Subgroups — C₃₁×#33713 #21490472

Order 7936.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#508	C₂	1	4	yes	maximal, complemented
2	C₃₁×#3147	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2249	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2248	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	—	2	2	no	—
9	C₂×C₃₁×#135	C₂²	1	4	yes	—
10	C₃₁×#532	—	2	2	no	—
11	C₃₁×#508	—	2	16	no	—
12	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
13	C₂×C₃₁×SD₃₂	C₂²	1	8	yes	—
14	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#495	—	2	2	no	—
17	C₃₁×#495	C₂²	1	4	yes	—
18	C₃₁×#494	C₂²	1	2	yes	—
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#492	—	2	2	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
25	C₂×#41×C₃₁	—	2	4	no	—
26	C₃₁×#135	—	2	16	no	—
27	C₃₁×#135	—	4	8	no	—
28	C₂×Q₁₆×C₃₁	—	2	2	no	—
29	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
30	C₂×SD₁₆×C₃₁	—	2	4	no	—
31	C₂×SD₁₆×C₃₁	—	2	4	no	—
32	C₂×D₈×C₃₁	—	2	2	no	—
33	C₂×D₈×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×#127	—	2	2	no	—
38	C₃₁×#124	—	2	4	no	—
39	C₃₁×#121	—	4	4	no	—
40	C₄×D₄×C₃₁	—	4	4	no	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₃₁×Q₃₂	—	2	8	no	—
43	C₃₁×SD₃₂	—	2	16	no	—
44	C₃₁×D₁₆	—	2	8	no	—
45	C₂×C₁₆×C₃₁	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
48	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	#41×C₃₁	—	4	8	no	—
53	#41×C₃₁	—	4	8	no	—
54	C₂×Q₈×C₃₁	—	2	4	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	4	4	no	—
57	C₂×D₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	D₈	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	Q₁₆×C₃₁	—	2	8	no	—
62	Q₁₆×C₃₁	—	4	4	no	—
63	SD₁₆×C₃₁	—	4	8	no	—
64	SD₁₆×C₃₁	—	4	8	no	—
65	D₈×C₃₁	—	2	8	no	—
66	D₈×C₃₁	—	4	4	no	—
67	C₂×C₈×C₃₁	D₈	1	4	yes	—
68	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₃₁	—	4	4	no	—
74	C₂²⋊C₄×C₃₁	—	4	8	no	—
75	C₄²×C₃₁	D₈	1	2	yes	—
76	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
79	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
80	#33713	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	—	4	4	no	—
82	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₃₁	—	4	4	no	—
84	Q₈×C₃₁	—	4	4	no	—
85	D₄×C₃₁	—	4	4	no	—
86	D₄×C₃₁	—	4	4	no	—
87	D₄×C₃₁	—	8	8	no	—
88	D₄×C₃₁	—	8	8	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	—	8	8	no	—
92	C₂×C₄×C₃₁	#127	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₃₁	#127	1	2	yes	—
99	C₈×C₃₁	C₂×D₈	1	2	yes	—
100	C₈×C₃₁	C₂×D₈	1	2	yes	—
101	C₈×C₃₁	C₂²×D₄	1	2	yes	—
102	C₂×#508	C₂×C₃₁	1	4	yes	complemented
103	#3147	C₂×C₃₁	1	2	yes	complemented
104	#2249	C₂×C₃₁	1	4	yes	complemented
105	#2248	C₂×C₃₁	1	2	yes	complemented
106	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
107	#2246	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	8	4	no	—
111	C₂²×C₃₁	—	8	8	no	—
112	C₂²×C₃₁	#493	1	2	yes	—
113	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	8	4	no	—
116	C₄×C₃₁	#493	1	2	yes	—
117	C₄×C₃₁	#508	1	4	yes	complemented
118	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
119	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
120	C₂×#135	—	2	2	no	—
121	C₂×#135	C₂²×C₃₁	1	4	yes	—
122	#532	—	2	2	no	—
123	#508	—	2	16	no	—
124	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
125	C₂×SD₃₂	C₂²×C₃₁	1	8	yes	—
126	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
127	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
128	#495	—	2	2	no	—
129	#495	C₂²×C₃₁	1	4	yes	—
130	#494	C₂²×C₃₁	1	2	yes	—
131	#493	C₂²×C₃₁	1	2	yes	—
132	#492	—	2	2	no	—
133	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
