Subgroups — #31009204

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#22569072	C₂	1	2	yes	maximal
3	#22562078	C₂	1	2	yes	maximal, complemented
4	#22562077	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562067	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561990	C₂	1	1	yes	maximal, characteristic
9	#22561927	C₂	1	2	yes	maximal
10	#22561804	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₂²	1	2	yes	—
13	#16863872	—	2	2	no	—
14	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
15	C₂×#14523989	C₂²	1	2	yes	—
16	Q₃₂×D₆₇	C₄	1	8	yes	complemented
17	C₂×#14523982	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	#16863501	—	2	2	no	—
20	#16863500	—	2	2	no	—
21	C₄×Dic₂₆₈	C₂²	1	2	yes	—
22	Q₁₆×Dic₆₇	C₂²	1	2	yes	—
23	#16863490	C₂²	1	2	yes	—
24	#16863433	C₂²	1	2	yes	—
25	#16863403	—	2	2	no	—
26	#16863350	C₂²	1	1	yes	characteristic
27	#16863223	C₂²	1	2	yes	—
28	#16863222	C₂²	1	1	yes	characteristic
29	#16863201	C₂²	1	2	yes	—
30	#16863200	C₂²	1	2	yes	—
31	#16863199	C₂²	1	2	yes	—
32	#16863172	C₂²	1	1	yes	characteristic
33	#16863171	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₆₇	—	2	2	no	—
41	C₂×Dic₂₆₈	C₂³	1	2	yes	—
42	C₂×#11761618	—	2	2	no	—
43	Q₁₆×D₆₇	—	2	4	no	—
44	Q₁₆×D₆₇	C₂×C₄	1	8	yes	—
45	C₂×#11761611	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	Q₈×Dic₆₇	—	2	2	no	—
48	C₄×Dic₁₃₄	—	2	2	no	—
49	C₄×Dic₁₃₄	—	4	2	no	—
50	#14524008	—	2	2	no	—
51	#14524008	—	4	2	no	—
52	#14524001	—	2	2	no	—
53	#14524001	—	4	2	no	—
54	#14524000	—	4	2	no	—
55	#14524000	D₄	1	1	yes	characteristic
56	Dic₅₃₆	C₂×C₄	1	4	yes	—
57	#14523989	C₂×C₄	1	8	yes	—
58	#14523982	—	2	2	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	—
62	#14523977	C₂³	1	2	yes	—
63	#14523974	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523967	—	2	2	no	—
66	#14523966	—	2	2	no	—
67	#14523954	D₄	1	1	yes	characteristic
68	#14523953	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
71	C₄×Q₈×C₆₇	—	2	2	no	—
72	Q₃₂×C₆₇	C₂×C₄	1	4	yes	—
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₃₄	—	2	2	no	—
79	C₂×Dic₁₃₄	—	4	2	no	—
80	Q₈×D₆₇	—	2	8	no	—
81	Q₈×D₆₇	—	4	4	no	—
82	C₂×C₄×D₆₇	—	4	2	no	—
83	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
84	Dic₂₆₈	—	2	2	no	—
85	Dic₂₆₈	C₂²×C₄	1	4	yes	—
86	#11761618	—	2	8	no	—
87	#11761611	—	2	2	no	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₆₇	C₂²×C₄	1	2	yes	—
91	C₈×D₆₇	#41	1	2	yes	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	4	2	no	—
94	Dic₆₇⋊C₄	—	4	2	no	—
95	Dic₆₇⋊C₄	—	4	2	no	—
96	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
97	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
98	(C₂×D₆₇)⋊C₄	—	4	2	no	—
99	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
100	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	—	4	2	no	—
103	C₄×Dic₆₇	—	4	2	no	—
104	C₄×Dic₆₇	—	4	2	no	—
