Subgroups — C₆₇×#32929 #30712180

Order 17152.

1708 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#465×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2848	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2847	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1893	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1892	C₂	1	3	yes	maximal, complemented
7	C₆₇×#1807	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₈×C₆₇	—	2	12	no	—
11	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
12	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	3	yes	—
14	C₂×#121×C₆₇	C₂²	1	3	yes	—
15	C₂×#120×C₆₇	—	2	1	no	—
16	#465×C₆₇	—	2	12	no	—
17	C₂×#106×C₆₇	C₂²	1	3	yes	complemented
18	C₂×#103×C₆₇	C₂²	1	3	yes	complemented
19	C₂×#102×C₆₇	C₂²	1	3	yes	complemented
20	#386×C₆₇	—	2	3	no	—
21	#385×C₆₇	—	2	3	no	—
22	#380×C₆₇	C₂²	1	3	yes	—
23	#380×C₆₇	C₂²	1	3	yes	—
24	#379×C₆₇	—	2	1	no	—
25	#340×C₆₇	C₂²	1	6	yes	complemented
26	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×D₄×C₆₇	—	2	3	no	—
29	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	—	2	6	no	—
31	C₂³×C₄×C₆₇	—	2	6	no	—
32	C₂³×C₄×C₆₇	D₄	1	3	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
34	C₂×SD₁₆×C₆₇	—	2	6	no	—
35	C₂²×C₈×C₆₇	—	2	32	no	—
36	C₂²×C₈×C₆₇	—	2	6	no	—
37	C₂²×C₈×C₆₇	—	2	12	no	—
38	C₂²×C₈×C₆₇	—	2	12	no	—
39	C₂²×C₈×C₆₇	D₄	1	12	yes	complemented
40	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Q₈×C₆₇	—	2	6	no	—
43	C₂²⋊Q₈×C₆₇	—	2	6	no	—
44	#121×C₆₇	—	2	6	no	—
45	#121×C₆₇	—	2	6	no	—
46	#120×C₆₇	—	4	4	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
53	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
54	#106×C₆₇	—	2	6	no	—
55	#103×C₆₇	—	2	6	no	—
56	#102×C₆₇	—	2	6	no	—
57	#95×C₆₇	—	2	3	no	—
58	#95×C₆₇	—	2	3	no	—
59	C₂⁴×C₆₇	—	2	6	no	—
60	C₂⁴×C₆₇	—	2	6	no	—
61	C₂⁴×C₆₇	—	2	1	no	—
62	C₂⁴×C₆₇	SD₁₆	1	4	yes	complemented
63	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
64	C₂×Q₈×C₆₇	—	2	3	no	—
65	C₂×Q₈×C₆₇	—	2	3	no	—
66	C₂×D₄×C₆₇	—	2	3	no	—
67	C₂×D₄×C₆₇	—	2	3	no	—
68	C₂×D₄×C₆₇	—	4	6	no	—
69	C₂×D₄×C₆₇	—	4	6	no	—
70	C₂×D₄×C₆₇	—	4	6	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	2	24	no	—
73	C₂²×C₄×C₆₇	—	2	6	no	—
74	C₂²×C₄×C₆₇	—	2	12	no	—
75	C₂²×C₄×C₆₇	—	2	6	no	—
76	C₂²×C₄×C₆₇	—	2	6	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	—	2	3	no	—
79	C₂²×C₄×C₆₇	—	2	3	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
82	C₂²×C₄×C₆₇	#41	1	6	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₆₇	—	4	4	no	—
85	C₂×C₈×C₆₇	—	2	24	no	—
86	C₂×C₈×C₆₇	—	2	6	no	—
87	C₂×C₈×C₆₇	—	2	24	no	—
