Subgroups — C₃₁×#33258 #21490017

Order 7936.

718 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	2	yes	maximal
2	C₃₁×#1942	C₂	1	8	yes	maximal
3	C₃₁×#1845	C₂	1	2	yes	maximal
4	C₃₁×#1834	C₂	1	2	yes	maximal
5	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#125	—	2	2	no	—
7	C₂×C₃₁×#107	C₂²	1	8	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#389	—	2	4	no	—
12	C₃₁×#389	C₂²	1	8	yes	—
13	C₃₁×#384	C₂²	1	4	yes	—
14	C₃₁×#382	—	2	2	no	—
15	C₄³×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#340	C₂²	1	8	yes	—
17	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
18	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
19	C₃₁×#125	—	4	8	no	—
20	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
21	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
22	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
23	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
24	C₂×C₄²×C₃₁	D₄	1	2	yes	—
25	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×#107	—	2	16	no	—
30	C₄×C₈×C₃₁	—	2	8	no	—
31	C₄×C₈×C₃₁	Q₈	1	4	yes	—
32	C₄×C₈×C₃₁	Q₈	1	4	yes	—
33	C₃₁×#95	—	2	8	no	—
34	C₂²×C₄×C₃₁	—	2	8	no	—
35	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
40	C₂×C₈×C₃₁	—	2	4	no	—
41	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
42	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
43	C₂×C₈×C₃₁	#41	1	8	yes	—
44	C₄⋊C₄×C₃₁	—	2	16	no	—
45	C₄⋊C₄×C₃₁	—	4	16	no	—
46	C₄⋊C₄×C₃₁	—	4	8	no	—
47	C₄⋊C₄×C₃₁	—	4	8	no	—
48	C₄²×C₃₁	—	2	4	no	—
49	C₄²×C₃₁	—	2	4	no	—
50	C₄²×C₃₁	D₈	1	4	yes	—
51	C₄²×C₃₁	Q₁₆	1	4	yes	—
52	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
53	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
54	#33258	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	4	8	no	—
61	C₂×C₄×C₃₁	—	4	8	no	—
62	C₂×C₄×C₃₁	—	4	8	no	—
63	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
66	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
68	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₃₁	#135	1	4	yes	—
70	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
73	C₈×C₃₁	—	2	4	no	—
74	C₈×C₃₁	#125	1	8	yes	complemented
75	#2859	C₂×C₃₁	1	2	yes	—
76	#1942	C₂×C₃₁	1	8	yes	—
77	#1845	C₂×C₃₁	1	2	yes	—
78	#1834	C₂×C₃₁	1	2	yes	—
79	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
80	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂²×C₃₁	#532	1	2	yes	—
82	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
83	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
84	C₂²×C₃₁	C₂×#135	1	2	yes	—
85	C₄×C₃₁	—	2	4	no	—
86	C₄×C₃₁	—	2	4	no	—
87	C₄×C₃₁	—	8	8	no	—
88	C₄×C₃₁	#499	1	4	yes	—
89	C₄×C₃₁	#500	1	4	yes	—
90	C₄×C₃₁	C₂×#125	1	2	yes	—
91	C₄×C₃₁	#533	1	2	yes	—
92	C₂×#125	—	2	2	no	—
93	C₂×#107	C₂²×C₃₁	1	8	yes	—
94	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
95	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
97	#389	—	2	4	no	—
98	#389	C₂²×C₃₁	1	8	yes	—
99	#384	C₂²×C₃₁	1	4	yes	—
100	#382	—	2	2	no	—
101	C₄³	C₂²×C₃₁	1	1	yes	characteristic
102	#340	C₂²×C₃₁	1	8	yes	—
103	C₂×C₃₁	#2859	1	1	yes	characteristic
104	C₂×C₃₁	C₂×#499	1	1	yes	characteristic
105	C₂×C₃₁	C₂×#500	1	1	yes	characteristic
106	C₂×C₃₁	#3169	1	2	yes	—
107	C₂×C₃₁	#3170	1	2	yes	—
108	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
109	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
110	#125	—	4	8	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	—	2	8	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
115	C₂×C₄²	D₄×C₃₁	1	2	yes	—
116	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
117	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
119	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
120	#107	—	2	16	no	—
121	C₄×C₈	—	2	8	no	—
122	C₄×C₈	Q₈×C₃₁	1	4	yes	—
123	C₄×C₈	Q₈×C₃₁	1	4	yes	—
124	#95	—	2	8	no	—
125	C₃₁	#33258	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
128	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
129	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
134	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
135	C₂×C₈	#41×C₃₁	1	8	yes	—
136	C₄⋊C₄	—	2	16	no	—
137	C₄⋊C₄	—	4	16	no	—
138	C₄⋊C₄	—	4	8	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄²	—	2	4	no	—
141	C₄²	—	2	4	no	—
142	C₄²	D₈×C₃₁	1	4	yes	—
143	C₄²	Q₁₆×C₃₁	1	4	yes	—
144	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
145	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
146	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	—	4	8	no	—
154	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
155	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
156	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
157	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
158	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
159	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
160	C₂×C₄	C₃₁×#135	1	4	yes	—
161	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
164	C₈	—	2	4	no	—
165	C₈	C₃₁×#125	1	8	yes	complemented
166	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
167	C₂²	C₃₁×#532	1	2	yes	—
168	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
169	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
170	C₂²	C₂×C₃₁×#135	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	4	no	—
173	C₄	—	8	8	no	—
174	C₄	C₃₁×#499	1	4	yes	—
175	C₄	C₃₁×#500	1	4	yes	—
176	C₄	C₂×C₃₁×#125	1	2	yes	—
177	C₄	C₃₁×#533	1	2	yes	—
178	C₂	C₃₁×#2859	1	1	yes	characteristic
179	C₂	C₂×C₃₁×#499	1	1	yes	characteristic
180	C₂	C₂×C₃₁×#500	1	1	yes	characteristic
181	C₂	C₃₁×#3169	1	2	yes	—
182	C₂	C₃₁×#3170	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
54. #33258	C₃₁	direct (N × H)
74. C₈×C₃₁	#125	semidirect
125. C₃₁	#33258	direct (N × H)
165. C₈	C₃₁×#125	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2859	2
C₃₁×#1942	2
C₃₁×#1845	2
C₃₁×#1834	2
C₄²×C₈×C₃₁	2
#33258	31

Subgroups — C31×#33258 #21490017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33258 #21490017