Subgroups — #16858355

Order 3968.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521768	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521758	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521403	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521402	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521358	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521357	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521337	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521262	C₂	1	1	yes	maximal, characteristic
10	#14521252	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521173	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#92480	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#463	C₂	1	1	yes	maximal, characteristic
16	C₂×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#124	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
21	#11758521	C₂²	1	1	yes	characteristic
22	#11758513	C₂²	1	1	yes	characteristic
23	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
24	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
25	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
26	C₂×#92480	C₂²	1	1	yes	characteristic
27	#11758501	—	2	1	no	—
28	C₂×Dic₃₁⋊C₄	—	2	1	no	—
29	#11758467	C₂²	1	1	yes	characteristic
30	#11758465	C₂²	1	1	yes	characteristic
31	#11758456	—	2	1	no	—
32	#11758456	—	2	1	no	—
33	#11758455	—	2	1	no	—
34	#11758455	—	2	1	no	—
35	#11758455	—	2	1	no	—
36	#11758454	—	2	1	no	—
37	#11758447	C₂²	1	1	yes	characteristic
38	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
39	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	#11758423	C₂²	1	1	yes	characteristic
44	#11758423	C₂²	1	1	yes	characteristic
45	#11758420	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758420	C₂²	1	1	yes	characteristic
48	#11758406	C₂²	1	1	yes	characteristic
49	#11758405	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
54	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
55	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
56	#11758398	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	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×Dic₃₁	—	2	1	no	—
64	C₂²×Dic₃₁	—	2	1	no	—
65	C₂²×Dic₃₁	—	2	1	no	—
66	C₂×Dic₆₂	—	2	1	no	—
67	C₂×Dic₆₂	D₄	1	1	yes	characteristic
68	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
69	SD₁₆×C₃₁	—	2	2	no	—
70	D₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂²⋊C₄×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	C₂²⋊Dic₃₁	—	2	1	no	—
82	C₂²⋊Dic₃₁	—	2	1	no	—
83	#92480	—	2	2	no	—
84	C₄⋊Dic₃₁	—	2	1	no	—
85	C₄⋊Dic₃₁	—	2	1	no	—
86	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
87	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
90	Dic₃₁⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
98	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
99	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	C₂³	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	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	Q₈×C₃₁	—	2	1	no	—
108	Q₈×C₃₁	—	2	1	no	—
109	D₄×C₃₁	—	2	1	no	—
110	D₄×C₃₁	—	2	1	no	—
111	D₄×C₃₁	—	2	1	no	—
112	D₄×C₃₁	#41	1	2	yes	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	8	1	no	—
116	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₃₁	—	2	1	no	—
120	C₂×Dic₃₁	—	2	1	no	—
121	C₂×Dic₃₁	—	2	1	no	—
122	C₂×Dic₃₁	—	2	1	no	—
123	C₂×Dic₃₁	—	4	2	no	—
124	C₂×Dic₃₁	—	4	2	no	—
125	C₂×Dic₃₁	—	8	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	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
130	Dic₆₂	—	2	2	no	—
131	Dic₆₂	—	2	1	no	—
132	Dic₆₂	—	2	1	no	—
133	Dic₆₂	—	8	1	no	—
134	C₈×C₃₁	—	2	1	no	—
135	C₈×C₃₁	—	2	1	no	—
136	C₃₁⋊C₈	—	2	1	no	—
137	C₃₁⋊C₈	—	2	1	no	—
138	#3360	—	31	1	no	maximal, Sylow
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	4	1	no	—
141	C₂²×C₃₁	—	4	1	no	—
142	C₂²×C₃₁	—	4	1	no	—
143	C₂²×C₃₁	—	4	1	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₃₁	—	2	1	no	—
147	C₄×C₃₁	—	2	1	no	—
148	C₄×C₃₁	—	4	1	no	—
149	C₄×C₃₁	—	4	1	no	—
150	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
151	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
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	Dic₃₁	#135	1	2	yes	—
