Subgroups — C₃₁×#13167 #21469926

Order 7936.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1942	C₂	1	2	yes	maximal
5	C₃₁×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#107	C₂²	1	2	yes	—
9	C₂×C₃₁×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#389	—	2	2	no	—
12	C₃₁×#367	C₄	1	4	yes	complemented
13	C₃₁×#366	—	2	2	no	—
14	C₃₁×#346	—	2	2	no	—
15	C₃₁×#340	—	2	2	no	—
16	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
21	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₁	—	2	1	no	—
23	C₂×C₁₆×C₃₁	—	2	1	no	—
24	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
27	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
28	C₃₁×#107	—	2	2	no	—
29	C₃₁×#107	—	2	2	no	—
30	C₃₁×#107	—	2	2	no	—
31	C₃₁×#107	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₃₁	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₃₁×#95	—	4	2	no	—
35	C₂²×C₄×C₃₁	—	4	2	no	—
36	C₂²×C₄×C₃₁	—	4	2	no	—
37	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₃₁	—	2	1	no	—
41	C₂×C₈×C₃₁	—	2	1	no	—
42	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
43	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
48	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₄⋊C₄×C₃₁	—	4	2	no	—
51	C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₄⋊C₄×C₃₁	—	4	2	no	—
53	C₄⋊C₄×C₃₁	—	8	4	no	—
54	C₄²×C₃₁	—	2	1	no	—
55	C₄²×C₃₁	—	2	1	no	—
56	C₁₆×C₃₁	—	2	2	no	—
57	C₁₆×C₃₁	C₄⋊C₄	1	4	yes	complemented
58	#13167	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	—	2	1	no	—
61	C₂×C₄×C₃₁	—	2	1	no	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	8	2	no	—
66	C₂×C₄×C₃₁	—	8	2	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	—	8	2	no	—
69	C₂×C₄×C₃₁	—	8	2	no	—
70	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
75	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
76	C₈×C₃₁	—	2	2	no	—
77	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
80	C₈×C₃₁	#125	1	1	yes	characteristic
81	C₂×#367	C₂×C₃₁	1	1	yes	characteristic
82	C₂×#366	C₂×C₃₁	1	1	yes	characteristic
83	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
84	#1942	C₂×C₃₁	1	2	yes	—
85	#1380	C₂×C₃₁	1	2	yes	—
86	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
87	C₂²×C₃₁	#466	1	1	yes	characteristic
88	C₂²×C₃₁	#500	1	1	yes	characteristic
89	C₂²×C₃₁	#509	1	2	yes	—
90	C₂²×C₃₁	#510	1	2	yes	—
91	C₄×C₃₁	—	4	1	no	—
92	C₄×C₃₁	—	4	1	no	—
93	C₄×C₃₁	—	8	4	no	—
94	C₄×C₃₁	—	16	2	no	—
95	C₄×C₃₁	#389	1	1	yes	characteristic
96	C₄×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
97	C₄×C₃₁	#467	1	1	yes	characteristic
98	C₄×C₃₁	#499	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
100	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
101	C₂×#107	C₂²×C₃₁	1	2	yes	—
102	C₂×#107	C₂²×C₃₁	1	2	yes	—
103	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
104	#389	—	2	2	no	—
105	#367	C₄×C₃₁	1	4	yes	complemented
106	#366	—	2	2	no	—
107	#346	—	2	2	no	—
108	#340	—	2	2	no	—
109	C₂×C₃₁	#1942	1	1	yes	characteristic
110	C₂×C₃₁	#2175	1	2	yes	—
111	C₂×C₃₁	#2218	1	1	yes	characteristic
112	C₂×C₃₁	#2219	1	1	yes	characteristic
113	C₂×C₃₁	#2255	1	2	yes	—
114	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
123	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
124	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
125	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
126	#107	—	2	2	no	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	C₂×C₄×C₃₁	1	4	yes	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	—	2	1	no	—
132	#95	—	4	2	no	—
133	C₃₁	#13167	1	1	yes	characteristic, direct product factor, Sylow
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	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
142	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
146	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
147	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	8	4	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	C₄⋊C₄×C₃₁	1	4	yes	complemented
157	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
170	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
173	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
176	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
177	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
178	C₈	C₃₁×#125	1	1	yes	characteristic
179	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
180	C₂²	C₃₁×#466	1	1	yes	characteristic
181	C₂²	C₃₁×#500	1	1	yes	characteristic
182	C₂²	C₃₁×#509	1	2	yes	—
183	C₂²	C₃₁×#510	1	2	yes	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	4	no	—
187	C₄	—	16	2	no	—
188	C₄	C₃₁×#389	1	1	yes	characteristic
189	C₄	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
190	C₄	C₃₁×#467	1	1	yes	characteristic
191	C₄	C₃₁×#499	1	1	yes	characteristic
192	C₂	C₃₁×#1942	1	1	yes	characteristic
193	C₂	C₃₁×#2175	1	2	yes	—
194	C₂	C₃₁×#2218	1	1	yes	characteristic
195	C₂	C₃₁×#2219	1	1	yes	characteristic
196	C₂	C₃₁×#2255	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
12. C₃₁×#367	C₄	semidirect
57. C₁₆×C₃₁	C₄⋊C₄	semidirect
58. #13167	C₃₁	direct (N × H)
105. #367	C₄×C₃₁	semidirect
133. C₃₁	#13167	direct (N × H)
156. C₁₆	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#367	2
C₂×C₃₁×#366	2
C₂×C₃₁×#346	2
C₃₁×#1942	2
C₃₁×#1380	2
#13167	31

Subgroups — C31×#13167 #21469926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13167 #21469926