Subgroups — C₆₇×#25946 #30705197

Order 17152.

1176 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#490×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2025	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2023	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1879	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1762	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
13	#554×C₆₇	—	2	4	no	—
14	#554×C₆₇	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
21	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
28	#490×C₆₇	—	2	4	no	—
29	#459×C₆₇	—	2	4	no	—
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	—
34	C₂×#102×C₆₇	C₂²	1	2	yes	—
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	#394×C₆₇	—	2	1	no	—
37	#394×C₆₇	—	2	1	no	—
38	#390×C₆₇	—	2	1	no	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#336×C₆₇	C₂²	1	2	yes	—
46	#336×C₆₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
52	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	—	2	1	no	—
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	2	4	no	—
57	C₂×D₈×C₆₇	—	2	4	no	—
58	C₂×D₈×C₆₇	—	2	4	no	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	C₄⋊Q₈×C₆₇	—	2	2	no	—
63	#127×C₆₇	—	2	2	no	—
64	#127×C₆₇	—	2	2	no	—
65	#125×C₆₇	—	2	2	no	—
66	#125×C₆₇	—	2	2	no	—
67	#125×C₆₇	—	2	2	no	—
68	#125×C₆₇	D₄	1	4	yes	—
69	#124×C₆₇	—	4	4	no	—
70	C₂²⋊Q₈×C₆₇	—	2	8	no	—
71	C₂²⋊Q₈×C₆₇	—	2	8	no	—
72	C₂²⋊Q₈×C₆₇	—	2	8	no	—
73	C₄×Q₈×C₆₇	—	2	2	no	—
74	C₄×Q₈×C₆₇	—	2	2	no	—
75	C₄×D₄×C₆₇	—	2	2	no	—
76	C₄×D₄×C₆₇	—	2	2	no	—
77	C₄×D₄×C₆₇	—	2	2	no	—
78	C₄×D₄×C₆₇	—	2	2	no	—
79	C₄×D₄×C₆₇	—	2	2	no	—
80	C₄×D₄×C₆₇	D₄	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₄²×C₆₇	—	2	1	no	—
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₈×C₆₇	—	2	2	no	—
106	#102×C₆₇	—	2	4	no	—
107	#102×C₆₇	—	2	4	no	—
108	#102×C₆₇	—	2	4	no	—
109	#102×C₆₇	—	2	4	no	—
110	#97×C₆₇	—	2	2	no	—
111	#95×C₆₇	—	2	1	no	—
112	C₂⁴×C₆₇	—	2	1	no	—
113	C₂⁴×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	2	4	no	—
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	4	1	no	—
118	C₂×Q₈×C₆₇	—	8	2	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	—	4	2	no	—
129	C₂×D₄×C₆₇	—	4	2	no	—
130	C₂×D₄×C₆₇	—	4	2	no	—
131	C₂×D₄×C₆₇	—	8	2	no	—
132	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×D₄×C₆₇	#41	1	4	yes	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	4	2	no	—
142	C₂²×C₄×C₆₇	—	4	2	no	—
143	C₂²×C₄×C₆₇	—	8	2	no	—
144	C₂²×C₄×C₆₇	—	8	2	no	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	D₈×C₆₇	—	4	8	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	4	no	—
160	C₄⋊C₄×C₆₇	—	2	4	no	—
161	C₄⋊C₄×C₆₇	—	2	4	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	1	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²⋊C₄×C₆₇	—	2	4	no	—
176	C₂²⋊C₄×C₆₇	—	2	4	no	—
177	C₂²⋊C₄×C₆₇	—	2	4	no	—
178	C₂²⋊C₄×C₆₇	—	4	4	no	—
179	C₂²⋊C₄×C₆₇	—	4	4	no	—
180	C₂²⋊C₄×C₆₇	—	8	2	no	—
181	C₂²⋊C₄×C₆₇	—	8	2	no	—
182	C₄²×C₆₇	—	2	1	no	—
183	C₄²×C₆₇	—	2	1	no	—
184	C₄²×C₆₇	—	2	1	no	—
185	C₄²×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	4	2	no	—
187	C₄²×C₆₇	C₂×D₄	1	2	yes	—
188	C₄²×C₆₇	C₂×D₄	1	2	yes	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	4	no	—
191	C₂³×C₆₇	—	4	1	no	—
192	C₂³×C₆₇	—	4	1	no	—
193	C₂³×C₆₇	—	4	2	no	—
194	C₂³×C₆₇	—	4	2	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	4	2	no	—
201	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₆₇	—	4	4	no	—
203	Q₈×C₆₇	—	8	4	no	—
204	D₄×C₆₇	—	2	2	no	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	4	2	no	—
207	D₄×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	4	2	no	—
209	D₄×C₆₇	—	4	2	no	—
210	D₄×C₆₇	—	8	4	no	—
211	D₄×C₆₇	—	8	4	no	—
212	D₄×C₆₇	#124	1	4	yes	complemented
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	4	8	no	—