135	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
136	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
137	C₂×C₃₁	—	2	2	no	—
138	C₂×C₃₁	—	16	4	no	—
139	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
140	C₂×C₃₁	C₂×#508	1	2	yes	—
141	C₂×#41	—	2	4	no	—
142	#135	—	2	16	no	—
143	#135	—	4	8	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
146	C₂×SD₁₆	—	2	4	no	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	C₂³×C₃₁	1	4	yes	—
150	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	#127	—	2	2	no	—
154	#124	—	2	4	no	—
155	#121	—	4	4	no	—
156	C₄×D₄	—	4	4	no	—
157	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
158	Q₃₂	—	2	8	no	—
159	SD₃₂	—	2	16	no	—
160	D₁₆	—	2	8	no	—
161	C₂×C₁₆	D₄×C₃₁	1	8	yes	complemented
162	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
163	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
164	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
166	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
167	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₃₁	#33713	1	1	yes	characteristic, direct product factor, Sylow
169	#41	—	4	8	no	—
170	#41	—	4	8	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	D₈×C₃₁	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
178	Q₁₆	—	2	8	no	—
179	Q₁₆	—	4	4	no	—
180	SD₁₆	—	4	8	no	—
181	SD₁₆	—	4	8	no	—
182	D₈	—	2	8	no	—
183	D₈	—	4	4	no	—
184	C₂×C₈	D₈×C₃₁	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
190	C₄⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₄²	D₈×C₃₁	1	2	yes	—
193	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
196	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
197	C₂³	—	4	4	no	—
198	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
199	Q₈	—	4	4	no	—
200	Q₈	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	8	no	—
204	D₄	—	8	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	C₃₁×#127	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
210	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
212	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
213	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
214	C₈	C₃₁×#127	1	2	yes	—
215	C₈	C₂×D₈×C₃₁	1	2	yes	—
216	C₈	C₂×D₈×C₃₁	1	2	yes	—
217	C₈	C₂²×D₄×C₃₁	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	8	8	no	—
221	C₂²	C₃₁×#493	1	2	yes	—
222	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	C₃₁×#493	1	2	yes	—
226	C₄	C₃₁×#508	1	4	yes	complemented
227	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
228	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	16	4	no	—
231	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
232	C₂	C₂×C₃₁×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#508	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2248	C₂	semidirect
5. C₃₁×#2247	C₂	semidirect
6. C₃₁×#2246	C₂	semidirect
7. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
45. C₂×C₁₆×C₃₁	D₄	semidirect
80. #33713	C₃₁	direct (N × H)
102. C₂×#508	C₂×C₃₁	semidirect
103. #3147	C₂×C₃₁	semidirect
104. #2249	C₂×C₃₁	semidirect
105. #2248	C₂×C₃₁	semidirect
106. #2247	C₂×C₃₁	semidirect
107. #2246	C₂×C₃₁	semidirect
108. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
117. C₄×C₃₁	#508	semidirect
161. C₂×C₁₆	D₄×C₃₁	semidirect
168. C₃₁	#33713	direct (N × H)
226. C₄	C₃₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#508	2
C₃₁×#3147	2
C₃₁×#2249	2
C₃₁×#2248	2
C₃₁×#2247	2
C₃₁×#2246	2
C₂×C₄×C₁₆×C₃₁	2
#33713	31

Subgroups — C31×#33713 #21490472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33713 #21490472