Subgroups — #16854370

Order 3968.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758449	C₂	1	1	yes	maximal, characteristic
2	#14521111	C₂	1	2	yes	maximal, complemented
3	#14521109	C₂	1	2	yes	maximal, complemented
4	#14521103	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521102	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521099	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521074	C₂	1	2	yes	maximal, complemented
8	#14521011	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
10	#14521006	C₂	1	2	yes	maximal, complemented
11	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₃₁)⋊C₄	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₃₁)⋊C₄	C₄	1	1	yes	characteristic
16	C₂×Dic₃₁⋊C₄	C₄	1	1	yes	characteristic
17	C₂×Dic₃₁⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
19	C₂×#92469	C₂²	1	2	yes	—
20	C₂×C₈×D₃₁	C₂²	1	2	yes	—
21	#11758449	C₄	1	1	yes	characteristic
22	#11758449	C₄	1	1	yes	characteristic
23	#11758449	C₄	1	1	yes	characteristic
24	#11758449	C₄	1	1	yes	characteristic
25	#11758449	C₂²	1	1	yes	characteristic
26	#11758449	C₂²	1	1	yes	characteristic
27	#11758449	C₂²	1	1	yes	characteristic
28	#11758449	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
30	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₃₁×#97	C₂²	1	2	yes	complemented
33	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
34	C₂²⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
35	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
36	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	—
37	#11758403	C₂²	1	2	yes	complemented
38	Dic₃₁⋊C₈	C₂²	1	2	yes	complemented
39	Dic₃₁⋊C₈	C₂²	1	2	yes	complemented
40	C₈×Dic₃₁	C₂²	1	2	yes	complemented
41	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
42	C₄×C₃₁⋊C₈	C₂²	1	2	yes	complemented
43	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₃₁	—	2	4	no	—
51	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂×C₈×C₃₁	C₂³	1	2	yes	—
57	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
62	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
63	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
64	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
65	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
66	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
67	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
68	(C₂×D₃₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
69	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
70	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
71	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
72	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
73	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
74	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
75	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
76	Dic₃₁⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
80	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
82	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
83	#92469	—	2	4	no	—
84	C₈×D₃₁	—	2	4	no	—
85	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₃₁	—	2	2	no	—
87	C₂²×D₃₁	—	2	2	no	—
88	C₂²×D₃₁	—	4	1	no	—
89	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	4	1	no	—
94	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₃₁	—	2	2	no	—
101	C₂×Dic₃₁	—	2	2	no	—
102	C₂×Dic₃₁	—	4	1	no	—
103	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₃₁	—	2	2	no	—
110	C₄×D₃₁	—	2	2	no	—
111	C₄×D₃₁	—	2	2	no	—
112	C₄×D₃₁	#41	1	4	yes	—
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	—	2	2	no	—
115	C₃₁⋊C₈	—	2	2	no	—
116	C₃₁⋊C₈	—	2	2	no	—
117	#2924	—	31	1	no	maximal, Sylow
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	#131	1	2	yes	—
121	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₃₁	—	2	1	no	—
123	C₂×D₃₁	—	2	1	no	—
124	C₂×D₃₁	—	2	1	no	—
125	C₂×D₃₁	—	2	1	no	—
126	C₂×D₃₁	—	2	1	no	—
127	C₂×D₃₁	—	4	4	no	—
128	C₂×D₃₁	#117	1	2	yes	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	2	1	no	—
134	C₄×C₃₁	C₂×#41	1	2	yes	—
135	Dic₃₁	—	2	1	no	—
136	Dic₃₁	—	2	1	no	—
137	Dic₃₁	—	2	1	no	—
138	Dic₃₁	—	2	1	no	—
139	Dic₃₁	—	2	1	no	—
140	Dic₃₁	#117	1	2	yes	—
141	C₂×#117	—	31	1	no	—
142	#433	—	31	2	no	—
143	#433	—	31	2	no	—
144	#432	—	31	1	no	—
145	#432	D₃₁	1	1	yes	characteristic, complemented
146	#431	—	31	1	no	—
