Subgroups — #16856313

Order 3968.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521603	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521581	C₂	1	1	yes	maximal, characteristic
3	#14521456	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521440	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521416	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521407	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521308	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521306	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521294	C₂	1	1	yes	maximal, characteristic
10	#14521292	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#92482	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#92480	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#92469	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	#11758523	C₂²	1	1	yes	characteristic
20	#11758521	C₂²	1	1	yes	characteristic
21	C₂×#92482	C₂²	1	1	yes	characteristic
22	#11758515	C₂²	1	1	yes	characteristic
23	#11758513	C₂²	1	1	yes	characteristic
24	C₂×#92480	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₃₁	—	2	1	no	—
26	#11758503	—	2	1	no	—
27	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
28	C₂×#92469	C₂²	1	1	yes	characteristic
29	C₂×#92469	C₂²	1	1	yes	characteristic
30	#11758474	C₂²	1	1	yes	characteristic
31	#11758474	C₂²	1	1	yes	characteristic
32	#11758473	—	2	1	no	—
33	#11758467	C₂²	1	1	yes	characteristic
34	#11758466	—	2	1	no	—
35	#11758465	C₂²	1	1	yes	characteristic
36	#11758460	—	2	1	no	—
37	#11758456	—	2	1	no	—
38	#11758449	—	2	1	no	—
39	C₄²×D₃₁	C₂²	1	1	yes	characteristic
40	C₄×Dic₆₂	—	2	1	no	—
41	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
42	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
43	C₃₁×#106	C₂²	1	1	yes	characteristic
44	C₃₁×#103	C₂²	1	1	yes	characteristic
45	C₃₁×#102	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
47	#11758423	C₂²	1	1	yes	characteristic
48	#11758420	C₂²	1	1	yes	characteristic
49	#11758406	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758403	C₂²	1	1	yes	characteristic
53	#11758403	C₂²	1	1	yes	characteristic
54	#11758398	C₂²	1	1	yes	characteristic
55	#11758398	C₂²	1	1	yes	characteristic
56	#11758396	C₂²	1	1	yes	characteristic
57	#11758395	C₂²	1	1	yes	characteristic
58	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	2	1	no	—
62	C₂×C₂²⋊D₃₁	—	2	1	no	—
63	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
64	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
66	C₂×Dic₆₂	—	2	1	no	—
67	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
68	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
69	SD₁₆×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₃₁	—	2	1	no	—
79	C₂²⋊Dic₃₁	—	2	1	no	—
80	C₂²⋊Dic₃₁	—	2	1	no	—
81	#92482	—	2	2	no	—
82	#92480	—	2	2	no	—
83	(C₂×D₃₁)⋊C₄	—	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	—	2	1	no	—
89	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
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	Dic₃₁⋊C₄	—	2	1	no	—
97	Dic₃₁⋊C₄	—	2	1	no	—
98	C₄×Dic₃₁	—	2	1	no	—
99	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	D₄	1	1	yes	characteristic
101	C₄×Dic₃₁	D₄	1	1	yes	characteristic
