Subgroups — C₃₁×#11940 #21468699

Order 7936.

334 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1364	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₃₁×M₃₂	—	2	1	no	—
7	C₃₁×#500	—	2	1	no	—
8	C₃₁×#500	—	2	1	no	—
9	C₃₁×#500	C₄	1	2	yes	—
10	C₃₁×#500	C₂²	1	2	yes	—
11	C₃₁×#431	—	2	1	no	—
12	C₂×C₃₁×#107	—	2	1	no	—
13	C₂×C₃₁×#107	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊C₁₆	—	2	1	no	—
16	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₃₁×#335	C₂²	1	2	yes	—
18	C₃₁×#334	—	2	1	no	—
19	C₂²×Q₈×C₃₁	—	4	1	no	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	—	2	2	no	—
23	C₃₁×C₄⋊Q₈	—	2	2	no	—
24	C₃₁×C₄⋊Q₈	—	4	1	no	—
25	C₃₁×C₄⋊Q₈	—	4	1	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
28	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
29	C₃₁×M₃₂	—	4	2	no	—
30	C₂×C₁₆×C₃₁	—	2	2	no	—
31	C₃₁×#107	—	2	2	no	—
32	C₃₁×#107	—	2	2	no	—
33	C₃₁×#107	—	2	2	no	—
34	C₃₁×#107	C₂×C₄	1	4	yes	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₂²⋊C₈	—	4	1	no	—
37	C₄×C₈×C₃₁	—	4	1	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	—	4	2	no	—
43	C₂×Q₈×C₃₁	—	4	2	no	—
44	C₂×Q₈×C₃₁	—	8	4	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	4	2	no	—
51	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
55	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄⋊C₄×C₃₁	—	4	2	no	—
61	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₃₁	D₈	1	1	yes	characteristic
63	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
64	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₃₁	—	4	2	no	—
66	#11940	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	#102	1	1	yes	characteristic
68	Q₈×C₃₁	—	8	4	no	—
69	Q₈×C₃₁	—	8	4	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	4	4	no	—
73	C₂×C₄×C₃₁	—	8	4	no	—
74	C₂×C₄×C₃₁	#101	1	2	yes	—
75	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₃₁	#103	1	2	yes	—
79	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₈×C₃₁	—	2	1	no	—
83	C₈×C₃₁	—	2	1	no	—
84	C₈×C₃₁	—	8	2	no	—
85	C₈×C₃₁	#103	1	2	yes	—
86	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₈×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
89	#2150	C₂×C₃₁	1	1	yes	characteristic
90	#1564	C₂×C₃₁	1	1	yes	characteristic
91	#1364	C₂×C₃₁	1	4	yes	complemented
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	—	2	1	no	—
94	C₂²×C₃₁	#332	1	2	yes	—
95	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
96	C₄×C₃₁	—	2	1	no	—
97	C₄×C₃₁	—	2	1	no	—
98	C₄×C₃₁	—	8	4	no	—
99	C₄×C₃₁	#332	1	2	yes	—
100	C₄×C₃₁	#362	1	2	yes	—
101	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
102	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
103	C₄×C₃₁	#509	1	2	yes	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×M₃₂	—	2	1	no	—
106	#500	—	2	1	no	—
107	#500	—	2	1	no	—
108	#500	C₄×C₃₁	1	2	yes	—
109	#500	C₂²×C₃₁	1	2	yes	—
110	#431	—	2	1	no	—
111	C₂×#107	—	2	1	no	—
112	C₂×#107	C₄×C₃₁	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	1	no	—
115	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
116	#335	C₂²×C₃₁	1	2	yes	—
117	#334	—	2	1	no	—
118	C₂×C₃₁	—	2	2	no	—
119	C₂×C₃₁	C₂×#332	1	1	yes	characteristic
120	C₂×C₃₁	#2140	1	1	yes	characteristic
121	C₂×C₃₁	C₂×#362	1	1	yes	characteristic
122	C₂²×Q₈	—	4	1	no	—
123	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
124	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	4	1	no	—
128	C₄⋊Q₈	—	4	1	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
132	M₃₂	—	4	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	C₂×C₄×C₃₁	1	4	yes	—
138	C₄⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	4	1	no	—
140	C₄×C₈	—	4	1	no	—
141	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
142	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
143	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
144	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
145	C₃₁	#11940	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	8	4	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
151	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
152	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
158	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
166	C₄²	D₈×C₃₁	1	1	yes	characteristic
167	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
169	C₁₆	—	4	2	no	—
170	C₂³	C₃₁×#102	1	1	yes	characteristic
171	Q₈	—	8	4	no	—
172	Q₈	—	8	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	C₃₁×#101	1	2	yes	—
178	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
179	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
180	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
181	C₂×C₄	C₃₁×#103	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	8	2	no	—
188	C₈	C₃₁×#103	1	2	yes	—
189	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
190	C₈	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₃₁×#332	1	2	yes	—
194	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₃₁×#332	1	2	yes	—
199	C₄	C₃₁×#362	1	2	yes	—
200	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
201	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
202	C₄	C₃₁×#509	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	C₂×C₃₁×#332	1	1	yes	characteristic
205	C₂	C₃₁×#2140	1	1	yes	characteristic
206	C₂	C₂×C₃₁×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
prime radical	C₂³×C₃₁	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
4. C₃₁×#1364	C₂	semidirect
66. #11940	C₃₁	direct (N × H)
91. #1364	C₂×C₃₁	semidirect
145. C₃₁	#11940	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₃₁×#2150	2
C₃₁×#1564	2
C₃₁×#1364	2
#11940	31

Subgroups — C31×#11940 #21468699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11940 #21468699