Subgroups — C₆₇×#21263 #30700514

Order 17152.

1726 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1983	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1982	C₂	1	4	yes	maximal, complemented
10	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#535×C₆₇	—	2	4	no	—
15	#535×C₆₇	C₄	1	8	yes	complemented
16	C₂×#127×C₆₇	C₂²	1	2	yes	—
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	—	2	2	no	—
19	C₂×#120×C₆₇	—	2	2	no	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	#424×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
25	#418×C₆₇	—	2	4	no	—
26	#417×C₆₇	—	2	4	no	—
27	#416×C₆₇	—	2	4	no	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
29	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#102×C₆₇	—	2	4	no	—
31	C₂×#102×C₆₇	C₂²	1	4	yes	—
32	C₂×#102×C₆₇	C₂²	1	4	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
35	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
37	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
38	C₂⁵×C₆₇	—	2	1	no	—
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	2	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	4	no	—
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂²×C₈×C₆₇	—	2	2	no	—
64	C₂²×C₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	—	2	2	no	—
66	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
68	#127×C₆₇	—	2	4	no	—
69	#127×C₆₇	C₂×C₄	1	8	yes	complemented
70	#124×C₆₇	—	2	4	no	—
71	#124×C₆₇	C₂×C₄	1	8	yes	complemented
72	#121×C₆₇	—	2	16	no	—
73	#120×C₆₇	—	2	16	no	—
74	#117×C₆₇	—	2	2	no	—
75	#117×C₆₇	C₂×C₄	1	4	yes	—
76	#117×C₆₇	D₄	1	4	yes	—
77	#117×C₆₇	D₄	1	4	yes	complemented
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
87	C₂×C₄²×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	—	2	4	no	—
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	C₄⋊C₈×C₆₇	—	2	8	no	—
91	#103×C₆₇	—	2	4	no	—
92	#103×C₆₇	—	2	4	no	—
93	#102×C₆₇	—	2	8	no	—
94	#102×C₆₇	—	2	8	no	—
95	#102×C₆₇	—	2	4	no	—
96	#102×C₆₇	—	2	4	no	—
97	#95×C₆₇	—	2	4	no	—
98	C₂⁴×C₆₇	—	2	2	no	—
99	C₂⁴×C₆₇	—	2	1	no	—
100	C₂⁴×C₆₇	—	2	4	no	—
101	C₂⁴×C₆₇	—	4	2	no	—
102	C₂⁴×C₆₇	—	4	2	no	—
103	C₂⁴×C₆₇	—	8	2	no	—
104	C₂⁴×C₆₇	—	8	2	no	—
105	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
106	#41×C₆₇	—	2	8	no	—
107	#41×C₆₇	—	2	8	no	—
108	#41×C₆₇	—	4	4	no	—
109	#41×C₆₇	—	4	4	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	—	2	16	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	2	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	1	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	1	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	4	2	no	—
132	C₂×D₄×C₆₇	—	4	2	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	4	2	no	—
135	C₂×D₄×C₆₇	—	4	4	no	—
136	C₂×D₄×C₆₇	—	4	4	no	—
137	C₂×D₄×C₆₇	—	8	4	no	—
138	C₂×D₄×C₆₇	—	8	4	no	—
139	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	2	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	8	2	no	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₆₇	#41	1	2	yes	—
156	C₂²×C₄×C₆₇	#41	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	M₁₆×C₆₇	—	4	4	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	4	4	no	—
166	C₄⋊C₄×C₆₇	—	2	4	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
170	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
171	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
172	C₂²⋊C₄×C₆₇	—	2	4	no	—
173	C₂²⋊C₄×C₆₇	—	2	16	no	—
174	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
175	C₄²×C₆₇	—	2	4	no	—
176	C₄²×C₆₇	—	8	4	no	—
177	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
178	C₄²×C₆₇	C₂×D₄	1	4	yes	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	4	no	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	4	1	no	—
184	C₂³×C₆₇	—	4	1	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	2	no	—