102	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
105	Dic₁₂₄	—	2	2	no	—
106	#92469	—	2	2	no	—
107	#92469	—	2	2	no	—
108	C₂³×C₃₁	—	2	1	no	—
109	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	1	no	—
111	Q₈×C₃₁	—	2	1	no	—
112	D₄×C₃₁	—	2	1	no	—
113	D₄×C₃₁	—	2	1	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	—	4	2	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	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₃₁	—	4	2	no	—
122	C₂×Dic₃₁	—	2	1	no	—
123	C₂×Dic₃₁	—	2	1	no	—
124	C₂×Dic₃₁	—	2	1	no	—
125	C₂×Dic₃₁	—	2	1	no	—
126	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄×D₃₁	—	2	2	no	—
130	C₄×D₃₁	—	2	1	no	—
131	C₄×D₃₁	—	2	1	no	—
132	C₄×D₃₁	—	4	1	no	—
133	Dic₆₂	—	2	1	no	—
134	Dic₆₂	—	2	1	no	—
135	Dic₆₂	—	2	1	no	—
136	Dic₆₂	—	2	1	no	—
137	Dic₆₂	—	4	2	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	#41	1	2	yes	—
140	C₃₁⋊C₈	—	2	1	no	—
141	C₃₁⋊C₈	#41	1	2	yes	—
142	#3172	—	31	1	no	maximal, Sylow
143	C₂²×C₃₁	—	4	1	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₃₁	—	2	1	no	—
148	C₂×D₃₁	—	2	1	no	—
149	C₂×D₃₁	—	2	1	no	—
150	C₄×C₃₁	—	2	1	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	—	4	1	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	#137	1	2	yes	—
155	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	Dic₃₁	—	2	1	no	—
158	Dic₃₁	—	2	1	no	—
159	Dic₃₁	—	2	1	no	—
160	Dic₃₁	—	4	1	no	—
161	Dic₃₁	—	4	1	no	—
162	Dic₃₁	—	4	1	no	—
163	Dic₃₁	—	4	1	no	—
164	#533	—	31	1	no	—
165	#529	—	31	1	no	—
166	#501	—	31	1	no	—
167	#497	—	31	1	no	—
168	#491	—	31	1	no	—
169	#489	—	31	1	no	—
170	#470	—	31	1	no	—
171	#470	—	31	1	no	—
172	#468	—	31	1	no	—
173	#468	—	31	1	no	—
174	C₄×Q₁₆	—	31	1	no	—
175	C₄×Q₁₆	—	31	1	no	—
176	C₄×SD₁₆	—	31	1	no	—
177	C₄×SD₁₆	D₃₁	1	1	yes	characteristic, complemented
178	C₄×M₁₆	—	31	1	no	—
179	C₂×C₃₁	—	8	1	no	—
180	C₂×C₃₁	#532	1	1	yes	characteristic
181	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
182	C₂×C₃₁	#575	1	1	yes	characteristic
183	D₃₁	—	4	1	no	—
184	C₂×Q₁₆	—	31	1	no	—
185	C₂×Q₁₆	—	31	1	no	—
186	C₂×SD₁₆	—	31	1	no	—
187	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
188	C₂×M₁₆	—	31	1	no	—
189	C₂×M₁₆	—	31	1	no	—
190	C₄⋊Q₈	—	31	1	no	—
191	C₄⋊Q₈	—	31	1	no	—
192	#126	—	62	1	no	—
193	#125	—	31	1	no	—
194	#125	—	62	1	no	—
195	#124	—	31	1	no	—
196	#123	—	62	1	no	—
197	C₂²⋊Q₈	—	31	1	no	—
198	C₂²⋊Q₈	—	31	1	no	—
199	C₂²⋊Q₈	—	31	1	no	—
200	C₂²⋊Q₈	—	62	1	no	—
201	#121	—	31	1	no	—
202	C₄×Q₈	—	31	1	no	—
203	C₄×Q₈	—	31	1	no	—
204	C₄×Q₈	—	62	1	no	—
205	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
206	C₄×D₄	—	62	1	no	—
207	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
208	#117	—	62	1	no	—
209	#117	—	62	1	no	—
210	C₂×C₄²	—	31	1	no	—
211	#107	—	31	1	no	—
212	#107	—	31	1	no	—
213	#106	—	31	1	no	—
214	#106	C₂×D₃₁	1	1	yes	characteristic
215	#103	—	31	1	no	—
216	#103	—	31	1	no	—
217	#103	—	31	1	no	—
218	#103	—	31	1	no	—
219	#103	—	31	1	no	—
220	#103	C₂×D₃₁	1	1	yes	characteristic
221	#102	—	31	1	no	—
222	#102	C₂×D₃₁	1	1	yes	characteristic
223	#97	—	31	1	no	—
224	#97	—	31	1	no	—
225	C₄×C₈	—	31	1	no	—
226	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
227	C₃₁	#3172	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₈	—	31	1	no	—
229	C₂×Q₈	—	31	1	no	—