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	Q₁₆×C₆₇	—	2	2	no	—
110	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄²×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₆₇	—	2	4	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	4	2	no	—
129	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
132	C₄×D₆₇	—	4	8	no	—
133	C₄×D₆₇	C₄×D₄	1	2	yes	—
134	C₄×D₆₇	#135	1	2	yes	—
135	Dic₁₃₄	—	2	4	no	—
136	Dic₁₃₄	—	4	2	no	—
137	Dic₁₃₄	—	4	8	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₆₇⋊C₈	—	2	1	no	—
142	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
143	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₆₇	—	2	1	no	—
145	C₂×D₆₇	—	2	1	no	—
146	C₂×D₆₇	C₄×D₈	1	2	yes	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	4	4	no	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
153	Dic₆₇	—	2	1	no	—
154	Dic₆₇	—	2	1	no	—
155	Dic₆₇	—	4	4	no	—
156	Dic₆₇	—	8	2	no	—
157	Dic₆₇	C₄×D₈	1	2	yes	—
158	#33458	—	67	1	no	maximal, Sylow
159	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
160	C₂×C₆₇	#3417	1	1	yes	characteristic
161	C₂×C₆₇	#3417	1	1	yes	characteristic
162	D₆₇	—	2	2	no	—
163	C₂²×Q₃₂	—	67	1	no	—
164	#2950	—	67	2	no	—
165	#2176	—	67	2	no	—
166	#2176	—	67	2	no	—
167	C₄×Q₃₂	—	67	1	no	—
168	C₄×Q₃₂	—	67	1	no	—
169	C₄×Q₃₂	—	67	1	no	—
170	C₄×Q₃₂	D₆₇	1	1	yes	characteristic, complemented
171	#2158	—	67	1	no	—
172	#2141	—	67	2	no	—
173	#2108	—	67	1	no	—
174	C₆₇	#33458	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	67	2	no	—
176	#519	—	134	2	no	—
177	C₂×Q₃₂	—	67	2	no	—
178	C₂×Q₃₂	—	67	8	no	—
179	C₂×Q₃₂	—	67	1	no	—
180	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
181	C₂×M₃₂	—	67	1	no	—
182	C₂²×C₁₆	—	67	1	no	—
183	#441	—	134	2	no	—
184	#441	—	134	2	no	—
185	C₄×Q₁₆	—	67	2	no	—
186	C₄×Q₁₆	—	67	2	no	—
187	C₄×Q₁₆	—	67	2	no	—
188	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
189	#427	—	67	2	no	—
190	#419	—	134	2	no	—
191	#405	—	67	1	no	—
192	#367	—	67	2	no	—
193	#366	—	67	1	no	—
194	#366	C₂×D₆₇	1	1	yes	characteristic
195	#358	—	67	2	no	—
196	#358	—	67	2	no	—
197	#358	—	67	2	no	—
198	#358	C₂×D₆₇	1	2	yes	—
199	#346	—	67	1	no	—
200	#346	—	67	1	no	—
201	C₄×C₁₆	—	67	1	no	—
202	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
203	C₂²×Q₈	—	134	2	no	—
204	C₂×Q₁₆	—	67	8	no	—
205	C₂×Q₁₆	—	67	2	no	—
206	C₂×Q₁₆	—	134	4	no	—
207	C₂×Q₁₆	—	134	2	no	—
208	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
209	C₂×M₁₆	—	67	1	no	—
210	C₂²×C₈	—	67	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	268	2	no	—
213	C₄×Q₈	—	268	2	no	—
214	C₄×Q₈	—	134	2	no	—
215	C₄×Q₈	—	134	2	no	—
216	C₄×Q₈	—	134	2	no	—
217	#117	—	268	2	no	—
218	#117	—	268	2	no	—
219	#117	—	67	1	no	—
220	#117	—	134	2	no	—
221	Q₃₂	—	67	8	no	—
222	Q₃₂	—	67	4	no	—
223	Q₃₂	C₄×D₆₇	1	4	yes	—
224	M₃₂	—	134	2	no	—
225	C₂×C₁₆	—	67	1	no	—
226	C₂×C₁₆	—	67	4	no	—
227	C₂×C₁₆	—	67	1	no	—
