C₆₇	1	4	yes	—
333	#391	C₂²×C₆₇	1	1	yes	characteristic
334	#391	C₂²×C₆₇	1	1	yes	characteristic
335	#391	C₂²×C₆₇	1	2	yes	—
336	#391	C₂²×C₆₇	1	2	yes	—
337	#391	C₂²×C₆₇	1	4	yes	—
338	#391	C₂²×C₆₇	1	1	yes	characteristic
339	#391	C₂²×C₆₇	1	1	yes	characteristic
340	#390	C₂²×C₆₇	1	2	yes	—
341	#389	C₂²×C₆₇	1	2	yes	—
342	#388	C₂²×C₆₇	1	4	yes	—
343	#388	C₂²×C₆₇	1	4	yes	—
344	#388	C₂²×C₆₇	1	4	yes	—
345	#388	C₂²×C₆₇	1	4	yes	—
346	#388	C₂²×C₆₇	1	4	yes	—
347	#388	C₂²×C₆₇	1	4	yes	—
348	#388	C₂²×C₆₇	1	4	yes	—
349	#387	C₂²×C₆₇	1	4	yes	—
350	#387	C₂²×C₆₇	1	4	yes	—
351	#386	C₂²×C₆₇	1	4	yes	—
352	#386	C₂²×C₆₇	1	4	yes	—
353	#386	C₂²×C₆₇	1	4	yes	—
354	#386	C₂²×C₆₇	1	4	yes	—
355	#386	C₂²×C₆₇	1	4	yes	—
356	#383	C₂²×C₆₇	1	4	yes	—
357	#381	C₂²×C₆₇	1	4	yes	—
358	#380	C₂²×C₆₇	1	2	yes	—
359	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
362	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
364	#376	C₂²×C₆₇	1	4	yes	—
365	C₂×#95	C₂²×C₆₇	1	4	yes	—
366	C₂²×Q₈	—	2	4	no	—
367	C₂²×Q₈	—	4	2	no	—
368	C₂²×Q₈	—	4	2	no	—
369	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
370	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
371	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
372	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
373	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
374	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
375	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
376	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
378	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
379	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
380	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
381	#124	—	2	8	no	—
382	#124	—	4	4	no	—
383	#121	—	2	8	no	—
384	#121	—	4	4	no	—
385	#121	—	4	4	no	—
386	C₄×D₄	—	2	16	no	—
387	C₄×D₄	—	4	4	no	—
388	C₂×C₄⋊C₄	—	4	2	no	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
409	C₂×C₄²	—	4	2	no	—
410	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
417	#95	C₂³×C₆₇	1	4	yes	—
418	#95	C₂³×C₆₇	1	4	yes	—
419	#95	C₂³×C₆₇	1	4	yes	—
420	#95	C₂³×C₆₇	1	4	yes	—
421	#95	C₂³×C₆₇	1	4	yes	—
422	#95	C₂³×C₆₇	1	4	yes	—
423	#95	C₂³×C₆₇	1	4	yes	—
424	#95	C₂³×C₆₇	1	2	yes	—
425	#95	C₂³×C₆₇	1	4	yes	—
426	#95	C₂³×C₆₇	1	2	yes	—
427	#95	C₂³×C₆₇	1	4	yes	—
428	#95	C₂³×C₆₇	1	4	yes	—
429	#95	C₂³×C₆₇	1	4	yes	—
430	#95	C₂³×C₆₇	1	4	yes	—
431	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
432	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
433	C₂×Q₈	—	2	4	no	—
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	4	1	no	—
437	C₂×Q₈	—	4	1	no	—
438	C₂×Q₈	—	4	1	no	—
439	C₂×Q₈	—	4	2	no	—
440	C₂×Q₈	—	4	1	no	—
441	C₂×Q₈	—	4	16	no	—
442	C₂×Q₈	#41×C₆₇	1	4	yes	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	4	2	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	—	4	2	no	—
451	C₂×D₄	—	4	2	no	—
452	C₂×D₄	—	4	2	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	8	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	—	4	2	no	—
463	C₂²×C₄	—	4	2	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	#41×C₆₇	1	8	yes	—
468	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
