Subgroups — C₆₇×#25891 #30705142

Order 17152.

1744 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#490×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#478×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2562	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2034	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2033	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1955	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1786	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1786	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1762	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₆₇	—	2	1	no	—
13	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
14	D₄²×C₆₇	—	2	4	no	—
15	D₄²×C₆₇	—	2	4	no	—
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×#121×C₆₇	—	2	1	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#120×C₆₇	—	2	2	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	#490×C₆₇	—	2	4	no	—
24	#478×C₆₇	—	2	4	no	—
25	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
35	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#400×C₆₇	—	2	1	no	—
39	#394×C₆₇	—	2	1	no	—
40	#388×C₆₇	—	2	2	no	—
41	#380×C₆₇	C₂²	1	2	yes	complemented
42	#380×C₆₇	C₂²	1	2	yes	complemented
43	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#340×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	8	no	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	—	2	4	no	—
49	C₂²×D₄×C₆₇	—	2	4	no	—
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	2	4	no	—
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	—	4	2	no	—
56	C₂²×D₄×C₆₇	—	4	2	no	—
57	C₂²×D₄×C₆₇	—	4	8	no	—
58	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	—	2	1	no	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	#127×C₆₇	—	2	2	no	—
67	#127×C₆₇	D₄	1	4	yes	—
68	#125×C₆₇	—	2	2	no	—
69	#125×C₆₇	—	2	2	no	—
70	#121×C₆₇	—	2	8	no	—
71	#121×C₆₇	—	2	4	no	—
72	#121×C₆₇	—	2	4	no	—
73	#120×C₆₇	—	2	16	no	—
74	C₄×D₄×C₆₇	—	2	2	no	—
75	C₄×D₄×C₆₇	—	2	2	no	—
76	C₄×D₄×C₆₇	—	2	2	no	—
77	C₄×D₄×C₆₇	D₄	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
90	C₂×C₄²×C₆₇	—	2	1	no	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	#107×C₆₇	—	2	2	no	—
93	#106×C₆₇	—	2	2	no	—
94	C₄⋊C₈×C₆₇	—	2	2	no	—
95	#102×C₆₇	—	2	2	no	—
96	#102×C₆₇	—	2	2	no	—
97	#102×C₆₇	—	2	4	no	—
98	#102×C₆₇	—	2	4	no	—
99	#102×C₆₇	—	2	4	no	—
100	#102×C₆₇	—	4	2	no	—
101	#102×C₆₇	—	4	2	no	—
102	C₂²⋊C₈×C₆₇	—	4	2	no	—
103	C₂²⋊C₈×C₆₇	—	4	2	no	—
104	#97×C₆₇	—	2	2	no	—
105	#95×C₆₇	—	2	1	no	—
106	#95×C₆₇	—	2	2	no	—
107	#95×C₆₇	—	2	2	no	—
108	C₂⁴×C₆₇	—	2	1	no	—
109	C₂⁴×C₆₇	—	2	8	no	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	4	4	no	—
112	C₂⁴×C₆₇	—	4	4	no	—
113	C₂⁴×C₆₇	—	4	4	no	—
114	C₂⁴×C₆₇	—	4	4	no	—
115	C₂⁴×C₆₇	—	8	2	no	—
116	C₂⁴×C₆₇	—	8	2	no	—
117	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₆₇	—	2	4	no	—
119	C₂×D₄×C₆₇	—	2	8	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	4	4	no	—
132	C₂×D₄×C₆₇	—	4	4	no	—
133	C₂×D₄×C₆₇	—	4	4	no	—
134	C₂×D₄×C₆₇	—	4	8	no	—
135	C₂×D₄×C₆₇	—	4	8	no	—
136	C₂×D₄×C₆₇	—	4	32	no	—
137	C₂×D₄×C₆₇	—	4	8	no	—
138	C₂×D₄×C₆₇	—	4	4	no	—
139	C₂×D₄×C₆₇	—	4	4	no	—
140	C₂×D₄×C₆₇	—	4	8	no	—
141	C₂×D₄×C₆₇	—	8	1	no	—
142	C₂×D₄×C₆₇	—	8	1	no	—
143	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
144	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂×D₄×C₆₇	#41	1	4	yes	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	—	4	4	no	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	1	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	—	2	2	no	—
168	C₂×C₈×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	1	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	4	2	no	—
178	C₄⋊C₄×C₆₇	—	4	2	no	—