228	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
230	#107	—	67	2	no	—
231	#107	C₂²×D₆₇	1	2	yes	—
232	#106	—	134	2	no	—
233	#103	—	2	2	no	—
234	#103	—	134	2	no	—
235	#103	—	134	2	no	—
236	#103	—	134	2	no	—
237	#97	—	67	1	no	—
238	#97	—	67	1	no	—
239	C₄×C₈	—	67	1	no	—
240	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	268	2	no	—
243	C₂×Q₈	—	268	4	no	—
244	C₂×Q₈	—	134	8	no	—
245	C₂×Q₈	—	134	2	no	—
246	C₂²×C₄	—	268	2	no	—
247	C₂²×C₄	—	67	1	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	67	4	no	—
250	Q₁₆	—	134	8	no	—
251	Q₁₆	—	134	2	no	—
252	Q₁₆	C₂×C₄×D₆₇	1	4	yes	—
253	M₁₆	—	134	2	no	—
254	C₂×C₈	—	67	1	no	—
255	C₂×C₈	—	67	2	no	—
256	C₂×C₈	—	67	2	no	—
257	C₂×C₈	—	67	1	no	—
258	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	268	2	no	—
263	C₄⋊C₄	—	268	2	no	—
264	C₄⋊C₄	—	268	2	no	—
265	C₄⋊C₄	—	67	1	no	—
266	C₄⋊C₄	—	67	1	no	—
267	C₄⋊C₄	—	134	2	no	—
268	C₂²⋊C₄	—	268	2	no	—
269	C₂²⋊C₄	—	67	1	no	—
270	C₂²⋊C₄	—	67	1	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	268	2	no	—
273	C₄²	—	268	2	no	—
274	C₄²	—	268	2	no	—
275	C₄²	—	268	2	no	—
276	C₄²	—	67	1	no	—
277	C₄²	D₄×D₆₇	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	67	2	no	—
280	C₁₆	—	134	1	no	—
281	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
282	C₂³	—	67	1	no	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	268	8	no	—
286	Q₈	—	268	2	no	—
287	Q₈	—	134	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	268	8	no	—
291	C₂×C₄	—	268	2	no	—
292	C₂×C₄	—	268	2	no	—
293	C₂×C₄	—	67	1	no	—
294	C₂×C₄	—	67	2	no	—
295	C₂×C₄	—	67	2	no	—
296	C₂×C₄	—	67	1	no	—
297	C₂×C₄	—	67	1	no	—
298	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
299	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	67	2	no	—
303	C₈	—	134	1	no	—
304	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
305	C₈	#41×D₆₇	1	1	yes	characteristic
306	C₂²	—	67	2	no	—
307	C₂²	—	134	1	no	—
308	C₂²	—	134	1	no	—
309	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	268	4	no	—
315	C₄	—	536	2	no	—
316	C₄	—	67	2	no	—
317	C₄	—	134	1	no	—
318	C₄	—	134	1	no	—
319	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
320	C₄	#135×D₆₇	1	1	yes	characteristic
321	C₂	—	134	2	no	—
322	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
323	C₂	#22573358	1	1	yes	characteristic
324	C₂	#22573359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×Q₃₂	D₆₇
hypocenter	C₆₇	#33458
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #22562078	C₂	semidirect
4. #22562077	C₂	semidirect
5. C₄×Dic₅₃₆	C₂	semidirect
6. Q₃₂×Dic₆₇	C₂	semidirect
7. #22562067	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
16. Q₃₂×D₆₇	C₄	semidirect
170. C₄×Q₃₂	D₆₇	semidirect
174. C₆₇	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₆₇	2
#22569072	2
#22562078	2
#22562077	2
C₄×Dic₅₃₆	2
Q₃₂×Dic₆₇	2
#22562067	2
#22561990	2
#22561927	2
#22561804	2
C₄×Q₃₂×C₆₇	2
#33458	67