Subgroups — #16856005

Order 3968.

358 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	#14521659	C₂	1	1	yes	maximal, characteristic
2	#14521527	C₂	1	1	yes	maximal, characteristic
3	#14521391	C₂	1	1	yes	maximal, characteristic
4	#14521387	C₂	1	1	yes	maximal, characteristic
5	#14521358	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521354	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521257	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521257	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521205	C₂	1	1	yes	maximal, characteristic
10	#14521156	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521153	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521141	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521107	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#451	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃₁	—	2	1	no	—
21	Q₈×Dic₃₁	—	2	1	no	—
22	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
23	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
25	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
26	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
27	#11758521	C₂²	1	1	yes	characteristic
28	C₂×#92482	C₂²	1	1	yes	characteristic
29	C₂×#92482	C₂²	1	1	yes	characteristic
30	#11758503	—	2	1	no	—
31	#11758503	—	2	1	no	—
32	#11758503	C₂²	1	1	yes	characteristic
33	#11758503	C₂²	1	1	yes	characteristic
34	#11758501	C₂²	1	1	yes	characteristic
35	#11758467	—	2	1	no	—
36	#11758467	C₂²	1	1	yes	characteristic
37	#11758466	—	2	1	no	—
38	#11758464	C₂²	1	1	yes	characteristic
39	#11758456	—	2	1	no	—
40	#11758456	—	2	1	no	—
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#103	C₂²	1	1	yes	characteristic
43	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#11758423	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758423	C₂²	1	1	yes	characteristic
47	#11758423	C₂²	1	1	yes	characteristic
48	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
49	#11758406	C₂²	1	1	yes	characteristic
50	#11758405	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	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
56	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
57	#11758396	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	—	2	1	no	—
62	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	—	2	1	no	—
66	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
68	Q₁₆×C₃₁	—	2	2	no	—
69	Q₁₆×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	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊Dic₃₁	—	2	1	no	—
76	C₂²⋊Dic₃₁	—	2	1	no	—
77	C₂²⋊Dic₃₁	D₄	1	1	yes	characteristic
78	C₂²⋊Dic₃₁	D₄	1	1	yes	characteristic
79	#92482	—	2	2	no	—
80	#92482	—	2	2	no	—
81	C₄⋊Dic₃₁	—	2	1	no	—
82	C₄⋊Dic₃₁	—	2	1	no	—
83	C₄⋊Dic₃₁	—	2	1	no	—
84	C₄⋊Dic₃₁	—	2	1	no	—
85	C₄⋊Dic₃₁	—	2	1	no	—
86	C₄⋊Dic₃₁	—	2	1	no	—
87	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
88	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
89	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
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	C₄×Dic₃₁	—	2	1	no	—
98	C₄×Dic₃₁	—	2	1	no	—
99	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	—	2	1	no	—
102	C₄×Dic₃₁	—	2	1	no	—
103	C₄×Dic₃₁	—	2	1	no	—
104	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
105	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₃₁	—	2	4	no	—
110	Q₈×C₃₁	—	2	1	no	—
111	Q₈×C₃₁	—	2	1	no	—
112	Q₈×C₃₁	—	2	1	no	—
113	Q₈×C₃₁	—	2	1	no	—
114	Q₈×C₃₁	#41	1	2	yes	—
115	Q₈×C₃₁	#41	1	2	yes	—
116	C₂×C₄×C₃₁	—	2	4	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	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	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
133	Dic₆₂	—	2	1	no	—
134	Dic₆₂	—	2	1	no	—
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	#3127	—	31	1	no	maximal, Sylow
140	C₂²×C₃₁	—	2	1	no	—
141	C₂²×C₃₁	—	2	1	no	—
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	2	2	no	—
146	C₄×C₃₁	—	2	1	no	—
147	C₄×C₃₁	—	4	1	no	—
148	C₄×C₃₁	—	4	1	no	—
149	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
151	Dic₃₁	—	2	1	no	—
152	Dic₃₁	—	2	1	no	—
153	Dic₃₁	—	2	1	no	—
154	Dic₃₁	—	2	1	no	—
