Subgroups — #16858393

Order 3968.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521776	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521768	C₂	1	1	yes	maximal, characteristic
3	#14521402	C₂	1	1	yes	maximal, characteristic
4	#14521398	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521358	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521353	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521327	C₂	1	1	yes	maximal, characteristic
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	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521141	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521140	C₂	1	1	yes	maximal, characteristic
15	C₃₁×#488	C₂	1	1	yes	maximal, characteristic
16	C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₃₁×#124	C₂²	1	1	yes	characteristic
18	Q₈×Dic₃₁	C₂²	1	1	yes	characteristic
19	#11758521	C₂²	1	1	yes	characteristic
20	C₂×#92482	C₂²	1	1	yes	characteristic
21	#11758513	C₂²	1	1	yes	characteristic
22	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
23	C₂×#92480	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₂₄	C₂²	1	1	yes	characteristic
25	#11758467	C₂²	1	1	yes	characteristic
26	#11758466	—	2	1	no	—
27	#11758466	—	2	1	no	—
28	#11758465	—	2	1	no	—
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	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	#11758423	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758406	C₂²	1	1	yes	characteristic
48	#11758404	C₂²	1	1	yes	characteristic
49	#11758404	C₂²	1	1	yes	characteristic
50	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₃₁	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	#11758395	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	C₂²×Dic₃₁	—	2	1	no	—
62	C₂×Dic₆₂	—	2	1	no	—
63	C₂×Dic₆₂	D₄	1	1	yes	characteristic
64	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
65	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₃₁	—	2	1	no	—
76	C₂²⋊Dic₃₁	—	2	1	no	—
77	C₂²⋊Dic₃₁	—	2	1	no	—
78	#92482	—	2	2	no	—
79	#92480	—	2	2	no	—
80	C₄⋊Dic₃₁	—	2	1	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	D₄×C₃₁	—	2	1	no	—
109	D₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₃₁	—	2	1	no	—
117	C₂×Dic₃₁	—	2	1	no	—
118	C₂×Dic₃₁	—	2	1	no	—
119	C₂×Dic₃₁	—	2	1	no	—
120	C₂×Dic₃₁	—	4	2	no	—
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	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
125	Dic₆₂	—	2	1	no	—
126	Dic₆₂	—	2	2	no	—
127	Dic₆₂	—	2	1	no	—
128	Dic₆₂	—	2	1	no	—
129	Dic₆₂	—	4	2	no	—
130	Dic₆₂	#41	1	2	yes	—
131	C₈×C₃₁	—	2	1	no	—
132	C₈×C₃₁	—	2	1	no	—
133	C₃₁⋊C₈	—	2	1	no	—
134	C₃₁⋊C₈	—	2	1	no	—
135	#3364	—	31	1	no	maximal, Sylow
136	C₂²×C₃₁	—	4	1	no	—
137	C₂²×C₃₁	—	4	1	no	—
138	C₂²×C₃₁	—	4	1	no	—
139	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	4	1	no	—
143	C₄×C₃₁	—	4	1	no	—
144	C₄×C₃₁	—	4	1	no	—
145	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
146	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₃₁	—	2	2	no	—
148	Dic₃₁	—	2	1	no	—
149	Dic₃₁	—	2	1	no	—
150	Dic₃₁	—	2	1	no	—
151	Dic₃₁	—	4	1	no	—
152	Dic₃₁	—	4	1	no	—
153	Dic₃₁	—	4	1	no	—
154	Dic₃₁	#135	1	2	yes	—
155	#557	—	31	1	no	—
156	#556	—	31	1	no	—
157	#488	—	31	1	no	—
158	#488	D₃₁	1	1	yes	characteristic
159	#487	—	31	1	no	—
160	#479	—	31	1	no	—
161	#479	—	31	1	no	—
162	#474	—	31	1	no	—
163	#464	—	31	1	no	—
164	#464	—	31	1	no	—
165	#462	—	31	1	no	—
166	C₈×Q₈	—	31	1	no	—
167	C₄×Q₁₆	—	31	1	no	—
168	C₄×Q₁₆	—	31	1	no	—
169	C₄×SD₁₆	—	31	1	no	—
170	C₂×C₃₁	—	8	1	no	—
171	C₂×C₃₁	#548	1	1	yes	characteristic
172	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
173	C₂×C₃₁	#578	1	1	yes	characteristic
174	C₂×Q₁₆	—	31	1	no	—