160	#556	—	31	1	no	—
161	#554	—	31	1	no	—
162	#488	—	31	1	no	—
163	#488	—	31	1	no	—
164	#487	—	31	1	no	—
165	#479	—	31	1	no	—
166	#479	—	31	1	no	—
167	#476	—	31	1	no	—
168	#463	—	31	1	no	—
169	#463	D₃₁	1	1	yes	characteristic
170	#461	—	31	1	no	—
171	C₈×Q₈	—	31	1	no	—
172	C₄×SD₁₆	—	31	1	no	—
173	C₄×SD₁₆	—	31	1	no	—
174	C₄×D₈	—	31	1	no	—
175	C₂×C₃₁	—	4	2	no	—
176	C₂×C₃₁	—	8	1	no	—
177	C₂×C₃₁	#548	1	1	yes	characteristic
178	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
179	C₂×C₃₁	#577	1	1	yes	characteristic
180	C₂×SD₁₆	—	31	1	no	—
181	C₂×SD₁₆	C₂×D₃₁	1	1	yes	characteristic
182	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
183	C₄⋊Q₈	—	31	1	no	—
184	C₄⋊Q₈	—	31	1	no	—
185	#126	—	62	1	no	—
186	#126	—	62	1	no	—
187	#125	—	31	1	no	—
188	#125	—	31	1	no	—
189	#124	—	31	1	no	—
190	#124	C₂×D₃₁	1	1	yes	characteristic
191	C₂²⋊Q₈	—	62	1	no	—
192	C₂²⋊Q₈	—	62	1	no	—
193	C₂²⋊Q₈	—	62	1	no	—
194	C₂²⋊Q₈	—	62	1	no	—
195	C₄×Q₈	—	31	1	no	—
196	C₄×Q₈	—	31	1	no	—
197	C₄×Q₈	—	31	1	no	—
198	C₄×D₄	—	31	1	no	—
199	C₄×D₄	—	31	1	no	—
200	C₄×D₄	—	31	1	no	—
201	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
202	#117	—	62	1	no	—
203	C₂×C₄⋊C₄	—	62	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	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
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	#102	—	31	1	no	—
217	#102	—	31	1	no	—
218	#102	—	31	1	no	—
219	#102	C₂×D₃₁	1	1	yes	characteristic
220	C₄×C₈	—	31	1	no	—
221	C₄×C₈	—	31	1	no	—
222	C₄×C₈	—	31	1	no	—
223	C₃₁	#3360	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	31	1	no	—
225	C₂×Q₈	—	31	1	no	—
226	C₂×Q₈	—	62	1	no	—
227	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
228	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
229	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	62	1	no	—
232	C₂²×C₄	—	62	1	no	—
233	C₂²×C₄	—	62	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	62	2	no	—
236	D₈	—	2	2	no	—
237	C₂×C₈	—	31	1	no	—
238	C₂×C₈	—	31	1	no	—
239	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
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	—	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	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	62	1	no	—
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	C₄²	—	31	1	no	—
266	C₄²	—	31	1	no	—
267	C₄²	—	62	1	no	—
268	C₄²	—	62	1	no	—
269	C₄²	C₂²×D₃₁	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	62	2	no	—
275	Q₈	—	62	1	no	—
276	Q₈	—	62	1	no	—
277	Q₈	—	248	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	#92497	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	31	1	no	—
286	C₂×C₄	—	31	1	no	—
287	C₂×C₄	—	31	1	no	—
288	C₂×C₄	—	31	1	no	—
289	C₂×C₄	—	62	1	no	—
290	C₂×C₄	—	62	1	no	—
291	C₂×C₄	—	62	1	no	—
292	C₂×C₄	—	62	1	no	—
293	C₂×C₄	—	124	2	no	—
294	C₂×C₄	—	124	2	no	—
295	C₂×C₄	—	248	1	no	—
296	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	62	1	no	—
302	C₈	—	62	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	31	2	no	—
315	C₄	—	62	1	no	—
316	C₄	—	62	1	no	—
317	C₄	—	62	1	no	—
318	C₄	—	124	1	no	—
319	C₄	—	124	1	no	—
320	C₄	—	124	1	no	—
321	C₄	—	124	1	no	—
322	C₄	C₂×#92497	1	1	yes	characteristic
323	C₄	#11758567	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	8	1	no	—
326	C₂	#14521705	1	1	yes	characteristic
327	C₂	#135×D₃₁	1	1	yes	characteristic
328	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×#463	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₈×C₃₁	C₂²
hypercenter	#463	D₃₁
hypocenter	C₃₁	#3360
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #14521768	C₂	semidirect
2. #14521758	C₂	semidirect
3. #14521403	C₂	semidirect
4. #14521402	C₂	semidirect
5. #14521398	C₂	semidirect
6. #14521358	C₂	semidirect
7. #14521357	C₂	semidirect
8. #14521337	C₂	semidirect
10. #14521252	C₂	semidirect
11. #14521173	C₂	semidirect
12. SD₁₆×Dic₃₁	C₂	semidirect
13. C₄×#92480	C₂	semidirect
223. C₃₁	#3360	semidirect

Maximal subgroups

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

subgroup	index
#14521768	2
#14521758	2
#14521403	2
#14521402	2
#14521398	2
#14521358	2
#14521357	2
#14521337	2
#14521262	2
#14521252	2
#14521173	2
SD₁₆×Dic₃₁	2
C₄×#92480	2
D₈×Dic₃₁	2
C₃₁×#463	2
#3360	31