179	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	2	4	no	—
182	C₂²⋊C₄×C₆₇	—	2	4	no	—
183	C₂²⋊C₄×C₆₇	—	4	4	no	—
184	C₂²⋊C₄×C₆₇	—	8	2	no	—
185	C₄²×C₆₇	—	8	1	no	—
186	C₄²×C₆₇	C₂×D₄	1	2	yes	—
187	C₄²×C₆₇	C₂×D₄	1	2	yes	—
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	2	2	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	4	no	—
197	C₂³×C₆₇	—	4	16	no	—
198	C₂³×C₆₇	—	4	16	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	—	4	2	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	—	4	2	no	—
206	C₂³×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	—	4	2	no	—
208	C₂³×C₆₇	—	8	8	no	—
209	C₂³×C₆₇	—	8	8	no	—
210	C₂³×C₆₇	—	8	8	no	—
211	C₂³×C₆₇	—	8	8	no	—
212	C₂³×C₆₇	—	8	8	no	—
213	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
214	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
215	D₄×C₆₇	—	2	8	no	—
216	D₄×C₆₇	—	2	2	no	—
217	D₄×C₆₇	—	2	4	no	—
218	D₄×C₆₇	—	2	4	no	—
219	D₄×C₆₇	—	2	4	no	—
220	D₄×C₆₇	—	4	8	no	—
221	D₄×C₆₇	—	4	16	no	—
222	D₄×C₆₇	—	4	16	no	—
223	D₄×C₆₇	—	4	2	no	—
224	D₄×C₆₇	—	4	2	no	—
225	D₄×C₆₇	—	8	4	no	—
226	D₄×C₆₇	#124	1	4	yes	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	4	8	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	1	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	2	no	—
243	C₂×C₄×C₆₇	—	4	2	no	—
244	C₂×C₄×C₆₇	—	4	2	no	—
245	C₂×C₄×C₆₇	—	4	2	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	#120	1	2	yes	—
249	C₂×C₄×C₆₇	#120	1	2	yes	—
250	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
253	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
254	C₈×C₆₇	—	4	1	no	—
255	C₈×C₆₇	—	4	1	no	—
256	C₈×C₆₇	—	4	2	no	—
257	C₂²×C₆₇	—	2	4	no	—
258	C₂²×C₆₇	—	2	4	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	2	4	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	2	no	—
263	C₂²×C₆₇	—	4	2	no	—
264	C₂²×C₆₇	—	4	2	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	4	2	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	—	8	32	no	—
273	C₂²×C₆₇	—	8	2	no	—
274	C₂²×C₆₇	—	8	2	no	—
275	C₂²×C₆₇	—	8	2	no	—
276	C₂²×C₆₇	—	8	2	no	—
277	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
278	C₂²×C₆₇	#546	1	2	yes	—
279	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
280	C₂²×C₆₇	#576	1	1	yes	characteristic
281	C₂²×C₆₇	#577	1	1	yes	characteristic
282	C₄×C₆₇	—	2	2	no	—
283	C₄×C₆₇	—	2	2	no	—
284	C₄×C₆₇	—	4	2	no	—
285	C₄×C₆₇	—	4	2	no	—
286	C₄×C₆₇	—	8	1	no	—
287	C₄×C₆₇	—	8	1	no	—
288	C₄×C₆₇	—	8	2	no	—
289	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
290	C₄×C₆₇	#535	1	1	yes	characteristic
291	C₄×C₆₇	#546	1	2	yes	—
292	#25891	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
293	C₂×C₆₇	—	4	4	no	—
294	C₂×C₆₇	—	4	4	no	—
295	C₂×C₆₇	—	8	4	no	—
296	C₂×C₆₇	#2435	1	1	yes	characteristic
297	C₂×C₆₇	#3003	1	1	yes	characteristic
298	C₂×C₆₇	#3012	1	1	yes	characteristic
299	C₂×C₆₇	#3310	1	2	yes	—
300	C₂×C₆₇	#3318	1	2	yes	—
301	C₂×D₄²	C₂×C₆₇	1	1	yes	characteristic
302	C₂×#490	C₂×C₆₇	1	1	yes	characteristic, complemented
303	C₂×#478	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#2562	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#2034	C₂×C₆₇	1	2	yes	complemented
306	#2033	C₂×C₆₇	1	2	yes	complemented
307	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#1955	C₂×C₆₇	1	1	yes	characteristic, complemented
309	#1786	C₂×C₆₇	1	2	yes	complemented
310	#1786	C₂×C₆₇	1	2	yes	complemented
311	#1762	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₆₇	#25891	1	1	yes	characteristic, direct product factor, Sylow
313	C₂³×D₄	—	2	1	no	—
314	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
315	D₄²	—	2	4	no	—
316	D₄²	—	2	4	no	—
317	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
319	C₂×#121	—	2	1	no	—
