Subgroups — #16858448

Order 3968.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521779	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521768	C₂	1	1	yes	maximal, characteristic
3	#14521407	C₂	1	1	yes	maximal, characteristic
4	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521337	C₂	1	1	yes	maximal, characteristic
6	#14521332	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521331	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521270	C₂	1	1	yes	maximal, characteristic
9	#14521268	C₂	1	1	yes	maximal, characteristic
10	#14521258	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521177	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521157	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521153	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521140	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#489	C₂	1	1	yes	maximal, characteristic
16	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×#124	C₂²	1	1	yes	characteristic
18	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
19	#11758521	—	2	1	no	—
20	#11758521	—	2	1	no	—
21	#11758521	C₂²	1	1	yes	characteristic
22	C₂×#92482	C₂²	1	1	yes	characteristic
23	#11758513	C₂²	1	1	yes	characteristic
24	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
25	C₂×#92480	C₂²	1	1	yes	characteristic
26	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
27	#11758465	—	2	1	no	—
28	#11758465	C₂²	1	1	yes	characteristic
29	#11758465	C₂²	1	1	yes	characteristic
30	#11758464	—	2	1	no	—
31	#11758464	C₂²	1	1	yes	characteristic
32	#11758464	C₂²	1	1	yes	characteristic
33	#11758456	—	2	1	no	—
34	#11758456	—	2	1	no	—
35	#11758455	—	2	1	no	—
36	#11758454	—	2	1	no	—
37	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
38	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
39	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
40	C₃₁×#103	C₂²	1	1	yes	characteristic
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	C₃₁×#102	C₂²	1	1	yes	characteristic
44	C₃₁×#97	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758405	C₂²	1	1	yes	characteristic
48	#11758404	C₂²	1	1	yes	characteristic
49	#11758404	C₂²	1	1	yes	characteristic
50	#11758403	C₂²	1	1	yes	characteristic
51	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
53	C₈×Dic₃₁	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	#11758396	C₂²	1	1	yes	characteristic
58	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₁	—	2	1	no	—
63	C₂×Dic₆₂	—	2	1	no	—
64	C₂×Dic₆₂	—	2	1	no	—
65	C₂×Dic₆₂	D₄	1	1	yes	characteristic
66	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
67	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₃₁	—	2	1	no	—
77	C₂²⋊Dic₃₁	—	2	1	no	—
78	C₂²⋊Dic₃₁	—	2	1	no	—
79	#92482	—	2	2	no	—
80	#92480	—	2	2	no	—
81	C₄⋊Dic₃₁	—	2	1	no	—
82	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