230	C₂×Q₈	—	62	1	no	—
231	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
232	C₂×D₄	—	62	1	no	—
233	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	31	1	no	—
236	C₂²×C₄	—	31	1	no	—
237	C₂²×C₄	—	31	1	no	—
238	Q₁₆	—	62	2	no	—
239	Q₁₆	—	62	2	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	62	2	no	—
242	M₁₆	—	62	2	no	—
243	M₁₆	—	62	2	no	—
244	C₂×C₈	—	31	1	no	—
245	C₂×C₈	—	31	1	no	—
246	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	31	1	no	—
250	C₄⋊C₄	—	31	1	no	—
251	C₄⋊C₄	—	62	1	no	—
252	C₄⋊C₄	—	62	1	no	—
253	C₄⋊C₄	—	62	1	no	—
254	C₄⋊C₄	—	62	1	no	—
255	C₄⋊C₄	—	62	1	no	—
256	C₄⋊C₄	—	62	1	no	—
257	C₄⋊C₄	—	62	1	no	—
258	C₄⋊C₄	—	62	1	no	—
259	C₄⋊C₄	—	62	1	no	—
260	C₄⋊C₄	—	62	1	no	—
261	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	62	1	no	—
265	C₂²⋊C₄	—	62	1	no	—
266	C₂²⋊C₄	—	62	1	no	—
267	C₂²⋊C₄	—	62	1	no	—
268	C₂²⋊C₄	—	62	1	no	—
269	C₂²⋊C₄	—	62	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	31	1	no	—
272	C₄²	—	31	1	no	—
273	C₄²	—	31	1	no	—
274	C₄²	—	62	1	no	—
275	C₄²	—	62	1	no	—
276	C₄²	C₂²×D₃₁	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	31	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	62	1	no	—
282	Q₈	—	62	1	no	—
283	Q₈	—	62	1	no	—
284	Q₈	—	62	1	no	—
285	Q₈	—	124	2	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	124	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	31	1	no	—
294	C₂×C₄	—	31	1	no	—
295	C₂×C₄	—	31	1	no	—
296	C₂×C₄	—	62	2	no	—
297	C₂×C₄	—	62	1	no	—
298	C₂×C₄	—	62	1	no	—
299	C₂×C₄	—	62	1	no	—
300	C₂×C₄	—	62	1	no	—
301	C₂×C₄	—	62	1	no	—
302	C₂×C₄	—	62	1	no	—
303	C₂×C₄	—	124	1	no	—
304	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
305	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	31	2	no	—
309	C₈	—	62	1	no	—
310	C₈	#92495	1	2	yes	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	62	1	no	—
315	C₂²	—	62	1	no	—
316	C₂²	—	62	1	no	—
317	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	62	1	no	—
323	C₄	—	62	1	no	—
324	C₄	—	62	1	no	—
325	C₄	—	124	1	no	—
326	C₄	—	124	1	no	—
327	C₄	—	124	1	no	—
328	C₄	—	124	1	no	—
329	C₄	#11758492	1	2	yes	—
330	C₄	C₂×#92495	1	1	yes	characteristic
331	C₄	#41×D₃₁	1	1	yes	characteristic
332	C₂	—	8	1	no	—
333	C₂	—	124	1	no	—
334	C₂	#14521596	1	1	yes	characteristic
335	C₂	C₂×#11758492	1	1	yes	characteristic
336	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	C₄×SD₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	#11758515	C₂²
hypercenter	C₄×SD₁₆	D₃₁
hypocenter	C₃₁	#3172
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #14521603	C₂	semidirect
3. #14521456	C₂	semidirect
4. #14521440	C₂	semidirect
5. #14521416	C₂	semidirect
6. #14521407	C₂	semidirect
7. #14521308	C₂	semidirect
8. #14521306	C₂	semidirect
11. C₄×Dic₁₂₄	C₂	semidirect
12. C₄×#92482	C₂	semidirect
13. C₄×#92480	C₂	semidirect
14. C₄×#92469	C₂	semidirect
15. C₄×SD₁₆×C₃₁	C₂	semidirect
177. C₄×SD₁₆	D₃₁	semidirect
227. C₃₁	#3172	semidirect

Maximal subgroups

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

subgroup	index
#14521603	2
#14521581	2
#14521456	2
#14521440	2
#14521416	2
#14521407	2
#14521308	2
#14521306	2
#14521294	2
#14521292	2
C₄×Dic₁₂₄	2
C₄×#92482	2
C₄×#92480	2
C₄×#92469	2
C₄×SD₁₆×C₃₁	2
#3172	31