Subgroups — #16858432

Order 3968.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521771	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521714	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521411	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521410	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521343	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521328	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521267	C₂	1	1	yes	maximal, characteristic
9	#14521265	C₂	1	1	yes	maximal, characteristic
10	#14521253	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521148	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521147	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521131	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₃₁	—	2	1	no	—
17	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	#11758524	C₂²	1	1	yes	characteristic
20	#11758523	—	2	1	no	—
21	#11758521	C₂²	1	1	yes	characteristic
22	C₂×#92481	C₂²	1	1	yes	characteristic
23	C₂×D₂₄₈	C₂²	1	1	yes	characteristic
24	C₂×#92470	C₂²	1	1	yes	characteristic
25	#11758475	—	2	1	no	—
26	#11758473	—	2	1	no	—
27	#11758473	—	2	1	no	—
28	#11758473	—	2	1	no	—
29	#11758470	C₂²	1	1	yes	characteristic
30	#11758470	C₂²	1	1	yes	characteristic
31	#11758470	C₂²	1	1	yes	characteristic
32	#11758465	C₂²	1	1	yes	characteristic
33	#11758465	C₂²	1	1	yes	characteristic
34	#11758464	C₂²	1	1	yes	characteristic
35	#11758453	—	2	1	no	—
36	#11758452	C₂²	1	1	yes	characteristic
37	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
38	#11758449	—	2	1	no	—
39	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
40	C₃₁×#107	C₂²	1	1	yes	characteristic
41	C₃₁×#107	C₂²	1	1	yes	characteristic
42	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×#103	C₂²	1	1	yes	characteristic
44	C₃₁×#103	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758409	C₂²	1	1	yes	characteristic
47	#11758404	C₂²	1	1	yes	characteristic
48	#11758403	C₂²	1	1	yes	characteristic
49	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
52	#11758398	C₂²	1	1	yes	characteristic
53	#11758398	C₂²	1	1	yes	characteristic
54	#11758397	C₂²	1	1	yes	characteristic
55	#11758397	C₂²	1	1	yes	characteristic
56	#11758397	C₂²	1	1	yes	characteristic
57	#11758395	C₂²	1	1	yes	characteristic
58	#11758391	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
61	Q₈×D₃₁	—	4	4	no	—
62	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
63	C₂×D₁₂₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₃₁	—	2	1	no	—
65	C₂×C₄×D₃₁	—	2	1	no	—
66	C₂×C₄×D₃₁	—	2	1	no	—
67	C₂×C₄×D₃₁	—	2	1	no	—
68	C₂×Dic₆₂	—	2	1	no	—
69	C₂×Dic₆₂	—	2	1	no	—
70	C₂×Dic₆₂	D₄	1	1	yes	characteristic
71	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₄²×C₃₁	—	2	1	no	—
80	C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	#92481	—	2	2	no	—
82	(C₂×D₃₁)⋊C₄	—	2	1	no	—
83	(C₂×D₃₁)⋊C₄	—	2	1	no	—
84	(C₂×D₃₁)⋊C₄	—	2	1	no	—
85	(C₂×D₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₃₁)⋊C₄	—	2	1	no	—
87	(C₂×D₃₁)⋊C₄	—	2	1	no	—
88	(C₂×D₃₁)⋊C₄	—	2	1	no	—
