Subgroups — #16854656

Order 3968.

356 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	#14521603	C₂	1	1	yes	maximal, characteristic
2	#14521180	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521177	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521173	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521074	C₂	1	1	yes	maximal, characteristic
10	#14521074	C₂	1	1	yes	maximal, characteristic
11	C₄⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
12	#14521006	C₂	1	1	yes	maximal, characteristic
13	#14521003	C₂	1	1	yes	maximal, characteristic
14	#14521001	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	#11758523	C₄	1	1	yes	characteristic
18	#11758521	C₄	1	1	yes	characteristic
19	C₂×#92469	C₂²	1	1	yes	characteristic
20	C₂×#92469	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
23	#11758474	C₄	1	1	yes	characteristic
24	#11758473	C₄	1	1	yes	characteristic
25	#11758473	C₄	1	1	yes	characteristic
26	#11758466	C₄	1	1	yes	characteristic
27	#11758466	C₄	1	1	yes	characteristic
28	#11758465	C₄	1	1	yes	characteristic
29	#11758449	C₂²	1	1	yes	characteristic
30	#11758449	C₂²	1	1	yes	characteristic
31	C₄²×D₃₁	C₂²	1	1	yes	characteristic
32	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
33	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
34	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
35	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃₁×#97	C₂²	1	1	yes	characteristic
39	C₃₁×#97	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	#11758403	C₂²	1	1	yes	characteristic
42	#11758403	C₂²	1	1	yes	characteristic
43	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
46	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
49	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
50	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
52	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
53	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
54	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
55	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
56	#11758391	C₂²	1	1	yes	characteristic
57	#11758391	C₂²	1	1	yes	characteristic
58	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
63	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₆₂	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₃₁	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	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
77	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
78	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
79	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
83	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
84	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
85	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
86	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
87	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
88	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
89	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
90	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
92	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
96	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
97	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
99	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
102	#92469	—	2	2	no	—
103	#92469	Q₈	1	4	yes	complemented
104	C₈×D₃₁	—	2	2	no	—
105	C₈×D₃₁	—	2	2	no	—
106	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
107	Q₈×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₄×D₃₁	—	2	2	no	—
123	C₄×D₃₁	—	4	1	no	—
124	C₄×D₃₁	C₂×Q₈	1	2	yes	—
125	C₄×D₃₁	#41	1	2	yes	—
126	Dic₆₂	—	2	2	no	—
127	Dic₆₂	—	2	2	no	—
128	Dic₆₂	—	4	1	no	—
129	C₈×C₃₁	—	2	1	no	—
130	C₈×C₃₁	—	2	1	no	—
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	C₂×Q₈	1	2	yes	—
133	C₃₁⋊C₈	—	2	1	no	—
134	C₃₁⋊C₈	—	2	1	no	—
135	C₃₁⋊C₈	—	2	1	no	—
136	C₃₁⋊C₈	C₂×Q₈	1	2	yes	—
137	#2962	—	31	1	no	maximal, Sylow
138	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
139	C₂×D₃₁	—	2	1	no	—
140	C₂×D₃₁	—	2	1	no	—
141	C₂×D₃₁	C₄×Q₈	1	2	yes	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	2	1	no	—
146	C₄×C₃₁	—	2	1	no	—
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	Dic₃₁	—	2	1	no	—
151	Dic₃₁	—	2	1	no	—
152	Dic₃₁	—	2	1	no	—
153	Dic₃₁	—	2	1	no	—
154	Dic₃₁	—	2	1	no	—
155	Dic₃₁	C₄×Q₈	1	2	yes	—
156	Dic₃₁	#131	1	2	yes	—
157	#533	—	31	1	no	—
