Subgroups — #16855497

Order 3968.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521602	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521517	C₂	1	1	yes	maximal, characteristic
3	#14521359	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521355	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521352	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521348	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521343	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521338	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521333	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521328	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521085	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521046	C₂	1	1	yes	maximal, characteristic
14	#14521043	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#92495	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₃₁×#117	C₂²	1	1	yes	characteristic
20	#11758503	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
23	#11758470	C₂²	1	1	yes	characteristic
24	#11758470	C₂²	1	1	yes	characteristic
25	#11758467	C₂²	1	1	yes	characteristic
26	#11758467	C₂²	1	1	yes	characteristic
27	#11758465	C₂²	1	1	yes	characteristic
28	#11758465	C₂²	1	1	yes	characteristic
29	#11758464	C₂²	1	1	yes	characteristic
30	#11758464	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₃₁)⋊C₄	—	2	1	no	—
32	(C₂²⋊D₃₁)⋊C₄	—	2	1	no	—
33	#11758455	—	2	1	no	—
34	#11758455	—	2	1	no	—
35	#11758449	—	2	1	no	—
36	C₄²×D₃₁	—	2	1	no	—
37	C₄×Dic₆₂	—	2	1	no	—
38	C₄×Dic₆₂	—	2	1	no	—
39	C₃₁×#107	C₂²	1	1	yes	characteristic
40	C₃₁×#107	C₂²	1	1	yes	characteristic
41	C₃₁×#106	C₂²	1	1	yes	characteristic
42	C₃₁×#106	C₂²	1	1	yes	characteristic
43	#11758409	C₂²	1	1	yes	characteristic
44	#11758409	C₂²	1	1	yes	characteristic
45	#11758404	C₂²	1	1	yes	characteristic
46	#11758404	C₂²	1	1	yes	characteristic
47	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
51	#11758398	C₂²	1	1	yes	characteristic
52	#11758398	C₂²	1	1	yes	characteristic
53	#11758397	C₂²	1	1	yes	characteristic
54	#11758397	C₂²	1	1	yes	characteristic
55	#11758396	C₂²	1	1	yes	characteristic
56	#11758396	C₂²	1	1	yes	characteristic
57	#11758395	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₃₁	—	2	1	no	—
61	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
62	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
63	#92495	—	2	2	no	—
64	#92495	Q₈	1	4	yes	complemented
65	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₃₁	—	2	1	no	—
67	C₂×C₄×D₃₁	—	2	1	no	—
68	C₂×C₄×D₃₁	—	2	1	no	—
69	C₂×Dic₆₂	—	2	1	no	—
70	C₂×Dic₆₂	—	2	1	no	—
71	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₄²×C₃₁	—	2	1	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₂²⋊Dic₃₁	—	2	1	no	—
84	(C₂×D₃₁)⋊C₄	—	2	1	no	—
85	(C₂×D₃₁)⋊C₄	—	2	1	no	—
86	C₄⋊Dic₃₁	—	2	1	no	—
87	C₄⋊Dic₃₁	—	2	1	no	—
88	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
90	Dic₃₁⋊C₄	—	2	1	no	—
91	Dic₃₁⋊C₄	—	2	1	no	—
92	Dic₃₁⋊C₄	—	2	1	no	—
93	Dic₃₁⋊C₄	—	2	1	no	—
94	Dic₃₁⋊C₄	—	2	1	no	—
95	Dic₃₁⋊C₄	—	2	1	no	—
96	C₄×Dic₃₁	—	2	1	no	—
97	C₄×Dic₃₁	—	2	1	no	—
