Subgroups — #16857757

Order 3968.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521733	C₂	1	1	yes	maximal, characteristic
2	#14521712	C₂	1	1	yes	maximal, characteristic
3	#14521416	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521387	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521381	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521359	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521289	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521257	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521204	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521203	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521153	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521118	C₂	1	1	yes	maximal, characteristic
13	#14521059	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₃₁	C₂²	1	1	yes	characteristic
17	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	#11758523	C₂²	1	1	yes	characteristic
23	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
24	#11758521	C₂²	1	1	yes	characteristic
25	C₂×#92482	C₂²	1	1	yes	characteristic
26	C₂×#92482	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
28	#11758503	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
30	#11758474	C₂²	1	1	yes	characteristic
31	#11758473	—	2	1	no	—
32	#11758469	—	2	1	no	—
33	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
34	#11758467	C₂²	1	1	yes	characteristic
35	#11758466	—	2	1	no	—
36	#11758465	—	2	1	no	—
37	#11758464	C₂²	1	1	yes	characteristic
38	#11758460	—	2	1	no	—
39	(C₂²⋊D₃₁)⋊C₄	—	2	1	no	—
40	#11758456	—	2	1	no	—
41	#11758455	—	2	1	no	—
42	C₃₁×#106	C₂²	1	1	yes	characteristic
43	C₃₁×#103	C₂²	1	1	yes	characteristic
44	C₃₁×#103	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758423	C₂²	1	1	yes	characteristic
47	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
48	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
49	#11758404	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758404	C₂²	1	1	yes	characteristic
53	#11758403	C₂²	1	1	yes	characteristic
54	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
55	#11758398	C₂²	1	1	yes	characteristic
56	#11758398	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₃₁	D₄	1	1	yes	characteristic
63	C₂²×Dic₃₁	—	2	1	no	—
64	Q₈×D₃₁	—	2	2	no	—
65	Q₈×D₃₁	—	4	1	no	—
66	C₂×C₄×D₃₁	—	2	1	no	—
67	C₂×C₄×D₃₁	—	2	1	no	—
68	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
69	C₂×Dic₆₂	—	2	2	no	—
70	C₂×Dic₆₂	—	2	1	no	—
71	C₂×Dic₆₂	—	2	1	no	—
72	C₂×Dic₆₂	—	4	1	no	—
73	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
74	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
75	Q₁₆×C₃₁	—	4	1	no	—
76	M₁₆×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂²⋊Dic₃₁	—	2	1	no	—
86	C₂²⋊Dic₃₁	D₄	1	1	yes	characteristic
87	#92482	—	2	2	no	—
88	#92482	—	4	1	no	—
89	(C₂×D₃₁)⋊C₄	—	2	1	no	—
90	(C₂×D₃₁)⋊C₄	—	2	1	no	—
91	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
92	C₄⋊Dic₃₁	—	2	1	no	—
93	C₄⋊Dic₃₁	—	2	1	no	—
94	C₄⋊Dic₃₁	—	2	1	no	—
95	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
97	Dic₃₁⋊C₄	—	2	1	no	—
98	Dic₃₁⋊C₄	—	2	1	no	—
99	Dic₃₁⋊C₄	—	2	1	no	—
100	Dic₃₁⋊C₄	—	2	1	no	—
101	Dic₃₁⋊C₄	—	2	1	no	—
