Subgroups — C₃₁×#33232 #21489991

Order 7936.

894 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×#491	C₂	1	4	yes	maximal
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1960	C₂	1	4	yes	maximal
5	C₃₁×#1956	C₂	1	4	yes	maximal
6	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₃₁×#125	C₂²	1	2	yes	—
11	C₃₁×#491	—	2	16	no	—
12	C₂×C₃₁×#103	C₂²	1	16	yes	—
13	C₂×C₃₁×#97	C₂²	1	2	yes	—
14	C₂×C₃₁×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#389	—	2	4	no	—
17	C₃₁×#384	—	2	4	no	—
18	C₃₁×#384	C₂²	1	2	yes	—
19	C₃₁×#382	C₂²	1	2	yes	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
22	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
23	C₃₁×C₄⋊Q₈	—	2	8	no	—
24	C₃₁×C₄⋊Q₈	—	2	8	no	—
25	C₃₁×C₄⋊Q₈	—	2	8	no	—
26	C₃₁×#125	—	2	8	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
33	C₂×C₄²×C₃₁	D₄	1	4	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#103	—	2	32	no	—
38	C₃₁×#97	—	2	4	no	—
39	C₃₁×#97	D₄	1	8	yes	—
40	C₄×C₈×C₃₁	—	2	4	no	—
41	C₃₁×#95	—	2	4	no	—
42	C₂×Q₈×C₃₁	—	2	8	no	—
43	C₂×Q₈×C₃₁	—	2	8	no	—
44	C₂×Q₈×C₃₁	—	2	8	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₃₁	—	2	4	no	—
52	C₂×C₈×C₃₁	—	2	4	no	—
53	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
54	C₄⋊C₄×C₃₁	—	2	8	no	—
55	C₄⋊C₄×C₃₁	—	4	8	no	—
56	C₄⋊C₄×C₃₁	—	4	8	no	—
57	C₄⋊C₄×C₃₁	—	4	8	no	—
58	C₄⋊C₄×C₃₁	—	4	8	no	—
59	C₄⋊C₄×C₃₁	—	4	8	no	—
60	C₄²×C₃₁	—	2	8	no	—
61	C₄²×C₃₁	C₂×D₄	1	4	yes	—
62	C₄²×C₃₁	#41	1	4	yes	—
63	C₄²×C₃₁	#41	1	4	yes	—
64	#33232	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
66	Q₈×C₃₁	—	4	16	no	—
67	C₂×C₄×C₃₁	—	2	4	no	—
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	4	4	no	—
71	C₂×C₄×C₃₁	—	4	4	no	—
72	C₂×C₄×C₃₁	—	4	4	no	—
73	C₂×C₄×C₃₁	—	4	8	no	—
74	C₂×C₄×C₃₁	#124	1	4	yes	—
75	C₂×C₄×C₃₁	#124	1	4	yes	—
76	C₂×C₄×C₃₁	#127	1	4	yes	—
77	C₂×C₄×C₃₁	#137	1	8	yes	—
78	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
81	C₈×C₃₁	—	2	8	no	—
82	C₂×#491	C₂×C₃₁	1	4	yes	—
83	#2868	C₂×C₃₁	1	1	yes	characteristic
84	#2859	C₂×C₃₁	1	1	yes	characteristic
85	#1960	C₂×C₃₁	1	4	yes	—
86	#1956	C₂×C₃₁	1	4	yes	—
87	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
88	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
89	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#137	1	4	yes	—
92	C₄×C₃₁	—	2	4	no	—
93	C₄×C₃₁	—	2	4	no	—
94	C₄×C₃₁	—	8	4	no	—
95	C₄×C₃₁	—	8	4	no	—
96	C₄×C₃₁	#491	1	8	yes	complemented
97	C₄×C₃₁	#532	1	4	yes	—
98	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
99	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
100	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
101	C₂×#125	C₂²×C₃₁	1	2	yes	—
102	#491	—	2	16	no	—
103	C₂×#103	C₂²×C₃₁	1	16	yes	—
104	C₂×#97	C₂²×C₃₁	1	2	yes	—
105	C₂×#97	C₂²×C₃₁	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
107	#389	—	2	4	no	—
108	#384	—	2	4	no	—
109	#384	C₂²×C₃₁	1	2	yes	—
110	#382	C₂²×C₃₁	1	2	yes	—
111	C₄³	C₂²×C₃₁	1	1	yes	characteristic
112	C₂×C₃₁	#2858	1	1	yes	characteristic
113	C₂×C₃₁	C₂×#491	1	2	yes	—
114	C₂×C₃₁	#3139	1	2	yes	—
115	C₂×C₃₁	#3152	1	2	yes	—
116	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
117	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
118	C₄⋊Q₈	—	2	8	no	—
119	C₄⋊Q₈	—	2	8	no	—
120	C₄⋊Q₈	—	2	8	no	—
121	#125	—	2	8	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
128	C₂×C₄²	D₄×C₃₁	1	4	yes	—
129	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
130	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
131	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
132	#103	—	2	32	no	—
133	#97	—	2	4	no	—
134	#97	D₄×C₃₁	1	8	yes	—
135	C₄×C₈	—	2	4	no	—
136	#95	—	2	4	no	—
137	C₃₁	#33232	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×Q₈	—	2	8	no	—
139	C₂×Q₈	—	2	8	no	—
140	C₂×Q₈	—	2	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
146	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
150	C₄⋊C₄	—	2	8	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄²	—	2	8	no	—
157	C₄²	C₂×D₄×C₃₁	1	4	yes	—
158	C₄²	#41×C₃₁	1	4	yes	—
159	C₄²	#41×C₃₁	1	4	yes	—
160	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
161	Q₈	—	4	16	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	C₃₁×#124	1	4	yes	—
170	C₂×C₄	C₃₁×#124	1	4	yes	—
171	C₂×C₄	C₃₁×#127	1	4	yes	—
172	C₂×C₄	C₃₁×#137	1	8	yes	—
173	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
174	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
175	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
176	C₈	—	2	8	no	—
177	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
178	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
179	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
180	C₂²	C₂×C₃₁×#137	1	4	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	8	4	no	—
185	C₄	C₃₁×#491	1	8	yes	complemented
186	C₄	C₃₁×#532	1	4	yes	—
187	C₂	C₃₁×#2858	1	1	yes	characteristic
188	C₂	C₂×C₃₁×#491	1	2	yes	—
189	C₂	C₃₁×#3139	1	2	yes	—
190	C₂	C₃₁×#3152	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
64. #33232	C₃₁	direct (N × H)
96. C₄×C₃₁	#491	semidirect
137. C₃₁	#33232	direct (N × H)
185. C₄	C₃₁×#491	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#491	2
C₃₁×#2868	2
C₃₁×#2859	2
C₃₁×#1960	2
C₃₁×#1956	2
C₄×C₃₁×#97	2
#33232	31

Subgroups — C31×#33232 #21489991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33232 #21489991