Subgroups — C₆₇×#17503 #30696754

Order 17152.

1066 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1494	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1489	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1485	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1484	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₆₇	—	2	1	no	—
9	#533×C₆₇	C₄	1	1	yes	characteristic
10	#533×C₆₇	C₄	1	1	yes	characteristic
11	#533×C₆₇	C₂²	1	1	yes	characteristic
12	#532×C₆₇	C₄	1	1	yes	characteristic
13	#532×C₆₇	C₄	1	1	yes	characteristic
14	#532×C₆₇	C₂²	1	1	yes	characteristic
15	#529×C₆₇	—	2	1	no	—
16	#529×C₆₇	—	2	1	no	—
17	#529×C₆₇	—	2	1	no	—
18	#529×C₆₇	—	2	1	no	—
19	#529×C₆₇	C₄	1	2	yes	—
20	#529×C₆₇	C₂²	1	2	yes	—
21	#525×C₆₇	—	2	1	no	—
22	#525×C₆₇	—	2	1	no	—
23	#525×C₆₇	C₄	1	2	yes	—
24	C₄×#41×C₆₇	—	2	1	no	—
25	C₄×#41×C₆₇	—	2	1	no	—
26	C₄×#41×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#432×C₆₇	—	2	1	no	—
29	#432×C₆₇	C₂²	1	2	yes	—
30	#431×C₆₇	—	2	1	no	—
31	#431×C₆₇	C₂²	1	2	yes	—
32	#423×C₆₇	C₂²	1	2	yes	—
33	#407×C₆₇	—	2	1	no	—
34	C₄×M₁₆×C₆₇	—	2	1	no	—
35	C₄×M₁₆×C₆₇	—	2	1	no	—
36	C₄×M₁₆×C₆₇	—	2	1	no	—
37	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
38	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
39	C₂×#97×C₆₇	—	2	1	no	—
40	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	8	yes	complemented
42	D₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	—	2	8	no	—
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
58	#126×C₆₇	—	2	2	no	—
59	#126×C₆₇	—	2	2	no	—
60	#125×C₆₇	—	2	1	no	—
61	#125×C₆₇	—	2	1	no	—
62	#125×C₆₇	C₂×C₄	1	2	yes	—
63	#124×C₆₇	—	2	1	no	—
64	#124×C₆₇	—	2	1	no	—
65	#124×C₆₇	C₂×C₄	1	2	yes	—
66	#123×C₆₇	—	2	2	no	—
67	#123×C₆₇	—	2	2	no	—
68	#123×C₆₇	—	2	2	no	—
69	C₂²⋊Q₈×C₆₇	—	2	1	no	—
70	C₂²⋊Q₈×C₆₇	—	2	1	no	—
71	C₂²⋊Q₈×C₆₇	—	2	1	no	—
72	C₂²⋊Q₈×C₆₇	—	2	1	no	—
73	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
74	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
75	#121×C₆₇	—	2	1	no	—
76	#121×C₆₇	—	2	1	no	—
77	#121×C₆₇	—	2	1	no	—
78	#121×C₆₇	—	2	1	no	—
79	#121×C₆₇	C₂×C₄	1	2	yes	—
80	#121×C₆₇	C₂×C₄	1	2	yes	—
81	#120×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	1	no	—
83	C₄×Q₈×C₆₇	—	2	1	no	—
84	C₄×Q₈×C₆₇	—	2	1	no	—
85	C₄×Q₈×C₆₇	—	2	1	no	—
86	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
87	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	2	2	no	—
92	C₄×D₄×C₆₇	—	2	2	no	—
93	C₄×D₄×C₆₇	—	2	2	no	—
94	C₄×D₄×C₆₇	—	2	1	no	—
95	C₄×D₄×C₆₇	—	2	1	no	—
96	C₄×D₄×C₆₇	—	2	1	no	—
97	C₄×D₄×C₆₇	—	2	1	no	—
98	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
99	C₄×D₄×C₆₇	C₂³	1	2	yes	—
100	#117×C₆₇	—	2	1	no	—
101	#117×C₆₇	—	2	1	no	—
102	#117×C₆₇	—	2	2	no	—
103	#117×C₆₇	—	2	2	no	—
104	#117×C₆₇	—	2	2	no	—
105	#117×C₆₇	C₂×C₄	1	2	yes	—
106	C₂×C₄²×C₆₇	—	2	2	no	—
107	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₄⋊C₈×C₆₇	—	2	2	no	—
120	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
121	C₂²⋊C₈×C₆₇	—	2	2	no	—
122	C₂²⋊C₈×C₆₇	—	2	2	no	—
123	#97×C₆₇	—	2	2	no	—
124	#97×C₆₇	—	2	2	no	—
125	#97×C₆₇	—	2	2	no	—
126	C₄×C₈×C₆₇	—	2	2	no	—
127	C₄×C₈×C₆₇	—	2	2	no	—
128	C₄×C₈×C₆₇	C₂³	1	4	yes	—
129	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
130	#41×C₆₇	—	4	4	no	—
131	#41×C₆₇	—	4	4	no	—
132	#41×C₆₇	—	4	4	no	—
133	C₂×Q₈×C₆₇	—	2	1	no	—
