Subgroups — #16854341

Order 3968.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521106	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521104	C₂	1	2	yes	maximal, complemented
4	#14521101	C₂	1	2	yes	maximal, complemented
5	#14521099	C₂	1	2	yes	maximal, complemented
6	#14521071	C₂	1	2	yes	maximal, complemented
7	#14521011	C₂	1	2	yes	maximal
8	#14521001	C₂	1	2	yes	maximal, complemented
9	C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×#92469	C₂²	1	4	yes	—
18	C₄²×D₃₁	C₄	1	1	yes	characteristic
19	C₄²×D₃₁	C₄	1	1	yes	characteristic
20	C₄²×D₃₁	C₄	1	1	yes	characteristic
21	C₄²×D₃₁	C₄	1	1	yes	characteristic
22	C₄²×D₃₁	C₂²	1	1	yes	characteristic
23	C₄²×D₃₁	C₂²	1	1	yes	characteristic
24	C₄²×D₃₁	C₂²	1	1	yes	characteristic
25	C₄²×D₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₃₁×#97	C₂²	1	2	yes	complemented
29	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
30	(C₂×D₃₁)⋊C₈	C₂²	1	4	yes	—
31	Dic₃₁⋊C₈	C₂²	1	4	yes	complemented
32	C₈×Dic₃₁	C₂²	1	4	yes	complemented
33	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
34	#11758391	C₂²	1	2	yes	complemented
35	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₃₁	—	2	4	no	—
43	C₂×C₄×D₃₁	—	2	4	no	—
44	C₂×C₄×D₃₁	—	2	4	no	—
45	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂³	1	4	yes	—
58	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
73	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
75	#92469	—	2	8	no	—
76	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₃₁	—	2	2	no	—
78	C₂²×D₃₁	—	2	2	no	—
79	C₂²×D₃₁	—	2	2	no	—
80	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₃₁	—	2	2	no	—
92	C₂×Dic₃₁	—	2	2	no	—
93	C₂×Dic₃₁	M₁₆	1	4	yes	complemented
94	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₃₁	—	2	2	no	—
101	C₄×D₃₁	—	2	2	no	—
102	C₄×D₃₁	—	2	2	no	—
103	C₄×D₃₁	—	2	2	no	—
104	C₄×D₃₁	—	2	2	no	—
105	C₄×D₃₁	—	2	2	no	—
106	C₄×D₃₁	—	2	2	no	—
107	C₄×D₃₁	—	2	2	no	—
108	C₄×D₃₁	—	2	2	no	—
109	C₄×D₃₁	—	2	2	no	—
110	C₄×D₃₁	M₁₆	1	4	yes	complemented
111	C₄×D₃₁	#41	1	4	yes	—
112	C₈×C₃₁	—	2	4	no	—
113	C₃₁⋊C₈	—	2	4	no	—
114	#2920	—	31	1	no	maximal, Sylow
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
118	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₃₁	—	2	8	no	—
120	C₂×D₃₁	—	2	1	no	—
121	C₂×D₃₁	—	2	1	no	—
122	C₂×D₃₁	—	2	1	no	—
123	C₂×D₃₁	—	2	1	no	—
124	C₂×D₃₁	#117	1	2	yes	—
125	C₂×D₃₁	C₂×M₁₆	1	2	yes	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	2	1	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	C₂×M₁₆	1	2	yes	—
131	C₄×C₃₁	C₂×#41	1	2	yes	—
132	Dic₃₁	—	2	1	no	—
133	Dic₃₁	—	2	1	no	—
134	Dic₃₁	#117	1	2	yes	—
135	Dic₃₁	C₂×M₁₆	1	2	yes	—
136	Dic₃₁	C₂×M₁₆	1	2	yes	—
137	Dic₃₁	C₂×M₁₆	1	2	yes	—
138	C₂²×C₄²	—	31	1	no	—
139	#432	—	31	1	no	—
140	#432	—	31	2	no	—
141	#432	D₃₁	1	1	yes	characteristic, complemented
142	#431	—	31	2	no	—
143	#431	—	31	2	no	—
144	#423	—	31	2	no	—
145	#407	—	31	2	no	—
146	C₄×M₁₆	—	31	2	no	—
147	C₂×C₃₁	—	2	2	no	—
148	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
149	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
150	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
151	D₃₁	—	2	2	no	—
152	D₃₁	—	2	2	no	—
153	C₂³×C₄	—	31	1	no	—
154	C₂³×C₄	—	31	1	no	—
155	C₂³×C₄	—	31	1	no	—
156	C₂×M₁₆	—	31	4	no	—
157	C₂×C₄²	—	31	1	no	—
158	C₂×C₄²	—	31	1	no	—
159	C₂×C₄²	—	31	1	no	—
160	C₂×C₄²	—	31	1	no	—
161	C₂×C₄²	—	31	1	no	—
162	C₂×C₄²	—	31	1	no	—
163	C₂×C₄²	—	31	1	no	—
164	C₂×C₄²	—	31	1	no	—
165	C₂×C₄²	—	31	1	no	—
166	C₂×C₄²	—	31	1	no	—