88	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
90	C₄⋊C₄×C₆₇	—	2	3	no	—
91	C₄⋊C₄×C₆₇	—	2	3	no	—
92	C₄⋊C₄×C₆₇	—	2	3	no	—
93	C₄⋊C₄×C₆₇	—	2	3	no	—
94	C₄⋊C₄×C₆₇	—	4	6	no	—
95	C₂²⋊C₄×C₆₇	—	4	6	no	—
96	C₂²⋊C₄×C₆₇	—	4	6	no	—
97	C₂²⋊C₄×C₆₇	—	4	6	no	—
98	C₂²⋊C₄×C₆₇	—	4	6	no	—
99	C₂³×C₆₇	—	2	8	no	—
100	C₂³×C₆₇	—	2	24	no	—
101	C₂³×C₆₇	—	2	6	no	—
102	C₂³×C₆₇	—	2	6	no	—
103	C₂³×C₆₇	—	2	24	no	—
104	C₂³×C₆₇	—	4	3	no	—
105	C₂³×C₆₇	—	4	1	no	—
106	C₂³×C₆₇	—	4	3	no	—
107	C₂³×C₆₇	#121	1	6	yes	—
108	C₂³×C₆₇	C₂×SD₁₆	1	6	yes	—
109	C₂³×C₆₇	#135	1	6	yes	—
110	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₆₇	—	4	1	no	—
112	Q₈×C₆₇	—	4	3	no	—
113	D₄×C₆₇	—	4	1	no	—
114	D₄×C₆₇	—	4	3	no	—
115	D₄×C₆₇	—	8	12	no	—
116	C₂×C₄×C₆₇	—	2	24	no	—
117	C₂×C₄×C₆₇	—	2	6	no	—
118	C₂×C₄×C₆₇	—	2	6	no	—
119	C₂×C₄×C₆₇	—	2	6	no	—
120	C₂×C₄×C₆₇	—	2	12	no	—
121	C₂×C₄×C₆₇	—	4	3	no	—
122	C₂×C₄×C₆₇	—	4	3	no	—
123	C₂×C₄×C₆₇	—	4	3	no	—
124	C₂×C₄×C₆₇	—	4	3	no	—
125	C₂×C₄×C₆₇	—	4	6	no	—
126	C₂×C₄×C₆₇	—	4	3	no	—
127	C₂×C₄×C₆₇	#121	1	6	yes	—
128	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
129	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
130	C₈×C₆₇	—	2	6	no	—
131	C₈×C₆₇	#120	1	4	yes	complemented
132	C₂²×C₆₇	—	2	32	no	—
133	C₂²×C₆₇	—	2	6	no	—
134	C₂²×C₆₇	—	2	12	no	—
135	C₂²×C₆₇	—	2	6	no	—
136	C₂²×C₆₇	—	2	12	no	—
137	C₂²×C₆₇	—	8	1	no	—
138	C₂²×C₆₇	—	8	3	no	—
139	C₂²×C₆₇	—	8	3	no	—
140	C₂²×C₆₇	#465	1	12	yes	complemented
141	C₂²×C₆₇	C₂×#121	1	3	yes	—
142	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#135	1	3	yes	—
144	C₄×C₆₇	—	2	6	no	—
145	C₄×C₆₇	—	8	1	no	—
146	C₄×C₆₇	—	8	3	no	—
147	C₄×C₆₇	—	8	3	no	—
148	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
149	C₄×C₆₇	#525	1	3	yes	—
150	#32929	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	—	2	12	no	—
152	C₂×C₆₇	—	16	1	no	—
153	C₂×C₆₇	#2847	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#465	1	3	yes	—
155	C₂×C₆₇	#3050	1	3	yes	—
156	C₂⁴×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
157	C₂×#465	C₂×C₆₇	1	3	yes	complemented
158	#2848	C₂×C₆₇	1	1	yes	characteristic, complemented
159	#2847	C₂×C₆₇	1	1	yes	characteristic
160	#1893	C₂×C₆₇	1	3	yes	complemented
161	#1892	C₂×C₆₇	1	3	yes	complemented
162	#1807	C₂×C₆₇	1	3	yes	complemented
163	C₆₇	#32929	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
165	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
166	C₂³×C₈	—	2	12	no	—
167	C₂³×C₈	C₂²×C₆₇	1	3	yes	—
168	C₂³×C₈	C₂²×C₆₇	1	3	yes	—
169	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	3	yes	—
170	C₂×#121	C₂²×C₆₇	1	3	yes	—
171	C₂×#120	—	2	1	no	—
172	#465	—	2	12	no	—