102	Dic₃₁⋊C₄	—	2	1	no	—
103	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₃₁	—	2	1	no	—
105	C₄×Dic₃₁	—	2	1	no	—
106	C₄×Dic₃₁	—	2	1	no	—
107	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
110	C₈×D₃₁	—	4	1	no	—
111	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₃₁	—	2	1	no	—
114	Q₈×C₃₁	—	2	1	no	—
115	Q₈×C₃₁	—	2	1	no	—
116	Q₈×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₃₁	—	2	2	no	—
126	C₂×Dic₃₁	—	2	1	no	—
127	C₂×Dic₃₁	—	2	1	no	—
128	C₂×Dic₃₁	—	2	1	no	—
129	C₂×Dic₃₁	—	2	1	no	—
130	C₂×Dic₃₁	—	4	2	no	—
131	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
134	C₄×D₃₁	—	2	1	no	—
135	C₄×D₃₁	—	2	1	no	—
136	C₄×D₃₁	—	4	2	no	—
137	C₄×D₃₁	—	4	2	no	—
138	Dic₆₂	—	2	1	no	—
139	Dic₆₂	—	2	1	no	—
140	Dic₆₂	—	2	1	no	—
141	Dic₆₂	—	4	2	no	—
142	Dic₆₂	—	4	2	no	—
143	Dic₆₂	#41	1	2	yes	—
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	—	2	1	no	—
146	C₃₁⋊C₈	—	2	1	no	—
147	C₃₁⋊C₈	—	2	1	no	—
148	#3312	—	31	1	no	maximal, Sylow
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₃₁	—	2	1	no	—
154	C₂×D₃₁	—	2	1	no	—
155	C₂×D₃₁	#137	1	2	yes	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	1	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	—	4	1	no	—
160	C₄×C₃₁	—	4	1	no	—
161	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
162	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
163	Dic₃₁	—	2	2	no	—
164	Dic₃₁	—	2	1	no	—
165	Dic₃₁	—	2	1	no	—
166	Dic₃₁	—	2	1	no	—
167	Dic₃₁	—	4	1	no	—
168	Dic₃₁	—	4	1	no	—
169	Dic₃₁	—	4	1	no	—
170	#552	—	31	1	no	—
171	D₄×Q₈	—	31	1	no	—
172	#491	—	31	1	no	—
173	#485	—	31	1	no	—
174	#483	—	31	1	no	—
175	#479	—	31	1	no	—
176	#470	D₃₁	1	1	yes	characteristic, complemented
177	#467	—	31	1	no	—
178	#462	—	31	1	no	—
179	#451	—	31	1	no	—
180	#451	—	31	1	no	—
181	#441	—	31	1	no	—
182	#435	—	31	1	no	—
183	#419	—	31	1	no	—
184	C₂×#103	—	31	1	no	—
185	C₂×C₃₁	—	4	1	no	—
186	C₂×C₃₁	#546	1	1	yes	characteristic
187	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
188	C₂×C₃₁	#578	1	1	yes	characteristic
189	D₃₁	—	2	2	no	—
190	C₂²×Q₈	—	31	1	no	—
191	C₂²×Q₈	—	31	1	no	—
192	C₂×Q₁₆	—	31	1	no	—
193	C₂×Q₁₆	—	31	1	no	—
194	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
195	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
196	C₂²×C₈	—	31	1	no	—
197	C₄⋊Q₈	—	31	1	no	—
198	C₄⋊Q₈	—	62	1	no	—
199	#126	—	62	1	no	—
200	#125	—	31	1	no	—
201	#125	—	62	1	no	—
202	#123	—	62	1	no	—
203	C₂²⋊Q₈	—	31	1	no	—
204	C₂²⋊Q₈	—	31	1	no	—
205	C₂²⋊Q₈	—	62	1	no	—
206	C₂²⋊Q₈	—	62	1	no	—
207	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
208	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
209	C₄×Q₈	—	31	1	no	—
210	C₄×Q₈	—	31	1	no	—
211	C₄×D₄	—	31	1	no	—
212	C₄×D₄	—	62	1	no	—
213	#117	—	31	1	no	—
214	#117	—	62	1	no	—
215	C₂×C₄⋊C₄	—	31	1	no	—
216	#107	—	31	1	no	—
217	#106	—	31	1	no	—
218	#106	C₂×D₃₁	1	1	yes	characteristic
219	C₄⋊C₈	—	31	1	no	—
220	#103	—	31	1	no	—
221	#103	—	31	1	no	—
222	#103	—	31	1	no	—
223	#103	—	31	1	no	—
224	#103	—	31	1	no	—
225	#103	—	31	1	no	—
226	#103	—	31	1	no	—
227	#103	—	31	1	no	—
228	#103	C₂×D₃₁	1	1	yes	characteristic
229	#103	C₂×D₃₁	1	1	yes	characteristic
230	C₂²⋊C₈	—	31	1	no	—
231	C₂²⋊C₈	—	31	1	no	—
232	#97	—	31	1	no	—
