Subgroups — C₃₁×#11972 #21468731

Order 7936.

330 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×#2162	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1365	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
6	C₃₁×#500	—	2	1	no	—
7	C₃₁×#500	—	2	1	no	—
8	C₃₁×#500	C₄	1	2	yes	—
9	C₃₁×#500	C₂²	1	2	yes	—
10	C₂×C₃₁×#107	—	2	1	no	—
11	C₂×C₃₁×#107	C₄	1	2	yes	—
12	C₃₁×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₃₁	—	2	1	no	—
17	C₃₁×#335	—	2	1	no	—
18	C₃₁×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₃₁	—	4	1	no	—
20	C₂×M₁₆×C₃₁	—	4	1	no	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	—	2	2	no	—
24	C₃₁×C₄⋊Q₈	—	2	2	no	—
25	C₃₁×C₄⋊Q₈	—	4	1	no	—
26	C₃₁×C₄⋊Q₈	—	4	1	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
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×C₄⋊C₈	—	4	1	no	—
37	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₃₁	—	4	2	no	—
42	C₂×Q₈×C₃₁	—	4	2	no	—
43	C₂×Q₈×C₃₁	—	8	4	no	—
44	C₂²×C₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
48	M₁₆×C₃₁	—	8	2	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	2	2	no	—
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	4	2	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₃₁	D₈	1	1	yes	characteristic
62	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
63	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₁₆×C₃₁	—	4	2	no	—
65	#11972	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	#102	1	1	yes	characteristic
67	Q₈×C₃₁	—	8	4	no	—
68	Q₈×C₃₁	—	8	4	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	1	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	4	4	no	—
73	C₂×C₄×C₃₁	—	8	4	no	—
74	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
75	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
78	C₂×C₄×C₃₁	Q₃₂	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	#101	1	2	yes	—
86	C₈×C₃₁	#137	1	2	yes	—
87	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
88	#2162	C₂×C₃₁	1	1	yes	characteristic
89	#1567	C₂×C₃₁	1	1	yes	characteristic
90	#1365	C₂×C₃₁	1	4	yes	complemented
91	C₂²×C₃₁	—	2	1	no	—
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	#358	1	2	yes	—
94	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	—	2	1	no	—
97	C₄×C₃₁	—	8	4	no	—
98	C₄×C₃₁	#358	1	2	yes	—
99	C₄×C₃₁	#361	1	2	yes	—
100	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
101	C₄×C₃₁	#419	1	1	yes	characteristic
102	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
103	C₄×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	#500	—	2	1	no	—
106	#500	—	2	1	no	—
107	#500	C₄×C₃₁	1	2	yes	—
108	#500	C₂²×C₃₁	1	2	yes	—
109	C₂×#107	—	2	1	no	—
110	C₂×#107	C₄×C₃₁	1	2	yes	—
111	#423	—	2	1	no	—
112	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
113	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
114	C₂²⋊C₁₆	—	2	1	no	—
115	C₄×C₁₆	—	2	1	no	—
116	#335	—	2	1	no	—
117	#335	C₂²×C₃₁	1	2	yes	—
118	C₂×C₃₁	—	2	2	no	—
119	C₂×C₃₁	#1546	1	1	yes	characteristic
120	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
121	C₂×C₃₁	C₂×#361	1	1	yes	characteristic
122	C₂²×Q₈	—	4	1	no	—
123	C₂×M₁₆	—	4	1	no	—
124	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
125	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	4	1	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
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	C₄⋊C₈	—	4	1	no	—
140	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
142	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
143	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
144	C₃₁	#11972	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
151	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
152	M₁₆	—	8	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
165	C₄²	D₈×C₃₁	1	1	yes	characteristic
166	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
167	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
168	C₁₆	—	4	2	no	—
169	C₂³	C₃₁×#102	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
178	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
179	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
180	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
181	C₂×C₄	C₃₁×Q₃₂	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×#101	1	2	yes	—
189	C₈	C₃₁×#137	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	C₃₁×#358	1	2	yes	—
193	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	C₃₁×#358	1	2	yes	—
198	C₄	C₃₁×#361	1	2	yes	—
199	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
200	C₄	C₃₁×#419	1	1	yes	characteristic
201	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
202	C₄	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₃₁×#1546	1	1	yes	characteristic
205	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
206	C₂	C₂×C₃₁×#361	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×#1365	C₂	semidirect
65. #11972	C₃₁	direct (N × H)
90. #1365	C₂×C₃₁	semidirect
144. C₃₁	#11972	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×#2162	2
C₃₁×#1567	2
C₃₁×#1365	2
#11972	31

Subgroups — C31×#11972 #21468731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11972 #21468731