173	C₂×#106	C₂²×C₆₇	1	3	yes	complemented
174	C₂×#103	C₂²×C₆₇	1	3	yes	complemented
175	C₂×#102	C₂²×C₆₇	1	3	yes	complemented
176	#386	—	2	3	no	—
177	#385	—	2	3	no	—
178	#380	C₂²×C₆₇	1	3	yes	—
179	#380	C₂²×C₆₇	1	3	yes	—
180	#379	—	2	1	no	—
181	#340	C₂²×C₆₇	1	6	yes	complemented
182	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
183	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
184	C₂²×D₄	—	2	3	no	—
185	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
186	C₂³×C₄	—	2	6	no	—
187	C₂³×C₄	—	2	6	no	—
188	C₂³×C₄	D₄×C₆₇	1	3	yes	—
189	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
190	C₂×SD₁₆	—	2	6	no	—
191	C₂²×C₈	—	2	32	no	—
192	C₂²×C₈	—	2	6	no	—
193	C₂²×C₈	—	2	12	no	—
194	C₂²×C₈	—	2	12	no	—
195	C₂²×C₈	D₄×C₆₇	1	12	yes	complemented
196	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
197	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
198	C₂²⋊Q₈	—	2	6	no	—
199	C₂²⋊Q₈	—	2	6	no	—
200	#121	—	2	6	no	—
201	#121	—	2	6	no	—
202	#120	—	4	4	no	—
203	C₂×C₄⋊C₄	—	2	3	no	—
204	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
205	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
206	C₂×C₂²⋊C₄	—	2	3	no	—
207	C₂×C₂²⋊C₄	—	2	3	no	—
208	C₂×C₂²⋊C₄	—	2	3	no	—
209	C₂×C₂²⋊C₄	—	2	3	no	—
210	#106	—	2	6	no	—
211	#103	—	2	6	no	—
212	#102	—	2	6	no	—
213	#95	—	2	3	no	—
214	#95	—	2	3	no	—
215	C₂⁴	—	2	6	no	—
216	C₂⁴	—	2	6	no	—
217	C₂⁴	—	2	1	no	—
218	C₂⁴	SD₁₆×C₆₇	1	4	yes	complemented
219	C₂⁴	C₂×D₄×C₆₇	1	3	yes	—
220	C₂×Q₈	—	2	3	no	—
221	C₂×Q₈	—	2	3	no	—
222	C₂×D₄	—	2	3	no	—
223	C₂×D₄	—	2	3	no	—
224	C₂×D₄	—	4	6	no	—
225	C₂×D₄	—	4	6	no	—
226	C₂×D₄	—	4	6	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	24	no	—
229	C₂²×C₄	—	2	6	no	—
230	C₂²×C₄	—	2	12	no	—
231	C₂²×C₄	—	2	6	no	—
232	C₂²×C₄	—	2	6	no	—
233	C₂²×C₄	—	2	3	no	—
234	C₂²×C₄	—	2	3	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
237	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
238	C₂²×C₄	#41×C₆₇	1	6	yes	—
239	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
240	SD₁₆	—	4	4	no	—
241	C₂×C₈	—	2	24	no	—
242	C₂×C₈	—	2	6	no	—
243	C₂×C₈	—	2	24	no	—
244	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
245	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
246	C₄⋊C₄	—	2	3	no	—
247	C₄⋊C₄	—	2	3	no	—
248	C₄⋊C₄	—	2	3	no	—
249	C₄⋊C₄	—	2	3	no	—
250	C₄⋊C₄	—	4	6	no	—
251	C₂²⋊C₄	—	4	6	no	—
252	C₂²⋊C₄	—	4	6	no	—
253	C₂²⋊C₄	—	4	6	no	—
254	C₂²⋊C₄	—	4	6	no	—
255	C₂³	—	2	8	no	—
256	C₂³	—	2	24	no	—
257	C₂³	—	2	24	no	—
258	C₂³	—	2	6	no	—
259	C₂³	—	2	6	no	—
260	C₂³	—	4	3	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	3	no	—