187	C₂³×C₆₇	—	4	2	no	—
188	C₂³×C₆₇	—	4	4	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	—	4	4	no	—
191	C₂³×C₆₇	—	4	8	no	—
192	C₂³×C₆₇	—	8	8	no	—
193	C₂³×C₆₇	—	8	4	no	—
194	C₂³×C₆₇	—	8	4	no	—
195	C₂³×C₆₇	—	8	8	no	—
196	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
197	C₂³×C₆₇	#121	1	2	yes	—
198	C₂³×C₆₇	#121	1	2	yes	—
199	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₆₇	—	2	2	no	—
201	Q₈×C₆₇	—	2	2	no	—
202	Q₈×C₆₇	—	4	1	no	—
203	Q₈×C₆₇	—	4	1	no	—
204	D₄×C₆₇	—	2	4	no	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	2	2	no	—
207	D₄×C₆₇	—	2	2	no	—
208	D₄×C₆₇	—	4	8	no	—
209	D₄×C₆₇	—	4	2	no	—
210	D₄×C₆₇	—	4	1	no	—
211	D₄×C₆₇	—	4	2	no	—
212	D₄×C₆₇	—	4	1	no	—
213	D₄×C₆₇	—	4	8	no	—
214	D₄×C₆₇	—	8	16	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	4	1	no	—
221	C₂×C₄×C₆₇	—	4	1	no	—
222	C₂×C₄×C₆₇	—	4	2	no	—
223	C₂×C₄×C₆₇	—	4	4	no	—
224	C₂×C₄×C₆₇	—	4	2	no	—
225	C₂×C₄×C₆₇	—	4	8	no	—
226	C₂×C₄×C₆₇	—	8	4	no	—
227	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
228	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
230	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
231	C₂×C₄×C₆₇	#121	1	2	yes	—
232	C₂×C₄×C₆₇	#121	1	2	yes	—
233	C₂×C₄×C₆₇	#124	1	4	yes	—
234	C₂×C₄×C₆₇	#127	1	4	yes	—
235	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
241	C₈×C₆₇	—	4	2	no	—
242	C₈×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	2	4	no	—
244	C₂²×C₆₇	—	2	2	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	4	no	—
247	C₂²×C₆₇	—	4	4	no	—
248	C₂²×C₆₇	—	4	8	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	4	no	—
251	C₂²×C₆₇	—	8	16	no	—
252	C₂²×C₆₇	—	8	1	no	—
253	C₂²×C₆₇	—	8	2	no	—
254	C₂²×C₆₇	—	8	1	no	—
255	C₂²×C₆₇	—	8	2	no	—
256	C₂²×C₆₇	—	8	2	no	—
257	C₂²×C₆₇	—	8	4	no	—
258	C₂²×C₆₇	#390	1	4	yes	—
259	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
260	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
261	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
262	C₂²×C₆₇	#576	1	2	yes	—
263	C₂²×C₆₇	#577	1	2	yes	—
264	C₄×C₆₇	—	2	8	no	—
265	C₄×C₆₇	—	4	4	no	—
266	C₄×C₆₇	—	4	4	no	—
267	C₄×C₆₇	—	4	2	no	—
268	C₄×C₆₇	#390	1	4	yes	—
269	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
270	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
271	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
272	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
273	#21263	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂×C₆₇	—	2	4	no	—
275	C₂×C₆₇	—	8	2	no	—
276	C₂×C₆₇	—	8	4	no	—
277	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
278	C₂×C₆₇	#2897	1	2	yes	—
279	C₂×C₆₇	#3041	1	1	yes	characteristic
280	C₂×C₆₇	#3043	1	1	yes	characteristic
281	C₂×C₆₇	#3045	1	2	yes	—
282	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
283	C₂×#424	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₂×#417	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
288	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#1983	C₂×C₆₇	1	4	yes	complemented
290	#1982	C₂×C₆₇	1	4	yes	complemented
291	C₆₇	#21263	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
293	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
294	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
295	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
296	#535	—	2	4	no	—
297	#535	C₄×C₆₇	1	8	yes	complemented
298	C₂×#127	C₂²×C₆₇	1	2	yes	—
299	C₂×#124	C₂²×C₆₇	1	2	yes	—
300	C₂×#121	—	2	2	no	—
301	C₂×#120	—	2	2	no	—
302	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
304	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
305	#424	—	2	4	no	—
306	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
307	#418	—	2	4	no	—
308	#417	—	2	4	no	—
309	#416	—	2	4	no	—
310	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