89	(C₂×D₃₁)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₃₁	D₄	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	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
98	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
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	D₂₄₈	—	2	2	no	—
105	#92470	—	2	2	no	—
106	C₂²×D₃₁	—	2	1	no	—
107	C₂²×D₃₁	—	2	1	no	—
108	Q₈×C₃₁	—	2	1	no	—
109	Q₈×C₃₁	—	2	1	no	—
110	Q₈×C₃₁	—	4	2	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	—	2	1	no	—
115	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₃₁	—	2	1	no	—
119	C₂×Dic₃₁	—	2	1	no	—
120	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
124	D₁₂₄	—	2	1	no	—
125	D₁₂₄	—	2	1	no	—
126	D₁₂₄	—	2	1	no	—
127	D₁₂₄	#41	1	2	yes	—
128	C₄×D₃₁	—	4	2	no	—
129	C₄×D₃₁	—	4	2	no	—
130	C₄×D₃₁	—	4	2	no	—
131	C₄×D₃₁	—	4	2	no	—
132	Dic₆₂	—	2	2	no	—
133	Dic₆₂	—	2	1	no	—
134	Dic₆₂	—	2	1	no	—
135	Dic₆₂	—	4	2	no	—
136	Dic₆₂	—	4	2	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₃₁⋊C₈	—	2	1	no	—
140	C₃₁⋊C₈	—	2	1	no	—
141	#3367	—	31	1	no	maximal, Sylow
142	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₃₁	—	2	2	no	—
144	C₂×D₃₁	—	4	1	no	—
145	C₂×D₃₁	—	4	1	no	—
146	C₂×D₃₁	—	4	1	no	—
147	C₂×D₃₁	—	4	1	no	—
148	C₂×D₃₁	—	4	1	no	—
149	C₄×C₃₁	—	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	—	4	1	no	—
155	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₄×C₃₁	2₋¹⁺⁴	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	AGL(1,ℤ/8)	1	2	yes	—
163	#556	—	31	1	no	—
164	D₄×Q₈	—	31	1	no	—
165	#489	—	31	1	no	—
166	#489	—	31	1	no	—
167	#487	—	31	1	no	—
168	#477	—	31	1	no	—
169	#477	D₃₁	1	1	yes	characteristic, complemented
170	#474	—	31	1	no	—
171	#463	—	31	1	no	—
172	#463	—	31	1	no	—
173	#461	—	31	1	no	—
174	#446	—	31	1	no	—
175	#440	—	31	1	no	—
176	#440	—	31	1	no	—
177	C₄×D₈	—	31	1	no	—
178	C₂×C₃₁	#548	1	1	yes	characteristic
179	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×C₃₁	#578	1	1	yes	characteristic
181	D₃₁	—	4	2	no	—
182	D₃₁	—	8	1	no	—
183	C₂²×Q₈	—	62	1	no	—
184	C₂×SD₁₆	—	31	1	no	—
185	C₂×SD₁₆	—	31	1	no	—
186	C₂×D₈	—	31	1	no	—
187	C₄⋊Q₈	—	31	1	no	—
188	C₄⋊Q₈	—	31	1	no	—
189	C₄⋊Q₈	—	31	1	no	—
190	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
191	#126	—	62	1	no	—
192	#126	—	62	1	no	—
193	#124	—	31	1	no	—
194	#124	—	31	1	no	—
195	C₂²⋊Q₈	—	62	1	no	—
196	C₂²⋊Q₈	—	62	1	no	—
197	C₂²⋊Q₈	—	62	1	no	—
198	C₂²⋊Q₈	—	62	1	no	—
199	C₄×Q₈	—	31	1	no	—
200	C₄×Q₈	—	31	1	no	—
201	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
202	C₄×D₄	—	31	1	no	—
203	C₄×D₄	—	31	1	no	—
204	C₄×D₄	—	31	1	no	—
205	C₄×D₄	—	31	1	no	—
206	#117	—	62	1	no	—
207	#107	—	31	1	no	—
208	#107	C₂×D₃₁	1	1	yes	characteristic
209	#107	C₂×D₃₁	1	1	yes	characteristic
210	C₄⋊C₈	—	31	1	no	—
211	C₄⋊C₈	—	31	1	no	—
212	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
213	#103	—	31	1	no	—
214	#103	—	31	1	no	—
215	#103	—	31	1	no	—
216	#103	—	31	1	no	—
217	#103	C₂×D₃₁	1	1	yes	characteristic
218	#103	C₂×D₃₁	1	1	yes	characteristic
219	#102	—	31	1	no	—
220	#102	—	31	1	no	—
221	#102	—	31	1	no	—
222	#102	—	31	1	no	—
223	#97	—	31	1	no	—
