Subgroups — #16857764

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	#14521739	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic
3	#14521418	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521391	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521380	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521357	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521307	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521290	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521260	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521207	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521207	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521157	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521059	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#11758404	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#435	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
17	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	#11758512	—	2	1	no	—
22	D₄×Dic₃₁	—	2	1	no	—
23	#11758503	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	#11758501	C₂²	1	1	yes	characteristic
26	#11758501	C₂²	1	1	yes	characteristic
27	#11758501	C₂²	1	1	yes	characteristic
28	#11758501	C₂²	1	1	yes	characteristic
29	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
32	#11758466	—	2	1	no	—
33	#11758465	—	2	1	no	—
34	#11758456	—	2	1	no	—
35	#11758455	—	2	1	no	—
36	#11758455	—	2	1	no	—
37	#11758454	—	2	1	no	—
38	#11758447	C₂²	1	1	yes	characteristic
39	#11758446	C₂²	1	1	yes	characteristic
40	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
41	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
42	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₃₁×#97	C₂²	1	1	yes	characteristic
46	#11758406	C₂²	1	1	yes	characteristic
47	#11758405	C₂²	1	1	yes	characteristic
48	#11758405	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	#11758404	C₂²	1	1	yes	characteristic
54	#11758404	C₂²	1	1	yes	characteristic
55	#11758404	C₂²	1	1	yes	characteristic
56	#11758404	C₂²	1	1	yes	characteristic
57	#11758404	C₂²	1	1	yes	characteristic
58	#11758404	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₁	—	2	1	no	—
63	C₂²×Dic₃₁	—	2	1	no	—
64	C₂²×Dic₃₁	—	2	1	no	—
65	C₂×Dic₆₂	—	2	2	no	—
66	C₂×Dic₆₂	—	2	2	no	—
67	C₂×Dic₆₂	—	2	1	no	—
68	C₂×Dic₆₂	—	2	1	no	—
69	C₂×Dic₆₂	—	4	1	no	—
70	C₂×Dic₆₂	—	4	1	no	—
71	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
72	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
73	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
74	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
75	M₁₆×C₃₁	—	4	1	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₃₁	—	2	1	no	—
86	C₂²⋊Dic₃₁	—	2	1	no	—
87	C₂²⋊Dic₃₁	—	2	1	no	—
88	C₂²⋊Dic₃₁	—	2	1	no	—
89	C₂²⋊Dic₃₁	—	2	1	no	—
90	C₄⋊Dic₃₁	—	2	1	no	—
91	C₄⋊Dic₃₁	—	2	1	no	—
92	C₄⋊Dic₃₁	—	2	1	no	—
93	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
104	Dic₃₁⋊C₄	—	2	1	no	—
105	C₄×Dic₃₁	—	2	1	no	—
106	C₄×Dic₃₁	—	2	1	no	—
107	C₄×Dic₃₁	—	2	1	no	—
108	Dic₁₂₄	—	2	2	no	—
109	Dic₁₂₄	—	4	1	no	—
110	Dic₁₂₄	—	4	1	no	—
111	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
113	D₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₃₁	—	2	1	no	—
124	C₂×Dic₃₁	—	2	1	no	—
125	C₂×Dic₃₁	—	2	1	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	—	2	1	no	—
131	C₂×Dic₃₁	—	4	2	no	—
132	C₂×Dic₃₁	—	4	2	no	—
133	C₂×Dic₃₁	—	4	2	no	—
134	Dic₆₂	—	2	1	no	—
135	Dic₆₂	—	2	1	no	—
136	Dic₆₂	—	2	1	no	—
137	Dic₆₂	—	2	1	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	C₈×C₃₁	—	2	1	no	—
147	C₈×C₃₁	—	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	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	#137	1	2	yes	—
155	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	2	1	no	—
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
161	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
162	Dic₃₁	—	2	2	no	—
163	Dic₃₁	—	4	1	no	—
164	Dic₃₁	—	4	1	no	—
165	Dic₃₁	—	4	1	no	—
166	Dic₃₁	—	4	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	—	31	1	no	—
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	D₃₁	1	1	yes	characteristic