147	#431	—	31	1	no	—
148	#424	—	31	2	no	—
149	#407	—	31	1	no	—
150	C₂×C₂²⋊C₈	—	31	1	no	—
151	#405	—	31	2	no	—
152	C₂×C₃₁	—	2	2	no	—
153	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
154	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
155	C₂×C₃₁	#568	1	1	yes	characteristic
156	D₃₁	—	2	2	no	—
157	D₃₁	—	4	1	no	—
158	C₂³×C₄	—	31	1	no	—
159	C₂×M₁₆	—	31	2	no	—
160	C₂²×C₈	—	31	2	no	—
161	#117	—	31	1	no	—
162	#117	—	31	1	no	—
163	#117	—	31	1	no	—
164	#117	—	31	1	no	—
165	#117	—	31	1	no	—
166	#117	—	31	1	no	—
167	#117	—	31	1	no	—
168	#117	—	31	1	no	—
169	C₂×C₄⋊C₄	—	31	1	no	—
170	C₂×C₄⋊C₄	—	31	1	no	—
171	C₂×C₂²⋊C₄	—	31	1	no	—
172	C₂×C₂²⋊C₄	—	31	1	no	—
173	C₂×C₄²	—	31	1	no	—
174	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
175	C₄⋊C₈	—	31	2	no	—
176	C₄⋊C₈	—	31	2	no	—
177	C₄⋊C₈	—	31	2	no	—
178	C₄⋊C₈	C₂×D₃₁	1	2	yes	complemented
179	C₂²⋊C₈	—	31	1	no	—
180	C₂²⋊C₈	—	31	1	no	—
181	C₂²⋊C₈	—	31	2	no	—
182	C₂²⋊C₈	—	31	2	no	—
183	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
184	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
185	#97	—	31	2	no	—
186	#97	C₂×D₃₁	1	2	yes	complemented
187	C₄×C₈	—	31	2	no	—
188	C₄×C₈	—	31	2	no	—
189	C₃₁	#2924	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	—	31	1	no	—
191	C₂²×C₄	—	31	1	no	—
192	C₂²×C₄	—	31	1	no	—
193	C₂²×C₄	—	31	1	no	—
194	C₂²×C₄	—	31	1	no	—
195	C₂²×C₄	—	31	1	no	—
196	C₂²×C₄	—	31	1	no	—
197	C₂²×C₄	—	31	1	no	—
198	C₂²×C₄	—	62	4	no	—
199	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
202	M₁₆	—	62	4	no	—
203	C₂×C₈	—	31	2	no	—
204	C₂×C₈	—	31	2	no	—
205	C₂×C₈	—	62	4	no	—
206	C₂×C₈	C₂²×D₃₁	1	2	yes	—
207	C₂×C₈	C₂²×D₃₁	1	2	yes	—
208	C₄⋊C₄	—	31	1	no	—
209	C₄⋊C₄	—	31	1	no	—
210	C₄⋊C₄	—	31	1	no	—
211	C₄⋊C₄	—	31	1	no	—
212	C₄⋊C₄	—	31	1	no	—
213	C₄⋊C₄	—	31	1	no	—
214	C₄⋊C₄	—	31	1	no	—
215	C₄⋊C₄	—	31	1	no	—
216	C₂²⋊C₄	—	31	1	no	—
217	C₂²⋊C₄	—	31	1	no	—
218	C₂²⋊C₄	—	31	1	no	—
219	C₂²⋊C₄	—	31	1	no	—
220	C₂²⋊C₄	—	31	1	no	—
221	C₂²⋊C₄	—	31	1	no	—
222	C₂²⋊C₄	—	31	1	no	—
223	C₂²⋊C₄	—	31	1	no	—
224	C₄²	—	31	1	no	—
225	C₄²	—	31	1	no	—
226	C₄²	—	31	1	no	—
227	C₄²	—	31	1	no	—
228	C₄²	C₄×D₃₁	1	1	yes	characteristic
229	C₄²	C₄×D₃₁	1	1	yes	characteristic
230	C₄²	C₂²×D₃₁	1	1	yes	characteristic
231	C₄²	C₂²×D₃₁	1	1	yes	characteristic
232	C₂³	—	31	1	no	—
233	C₂³	—	31	1	no	—
234	C₂³	—	62	2	no	—
235	C₂³	—	62	2	no	—
236	C₂³	—	124	1	no	—
237	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	31	1	no	—
242	C₂×C₄	—	31	4	no	—
243	C₂×C₄	—	31	1	no	—
244	C₂×C₄	—	31	1	no	—
245	C₂×C₄	—	31	1	no	—
246	C₂×C₄	—	31	1	no	—
247	C₂×C₄	—	31	1	no	—
248	C₂×C₄	—	62	2	no	—
249	C₂×C₄	—	62	2	no	—
250	C₂×C₄	—	62	2	no	—
251	C₂×C₄	—	62	2	no	—
252	C₂×C₄	—	62	2	no	—
253	C₂×C₄	—	124	1	no	—
254	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₈	—	62	2	no	—
263	C₈	—	62	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	31	2	no	—
267	C₂²	—	62	1	no	—
268	C₂²	—	62	1	no	—
269	C₂²	—	62	1	no	—
270	C₂²	—	62	1	no	—
271	C₂²	—	62	1	no	—
272	C₂²	—	124	4	no	—
273	C₂²	#11758484	1	2	yes	—
274	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	31	2	no	—
281	C₄	—	62	1	no	—
282	C₄	—	62	1	no	—
283	C₄	—	62	1	no	—
284	C₄	—	62	1	no	—
285	C₄	—	62	1	no	—
286	C₄	#41×D₃₁	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	62	2	no	—
289	C₂	—	124	1	no	—
290	C₂	#117×D₃₁	1	1	yes	characteristic
291	C₂	C₂×#11758484	1	1	yes	characteristic
292	C₂	#14521815	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#432	C₂
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#432	D₃₁
hypocenter	C₃₁	#2924
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #14521111	C₂	semidirect
3. #14521109	C₂	semidirect
4. #14521103	C₂	semidirect
5. #14521102	C₂	semidirect
6. #14521099	C₂	semidirect
7. #14521074	C₂	semidirect
10. #14521006	C₂	semidirect
11. C₃₁×#432	C₂	semidirect
29. C₃₁×C₄⋊C₈	C₂²	semidirect
32. C₃₁×#97	C₂²	semidirect
37. #11758403	C₂²	semidirect
38. Dic₃₁⋊C₈	C₂²	semidirect
39. Dic₃₁⋊C₈	C₂²	semidirect
40. C₈×Dic₃₁	C₂²	semidirect
41. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
42. C₄×C₃₁⋊C₈	C₂²	semidirect
145. #432	D₃₁	semidirect
178. C₄⋊C₈	C₂×D₃₁	semidirect
186. #97	C₂×D₃₁	semidirect
189. C₃₁	#2924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758449	2
#14521111	2
#14521109	2
#14521103	2
#14521102	2
#14521099	2
#14521074	2
#14521011	2
C₂²⋊C₈×D₃₁	2
#14521006	2
C₃₁×#432	2
#2924	31