233	C₃₁	#3312	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₈	—	31	1	no	—
235	C₂×Q₈	—	31	1	no	—
236	C₂×Q₈	—	62	1	no	—
237	C₂×Q₈	—	62	2	no	—
238	C₂×Q₈	—	62	1	no	—
239	C₂×Q₈	—	62	2	no	—
240	C₂×Q₈	—	124	1	no	—
241	C₂×Q₈	—	124	1	no	—
242	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
244	C₂×D₄	—	31	1	no	—
245	C₂²×C₄	—	31	1	no	—
246	C₂²×C₄	—	62	1	no	—
247	C₂²×C₄	—	62	1	no	—
248	C₂²×C₄	—	62	1	no	—
249	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
250	Q₁₆	—	4	1	no	—
251	Q₁₆	—	62	2	no	—
252	Q₁₆	—	124	1	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	31	1	no	—
255	C₂×C₈	—	31	1	no	—
256	C₂×C₈	—	124	1	no	—
257	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	31	1	no	—
262	C₄⋊C₄	—	31	1	no	—
263	C₄⋊C₄	—	31	1	no	—
264	C₄⋊C₄	—	62	1	no	—
265	C₄⋊C₄	—	62	1	no	—
266	C₄⋊C₄	—	62	1	no	—
267	C₄⋊C₄	—	62	1	no	—
268	C₄⋊C₄	—	62	1	no	—
269	C₄⋊C₄	—	62	1	no	—
270	C₄⋊C₄	—	62	1	no	—
271	C₄⋊C₄	—	62	1	no	—
272	C₄⋊C₄	—	62	1	no	—
273	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	31	1	no	—
278	C₂²⋊C₄	—	31	1	no	—
279	C₂²⋊C₄	—	62	1	no	—
280	C₂²⋊C₄	—	62	1	no	—
281	C₂²⋊C₄	—	62	1	no	—
282	C₄²	—	31	1	no	—
283	C₄²	—	62	1	no	—
284	C₄²	—	62	1	no	—
285	C₄²	—	62	1	no	—
286	C₂³	—	31	1	no	—
287	C₂³	D₄×D₃₁	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	31	2	no	—
293	Q₈	—	62	1	no	—
294	Q₈	—	62	1	no	—
295	Q₈	—	62	1	no	—
296	Q₈	—	124	2	no	—
297	Q₈	—	124	2	no	—
298	D₄	—	62	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	31	1	no	—
306	C₂×C₄	—	31	1	no	—
307	C₂×C₄	—	31	1	no	—
308	C₂×C₄	—	62	1	no	—
309	C₂×C₄	—	62	1	no	—
310	C₂×C₄	—	62	1	no	—
311	C₂×C₄	—	62	1	no	—
312	C₂×C₄	—	62	1	no	—
313	C₂×C₄	—	62	1	no	—
314	C₂×C₄	—	124	2	no	—
315	C₂×C₄	—	124	2	no	—
316	C₂×C₄	—	124	2	no	—
317	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	62	1	no	—
322	C₈	—	62	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	31	2	no	—
327	C₂²	—	62	1	no	—
328	C₂²	—	62	1	no	—
329	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	62	1	no	—
336	C₄	—	62	1	no	—
337	C₄	—	62	2	no	—
338	C₄	—	62	1	no	—
339	C₄	—	124	1	no	—
340	C₄	—	124	1	no	—
341	C₄	—	124	1	no	—
342	C₄	#11758561	1	1	yes	characteristic
343	C₄	#41×D₃₁	1	1	yes	characteristic
344	C₂	—	4	1	no	—
345	C₂	—	62	2	no	—
346	C₂	#14521688	1	1	yes	characteristic
347	C₂	#137×D₃₁	1	1	yes	characteristic
348	C₂	#14521865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#470	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×C₂²⋊D₃₁	D₄
hypercenter	#470	D₃₁
hypocenter	C₃₁	#3312
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
3. #14521416	C₂	semidirect
4. #14521387	C₂	semidirect
5. #14521381	C₂	semidirect
6. #14521359	C₂	semidirect
7. #14521289	C₂	semidirect
8. #14521257	C₂	semidirect
9. #14521204	C₂	semidirect
10. #14521203	C₂	semidirect
11. #14521153	C₂	semidirect
13. #14521059	C₂	semidirect
14. #103×D₃₁	C₂	semidirect
15. C₃₁×#470	C₂	semidirect
176. #470	D₃₁	semidirect
233. C₃₁	#3312	semidirect

Maximal subgroups

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

subgroup	index
#14521733	2
#14521712	2
#14521416	2
#14521387	2
#14521381	2
#14521359	2
#14521289	2
#14521257	2
#14521204	2
#14521203	2
#14521153	2
#14521118	2
#14521059	2
#103×D₃₁	2
C₃₁×#470	2
#3312	31