475	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	2	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₄⋊C₄	—	4	2	no	—
485	C₄⋊C₄	—	4	2	no	—
486	C₄⋊C₄	—	4	2	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	8	no	—
493	C₂²⋊C₄	—	2	8	no	—
494	C₂²⋊C₄	—	4	4	no	—
495	C₂²⋊C₄	—	4	4	no	—
496	C₂²⋊C₄	—	4	4	no	—
497	C₂²⋊C₄	—	4	4	no	—
498	C₂²⋊C₄	—	4	4	no	—
499	C₄²	—	2	8	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	4	1	no	—
503	C₄²	—	4	1	no	—
504	C₄²	—	4	2	no	—
505	C₄²	—	4	4	no	—
506	C₂³	—	2	1	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	2	4	no	—
512	C₂³	—	2	4	no	—
513	C₂³	—	2	4	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	C₂×#41×C₆₇	1	4	yes	—
517	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
518	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
519	Q₈	—	4	4	no	—
520	Q₈	—	4	8	no	—
521	Q₈	—	4	4	no	—
522	D₄	—	4	8	no	—
523	D₄	—	4	16	no	—
524	D₄	—	8	4	no	—
525	D₄	—	8	4	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	—	8	8	no	—
556	C₂×C₄	—	8	4	no	—
557	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
558	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
559	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
560	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
561	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	4	4	no	—
573	C₂²	#555×C₆₇	1	4	yes	—
574	C₂²	C₂²×#41×C₆₇	1	2	yes	—
575	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
576	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
577	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
578	C₂²	#585×C₆₇	1	2	yes	—
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	4	no	—
582	C₄	—	2	4	no	—
583	C₄	—	2	4	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	4	no	—
587	C₄	—	4	4	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	4	no	—
590	C₂	—	2	4	no	—
591	C₂	—	8	4	no	—
592	C₂	C₂×#555×C₆₇	1	1	yes	characteristic
593	C₂	C₆₇×#3480	1	2	yes	—
594	C₂	C₆₇×#3490	1	1	yes	characteristic
595	C₂	C₆₇×#3505	1	2	yes	—
596	C₂	C₆₇×#3545	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
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×#2809	C₂	semidirect
2. C₆₇×#2808	C₂	semidirect
3. C₆₇×#2733	C₂	semidirect
4. C₆₇×#2591	C₂	semidirect
5. C₆₇×#2572	C₂	semidirect
6. C₆₇×#2543	C₂	semidirect
7. C₆₇×#2532	C₂	semidirect
8. C₆₇×#2459	C₂	semidirect
9. C₆₇×#2436	C₂	semidirect
10. C₆₇×#2372	C₂	semidirect
11. C₆₇×#2353	C₂	semidirect
12. C₄×#121×C₆₇	C₂	semidirect
13. C₂×#391×C₆₇	C₂	semidirect
291. #59885	C₆₇	direct (N × H)
299. #2809	C₂×C₆₇	semidirect
300. #2808	C₂×C₆₇	semidirect
301. #2733	C₂×C₆₇	semidirect
302. #2591	C₂×C₆₇	semidirect
303. #2572	C₂×C₆₇	semidirect
304. #2543	C₂×C₆₇	semidirect
305. #2532	C₂×C₆₇	semidirect
306. #2459	C₂×C₆₇	semidirect
307. #2436	C₂×C₆₇	semidirect
308. #2372	C₂×C₆₇	semidirect
309. #2353	C₂×C₆₇	semidirect
310. C₄×#121	C₂×C₆₇	semidirect
311. C₂×#391	C₂×C₆₇	semidirect
312. C₆₇	#59885	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×#2809	2
C₆₇×#2808	2
C₆₇×#2733	2
C₆₇×#2591	2
C₆₇×#2572	2
C₆₇×#2543	2
C₆₇×#2532	2
C₆₇×#2459	2
C₆₇×#2436	2
C₆₇×#2372	2
C₆₇×#2353	2
C₄×#121×C₆₇	2
C₂×#391×C₆₇	2
#59885	67

Subgroups — C67×#59885 #30739136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59885 #30739136