320	C₂×#121	C₂²×C₆₇	1	2	yes	—
321	C₂×#120	—	2	2	no	—
322	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
324	#490	—	2	4	no	—
325	#478	—	2	4	no	—
326	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
327	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
328	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
329	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#102	C₂²×C₆₇	1	2	yes	—
332	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
333	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
334	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
335	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
336	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
337	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
338	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic, complemented
339	#400	—	2	1	no	—
340	#394	—	2	1	no	—
341	#388	—	2	2	no	—
342	#380	C₂²×C₆₇	1	2	yes	complemented
343	#380	C₂²×C₆₇	1	2	yes	complemented
344	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
345	#340	C₂²×C₆₇	1	2	yes	complemented
346	C₂⁵	—	2	2	no	—
347	C₂²×D₄	—	2	8	no	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	—	2	4	no	—
350	C₂²×D₄	—	2	4	no	—
351	C₂²×D₄	—	2	4	no	—
352	C₂²×D₄	—	2	4	no	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	1	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	—	4	2	no	—
357	C₂²×D₄	—	4	2	no	—
358	C₂²×D₄	—	4	8	no	—
359	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
360	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
361	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
362	C₂³×C₄	—	2	1	no	—
363	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
364	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
365	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
367	#127	—	2	2	no	—
368	#127	D₄×C₆₇	1	4	yes	—
369	#125	—	2	2	no	—
370	#125	—	2	2	no	—
371	#121	—	2	8	no	—
372	#121	—	2	4	no	—
373	#121	—	2	4	no	—
374	#120	—	2	16	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	D₄×C₆₇	1	4	yes	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	#107	—	2	2	no	—
394	#106	—	2	2	no	—
395	C₄⋊C₈	—	2	2	no	—
396	#102	—	2	2	no	—
397	#102	—	2	2	no	—
398	#102	—	2	4	no	—
399	#102	—	2	4	no	—
400	#102	—	2	4	no	—
401	#102	—	4	2	no	—
402	#102	—	4	2	no	—
403	C₂²⋊C₈	—	4	2	no	—
404	C₂²⋊C₈	—	4	2	no	—
405	#97	—	2	2	no	—
406	#95	—	2	1	no	—
407	#95	—	2	2	no	—
408	#95	—	2	2	no	—
409	C₂⁴	—	2	1	no	—
410	C₂⁴	—	2	8	no	—
411	C₂⁴	—	2	2	no	—
412	C₂⁴	—	4	4	no	—
413	C₂⁴	—	4	4	no	—
414	C₂⁴	—	4	4	no	—
415	C₂⁴	—	4	4	no	—
416	C₂⁴	—	8	2	no	—
417	C₂⁴	—	8	2	no	—
418	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	4	4	no	—
433	C₂×D₄	—	4	4	no	—
434	C₂×D₄	—	4	4	no	—
435	C₂×D₄	—	4	8	no	—
436	C₂×D₄	—	4	8	no	—
437	C₂×D₄	—	4	32	no	—
438	C₂×D₄	—	4	8	no	—
439	C₂×D₄	—	4	4	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	4	8	no	—
442	C₂×D₄	—	8	1	no	—
443	C₂×D₄	—	8	1	no	—
444	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
445	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
446	C₂×D₄	#41×C₆₇	1	4	yes	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	4	4	no	—
459	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	4	2	no	—
479	C₄⋊C₄	—	4	2	no	—
480	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	4	4	no	—
485	C₂²⋊C₄	—	8	2	no	—
486	C₄²	—	8	1	no	—
487	C₄²	C₂×D₄×C₆₇	1	2	yes	—
488	C₄²	C₂×D₄×C₆₇	1	2	yes	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	4	16	no	—
499	C₂³	—	4	16	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	4	2	no	—
504	C₂³	—	4	2	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	2	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	4	2	no	—
509	C₂³	—	8	8	no	—
510	C₂³	—	8	8	no	—
511	C₂³	—	8	8	no	—
512	C₂³	—	8	8	no	—