134	C₂×Q₈×C₆₇	—	2	1	no	—
135	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	—	2	2	no	—
138	C₂×D₄×C₆₇	—	2	2	no	—
139	C₂×D₄×C₆₇	—	2	1	no	—
140	C₂×D₄×C₆₇	—	2	1	no	—
141	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	M₁₆×C₆₇	—	2	8	no	—
167	M₁₆×C₆₇	—	2	8	no	—
168	M₁₆×C₆₇	—	4	4	no	—
169	C₂×C₈×C₆₇	—	2	8	no	—
170	C₂×C₈×C₆₇	—	2	4	no	—
171	C₂×C₈×C₆₇	—	2	4	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	1	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
183	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
184	C₂²⋊C₄×C₆₇	—	2	2	no	—
185	C₂²⋊C₄×C₆₇	—	2	2	no	—
186	C₂²⋊C₄×C₆₇	—	2	2	no	—
187	C₂²⋊C₄×C₆₇	—	2	2	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	2	2	no	—
190	C₂²⋊C₄×C₆₇	—	2	2	no	—
191	C₂²⋊C₄×C₆₇	—	2	2	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	—	2	2	no	—
195	C₄²×C₆₇	—	2	2	no	—
196	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
197	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
198	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
203	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	4	1	no	—
215	C₂³×C₆₇	—	4	1	no	—
216	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	Q₈×C₆₇	—	2	4	no	—
220	Q₈×C₆₇	—	4	2	no	—
221	Q₈×C₆₇	—	4	2	no	—
222	D₄×C₆₇	—	2	4	no	—
223	D₄×C₆₇	—	4	2	no	—
224	D₄×C₆₇	—	4	4	no	—
225	D₄×C₆₇	—	4	4	no	—
226	D₄×C₆₇	—	4	4	no	—
227	D₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	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	—	2	2	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	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	4	no	—
249	C₂×C₄×C₆₇	—	4	4	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	#104	1	4	yes	—
252	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
264	C₈×C₆₇	—	2	8	no	—
265	C₈×C₆₇	—	4	4	no	—
266	C₂²×C₆₇	—	2	1	no	—
267	C₂²×C₆₇	—	2	1	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	2	1	no	—
271	C₂²×C₆₇	—	2	1	no	—
272	C₂²×C₆₇	—	2	1	no	—
273	C₂²×C₆₇	—	2	1	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	4	2	no	—
277	C₂²×C₆₇	—	4	2	no	—
278	C₂²×C₆₇	C₂×#104	1	2	yes	—
279	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
280	C₄×C₆₇	—	2	1	no	—
281	C₄×C₆₇	—	2	1	no	—
282	C₄×C₆₇	—	2	1	no	—
283	C₄×C₆₇	—	2	1	no	—
284	C₄×C₆₇	—	2	1	no	—
285	C₄×C₆₇	—	2	1	no	—
286	C₄×C₆₇	—	2	1	no	—
287	C₄×C₆₇	—	2	1	no	—
288	C₄×C₆₇	—	4	2	no	—
289	C₄×C₆₇	—	4	2	no	—
290	C₄×C₆₇	—	4	2	no	—
291	C₄×C₆₇	C₂×#104	1	2	yes	—
292	C₄×C₆₇	C₂×#104	1	2	yes	—
293	C₄×C₆₇	#568	1	2	yes	—
294	C₄×C₆₇	#575	1	2	yes	—
295	#17503	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	2	no	—
297	C₂×C₆₇	—	4	1	no	—
298	C₂×C₆₇	—	4	1	no	—
299	C₂×C₆₇	—	8	2	no	—
300	C₂×C₆₇	#2879	1	1	yes	characteristic
301	C₂×C₆₇	#2896	1	1	yes	characteristic
302	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
303	#3444	C₂×C₆₇	1	1	yes	characteristic
304	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
306	#1494	C₂×C₆₇	1	4	yes	complemented
307	#1489	C₂×C₆₇	1	4	yes	complemented
308	#1485	C₂×C₆₇	1	2	yes	complemented
309	#1484	C₂×C₆₇	1	2	yes	complemented
310	C₆₇	#17503	1	1	yes	characteristic, direct product factor, Sylow
311	C₂²×M₁₆	—	2	1	no	—
312	#533	C₄×C₆₇	1	1	yes	characteristic
313	#533	C₄×C₆₇	1	1	yes	characteristic
314	#533	C₂²×C₆₇	1	1	yes	characteristic
315	#532	C₄×C₆₇	1	1	yes	characteristic