167	C₂×C₄²	—	31	1	no	—
168	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
169	C₄⋊C₈	—	31	4	no	—
170	C₄⋊C₈	—	31	2	no	—
171	C₄⋊C₈	C₂×D₃₁	1	2	yes	complemented
172	C₂²⋊C₈	—	31	2	no	—
173	C₂²⋊C₈	—	31	4	no	—
174	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
175	#97	—	31	2	no	—
176	#97	C₂×D₃₁	1	2	yes	complemented
177	C₄×C₈	—	31	4	no	—
178	C₃₁	#2920	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	31	1	no	—
180	C₂²×C₄	—	31	1	no	—
181	C₂²×C₄	—	31	1	no	—
182	C₂²×C₄	—	31	1	no	—
183	C₂²×C₄	—	31	1	no	—
184	C₂²×C₄	—	31	1	no	—
185	C₂²×C₄	—	31	1	no	—
186	C₂²×C₄	—	31	1	no	—
187	C₂²×C₄	—	31	1	no	—
188	C₂²×C₄	—	31	1	no	—
189	C₂²×C₄	—	31	1	no	—
190	C₂²×C₄	—	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	—	62	4	no	—
196	C₂²×C₄	—	62	4	no	—
197	C₂²×C₄	—	62	4	no	—
198	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
201	M₁₆	—	62	8	no	—
202	C₂×C₈	—	31	4	no	—
203	C₂×C₈	C₂²×D₃₁	1	4	yes	—
204	C₄²	—	31	1	no	—
205	C₄²	—	31	1	no	—
206	C₄²	—	31	1	no	—
207	C₄²	—	31	1	no	—
208	C₄²	—	31	1	no	—
209	C₄²	—	31	1	no	—
210	C₄²	—	31	1	no	—
211	C₄²	—	31	1	no	—
212	C₄²	—	31	1	no	—
213	C₄²	—	31	1	no	—
214	C₄²	—	31	1	no	—
215	C₄²	—	31	1	no	—
216	C₄²	C₄×D₃₁	1	1	yes	characteristic
217	C₄²	C₄×D₃₁	1	1	yes	characteristic
218	C₄²	C₂²×D₃₁	1	1	yes	characteristic
219	C₄²	C₂²×D₃₁	1	1	yes	characteristic
220	C₂³	—	31	1	no	—
221	C₂³	—	31	1	no	—
222	C₂³	—	62	2	no	—
223	C₂³	—	62	2	no	—
224	C₂³	—	62	2	no	—
225	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	31	4	no	—
230	C₂×C₄	—	31	1	no	—
231	C₂×C₄	—	31	1	no	—
232	C₂×C₄	—	31	1	no	—
233	C₂×C₄	—	31	1	no	—
234	C₂×C₄	—	31	4	no	—
235	C₂×C₄	—	31	4	no	—
236	C₂×C₄	—	31	1	no	—
237	C₂×C₄	—	31	1	no	—
238	C₂×C₄	—	62	2	no	—
239	C₂×C₄	—	62	2	no	—
240	C₂×C₄	—	62	2	no	—
241	C₂×C₄	—	62	2	no	—
242	C₂×C₄	—	62	2	no	—
243	C₂×C₄	—	62	2	no	—
244	C₂×C₄	—	62	2	no	—
245	C₂×C₄	—	62	2	no	—
246	C₂×C₄	—	62	2	no	—
247	C₂×C₄	—	62	2	no	—
248	C₂×C₄	—	62	2	no	—
249	C₂×C₄	—	62	2	no	—
250	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	62	4	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	31	2	no	—
261	C₂²	—	31	2	no	—
262	C₂²	—	62	8	no	—
263	C₂²	—	62	1	no	—
264	C₂²	—	62	1	no	—
265	C₂²	—	62	1	no	—
266	C₂²	—	62	1	no	—
267	C₂²	M₁₆×D₃₁	1	2	yes	—
268	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	31	2	no	—
274	C₄	—	31	2	no	—
275	C₄	—	31	2	no	—
276	C₄	—	31	2	no	—
277	C₄	—	62	1	no	—
278	C₄	—	62	1	no	—
279	C₄	M₁₆×D₃₁	1	2	yes	—
280	C₄	#41×D₃₁	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	62	2	no	—
283	C₂	—	62	2	no	—
284	C₂	#117×D₃₁	1	1	yes	characteristic
285	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
286	C₂	C₂×M₁₆×D₃₁	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	#2920
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. #14521106	C₂	semidirect
3. #14521104	C₂	semidirect
4. #14521101	C₂	semidirect
5. #14521099	C₂	semidirect
6. #14521071	C₂	semidirect
8. #14521001	C₂	semidirect
9. C₃₁×#432	C₂	semidirect
26. C₃₁×C₄⋊C₈	C₂²	semidirect
28. C₃₁×#97	C₂²	semidirect
31. Dic₃₁⋊C₈	C₂²	semidirect
32. C₈×Dic₃₁	C₂²	semidirect
33. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
34. #11758391	C₂²	semidirect
93. C₂×Dic₃₁	M₁₆	semidirect
110. C₄×D₃₁	M₁₆	semidirect
141. #432	D₃₁	semidirect
171. C₄⋊C₈	C₂×D₃₁	semidirect
176. #97	C₂×D₃₁	semidirect
178. C₃₁	#2920	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×D₃₁	2
#14521106	2
#14521104	2
#14521101	2
#14521099	2
#14521071	2
#14521011	2
#14521001	2
C₃₁×#432	2
#2920	31