224	#97	—	31	1	no	—
225	C₄×C₈	—	31	1	no	—
226	C₃₁	#3367	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	31	1	no	—
229	C₂×Q₈	—	31	1	no	—
230	C₂×Q₈	—	62	1	no	—
231	C₂×Q₈	—	62	1	no	—
232	C₂×Q₈	—	124	4	no	—
233	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
234	C₂×D₄	—	31	1	no	—
235	C₂×D₄	—	31	1	no	—
236	C₂²×C₄	—	62	1	no	—
237	C₂²×C₄	—	62	1	no	—
238	C₂²×C₄	—	62	1	no	—
239	C₂²×C₄	—	62	1	no	—
240	SD₁₆	—	62	2	no	—
241	SD₁₆	—	62	2	no	—
242	D₈	—	62	2	no	—
243	C₂×C₈	—	31	1	no	—
244	C₂×C₈	—	31	1	no	—
245	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	31	1	no	—
250	C₄⋊C₄	—	31	1	no	—
251	C₄⋊C₄	—	31	1	no	—
252	C₄⋊C₄	—	31	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	C₂²×D₃₁	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	62	1	no	—
262	C₂²⋊C₄	—	62	1	no	—
263	C₂²⋊C₄	—	62	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	C₄²	—	2	1	no	—
270	C₄²	—	31	1	no	—
271	C₄²	—	31	1	no	—
272	C₄²	—	62	1	no	—
273	C₄²	C₂²×D₃₁	1	1	yes	characteristic
274	C₂³	—	62	1	no	—
275	C₂³	—	62	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	62	2	no	—
280	Q₈	—	62	1	no	—
281	Q₈	—	62	1	no	—
282	Q₈	—	124	2	no	—
283	Q₈	—	124	2	no	—
284	D₄	—	31	2	no	—
285	D₄	—	62	1	no	—
286	D₄	—	62	1	no	—
287	D₄	—	62	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	31	1	no	—
293	C₂×C₄	—	31	1	no	—
294	C₂×C₄	—	31	1	no	—
295	C₂×C₄	—	31	1	no	—
296	C₂×C₄	—	62	1	no	—
297	C₂×C₄	—	62	1	no	—
298	C₂×C₄	—	124	2	no	—
299	C₂×C₄	—	124	2	no	—
300	C₂×C₄	—	124	2	no	—
301	C₂×C₄	—	124	2	no	—
302	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
303	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	62	1	no	—
308	C₈	—	62	1	no	—
309	C₂²	—	62	2	no	—
310	C₂²	—	124	1	no	—
311	C₂²	—	124	1	no	—
312	C₂²	—	124	1	no	—
313	C₂²	—	124	1	no	—
314	C₂²	—	124	1	no	—
315	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	31	2	no	—
323	C₄	—	62	1	no	—
324	C₄	—	62	1	no	—
325	C₄	—	62	1	no	—
326	C₄	—	124	1	no	—
327	C₄	—	124	1	no	—
328	C₄	#11758564	1	1	yes	characteristic
329	C₄	#41×D₃₁	1	1	yes	characteristic
330	C₂	—	124	2	no	—
331	C₂	—	248	1	no	—
332	C₂	#14521709	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₃₁	1	1	yes	characteristic
334	C₂	#14521862	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×#477	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₂₄₈	C₂²
hypercenter	#477	D₃₁
hypocenter	C₃₁	#3367
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #14521771	C₂	semidirect
2. #14521714	C₂	semidirect
3. #14521411	C₂	semidirect
4. #14521410	C₂	semidirect
5. #14521398	C₂	semidirect
6. #14521343	C₂	semidirect
7. #14521328	C₂	semidirect
10. #14521253	C₂	semidirect
11. #14521178	C₂	semidirect
12. #14521148	C₂	semidirect
13. #14521147	C₂	semidirect
15. C₃₁×#477	C₂	semidirect
169. #477	D₃₁	semidirect
226. C₃₁	#3367	semidirect

Maximal subgroups

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

subgroup	index
#14521771	2
#14521714	2
#14521411	2
#14521410	2
#14521398	2
#14521343	2
#14521328	2
#14521267	2
#14521265	2
#14521253	2
#14521178	2
#14521148	2
#14521147	2
#14521131	2
C₃₁×#477	2
#3367	31