316	#532	C₄×C₆₇	1	1	yes	characteristic
317	#532	C₂²×C₆₇	1	1	yes	characteristic
318	#529	—	2	1	no	—
319	#529	—	2	1	no	—
320	#529	—	2	1	no	—
321	#529	—	2	1	no	—
322	#529	C₄×C₆₇	1	2	yes	—
323	#529	C₂²×C₆₇	1	2	yes	—
324	#525	—	2	1	no	—
325	#525	—	2	1	no	—
326	#525	C₄×C₆₇	1	2	yes	—
327	C₄×#41	—	2	1	no	—
328	C₄×#41	—	2	1	no	—
329	C₄×#41	C₂²×C₆₇	1	2	yes	—
330	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
331	#432	—	2	1	no	—
332	#432	C₂²×C₆₇	1	2	yes	—
333	#431	—	2	1	no	—
334	#431	C₂²×C₆₇	1	2	yes	—
335	#423	C₂²×C₆₇	1	2	yes	—
336	#407	—	2	1	no	—
337	C₄×M₁₆	—	2	1	no	—
338	C₄×M₁₆	—	2	1	no	—
339	C₄×M₁₆	—	2	1	no	—
340	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
341	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
342	C₂×#97	—	2	1	no	—
343	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
344	Q₈⋊C₈	C₂²×C₆₇	1	8	yes	complemented
345	D₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
346	C₂×#41	—	2	1	no	—
347	C₂×#41	—	2	1	no	—
348	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
349	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
350	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
351	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
352	C₂×M₁₆	—	2	8	no	—
353	C₂×M₁₆	—	2	2	no	—
354	C₂×M₁₆	—	2	2	no	—
355	C₂×M₁₆	—	2	2	no	—
356	C₂²×C₈	—	2	2	no	—
357	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
359	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
360	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
361	#126	—	2	2	no	—
362	#126	—	2	2	no	—
363	#125	—	2	1	no	—
364	#125	—	2	1	no	—
365	#125	C₂×C₄×C₆₇	1	2	yes	—
366	#124	—	2	1	no	—
367	#124	—	2	1	no	—
368	#124	C₂×C₄×C₆₇	1	2	yes	—
369	#123	—	2	2	no	—
370	#123	—	2	2	no	—
371	#123	—	2	2	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
377	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
378	#121	—	2	1	no	—
379	#121	—	2	1	no	—
380	#121	—	2	1	no	—
381	#121	—	2	1	no	—
382	#121	C₂×C₄×C₆₇	1	2	yes	—
383	#121	C₂×C₄×C₆₇	1	2	yes	—
384	#120	—	2	2	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
390	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	—	2	1	no	—
399	C₄×D₄	—	2	1	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
402	C₄×D₄	C₂³×C₆₇	1	2	yes	—
403	#117	—	2	1	no	—
404	#117	—	2	1	no	—
405	#117	—	2	2	no	—
406	#117	—	2	2	no	—
407	#117	—	2	2	no	—
408	#117	C₂×C₄×C₆₇	1	2	yes	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
413	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
415	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
418	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
422	C₄⋊C₈	—	2	2	no	—
423	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
424	C₂²⋊C₈	—	2	2	no	—
425	C₂²⋊C₈	—	2	2	no	—
426	#97	—	2	2	no	—
427	#97	—	2	2	no	—
428	#97	—	2	2	no	—
429	C₄×C₈	—	2	2	no	—
430	C₄×C₈	—	2	2	no	—
431	C₄×C₈	C₂³×C₆₇	1	4	yes	—
432	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
433	#41	—	4	4	no	—
434	#41	—	4	4	no	—
435	#41	—	4	4	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
458	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
459	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
469	M₁₆	—	2	8	no	—
470	M₁₆	—	2	8	no	—
471	M₁₆	—	4	4	no	—
472	C₂×C₈	—	2	8	no	—
473	C₂×C₈	—	2	4	no	—
474	C₂×C₈	—	2	4	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