98	C₄×Dic₃₁	—	2	1	no	—
99	C₄×Dic₃₁	D₄	1	1	yes	characteristic
100	C₄×Dic₃₁	D₄	1	1	yes	characteristic
101	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₃₁⋊C₈	—	2	2	no	—
104	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
114	C₂×C₄×C₃₁	#41	1	2	yes	—
115	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₃₁	—	2	4	no	—
117	C₂×Dic₃₁	—	2	1	no	—
118	C₂×Dic₃₁	—	2	2	no	—
119	C₂×Dic₃₁	—	2	1	no	—
120	C₂×Dic₃₁	—	2	1	no	—
121	C₂×Dic₃₁	—	4	1	no	—
122	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
126	D₁₂₄	—	2	1	no	—
127	D₁₂₄	C₂×Q₈	1	2	yes	—
128	C₄×D₃₁	—	2	4	no	—
129	C₄×D₃₁	—	4	2	no	—
130	C₄×D₃₁	—	4	2	no	—
131	Dic₆₂	—	2	1	no	—
132	Dic₆₂	—	4	2	no	—
133	Dic₆₂	—	4	2	no	—
134	Dic₆₂	C₂×Q₈	1	2	yes	—
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₃₁⋊C₈	—	2	1	no	—
138	C₃₁⋊C₈	—	2	1	no	—
139	#3076	—	31	1	no	maximal, Sylow
140	C₂²×C₃₁	—	2	1	no	—
141	C₂²×C₃₁	—	2	1	no	—
142	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
143	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₃₁	—	2	2	no	—
145	C₂×D₃₁	—	4	1	no	—
146	C₂×D₃₁	—	4	1	no	—
147	C₄×C₃₁	—	4	1	no	—
148	C₄×C₃₁	—	4	1	no	—
149	C₄×C₃₁	—	4	1	no	—
150	C₄×C₃₁	—	4	1	no	—
151	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
152	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
154	Dic₃₁	—	2	2	no	—
155	Dic₃₁	—	2	1	no	—
156	Dic₃₁	—	2	1	no	—
157	Dic₃₁	—	2	1	no	—
158	Dic₃₁	—	4	1	no	—
159	Dic₃₁	—	4	1	no	—
160	Dic₃₁	#135	1	2	yes	—
161	#533	—	31	1	no	—
162	C₄×#41	—	31	1	no	—
163	#479	—	31	1	no	—
164	#479	—	31	1	no	—
165	#478	—	31	1	no	—
166	#478	—	31	1	no	—
167	#477	—	31	1	no	—
168	#476	—	31	1	no	—
169	#475	—	31	1	no	—
170	#474	—	31	1	no	—
171	#427	—	31	1	no	—
172	#427	D₃₁	1	1	yes	characteristic, complemented
173	C₂×C₄⋊C₈	—	31	1	no	—
174	#416	—	31	1	no	—
175	#416	—	31	1	no	—
176	C₂×C₃₁	—	2	2	no	—
177	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
179	C₂×C₃₁	#575	1	1	yes	characteristic
180	D₃₁	—	4	2	no	—
181	C₂×#41	—	31	1	no	—
182	C₂²×C₈	—	31	1	no	—
183	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
184	C₄⋊Q₈	—	31	1	no	—
185	C₄⋊Q₈	—	31	1	no	—
186	#125	—	31	1	no	—
187	#125	—	31	1	no	—
188	C₂²⋊Q₈	—	62	1	no	—
189	C₂²⋊Q₈	—	62	1	no	—
190	C₄×Q₈	—	31	1	no	—
191	C₄×Q₈	—	31	1	no	—
192	C₄×Q₈	—	62	1	no	—
193	C₄×Q₈	—	62	1	no	—
194	C₄×D₄	—	31	1	no	—
195	C₄×D₄	—	31	1	no	—
196	C₄×D₄	—	62	1	no	—
197	C₄×D₄	—	62	1	no	—
198	#117	—	31	1	no	—
199	#117	—	62	1	no	—
200	#117	C₂×D₃₁	1	1	yes	characteristic
201	#117	C₂×D₃₁	1	1	yes	characteristic
202	C₂×C₄²	—	31	1	no	—
203	C₂×C₄²	—	62	1	no	—
204	#107	—	31	1	no	—
205	#107	—	31	1	no	—
206	#107	C₂×D₃₁	1	1	yes	characteristic
207	#107	C₂×D₃₁	1	1	yes	characteristic
208	#106	—	31	1	no	—
209	#106	—	31	1	no	—
210	#106	C₂×D₃₁	1	1	yes	characteristic
211	#106	C₂×D₃₁	1	1	yes	characteristic
212	C₄⋊C₈	—	31	1	no	—
213	C₄⋊C₈	—	31	1	no	—
214	C₄⋊C₈	—	31	1	no	—
215	C₄⋊C₈	—	31	1	no	—
216	#103	—	31	1	no	—
217	#103	—	31	1	no	—
218	#103	—	31	1	no	—
219	#103	—	31	1	no	—
220	#102	—	31	1	no	—
221	#102	—	31	1	no	—
222	#102	—	31	1	no	—
223	#102	—	31	1	no	—
224	C₃₁	#3076	1	1	yes	characteristic, complemented, Sylow
225	#41	—	31	4	no	—
226	#41	—	62	2	no	—