83	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
85	Dic₃₁⋊C₄	—	2	1	no	—
86	Dic₃₁⋊C₄	—	2	1	no	—
87	Dic₃₁⋊C₄	—	2	1	no	—
88	Dic₃₁⋊C₄	—	2	1	no	—
89	Dic₃₁⋊C₄	—	2	1	no	—
90	Dic₃₁⋊C₄	—	2	1	no	—
91	Dic₃₁⋊C₄	—	2	1	no	—
92	Dic₃₁⋊C₄	—	2	1	no	—
93	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
94	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₃₁	—	2	1	no	—
96	C₄×Dic₃₁	—	2	1	no	—
97	C₄×Dic₃₁	—	2	1	no	—
98	C₄×Dic₃₁	—	2	1	no	—
99	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
101	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
104	Dic₁₂₄	—	2	2	no	—
105	C₂³×C₃₁	—	2	1	no	—
106	Q₈×C₃₁	—	2	1	no	—
107	Q₈×C₃₁	—	2	1	no	—
108	Q₈×C₃₁	—	8	1	no	—
109	D₄×C₃₁	—	2	1	no	—
110	D₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₃₁	—	2	1	no	—
118	C₂×Dic₃₁	—	2	1	no	—
119	C₂×Dic₃₁	—	2	1	no	—
120	C₂×Dic₃₁	—	2	1	no	—
121	C₂×Dic₃₁	—	4	2	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	Dic₆₂	—	2	1	no	—
127	Dic₆₂	—	2	2	no	—
128	Dic₆₂	—	2	1	no	—
129	Dic₆₂	—	2	1	no	—
130	Dic₆₂	—	8	1	no	—
131	Dic₆₂	—	8	1	no	—
132	Dic₆₂	#41	1	2	yes	—
133	C₈×C₃₁	—	2	1	no	—
134	C₈×C₃₁	—	2	1	no	—
135	C₃₁⋊C₈	—	2	1	no	—
136	C₃₁⋊C₈	—	2	1	no	—
137	#3369	—	31	1	no	maximal, Sylow
138	C₂²×C₃₁	—	4	1	no	—
139	C₂²×C₃₁	—	4	1	no	—
140	C₂²×C₃₁	—	4	1	no	—
141	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	4	1	no	—
145	C₄×C₃₁	—	4	1	no	—
146	C₄×C₃₁	—	4	1	no	—
147	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₃₁	—	2	2	no	—
150	Dic₃₁	—	2	1	no	—
151	Dic₃₁	—	2	1	no	—
152	Dic₃₁	—	2	1	no	—
153	Dic₃₁	—	4	1	no	—
154	Dic₃₁	—	4	1	no	—
155	Dic₃₁	—	4	1	no	—
156	Dic₃₁	#137	1	2	yes	—
157	Q₈²	—	31	1	no	—
158	#556	—	31	1	no	—
159	#489	—	31	1	no	—
160	#489	D₃₁	1	1	yes	characteristic
161	#487	—	31	1	no	—
162	#476	—	31	1	no	—
163	#475	—	31	1	no	—
164	#475	—	31	1	no	—
165	#464	—	31	1	no	—
166	#464	—	31	1	no	—
167	#462	—	31	1	no	—
168	#446	—	31	1	no	—
169	#441	—	31	1	no	—
170	#441	—	31	1	no	—
171	C₄×SD₁₆	—	31	1	no	—
172	C₂×C₃₁	—	8	1	no	—
173	C₂×C₃₁	#548	1	1	yes	characteristic
174	C₂×C₃₁	C₂×#137	1	1	yes	characteristic
175	C₂×C₃₁	#577	1	1	yes	characteristic
176	C₂×Q₁₆	—	31	1	no	—
177	C₂×Q₁₆	—	31	1	no	—
178	C₂×SD₁₆	—	31	1	no	—
179	C₄⋊Q₈	—	31	1	no	—
180	C₄⋊Q₈	—	31	1	no	—
181	C₄⋊Q₈	—	31	1	no	—
182	C₄⋊Q₈	—	62	1	no	—
183	C₄⋊Q₈	—	62	1	no	—
184	C₄⋊Q₈	—	62	1	no	—
185	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
186	#126	—	62	1	no	—
187	#126	—	62	1	no	—
188	#124	—	31	1	no	—
189	#124	C₂×D₃₁	1	1	yes	characteristic
190	C₂²⋊Q₈	—	62	1	no	—
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×D₄	—	31	1	no	—
199	#117	—	62	1	no	—
200	#106	—	31	1	no	—
201	#106	—	31	1	no	—
202	#106	—	31	1	no	—
203	C₄⋊C₈	—	31	1	no	—
204	C₄⋊C₈	—	31	1	no	—
205	C₄⋊C₈	—	31	1	no	—
206	#103	—	31	1	no	—
207	#103	—	31	1	no	—
208	#103	—	31	1	no	—
209	#103	—	31	1	no	—
210	#103	—	31	1	no	—
211	#103	C₂×D₃₁	1	1	yes	characteristic
212	#103	C₂×D₃₁	1	1	yes	characteristic
213	#102	—	31	1	no	—
