Subgroups — C₃₁×#11583 #21468342

Order 7936.

450 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×D₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₃₁	—	2	1	no	—
6	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	—	2	1	no	—
8	C₂×C₃₁×M₃₂	—	2	1	no	—
9	C₈×D₄×C₃₁	—	2	1	no	—
10	C₈×D₄×C₃₁	—	2	1	no	—
11	C₈×D₄×C₃₁	C₄	1	2	yes	—
12	C₈×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	—	2	1	no	—
17	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₃₁×#346	—	2	1	no	—
19	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₃₁	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	—	2	1	no	—
22	C₂²×C₈×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	4	2	no	—
26	C₂²×C₈×C₃₁	—	4	2	no	—
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	—	2	1	no	—
31	C₄×D₄×C₃₁	C₈	1	2	yes	—
32	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₃₁×M₃₂	—	2	4	no	—
37	C₃₁×M₃₂	—	4	2	no	—
38	C₂×C₁₆×C₃₁	—	2	2	no	—
39	C₂×C₁₆×C₃₁	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₃₁×C₂²⋊C₈	—	2	2	no	—
43	C₃₁×C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂⁴×C₃₁	—	2	1	no	—
49	C₂×D₄×C₃₁	—	2	4	no	—
50	C₂×D₄×C₃₁	—	2	1	no	—
51	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	4	2	no	—
57	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₂×C₈×C₃₁	—	4	4	no	—
64	C₂×C₈×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	D₈	1	2	yes	—
69	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
70	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
74	C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₃₁	M₁₆	1	1	yes	characteristic
79	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₁₆×C₃₁	—	2	4	no	—
81	C₁₆×C₃₁	—	4	2	no	—
82	#11583	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	2	4	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	4	2	no	—
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
88	D₄×C₃₁	—	2	4	no	—
89	D₄×C₃₁	—	2	2	no	—
90	D₄×C₃₁	M₃₂	1	4	yes	complemented
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	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#102	1	2	yes	—
102	C₂×C₄×C₃₁	#104	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	4	2	no	—
109	C₈×C₃₁	#102	1	2	yes	—
110	C₈×C₃₁	C₂×D₈	1	1	yes	characteristic
111	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
113	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
114	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
115	D₄⋊C₁₆	C₂×C₃₁	1	4	yes	complemented
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	—	2	4	no	—
119	C₂²×C₃₁	—	4	8	no	—
120	C₂²×C₃₁	—	4	2	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	D₄⋊C₈	1	2	yes	—
123	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	D₄⋊C₈	1	2	yes	—
129	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
130	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
131	C₄×C₃₁	C₂×M₃₂	1	2	yes	—
132	C₂³×C₈	—	2	1	no	—
133	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
134	C₂×M₃₂	—	2	1	no	—
135	C₂×M₃₂	—	2	1	no	—
136	C₈×D₄	—	2	1	no	—
137	C₈×D₄	—	2	1	no	—
138	C₈×D₄	C₄×C₃₁	1	2	yes	—
139	C₈×D₄	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
141	C₂×C₂²⋊C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₄⋊C₁₆	—	2	1	no	—
144	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
145	#346	—	2	1	no	—
146	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
147	C₂×C₃₁	—	2	2	no	—
148	C₂×C₃₁	—	4	4	no	—
149	C₂×C₃₁	C₂×D₄⋊C₈	1	1	yes	characteristic
150	C₂×C₃₁	#2112	1	1	yes	characteristic
151	C₂×C₃₁	#2117	1	1	yes	characteristic
152	C₂²×D₄	C₈×C₃₁	1	1	yes	characteristic
153	C₂³×C₄	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	4	2	no	—
158	C₂²×C₈	—	4	2	no	—
159	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	C₈×C₃₁	1	2	yes	—
164	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
165	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	—	2	1	no	—
167	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
168	M₃₂	—	2	4	no	—
169	M₃₂	—	4	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
178	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
179	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
180	C₃₁	#11583	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	—	2	1	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₈	D₈×C₃₁	1	2	yes	—
202	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
211	C₄²	M₁₆×C₃₁	1	1	yes	characteristic
212	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₁₆	—	2	4	no	—
214	C₁₆	—	4	2	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
220	D₄	—	2	4	no	—
221	D₄	—	2	2	no	—
222	D₄	C₃₁×M₃₂	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
231	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
232	C₂×C₄	C₃₁×C₂²⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#102	1	2	yes	—
234	C₂×C₄	C₃₁×#104	1	2	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	C₃₁×#102	1	2	yes	—
242	C₈	C₂×D₈×C₃₁	1	1	yes	characteristic
243	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	C₃₁×D₄⋊C₈	1	2	yes	—
251	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	C₃₁×D₄⋊C₈	1	2	yes	—
257	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
258	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
259	C₄	C₂×C₃₁×M₃₂	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×C₃₁×D₄⋊C₈	1	1	yes	characteristic
263	C₂	C₃₁×#2112	1	1	yes	characteristic
264	C₂	C₃₁×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₃₁	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2150	C₂	semidirect
3. C₄×C₃₁×M₃₂	C₂	semidirect
4. C₃₁×D₄⋊C₁₆	C₂	semidirect
17. C₃₁×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₁₆×C₃₁	C₂²	semidirect
82. #11583	C₃₁	direct (N × H)
90. D₄×C₃₁	M₃₂	semidirect
113. #2150	C₂×C₃₁	semidirect
114. C₄×M₃₂	C₂×C₃₁	semidirect
115. D₄⋊C₁₆	C₂×C₃₁	semidirect
144. C₄⋊C₁₆	C₂²×C₃₁	semidirect
146. C₄×C₁₆	C₂²×C₃₁	semidirect
180. C₃₁	#11583	direct (N × H)
222. D₄	C₃₁×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₁	2
C₃₁×#2150	2
C₄×C₃₁×M₃₂	2
C₃₁×D₄⋊C₁₆	2
#11583	31

Subgroups — C31×#11583 #21468342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11583 #21468342