227	C₂×Q₈	—	31	1	no	—
228	C₂×Q₈	—	62	1	no	—
229	C₂×Q₈	—	62	1	no	—
230	C₂×D₄	—	31	1	no	—
231	C₂×D₄	—	62	1	no	—
232	C₂²×C₄	—	31	1	no	—
233	C₂²×C₄	—	31	1	no	—
234	C₂²×C₄	—	62	1	no	—
235	C₂²×C₄	—	62	1	no	—
236	C₂²×C₄	—	62	1	no	—
237	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	31	1	no	—
240	C₂×C₈	—	31	1	no	—
241	C₂×C₈	—	62	2	no	—
242	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
244	C₄⋊C₄	—	31	1	no	—
245	C₄⋊C₄	—	31	1	no	—
246	C₄⋊C₄	—	62	1	no	—
247	C₄⋊C₄	—	62	1	no	—
248	C₄⋊C₄	—	62	1	no	—
249	C₄⋊C₄	—	62	1	no	—
250	C₄⋊C₄	—	62	1	no	—
251	C₄⋊C₄	—	62	1	no	—
252	C₄⋊C₄	—	62	1	no	—
253	C₄⋊C₄	—	62	1	no	—
254	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	62	1	no	—
261	C₂²⋊C₄	—	62	1	no	—
262	C₂²⋊C₄	—	62	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	31	1	no	—
266	C₄²	—	31	1	no	—
267	C₄²	—	31	1	no	—
268	C₄²	—	31	1	no	—
269	C₄²	—	62	1	no	—
270	C₄²	—	62	1	no	—
271	C₄²	—	62	1	no	—
272	C₂³	—	62	1	no	—
273	C₂³	D₄×D₃₁	1	1	yes	characteristic
274	Q₈	—	31	2	no	—
275	Q₈	—	62	1	no	—
276	Q₈	—	124	2	no	—
277	Q₈	—	124	2	no	—
278	D₄	—	31	2	no	—
279	D₄	—	62	4	no	—
280	D₄	—	62	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	31	1	no	—
286	C₂×C₄	—	31	1	no	—
287	C₂×C₄	—	31	1	no	—
288	C₂×C₄	—	31	1	no	—
289	C₂×C₄	—	62	2	no	—
290	C₂×C₄	—	62	1	no	—
291	C₂×C₄	—	62	4	no	—
292	C₂×C₄	—	62	1	no	—
293	C₂×C₄	—	62	1	no	—
294	C₂×C₄	—	124	2	no	—
295	C₂×C₄	—	124	2	no	—
296	C₂×C₄	—	124	1	no	—
297	C₂×C₄	#92495	1	2	yes	—
298	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
299	C₂×C₄	Q₈×D₃₁	1	2	yes	—
300	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	62	1	no	—
304	C₈	—	62	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	62	2	no	—
308	C₂²	—	124	1	no	—
309	C₂²	—	124	1	no	—
310	C₂²	#11758473	1	2	yes	—
311	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	31	2	no	—
317	C₄	—	62	2	no	—
318	C₄	—	62	1	no	—
319	C₄	—	62	1	no	—
320	C₄	—	62	1	no	—
321	C₄	—	124	1	no	—
322	C₄	—	124	1	no	—
323	C₄	#11758473	1	2	yes	—
324	C₄	C₂×#92495	1	1	yes	characteristic
325	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	124	2	no	—
328	C₂	C₂×#11758473	1	1	yes	characteristic
329	C₂	#135×D₃₁	1	1	yes	characteristic
330	C₂	#14521846	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#427	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#92495	C₂²
hypercenter	#427	D₃₁
hypocenter	C₃₁	#3076
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
1. #14521602	C₂	semidirect
3. #14521359	C₂	semidirect
4. #14521355	C₂	semidirect
5. #14521352	C₂	semidirect
6. #14521348	C₂	semidirect
7. #14521343	C₂	semidirect
8. #14521338	C₂	semidirect
9. #14521333	C₂	semidirect
10. #14521328	C₂	semidirect
11. #14521085	C₂	semidirect
12. C₂×Dic₃₁⋊C₈	C₂	semidirect
15. C₃₁×#427	C₂	semidirect
64. #92495	Q₈	semidirect
172. #427	D₃₁	semidirect
224. C₃₁	#3076	semidirect

Maximal subgroups

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

subgroup	index
#14521602	2
#14521517	2
#14521359	2
#14521355	2
#14521352	2
#14521348	2
#14521343	2
#14521338	2
#14521333	2
#14521328	2
#14521085	2
C₂×Dic₃₁⋊C₈	2
#14521046	2
#14521043	2
C₃₁×#427	2
#3076	31