158	#446	—	31	1	no	—
159	#446	—	31	1	no	—
160	#446	—	31	1	no	—
161	#446	—	31	1	no	—
162	#446	—	31	1	no	—
163	#446	D₃₁	1	1	yes	characteristic, complemented
164	C₈×Q₈	—	31	1	no	—
165	C₈×Q₈	—	31	1	no	—
166	#424	—	31	1	no	—
167	#424	—	31	1	no	—
168	C₂×C₄⋊C₈	—	31	1	no	—
169	#405	—	31	1	no	—
170	#405	—	31	1	no	—
171	C₄×M₁₆	—	31	1	no	—
172	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
174	C₂×C₃₁	#568	1	1	yes	characteristic
175	D₃₁	—	2	2	no	—
176	C₂×M₁₆	—	31	1	no	—
177	C₂×M₁₆	—	31	1	no	—
178	C₂²×C₈	—	31	1	no	—
179	C₂²×C₈	—	31	1	no	—
180	C₄⋊Q₈	—	31	1	no	—
181	C₄⋊Q₈	—	31	1	no	—
182	#125	—	31	1	no	—
183	#125	—	31	1	no	—
184	C₂²⋊Q₈	—	31	1	no	—
185	C₂²⋊Q₈	—	31	1	no	—
186	C₂²⋊Q₈	—	31	1	no	—
187	C₂²⋊Q₈	—	31	1	no	—
188	C₄×Q₈	—	31	1	no	—
189	C₄×Q₈	—	31	1	no	—
190	C₄×Q₈	—	31	1	no	—
191	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
192	#117	—	31	1	no	—
193	#117	—	31	1	no	—
194	C₂×C₄²	—	31	1	no	—
195	C₄⋊C₈	—	31	1	no	—
196	C₄⋊C₈	—	31	1	no	—
197	C₄⋊C₈	—	31	1	no	—
198	C₄⋊C₈	—	31	1	no	—
199	C₄⋊C₈	—	31	1	no	—
200	C₄⋊C₈	—	31	1	no	—
201	C₄⋊C₈	—	31	1	no	—
202	C₄⋊C₈	—	31	1	no	—
203	C₄⋊C₈	—	31	1	no	—
204	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
205	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
207	#97	—	31	1	no	—
208	#97	—	31	1	no	—
209	#97	—	31	1	no	—
210	#97	—	31	1	no	—
211	#97	C₂×D₃₁	1	1	yes	characteristic
212	#97	C₂×D₃₁	1	1	yes	characteristic
213	C₄×C₈	—	31	1	no	—
214	C₄×C₈	—	31	1	no	—
215	C₄×C₈	—	31	1	no	—
216	C₄×C₈	—	31	1	no	—
217	C₄×C₈	—	31	1	no	—
218	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
219	C₃₁	#2962	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	31	1	no	—
221	C₂×Q₈	—	31	1	no	—
222	C₂×Q₈	—	31	1	no	—
223	C₂×Q₈	C₄×D₃₁	1	1	yes	characteristic
224	C₂²×C₄	—	31	1	no	—
225	C₂²×C₄	—	31	1	no	—
226	C₂²×C₄	—	31	1	no	—
227	M₁₆	—	31	4	no	—
228	M₁₆	—	62	2	no	—
229	C₂×C₈	—	31	1	no	—
230	C₂×C₈	—	31	1	no	—
231	C₂×C₈	—	31	1	no	—
232	C₂×C₈	—	31	1	no	—
233	C₂×C₈	—	62	2	no	—
234	C₂×C₈	—	62	2	no	—
235	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
239	C₄⋊C₄	—	31	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	—	31	1	no	—
246	C₄⋊C₄	—	31	1	no	—
247	C₄⋊C₄	—	31	1	no	—
248	C₄⋊C₄	—	31	1	no	—
249	C₄⋊C₄	—	31	1	no	—
250	C₄⋊C₄	—	31	1	no	—
251	C₄⋊C₄	—	31	1	no	—
252	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
255	C₂²⋊C₄	—	31	1	no	—
256	C₂²⋊C₄	—	31	1	no	—
257	C₂²⋊C₄	—	31	1	no	—
258	C₂²⋊C₄	—	31	1	no	—
259	C₄²	—	31	1	no	—
260	C₄²	—	31	1	no	—
261	C₄²	—	31	1	no	—
262	C₄²	—	31	1	no	—
263	C₄²	—	31	1	no	—
264	C₄²	C₂²×D₃₁	1	1	yes	characteristic
265	C₄²	C₂²×D₃₁	1	1	yes	characteristic
266	C₄²	C₂²×D₃₁	1	1	yes	characteristic
267	C₂³	—	31	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	62	2	no	—
270	Q₈	—	62	2	no	—
271	Q₈	—	124	1	no	—
272	C₂×C₄	—	31	1	no	—
273	C₂×C₄	—	31	1	no	—
274	C₂×C₄	—	31	2	no	—
275	C₂×C₄	—	31	2	no	—
276	C₂×C₄	—	31	1	no	—
277	C₂×C₄	—	31	1	no	—
278	C₂×C₄	—	31	1	no	—
279	C₂×C₄	—	31	1	no	—
280	C₂×C₄	—	31	1	no	—
281	C₂×C₄	—	62	2	no	—
282	C₂×C₄	—	124	1	no	—
283	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	31	2	no	—
294	C₈	—	62	1	no	—
295	C₈	—	62	1	no	—
296	C₈	—	62	1	no	—
297	C₈	Q₈×D₃₁	1	2	yes	—
298	C₂²	—	31	2	no	—
299	C₂²	—	62	1	no	—
300	C₂²	—	62	1	no	—
301	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	31	2	no	—
309	C₄	—	31	2	no	—
310	C₄	—	62	1	no	—
311	C₄	—	62	1	no	—
312	C₄	—	62	1	no	—
313	C₄	—	62	1	no	—
314	C₄	—	62	1	no	—
315	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
316	C₄	#41×D₃₁	1	1	yes	characteristic
317	C₂	—	62	2	no	—
318	C₂	C₄×Q₈×D₃₁	1	1	yes	characteristic
319	C₂	#131×D₃₁	1	1	yes	characteristic
320	C₂	#14521815	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#446	C₂
prime radical	C₂²×D₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#446	D₃₁
hypocenter	C₃₁	#2962
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #14521180	C₂	semidirect
3. #14521178	C₂	semidirect
4. #14521178	C₂	semidirect
5. #14521177	C₂	semidirect
6. #14521177	C₂	semidirect
7. C₈×Dic₆₂	C₂	semidirect
8. #14521173	C₂	semidirect
15. C₃₁×#446	C₂	semidirect
103. #92469	Q₈	semidirect
163. #446	D₃₁	semidirect
219. C₃₁	#2962	semidirect

Maximal subgroups

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

subgroup	index
#14521603	2
#14521180	2
#14521178	2
#14521177	2
C₈×Dic₆₂	2
#14521173	2
#14521074	2
C₄⋊C₈×D₃₁	2
#14521006	2
#14521003	2
#14521001	2
C₃₁×#446	2
#2962	31