155	Dic₃₁	—	4	1	no	—
156	Dic₃₁	—	4	1	no	—
157	Dic₃₁	—	4	1	no	—
158	Dic₃₁	—	4	1	no	—
159	#541	—	31	1	no	—
160	#519	—	31	1	no	—
161	#485	—	31	1	no	—
162	#485	—	31	1	no	—
163	#479	—	31	1	no	—
164	#479	—	31	1	no	—
165	#462	—	31	1	no	—
166	#462	—	31	1	no	—
167	#451	—	31	1	no	—
168	#451	D₃₁	1	1	yes	characteristic
169	#441	—	31	1	no	—
170	#441	—	31	1	no	—
171	C₄×Q₁₆	—	31	1	no	—
172	C₄×Q₁₆	—	31	1	no	—
173	#433	—	31	1	no	—
174	C₂×C₃₁	—	4	1	no	—
175	C₂×C₃₁	#525	1	1	yes	characteristic
176	C₂×C₃₁	#578	1	1	yes	characteristic
177	C₂×C₃₁	#578	1	1	yes	characteristic
178	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
179	C₂×Q₁₆	—	31	1	no	—
180	C₂×Q₁₆	—	31	1	no	—
181	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
182	C₂×Q₁₆	C₂×D₃₁	1	1	yes	characteristic
183	C₄⋊Q₈	—	31	1	no	—
184	#126	—	62	1	no	—
185	#126	—	62	1	no	—
186	#125	—	31	1	no	—
187	#125	—	62	1	no	—
188	#125	—	62	1	no	—
189	C₂²⋊Q₈	—	31	1	no	—
190	C₂²⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
191	C₄×Q₈	—	31	1	no	—
192	C₄×Q₈	—	31	1	no	—
193	C₄×Q₈	—	31	1	no	—
194	C₄×Q₈	—	31	1	no	—
195	C₄×Q₈	—	31	1	no	—
196	C₄×Q₈	—	31	1	no	—
197	C₄×Q₈	—	62	1	no	—
198	C₄×Q₈	—	62	1	no	—
199	#117	—	31	1	no	—
200	#117	—	31	1	no	—
201	#117	—	62	1	no	—
202	#117	—	62	1	no	—
203	#107	—	31	1	no	—
204	#107	—	31	1	no	—
205	#106	—	31	1	no	—
206	#106	—	31	1	no	—
207	C₄⋊C₈	—	31	1	no	—
208	C₄⋊C₈	—	31	1	no	—
209	#103	—	31	1	no	—
210	#103	—	31	1	no	—
211	#103	—	31	1	no	—
212	#103	—	31	1	no	—
213	#103	—	31	1	no	—
214	#103	—	31	1	no	—
215	#103	C₂×D₃₁	1	1	yes	characteristic
216	#103	C₂×D₃₁	1	1	yes	characteristic
217	C₂²⋊C₈	—	31	1	no	—
218	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
219	#97	—	31	1	no	—
220	C₄×C₈	—	31	1	no	—
221	C₃₁	#3127	1	1	yes	characteristic, complemented, Sylow
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	31	1	no	—
226	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	62	2	no	—
234	Q₁₆	—	62	2	no	—
235	C₂×C₈	—	31	1	no	—
236	C₂×C₈	—	31	1	no	—
237	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	31	1	no	—
241	C₄⋊C₄	—	31	1	no	—
242	C₄⋊C₄	—	31	1	no	—
243	C₄⋊C₄	—	31	1	no	—
244	C₄⋊C₄	—	31	1	no	—
245	C₄⋊C₄	—	62	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	—	62	1	no	—
255	C₄⋊C₄	—	62	1	no	—
256	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	31	1	no	—
259	C₂²⋊C₄	—	31	1	no	—
260	C₂²⋊C₄	—	62	1	no	—
261	C₂²⋊C₄	—	62	1	no	—
262	C₄²	—	31	1	no	—
263	C₄²	—	31	1	no	—
264	C₄²	—	62	1	no	—
265	C₄²	—	62	1	no	—
266	C₄²	—	62	1	no	—
267	C₄²	—	62	1	no	—
268	C₄²	—	62	1	no	—
269	C₄²	—	62	1	no	—
270	C₄²	—	62	1	no	—
271	C₂³	D₄×D₃₁	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	62	1	no	—
278	Q₈	—	62	1	no	—
279	Q₈	#92497	1	2	yes	—
280	Q₈	#92497	1	2	yes	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	31	1	no	—
289	C₂×C₄	—	31	1	no	—
290	C₂×C₄	—	31	1	no	—
291	C₂×C₄	—	31	1	no	—
292	C₂×C₄	—	62	1	no	—
293	C₂×C₄	—	62	1	no	—
294	C₂×C₄	—	62	1	no	—
295	C₂×C₄	—	62	1	no	—
296	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	62	1	no	—
301	C₈	—	62	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	62	1	no	—
312	C₄	—	62	1	no	—
313	C₄	—	62	1	no	—
314	C₄	—	62	1	no	—
315	C₄	—	124	1	no	—
316	C₄	—	124	1	no	—
317	C₄	—	124	1	no	—
318	C₄	—	124	1	no	—
319	C₄	C₂×#92497	1	1	yes	characteristic
320	C₄	C₂×#92497	1	1	yes	characteristic
321	C₂	—	4	1	no	—
322	C₂	#14521555	1	1	yes	characteristic
323	C₂	#14521862	1	1	yes	characteristic
324	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×#451	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#451	D₃₁
hypocenter	C₃₁	#3127
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
5. #14521358	C₂	semidirect
6. #14521354	C₂	semidirect
7. #14521257	C₂	semidirect
8. #14521257	C₂	semidirect
10. #14521156	C₂	semidirect
11. #14521153	C₂	semidirect
12. Q₁₆×Dic₃₁	C₂	semidirect
13. #14521141	C₂	semidirect
221. C₃₁	#3127	semidirect

Maximal subgroups

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

subgroup	index
#14521659	2
#14521527	2
#14521391	2
#14521387	2
#14521358	2
#14521354	2
#14521257	2
#14521205	2
#14521156	2
#14521153	2
Q₁₆×Dic₃₁	2
#14521141	2
#14521107	2
C₃₁×#451	2
#3127	31