486	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	2	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
500	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
501	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
502	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
503	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
504	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
505	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
506	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
507	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
508	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
509	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
510	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
514	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
515	C₂³	—	2	2	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
520	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
521	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	4	2	no	—
524	Q₈	—	4	2	no	—
525	D₄	—	2	4	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	4	no	—
530	D₄	—	4	2	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	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	#104×C₆₇	1	4	yes	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
556	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
557	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
558	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
559	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
560	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
561	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
562	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
563	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
564	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
567	C₈	—	2	8	no	—
568	C₈	—	4	4	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	1	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	4	4	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	C₂×#104×C₆₇	1	2	yes	—
582	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	2	no	—
594	C₄	C₂×#104×C₆₇	1	2	yes	—
595	C₄	C₂×#104×C₆₇	1	2	yes	—
596	C₄	#568×C₆₇	1	2	yes	—
597	C₄	#575×C₆₇	1	2	yes	—
598	C₂	—	2	2	no	—
599	C₂	—	4	1	no	—
600	C₂	—	4	1	no	—
601	C₂	—	8	2	no	—
602	C₂	C₆₇×#2879	1	1	yes	characteristic
603	C₂	C₆₇×#2896	1	1	yes	characteristic
604	C₂	C₂²×#104×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2920	C₂	semidirect
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1494	C₂	semidirect
5. C₆₇×#1489	C₂	semidirect
6. C₆₇×#1485	C₂	semidirect
7. C₆₇×#1484	C₂	semidirect
41. Q₈⋊C₈×C₆₇	C₂²	semidirect
42. D₄⋊C₈×C₆₇	C₂²	semidirect
295. #17503	C₆₇	direct (N × H)
304. #2920	C₂×C₆₇	semidirect
305. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
306. #1494	C₂×C₆₇	semidirect
307. #1489	C₂×C₆₇	semidirect
308. #1485	C₂×C₆₇	semidirect
309. #1484	C₂×C₆₇	semidirect
310. C₆₇	#17503	direct (N × H)
344. Q₈⋊C₈	C₂²×C₆₇	semidirect
345. D₄⋊C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3444	2
C₆₇×#2920	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#1494	2
C₆₇×#1489	2
C₆₇×#1485	2
C₆₇×#1484	2
#17503	67

Subgroups — C67×#17503 #30696754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17503 #30696754