229	C₂×C₄×C₆₇	—	4	1	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	2	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	8	4	no	—
237	C₂×C₄×C₆₇	—	8	4	no	—
238	C₂×C₄×C₆₇	#120	1	2	yes	—
239	C₂×C₄×C₆₇	#120	1	2	yes	—
240	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
244	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
245	C₈×C₆₇	—	4	2	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₂²×C₆₇	—	2	4	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	4	4	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	8	2	no	—
253	C₂²×C₆₇	—	8	2	no	—
254	C₂²×C₆₇	—	8	2	no	—
255	C₂²×C₆₇	—	8	2	no	—
256	C₂²×C₆₇	—	8	2	no	—
257	C₂²×C₆₇	—	8	2	no	—
258	C₂²×C₆₇	—	8	2	no	—
259	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
260	C₂²×C₆₇	#548	1	2	yes	—
261	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₂²×C₆₇	#575	1	1	yes	characteristic
263	C₂²×C₆₇	#576	1	2	yes	—
264	C₄×C₆₇	—	2	2	no	—
265	C₄×C₆₇	—	2	2	no	—
266	C₄×C₆₇	—	2	2	no	—
267	C₄×C₆₇	—	4	1	no	—
268	C₄×C₆₇	—	4	2	no	—
269	C₄×C₆₇	—	4	1	no	—
270	C₄×C₆₇	—	4	1	no	—
271	C₄×C₆₇	—	4	2	no	—
272	C₄×C₆₇	—	4	2	no	—
273	C₄×C₆₇	—	8	2	no	—
274	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
275	C₄×C₆₇	#536	1	1	yes	characteristic
276	C₄×C₆₇	#548	1	2	yes	—
277	#25946	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂×C₆₇	—	4	4	no	—
279	C₂×C₆₇	—	16	2	no	—
280	C₂×C₆₇	#2436	1	1	yes	characteristic
281	C₂×C₆₇	#3006	1	1	yes	characteristic
282	C₂×C₆₇	#3008	1	1	yes	characteristic
283	C₂×C₆₇	#3366	1	2	yes	—
284	C₂×C₆₇	#3370	1	2	yes	—
285	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₂×#490	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
288	#2548	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#2025	C₂×C₆₇	1	2	yes	complemented
290	#2023	C₂×C₆₇	1	2	yes	complemented
291	#1989	C₂×C₆₇	1	1	yes	characteristic, complemented
292	#1985	C₂×C₆₇	1	1	yes	characteristic, complemented
293	#1879	C₂×C₆₇	1	2	yes	complemented
294	#1879	C₂×C₆₇	1	2	yes	complemented
295	#1762	C₂×C₆₇	1	1	yes	characteristic, complemented
296	C₆₇	#25946	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
298	#554	—	2	4	no	—
299	#554	—	2	4	no	—
300	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#124	—	2	1	no	—
305	C₂×C₂²⋊Q₈	—	2	1	no	—
306	C₂×C₂²⋊Q₈	—	2	1	no	—
307	C₂×C₂²⋊Q₈	—	2	1	no	—
308	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×C₄⋊C₄	—	2	1	no	—
313	#490	—	2	4	no	—
314	#459	—	2	4	no	—
315	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#102	C₂²×C₆₇	1	2	yes	—
317	C₂×#102	C₂²×C₆₇	1	2	yes	—
318	C₂×#102	C₂²×C₆₇	1	2	yes	—
319	C₂×#102	C₂²×C₆₇	1	2	yes	—
320	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
321	#394	—	2	1	no	—
322	#394	—	2	1	no	—
323	#390	—	2	1	no	—
324	#390	C₂²×C₆₇	1	2	yes	—
325	#390	C₂²×C₆₇	1	2	yes	—
326	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
327	#340	C₂²×C₆₇	1	2	yes	—
328	#340	C₂²×C₆₇	1	2	yes	—
329	#340	C₂²×C₆₇	1	2	yes	—
330	#336	C₂²×C₆₇	1	2	yes	—
331	#336	C₂²×C₆₇	1	2	yes	—
332	C₂²×Q₈	—	2	1	no	—
333	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
337	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
338	C₂³×C₄	—	2	1	no	—
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂×D₈	—	2	4	no	—
342	C₂×D₈	—	2	4	no	—
343	C₂×D₈	—	2	4	no	—
344	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
345	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
346	C₄⋊Q₈	—	2	2	no	—
347	C₄⋊Q₈	—	2	2	no	—
348	#127	—	2	2	no	—
349	#127	—	2	2	no	—
350	#125	—	2	2	no	—
351	#125	—	2	2	no	—
352	#125	—	2	2	no	—
353	#125	D₄×C₆₇	1	4	yes	—
354	#124	—	4	4	no	—
355	C₂²⋊Q₈	—	2	8	no	—
356	C₂²⋊Q₈	—	2	8	no	—
357	C₂²⋊Q₈	—	2	8	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	D₄×C₆₇	1	4	yes	—