175	C₂×Q₁₆	—	31	1	no	—
176	C₂×SD₁₆	—	31	1	no	—
177	C₄⋊Q₈	—	31	1	no	—
178	C₄⋊Q₈	—	31	1	no	—
179	C₄⋊Q₈	—	62	1	no	—
180	#126	—	62	1	no	—
181	#126	—	62	1	no	—
182	#125	—	31	1	no	—
183	#125	—	62	1	no	—
184	#125	—	62	1	no	—
185	#125	C₂×D₃₁	1	1	yes	characteristic
186	#124	—	31	1	no	—
187	#124	C₂×D₃₁	1	1	yes	characteristic
188	C₂²⋊Q₈	—	62	1	no	—
189	C₄×Q₈	—	31	1	no	—
190	C₄×Q₈	—	31	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	—	62	1	no	—
196	C₄×D₄	—	31	1	no	—
197	#117	—	62	1	no	—
198	#107	—	31	1	no	—
199	#107	—	31	1	no	—
200	#106	—	31	1	no	—
201	C₄⋊C₈	—	31	1	no	—
202	C₄⋊C₈	—	31	1	no	—
203	C₄⋊C₈	—	31	1	no	—
204	#103	—	31	1	no	—
205	#103	—	31	1	no	—
206	#103	—	31	1	no	—
207	#103	—	31	1	no	—
208	#103	—	31	1	no	—
209	#103	C₂×D₃₁	1	1	yes	characteristic
210	#103	C₂×D₃₁	1	1	yes	characteristic
211	#102	—	31	1	no	—
212	#102	C₂×D₃₁	1	1	yes	characteristic
213	#102	C₂×D₃₁	1	1	yes	characteristic
214	C₄×C₈	—	31	1	no	—
215	C₄×C₈	—	31	1	no	—
216	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
217	C₃₁	#3364	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	31	1	no	—
219	C₂×Q₈	—	31	1	no	—
220	C₂×Q₈	—	31	1	no	—
221	C₂×Q₈	—	62	1	no	—
222	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
223	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
224	C₂²×C₄	—	62	1	no	—
225	Q₁₆	—	62	2	no	—
226	Q₁₆	—	62	2	no	—
227	SD₁₆	—	62	2	no	—
228	C₂×C₈	—	31	1	no	—
229	C₂×C₈	—	31	1	no	—
230	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	31	1	no	—
235	C₄⋊C₄	—	31	1	no	—
236	C₄⋊C₄	—	31	1	no	—
237	C₄⋊C₄	—	31	1	no	—
238	C₄⋊C₄	—	31	1	no	—
239	C₄⋊C₄	—	62	1	no	—
240	C₄⋊C₄	—	62	1	no	—
241	C₄⋊C₄	—	62	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	C₂²×D₃₁	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	62	1	no	—
254	C₂²⋊C₄	—	62	1	no	—
255	C₂²⋊C₄	—	62	1	no	—
256	C₄²	—	31	1	no	—
257	C₄²	—	31	1	no	—
258	C₄²	—	62	1	no	—
259	C₄²	—	62	1	no	—
260	C₄²	—	62	1	no	—
261	C₄²	—	62	1	no	—
262	C₄²	—	62	1	no	—
263	C₄²	C₂²×D₃₁	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	31	2	no	—
268	Q₈	—	62	2	no	—
269	Q₈	—	62	1	no	—
270	Q₈	—	62	1	no	—
271	Q₈	—	62	1	no	—
272	Q₈	—	124	2	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	31	1	no	—
279	C₂×C₄	—	31	1	no	—
280	C₂×C₄	—	31	1	no	—
281	C₂×C₄	—	31	1	no	—
282	C₂×C₄	—	62	1	no	—
283	C₂×C₄	—	62	1	no	—
284	C₂×C₄	—	62	1	no	—
285	C₂×C₄	—	62	1	no	—
286	C₂×C₄	—	124	2	no	—
287	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
288	C₂×C₄	D₄×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	—	62	1	no	—
293	C₈	—	62	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	31	2	no	—
304	C₄	—	62	1	no	—
305	C₄	—	62	1	no	—
306	C₄	—	62	2	no	—
307	C₄	—	62	1	no	—
308	C₄	—	124	1	no	—
309	C₄	—	124	1	no	—
310	C₄	—	124	1	no	—
311	C₄	#41×D₃₁	1	1	yes	characteristic
312	C₄	#11758567	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	#14521707	1	1	yes	characteristic
315	C₂	#135×D₃₁	1	1	yes	characteristic
316	C₂	#14521863	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×#488	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×D₄×C₃₁	C₂³
hypercenter	#488	D₃₁
hypocenter	C₃₁	#3364
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #14521776	C₂	semidirect
4. #14521398	C₂	semidirect
5. #14521358	C₂	semidirect
6. #14521353	C₂	semidirect
10. #14521258	C₂	semidirect
11. C₈×Dic₆₂	C₂	semidirect
12. C₄×Dic₁₂₄	C₂	semidirect
13. #14521141	C₂	semidirect
217. C₃₁	#3364	semidirect

Maximal subgroups

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

subgroup	index
#14521776	2
#14521768	2
#14521402	2
#14521398	2
#14521358	2
#14521353	2
#14521327	2
#14521270	2
#14521268	2
#14521258	2
C₈×Dic₆₂	2
C₄×Dic₁₂₄	2
#14521141	2
#14521140	2
C₃₁×#488	2
#3364	31