311	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
312	C₂×#102	—	2	4	no	—
313	C₂×#102	C₂²×C₆₇	1	4	yes	—
314	C₂×#102	C₂²×C₆₇	1	4	yes	—
315	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
316	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
317	C₄×C₄⋊C₄	—	2	2	no	—
318	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
319	C₄×C₂²⋊C₄	—	2	2	no	—
320	C₂⁵	—	2	1	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	2	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
327	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	—	2	2	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	—	2	4	no	—
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
337	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
338	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂×M₁₆	—	2	2	no	—
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	—	2	2	no	—
345	C₂²×C₈	—	2	2	no	—
346	C₂²×C₈	—	2	2	no	—
347	C₂²×C₈	—	2	2	no	—
348	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
349	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
350	#127	—	2	4	no	—
351	#127	C₂×C₄×C₆₇	1	8	yes	complemented
352	#124	—	2	4	no	—
353	#124	C₂×C₄×C₆₇	1	8	yes	complemented
354	#121	—	2	16	no	—
355	#120	—	2	16	no	—
356	#117	—	2	2	no	—
357	#117	C₂×C₄×C₆₇	1	4	yes	—
358	#117	D₄×C₆₇	1	4	yes	—
359	#117	D₄×C₆₇	1	4	yes	complemented
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	—	2	2	no	—
362	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
363	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
366	C₂×C₂²⋊C₄	—	2	1	no	—
367	C₂×C₂²⋊C₄	—	2	4	no	—
368	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	—	2	4	no	—
371	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
372	C₄⋊C₈	—	2	8	no	—
373	#103	—	2	4	no	—
374	#103	—	2	4	no	—
375	#102	—	2	8	no	—
376	#102	—	2	8	no	—
377	#102	—	2	4	no	—
378	#102	—	2	4	no	—
379	#95	—	2	4	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	2	1	no	—
382	C₂⁴	—	2	4	no	—
383	C₂⁴	—	4	2	no	—
384	C₂⁴	—	4	2	no	—
385	C₂⁴	—	8	2	no	—
386	C₂⁴	—	8	2	no	—
387	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
388	#41	—	2	8	no	—
389	#41	—	2	8	no	—
390	#41	—	4	4	no	—
391	#41	—	4	4	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	16	no	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	4	2	no	—
414	C₂×D₄	—	4	2	no	—
415	C₂×D₄	—	4	2	no	—
416	C₂×D₄	—	4	2	no	—
417	C₂×D₄	—	4	4	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	—	8	4	no	—
420	C₂×D₄	—	8	4	no	—
421	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
422	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	4	2	no	—
429	C₂²×C₄	—	4	2	no	—
430	C₂²×C₄	—	8	2	no	—
431	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
432	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
437	C₂²×C₄	#41×C₆₇	1	2	yes	—
438	C₂²×C₄	#41×C₆₇	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
440	M₁₆	—	4	4	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	4	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
452	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
453	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	16	no	—
456	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
457	C₄²	—	2	4	no	—
458	C₄²	—	8	4	no	—
459	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
460	C₄²	C₂×D₄×C₆₇	1	4	yes	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	2	4	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	4	1	no	—
466	C₂³	—	4	1	no	—
467	C₂³	—	4	2	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	2	no	—
472	C₂³	—	4	4	no	—
473	C₂³	—	4	8	no	—
474	C₂³	—	8	8	no	—
475	C₂³	—	8	4	no	—
476	C₂³	—	8	4	no	—
477	C₂³	—	8	8	no	—
478	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
479	C₂³	#121×C₆₇	1	2	yes	—
480	C₂³	#121×C₆₇	1	2	yes	—