513	C₂³	—	8	8	no	—
514	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
515	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
516	D₄	—	2	8	no	—
517	D₄	—	2	2	no	—
518	D₄	—	2	4	no	—
519	D₄	—	2	4	no	—
520	D₄	—	2	4	no	—
521	D₄	—	4	8	no	—
522	D₄	—	4	16	no	—
523	D₄	—	4	16	no	—
524	D₄	—	4	2	no	—
525	D₄	—	4	2	no	—
526	D₄	—	8	4	no	—
527	D₄	#124×C₆₇	1	4	yes	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	4	8	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	#120×C₆₇	1	2	yes	—
550	C₂×C₄	#120×C₆₇	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
554	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	2	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	8	32	no	—
574	C₂²	—	8	2	no	—
575	C₂²	—	8	2	no	—
576	C₂²	—	8	2	no	—
577	C₂²	—	8	2	no	—
578	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
579	C₂²	#546×C₆₇	1	2	yes	—
580	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
581	C₂²	#576×C₆₇	1	1	yes	characteristic
582	C₂²	#577×C₆₇	1	1	yes	characteristic
583	C₄	—	2	2	no	—
584	C₄	—	2	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	2	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	2	no	—
589	C₄	—	8	1	no	—
590	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
591	C₄	#535×C₆₇	1	1	yes	characteristic
592	C₄	#546×C₆₇	1	2	yes	—
593	C₂	—	4	4	no	—
594	C₂	—	4	4	no	—
595	C₂	—	8	4	no	—
596	C₂	C₆₇×#2435	1	1	yes	characteristic
597	C₂	C₆₇×#3003	1	1	yes	characteristic
598	C₂	C₆₇×#3012	1	1	yes	characteristic
599	C₂	C₆₇×#3310	1	2	yes	—
600	C₂	C₆₇×#3318	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×D₄²×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#490×C₆₇	C₂	semidirect
3. C₂×#478×C₆₇	C₂	semidirect
4. C₆₇×#2562	C₂	semidirect
5. C₆₇×#2034	C₂	semidirect
6. C₆₇×#2033	C₂	semidirect
7. C₆₇×#1982	C₂	semidirect
8. C₆₇×#1955	C₂	semidirect
9. C₆₇×#1786	C₂	semidirect
10. C₆₇×#1786	C₂	semidirect
11. C₆₇×#1762	C₂	semidirect
17. C₂×#125×C₆₇	C₂²	semidirect
25. C₂×#107×C₆₇	C₂²	semidirect
26. C₂×#106×C₆₇	C₂²	semidirect
27. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
31. C₂×#102×C₆₇	C₂²	semidirect
32. C₂×#102×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
34. C₂×#102×C₆₇	C₂²	semidirect
35. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
36. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
37. C₂×#97×C₆₇	C₂²	semidirect
41. #380×C₆₇	C₂²	semidirect
42. #380×C₆₇	C₂²	semidirect
43. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
44. #340×C₆₇	C₂²	semidirect
292. #25891	C₆₇	direct (N × H)
302. C₂×#490	C₂×C₆₇	semidirect
303. C₂×#478	C₂×C₆₇	semidirect
304. #2562	C₂×C₆₇	semidirect
305. #2034	C₂×C₆₇	semidirect
306. #2033	C₂×C₆₇	semidirect
307. #1982	C₂×C₆₇	semidirect
308. #1955	C₂×C₆₇	semidirect
309. #1786	C₂×C₆₇	semidirect
310. #1786	C₂×C₆₇	semidirect
311. #1762	C₂×C₆₇	semidirect
312. C₆₇	#25891	direct (N × H)
318. C₂×#125	C₂²×C₆₇	semidirect
326. C₂×#107	C₂²×C₆₇	semidirect
327. C₂×#106	C₂²×C₆₇	semidirect
328. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
332. C₂×#102	C₂²×C₆₇	semidirect
333. C₂×#102	C₂²×C₆₇	semidirect
334. C₂×#102	C₂²×C₆₇	semidirect
335. C₂×#102	C₂²×C₆₇	semidirect
336. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
337. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
338. C₂×#97	C₂²×C₆₇	semidirect
342. #380	C₂²×C₆₇	semidirect
343. #380	C₂²×C₆₇	semidirect
344. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
345. #340	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄²×C₆₇	2
C₂×#490×C₆₇	2
C₂×#478×C₆₇	2
C₆₇×#2562	2
C₆₇×#2034	2
C₆₇×#2033	2
C₆₇×#1982	2
C₆₇×#1955	2
C₆₇×#1786	2
C₆₇×#1762	2
#25891	67

Subgroups — C67×#25891 #30705142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25891 #30705142