263	C₂³	#121×C₆₇	1	6	yes	—
264	C₂³	C₂×SD₁₆×C₆₇	1	6	yes	—
265	C₂³	#135×C₆₇	1	6	yes	—
266	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
267	Q₈	—	4	1	no	—
268	Q₈	—	4	3	no	—
269	D₄	—	4	1	no	—
270	D₄	—	4	3	no	—
271	D₄	—	8	12	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	6	no	—
275	C₂×C₄	—	2	6	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	4	3	no	—
278	C₂×C₄	—	4	3	no	—
279	C₂×C₄	—	4	3	no	—
280	C₂×C₄	—	4	3	no	—
281	C₂×C₄	—	4	6	no	—
282	C₂×C₄	—	4	3	no	—
283	C₂×C₄	#121×C₆₇	1	6	yes	—
284	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
285	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
286	C₈	—	2	6	no	—
287	C₈	#120×C₆₇	1	4	yes	complemented
288	C₂²	—	2	32	no	—
289	C₂²	—	2	12	no	—
290	C₂²	—	2	6	no	—
291	C₂²	—	2	12	no	—
292	C₂²	—	2	6	no	—
293	C₂²	—	8	3	no	—
294	C₂²	—	8	3	no	—
295	C₂²	—	8	1	no	—
296	C₂²	#465×C₆₇	1	12	yes	complemented
297	C₂²	C₂×#121×C₆₇	1	3	yes	—
298	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
299	C₂²	C₂×#135×C₆₇	1	3	yes	—
300	C₄	—	2	6	no	—
301	C₄	—	8	3	no	—
302	C₄	—	8	3	no	—
303	C₄	—	8	1	no	—
304	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
305	C₄	#525×C₆₇	1	3	yes	—
306	C₂	—	2	12	no	—
307	C₂	—	16	1	no	—
308	C₂	C₆₇×#2847	1	1	yes	characteristic
309	C₂	C₂×#465×C₆₇	1	3	yes	—
310	C₂	C₆₇×#3050	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2847	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₈×C₆₇	C₂	semidirect
2. C₂×#465×C₆₇	C₂	semidirect
3. C₆₇×#2848	C₂	semidirect
5. C₆₇×#1893	C₂	semidirect
6. C₆₇×#1892	C₂	semidirect
7. C₆₇×#1807	C₂	semidirect
9. C₂²×SD₁₆×C₆₇	C₂²	semidirect
17. C₂×#106×C₆₇	C₂²	semidirect
18. C₂×#103×C₆₇	C₂²	semidirect
19. C₂×#102×C₆₇	C₂²	semidirect
25. #340×C₆₇	C₂²	semidirect
39. C₂²×C₈×C₆₇	D₄	semidirect
62. C₂⁴×C₆₇	SD₁₆	semidirect
131. C₈×C₆₇	#120	semidirect
140. C₂²×C₆₇	#465	semidirect
150. #32929	C₆₇	direct (N × H)
156. C₂⁴×C₈	C₂×C₆₇	semidirect
157. C₂×#465	C₂×C₆₇	semidirect
158. #2848	C₂×C₆₇	semidirect
160. #1893	C₂×C₆₇	semidirect
161. #1892	C₂×C₆₇	semidirect
162. #1807	C₂×C₆₇	semidirect
163. C₆₇	#32929	direct (N × H)
165. C₂²×SD₁₆	C₂²×C₆₇	semidirect
173. C₂×#106	C₂²×C₆₇	semidirect
174. C₂×#103	C₂²×C₆₇	semidirect
175. C₂×#102	C₂²×C₆₇	semidirect
181. #340	C₂²×C₆₇	semidirect
195. C₂²×C₈	D₄×C₆₇	semidirect
218. C₂⁴	SD₁₆×C₆₇	semidirect
287. C₈	#120×C₆₇	semidirect
296. C₂²	#465×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₆₇	2
C₂×#465×C₆₇	2
C₆₇×#2848	2
C₆₇×#2847	2
C₆₇×#1893	2
C₆₇×#1892	2
C₆₇×#1807	2
#32929	67

Subgroups — C67×#32929 #30712180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32929 #30712180