481	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
482	Q₈	—	2	2	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	4	1	no	—
485	Q₈	—	4	1	no	—
486	D₄	—	2	4	no	—
487	D₄	—	2	4	no	—
488	D₄	—	2	2	no	—
489	D₄	—	2	2	no	—
490	D₄	—	4	8	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	1	no	—
493	D₄	—	4	2	no	—
494	D₄	—	4	1	no	—
495	D₄	—	4	8	no	—
496	D₄	—	8	16	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	8	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
510	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
511	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
512	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
513	C₂×C₄	#121×C₆₇	1	2	yes	—
514	C₂×C₄	#121×C₆₇	1	2	yes	—
515	C₂×C₄	#124×C₆₇	1	4	yes	—
516	C₂×C₄	#127×C₆₇	1	4	yes	—
517	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
518	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
521	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
523	C₈	—	4	2	no	—
524	C₈	—	4	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	8	no	—
533	C₂²	—	8	16	no	—
534	C₂²	—	8	2	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	2	no	—
537	C₂²	—	8	1	no	—
538	C₂²	—	8	4	no	—
539	C₂²	—	8	2	no	—
540	C₂²	#390×C₆₇	1	4	yes	—
541	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
542	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
543	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
544	C₂²	#576×C₆₇	1	2	yes	—
545	C₂²	#577×C₆₇	1	2	yes	—
546	C₄	—	2	8	no	—
547	C₄	—	4	4	no	—
548	C₄	—	4	4	no	—
549	C₄	—	4	2	no	—
550	C₄	#390×C₆₇	1	4	yes	—
551	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
552	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
553	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
554	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
555	C₂	—	2	4	no	—
556	C₂	—	8	4	no	—
557	C₂	—	8	2	no	—
558	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
559	C₂	C₆₇×#2897	1	2	yes	—
560	C₂	C₆₇×#3041	1	1	yes	characteristic
561	C₂	C₆₇×#3043	1	1	yes	characteristic
562	C₂	C₆₇×#3045	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#535×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#424×C₆₇	C₂	semidirect
3. C₂×#417×C₆₇	C₂	semidirect
4. C₂×#418×C₆₇	C₂	semidirect
5. C₂×#416×C₆₇	C₂	semidirect
6. C₂²×#102×C₆₇	C₂	semidirect
7. C₆₇×#2274	C₂	semidirect
8. C₆₇×#1983	C₂	semidirect
9. C₆₇×#1982	C₂	semidirect
15. #535×C₆₇	C₄	semidirect
24. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
28. C₂×#103×C₆₇	C₂²	semidirect
29. C₂×#103×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
34. C₂×#102×C₆₇	C₂²	semidirect
36. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
69. #127×C₆₇	C₂×C₄	semidirect
71. #124×C₆₇	C₂×C₄	semidirect
77. #117×C₆₇	D₄	semidirect
81. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
273. #21263	C₆₇	direct (N × H)
283. C₂×#424	C₂×C₆₇	semidirect
284. C₂×#417	C₂×C₆₇	semidirect
285. C₂×#418	C₂×C₆₇	semidirect
286. C₂×#416	C₂×C₆₇	semidirect
287. C₂²×#102	C₂×C₆₇	semidirect
288. #2274	C₂×C₆₇	semidirect
289. #1983	C₂×C₆₇	semidirect
290. #1982	C₂×C₆₇	semidirect
291. C₆₇	#21263	direct (N × H)
297. #535	C₄×C₆₇	semidirect
306. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
310. C₂×#103	C₂²×C₆₇	semidirect
311. C₂×#103	C₂²×C₆₇	semidirect
315. C₂×#102	C₂²×C₆₇	semidirect
316. C₂×#102	C₂²×C₆₇	semidirect
318. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
351. #127	C₂×C₄×C₆₇	semidirect
353. #124	C₂×C₄×C₆₇	semidirect
359. #117	D₄×C₆₇	semidirect
363. C₂×C₄⋊C₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₆₇	2
C₂×#424×C₆₇	2
C₂×#417×C₆₇	2
C₂×#418×C₆₇	2
C₂×#416×C₆₇	2
C₂²×#102×C₆₇	2
C₆₇×#2274	2
C₆₇×#1983	2
C₆₇×#1982	2
#21263	67

Subgroups — C67×#21263 #30700514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21263 #30700514