183	#419	—	31	1	no	—
184	C₂×#103	—	31	1	no	—
185	C₂×C₃₁	—	2	2	no	—
186	C₂×C₃₁	—	4	1	no	—
187	C₂×C₃₁	#546	1	1	yes	characteristic
188	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
189	C₂×C₃₁	#578	1	1	yes	characteristic
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	—	31	1	no	—
195	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
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	—	31	1	no	—
206	C₂²⋊Q₈	—	31	1	no	—
207	C₂²⋊Q₈	—	62	1	no	—
208	C₂²⋊Q₈	—	62	1	no	—
209	C₄×Q₈	—	31	1	no	—
210	C₄×Q₈	—	31	1	no	—
211	C₄×D₄	—	62	1	no	—
212	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
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	—	31	1	no	—
219	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
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	—	31	1	no	—
229	#103	—	31	1	no	—
230	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
231	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
232	#97	C₂×D₃₁	1	1	yes	characteristic
233	C₃₁	#3312	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₈	—	31	1	no	—
235	C₂×Q₈	—	31	1	no	—
236	C₂×Q₈	—	31	1	no	—
237	C₂×Q₈	—	31	1	no	—
238	C₂×Q₈	—	62	2	no	—
239	C₂×Q₈	—	62	2	no	—
240	C₂×Q₈	—	62	1	no	—
241	C₂×Q₈	—	62	1	no	—
242	C₂×Q₈	—	124	1	no	—
243	C₂×Q₈	—	124	1	no	—
244	C₂×D₄	D₁₂₄	1	1	yes	characteristic
245	C₂²×C₄	—	62	1	no	—
246	C₂²×C₄	—	62	1	no	—
247	C₂²×C₄	—	62	1	no	—
248	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
250	Q₁₆	—	62	2	no	—
251	Q₁₆	—	124	1	no	—
252	Q₁₆	—	124	1	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
259	C₄⋊C₄	—	31	1	no	—
260	C₄⋊C₄	—	31	1	no	—
261	C₄⋊C₄	—	31	1	no	—
262	C₄⋊C₄	—	31	1	no	—
263	C₄⋊C₄	—	62	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	—	62	1	no	—
274	C₄⋊C₄	D₁₂₄	1	1	yes	characteristic
275	C₂²⋊C₄	—	62	1	no	—
276	C₂²⋊C₄	—	62	1	no	—
277	C₂²⋊C₄	—	62	1	no	—
278	C₂²⋊C₄	—	62	1	no	—
279	C₂²⋊C₄	—	62	1	no	—
280	C₂²⋊C₄	D₁₂₄	1	1	yes	characteristic
281	C₂²⋊C₄	D₁₂₄	1	1	yes	characteristic
282	C₄²	—	62	1	no	—
283	C₄²	—	62	1	no	—
284	C₄²	—	62	1	no	—
285	C₄²	C₂²×D₃₁	1	1	yes	characteristic
286	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
287	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
288	Q₈	—	31	2	no	—
289	Q₈	—	62	1	no	—
290	Q₈	—	62	1	no	—
291	Q₈	—	62	1	no	—
292	Q₈	—	62	1	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	—	2	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	—	62	1	no	—
304	C₂×C₄	—	62	1	no	—
305	C₂×C₄	—	62	1	no	—
306	C₂×C₄	—	62	1	no	—
307	C₂×C₄	—	62	1	no	—
308	C₂×C₄	—	62	1	no	—
309	C₂×C₄	—	62	1	no	—
310	C₂×C₄	—	62	1	no	—
311	C₂×C₄	—	124	2	no	—
312	C₂×C₄	—	124	2	no	—
313	C₂×C₄	—	124	2	no	—
314	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
315	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
316	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
317	C₂×C₄	C₂×D₁₂₄	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	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	#11758486	1	2	yes	—
329	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	62	2	no	—
335	C₄	—	124	1	no	—
336	C₄	—	124	1	no	—
337	C₄	—	124	1	no	—
338	C₄	—	124	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	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	#14521694	1	1	yes	characteristic
347	C₂	C₂×#11758486	1	1	yes	characteristic
348	C₂	#14521866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#435	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₄×C₃₁	D₄
hypercenter	#435	D₃₁
hypocenter	C₃₁	#3312
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
3. #14521418	C₂	semidirect
4. #14521391	C₂	semidirect
5. #14521380	C₂	semidirect
6. #14521357	C₂	semidirect
7. #14521307	C₂	semidirect
8. #14521290	C₂	semidirect
9. #14521260	C₂	semidirect
10. #14521207	C₂	semidirect
11. #14521207	C₂	semidirect
12. #14521157	C₂	semidirect
13. #14521059	C₂	semidirect
14. C₂×#11758404	C₂	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
#14521739	2
D₄×Dic₆₂	2
#14521418	2
#14521391	2
#14521380	2
#14521357	2
#14521307	2
#14521290	2
#14521260	2
#14521207	2
#14521157	2
#14521059	2
C₂×#11758404	2
C₃₁×#435	2
#3312	31