214	#102	C₂×D₃₁	1	1	yes	characteristic
215	#102	C₂×D₃₁	1	1	yes	characteristic
216	#97	—	31	1	no	—
217	#97	C₂×D₃₁	1	1	yes	characteristic
218	C₄×C₈	—	31	1	no	—
219	C₃₁	#3369	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	31	1	no	—
222	C₂×Q₈	—	31	1	no	—
223	C₂×Q₈	—	31	1	no	—
224	C₂×Q₈	—	62	1	no	—
225	C₂×Q₈	—	62	1	no	—
226	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
227	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
228	C₂²×C₄	—	62	1	no	—
229	Q₁₆	—	62	2	no	—
230	Q₁₆	—	62	2	no	—
231	SD₁₆	—	62	2	no	—
232	C₂×C₈	—	31	1	no	—
233	C₂×C₈	—	31	1	no	—
234	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	31	1	no	—
238	C₄⋊C₄	—	31	1	no	—
239	C₄⋊C₄	—	31	1	no	—
240	C₄⋊C₄	—	31	1	no	—
241	C₄⋊C₄	—	31	1	no	—
242	C₄⋊C₄	—	62	1	no	—
243	C₄⋊C₄	—	62	1	no	—
244	C₄⋊C₄	—	62	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	C₂²×D₃₁	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	62	1	no	—
256	C₂²⋊C₄	—	62	1	no	—
257	C₂²⋊C₄	—	62	1	no	—
258	C₄²	—	31	1	no	—
259	C₄²	—	31	1	no	—
260	C₄²	—	62	1	no	—
261	C₄²	—	62	1	no	—
262	C₄²	—	62	1	no	—
263	C₄²	—	62	1	no	—
264	C₄²	—	62	1	no	—
265	C₄²	C₂²×D₃₁	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	8	1	no	—
270	Q₈	—	31	2	no	—
271	Q₈	—	62	2	no	—
272	Q₈	—	62	1	no	—
273	Q₈	—	62	1	no	—
274	Q₈	—	62	1	no	—
275	Q₈	—	248	1	no	—
276	Q₈	—	248	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	31	1	no	—
283	C₂×C₄	—	31	1	no	—
284	C₂×C₄	—	31	1	no	—
285	C₂×C₄	—	31	1	no	—
286	C₂×C₄	—	62	1	no	—
287	C₂×C₄	—	62	1	no	—
288	C₂×C₄	—	62	1	no	—
289	C₂×C₄	—	62	1	no	—
290	C₂×C₄	—	124	2	no	—
291	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
292	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	62	1	no	—
297	C₈	—	62	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	31	2	no	—
308	C₄	—	62	1	no	—
309	C₄	—	62	1	no	—
310	C₄	—	62	2	no	—
311	C₄	—	62	1	no	—
312	C₄	—	124	1	no	—
313	C₄	—	124	1	no	—
314	C₄	—	124	1	no	—
315	C₄	#41×D₃₁	1	1	yes	characteristic
316	C₄	#11758567	1	1	yes	characteristic
317	C₂	—	8	1	no	—
318	C₂	#14521707	1	1	yes	characteristic
319	C₂	#137×D₃₁	1	1	yes	characteristic
320	C₂	#14521859	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×#489	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₄×C₃₁	C₂³
hypercenter	#489	D₃₁
hypocenter	C₃₁	#3369
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #14521779	C₂	semidirect
4. #14521398	C₂	semidirect
6. #14521332	C₂	semidirect
7. #14521331	C₂	semidirect
10. #14521258	C₂	semidirect
11. #14521177	C₂	semidirect
12. #14521157	C₂	semidirect
13. #14521153	C₂	semidirect
219. C₃₁	#3369	semidirect

Maximal subgroups

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

subgroup	index
#14521779	2
#14521768	2
#14521407	2
#14521398	2
#14521337	2
#14521332	2
#14521331	2
#14521270	2
#14521268	2
#14521258	2
#14521177	2
#14521157	2
#14521153	2
#14521140	2
C₃₁×#489	2
#3369	31