366	C₂×C₄⋊C₄	—	2	8	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
390	C₄⋊C₈	—	2	2	no	—
391	#102	—	2	4	no	—
392	#102	—	2	4	no	—
393	#102	—	2	4	no	—
394	#102	—	2	4	no	—
395	#97	—	2	2	no	—
396	#95	—	2	1	no	—
397	C₂⁴	—	2	1	no	—
398	C₂⁴	—	2	2	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	4	1	no	—
403	C₂×Q₈	—	8	2	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	2	no	—
414	C₂×D₄	—	4	2	no	—
415	C₂×D₄	—	4	2	no	—
416	C₂×D₄	—	8	2	no	—
417	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
418	C₂×D₄	#41×C₆₇	1	4	yes	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	—	8	2	no	—
429	C₂²×C₄	—	8	2	no	—
430	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
437	D₈	—	4	8	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
458	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
459	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	4	4	no	—
464	C₂²⋊C₄	—	4	4	no	—
465	C₂²⋊C₄	—	8	2	no	—
466	C₂²⋊C₄	—	8	2	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	4	2	no	—
472	C₄²	C₂×D₄×C₆₇	1	2	yes	—
473	C₄²	C₂×D₄×C₆₇	1	2	yes	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	4	1	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	2	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
487	Q₈	—	4	4	no	—
488	Q₈	—	8	4	no	—
489	D₄	—	2	2	no	—
490	D₄	—	2	4	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	2	no	—
494	D₄	—	4	2	no	—
495	D₄	—	8	4	no	—
496	D₄	—	8	4	no	—
497	D₄	#124×C₆₇	1	4	yes	complemented
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	8	4	no	—
522	C₂×C₄	—	8	4	no	—
523	C₂×C₄	#120×C₆₇	1	2	yes	—
524	C₂×C₄	#120×C₆₇	1	2	yes	—
525	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
526	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
528	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
529	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
530	C₈	—	4	2	no	—
531	C₈	—	4	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	8	2	no	—
538	C₂²	—	8	2	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	2	no	—
541	C₂²	—	8	2	no	—
542	C₂²	—	8	2	no	—
543	C₂²	—	8	2	no	—
544	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
545	C₂²	#548×C₆₇	1	2	yes	—
546	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
547	C₂²	#575×C₆₇	1	1	yes	characteristic
548	C₂²	#576×C₆₇	1	2	yes	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	2	no	—
558	C₄	—	8	2	no	—
559	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
560	C₄	#536×C₆₇	1	1	yes	characteristic
561	C₄	#548×C₆₇	1	2	yes	—
562	C₂	—	4	4	no	—
563	C₂	—	16	2	no	—
564	C₂	C₆₇×#2436	1	1	yes	characteristic
565	C₂	C₆₇×#3006	1	1	yes	characteristic
566	C₂	C₆₇×#3008	1	1	yes	characteristic
567	C₂	C₆₇×#3366	1	2	yes	—
568	C₂	C₆₇×#3370	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#459×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#554×C₆₇	C₂	semidirect
2. C₂×#490×C₆₇	C₂	semidirect
4. C₆₇×#2548	C₂	semidirect
5. C₆₇×#2025	C₂	semidirect
6. C₆₇×#2023	C₂	semidirect
7. C₆₇×#1989	C₂	semidirect
8. C₆₇×#1985	C₂	semidirect
9. C₆₇×#1879	C₂	semidirect
10. C₆₇×#1879	C₂	semidirect
11. C₆₇×#1762	C₂	semidirect
212. D₄×C₆₇	#124	semidirect
277. #25946	C₆₇	direct (N × H)
285. C₂×#554	C₂×C₆₇	semidirect
286. C₂×#490	C₂×C₆₇	semidirect
288. #2548	C₂×C₆₇	semidirect
289. #2025	C₂×C₆₇	semidirect
290. #2023	C₂×C₆₇	semidirect
291. #1989	C₂×C₆₇	semidirect
292. #1985	C₂×C₆₇	semidirect
293. #1879	C₂×C₆₇	semidirect
294. #1879	C₂×C₆₇	semidirect
295. #1762	C₂×C₆₇	semidirect
296. C₆₇	#25946	direct (N × H)
497. D₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
C₂×#490×C₆₇	2
C₂×#459×C₆₇	2
C₆₇×#2548	2
C₆₇×#2025	2
C₆₇×#2023	2
C₆₇×#1989	2
C₆₇×#1985	2
C₆₇×#1879	2
C₆₇×#1762	2
#25946	67

Subgroups — C67×#25946 #30705197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25946 #30705197