Subgroups — C₃₁×#13276 #21470035

Order 7936.

398 conjugacy classes of subgroups.

The subgroup lattice has 192 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	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1380	C₂	1	4	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#500	—	2	2	no	—
8	C₃₁×#500	C₄	1	4	yes	complemented
9	C₂×C₃₁×#107	—	2	1	no	—
10	C₂×C₃₁×#107	C₂²	1	2	yes	—
11	C₂×C₃₁×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
14	C₃₁×C₄⋊C₁₆	—	2	2	no	—
15	C₃₁×#340	—	2	1	no	—
16	C₂²×Q₈×C₃₁	—	4	1	no	—
17	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁×C₄⋊Q₈	—	2	4	no	—
20	C₃₁×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
24	C₂×C₄²×C₃₁	—	4	1	no	—
25	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₃₁	—	2	2	no	—
27	C₂×C₁₆×C₃₁	—	2	4	no	—
28	C₃₁×#107	—	2	4	no	—
29	C₃₁×#107	—	2	2	no	—
30	C₃₁×#107	—	2	2	no	—
31	C₃₁×#107	C₂×C₄	1	4	yes	—
32	C₃₁×#107	Q₈	1	4	yes	complemented
33	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₃₁	D₄	1	2	yes	—
35	C₃₁×#95	—	4	1	no	—
36	C₂×Q₈×C₃₁	—	4	4	no	—
37	C₂×Q₈×C₃₁	—	8	4	no	—
38	C₂²×C₄×C₃₁	—	4	2	no	—
39	C₂²×C₄×C₃₁	—	4	2	no	—
40	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₃₁	#41	1	2	yes	—
49	C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₄⋊C₄×C₃₁	—	4	4	no	—
52	C₄⋊C₄×C₃₁	—	4	2	no	—
53	C₄⋊C₄×C₃₁	—	4	2	no	—
54	C₄²×C₃₁	—	8	2	no	—
55	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₃₁	SD₁₆	1	2	yes	—
57	C₁₆×C₃₁	—	4	2	no	—
58	#13276	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₃₁	—	8	8	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	4	4	no	—
63	C₂×C₄×C₃₁	—	4	4	no	—
64	C₂×C₄×C₃₁	—	8	2	no	—
65	C₂×C₄×C₃₁	—	8	2	no	—
66	C₂×C₄×C₃₁	—	8	4	no	—
67	C₂×C₄×C₃₁	#102	1	2	yes	—
68	C₂×C₄×C₃₁	#102	1	2	yes	—
69	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
70	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
71	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₃₁	#137	1	2	yes	—
75	C₈×C₃₁	—	2	2	no	—
76	C₈×C₃₁	C₄×Q₈	1	2	yes	—
77	C₈×C₃₁	#124	1	1	yes	characteristic
78	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
81	C₄×#107	C₂×C₃₁	1	1	yes	characteristic
82	#1380	C₂×C₃₁	1	4	yes	—
83	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
84	C₂²×C₃₁	#474	1	2	yes	—
85	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
86	C₂²×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₃₁	#509	1	2	yes	—
88	C₄×C₃₁	—	2	2	no	—
89	C₄×C₃₁	—	8	4	no	—
90	C₄×C₃₁	—	8	4	no	—
91	C₄×C₃₁	#358	1	4	yes	—
92	C₄×C₃₁	#391	1	1	yes	characteristic
93	C₄×C₃₁	#419	1	1	yes	characteristic
94	C₄×C₃₁	#475	1	2	yes	—
95	C₂×C₄⋊Q₈	—	2	1	no	—
96	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
97	#500	—	2	2	no	—
98	#500	C₄×C₃₁	1	4	yes	complemented
99	C₂×#107	—	2	1	no	—
100	C₂×#107	C₂²×C₃₁	1	2	yes	—
101	C₂×#107	C₂²×C₃₁	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
103	C₄×C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₁₆	—	2	2	no	—
105	#340	—	2	1	no	—
106	C₂×C₃₁	#1988	1	1	yes	characteristic
107	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
108	C₂×C₃₁	#2140	1	1	yes	characteristic
109	C₂×C₃₁	#2226	1	2	yes	—
110	C₂×C₃₁	#2227	1	2	yes	—
111	C₂²×Q₈	—	4	1	no	—
112	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	4	no	—
115	C₄⋊Q₈	—	4	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄²	—	4	1	no	—
120	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	4	no	—
123	#107	—	2	4	no	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	C₂×C₄×C₃₁	1	4	yes	—
127	#107	Q₈×C₃₁	1	4	yes	complemented
128	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
129	C₄×C₈	D₄×C₃₁	1	2	yes	—
130	#95	—	4	1	no	—
131	C₃₁	#13276	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	8	4	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
144	C₂×C₈	#41×C₃₁	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	8	2	no	—
151	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
152	C₄²	SD₁₆×C₃₁	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	C₃₁×#102	1	2	yes	—
163	C₂×C₄	C₃₁×#102	1	2	yes	—
164	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
165	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₃₁×#137	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	C₄×Q₈×C₃₁	1	2	yes	—
172	C₈	C₃₁×#124	1	1	yes	characteristic
173	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
174	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
175	C₂²	C₃₁×#474	1	2	yes	—
176	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
177	C₂²	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
178	C₂²	C₃₁×#509	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₃₁×#358	1	4	yes	—
183	C₄	C₃₁×#391	1	1	yes	characteristic
184	C₄	C₃₁×#419	1	1	yes	characteristic
185	C₄	C₃₁×#475	1	2	yes	—
186	C₂	C₃₁×#1988	1	1	yes	characteristic
187	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
188	C₂	C₃₁×#2140	1	1	yes	characteristic
189	C₂	C₃₁×#2226	1	2	yes	—
190	C₂	C₃₁×#2227	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
8. C₃₁×#500	C₄	semidirect
32. C₃₁×#107	Q₈	semidirect
58. #13276	C₃₁	direct (N × H)
98. #500	C₄×C₃₁	semidirect
127. #107	Q₈×C₃₁	semidirect
131. C₃₁	#13276	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
C₂×C₃₁×C₄⋊C₁₆	2
C₄×C₃₁×#107	2
C₃₁×#1380	2
#13276	31

Subgroups — C31×#13276 #21470035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13276 #21470035