Subgroups — C₃₁×#29294 #21486053

Order 7936.

630 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2330	C₂	1	2	yes	maximal
2	C₃₁×#2087	C₂	1	2	yes	maximal
3	C₃₁×#2086	C₂	1	2	yes	maximal
4	C₃₁×#1989	C₂	1	4	yes	maximal
5	C₃₁×#1947	C₂	1	4	yes	maximal
6	C₃₁×#1843	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#125	C₂²	1	2	yes	—
8	C₂×C₃₁×#107	C₂²	1	2	yes	—
9	C₂×C₃₁×#107	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃₁×#389	—	2	4	no	—
14	C₃₁×#389	C₂²	1	4	yes	—
15	C₃₁×#387	—	2	4	no	—
16	C₃₁×#384	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
18	C₃₁×#376	C₂²	1	1	yes	characteristic
19	C₃₁×#340	C₂²	1	4	yes	—
20	C₃₁×#340	C₂²	1	2	yes	—
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₃₁×#340	C₂²	1	2	yes	—
23	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
25	C₃₁×#125	—	2	4	no	—
26	C₃₁×#125	Q₈	1	8	yes	—
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	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₃₁	—	2	4	no	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
35	C₃₁×#107	—	4	2	no	—
36	C₃₁×#107	—	4	2	no	—
37	C₃₁×C₄⋊C₈	—	2	8	no	—
38	C₃₁×C₄⋊C₈	—	4	1	no	—
39	C₃₁×C₄⋊C₈	—	4	1	no	—
40	C₃₁×#95	—	2	4	no	—
41	C₃₁×#95	—	2	4	no	—
42	C₃₁×#95	D₄	1	2	yes	—
43	C₃₁×#95	D₄	1	2	yes	—
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₃₁	—	2	4	no	—
51	C₂×C₈×C₃₁	—	2	4	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₄⋊C₄×C₃₁	—	2	16	no	—
55	C₄⋊C₄×C₃₁	—	2	4	no	—
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	8	4	no	—
59	C₄⋊C₄×C₃₁	—	8	4	no	—
60	C₄⋊C₄×C₃₁	C₂×Q₈	1	8	yes	—
61	C₄²×C₃₁	—	2	1	no	—
62	C₄²×C₃₁	—	2	1	no	—
63	C₄²×C₃₁	—	8	4	no	—
64	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
65	C₄²×C₃₁	#41	1	4	yes	—
66	#29294	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	—	2	8	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	2	1	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	4	4	no	—
76	C₂×C₄×C₃₁	—	4	4	no	—
77	C₂×C₄×C₃₁	—	4	4	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	4	4	no	—
80	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
81	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
82	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
83	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
85	C₈×C₃₁	—	4	2	no	—
86	C₈×C₃₁	—	4	2	no	—
87	#2330	C₂×C₃₁	1	2	yes	—
88	#2087	C₂×C₃₁	1	2	yes	—
89	#2086	C₂×C₃₁	1	2	yes	—
90	#1989	C₂×C₃₁	1	4	yes	—
91	#1947	C₂×C₃₁	1	4	yes	—
92	#1843	C₂×C₃₁	1	1	yes	characteristic
93	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
94	C₂²×C₃₁	#537	1	1	yes	characteristic
95	C₂²×C₃₁	#576	1	2	yes	—
96	C₂²×C₃₁	#578	1	2	yes	—
97	C₄×C₃₁	—	2	8	no	—
98	C₄×C₃₁	—	4	8	no	—
99	C₄×C₃₁	—	4	1	no	—
100	C₄×C₃₁	—	4	1	no	—
101	C₄×C₃₁	—	8	4	no	—
102	C₄×C₃₁	#557	1	2	yes	—
103	C₄×C₃₁	Q₈²	1	1	yes	characteristic
104	C₄×C₃₁	#559	1	1	yes	characteristic
105	C₂×#125	C₂²×C₃₁	1	2	yes	—
106	C₂×#107	C₂²×C₃₁	1	2	yes	—
107	C₂×#107	C₂²×C₃₁	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
109	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
110	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
111	#389	—	2	4	no	—
112	#389	C₂²×C₃₁	1	4	yes	—
113	#387	—	2	4	no	—
114	#384	C₂²×C₃₁	1	2	yes	—
115	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
116	#376	C₂²×C₃₁	1	1	yes	characteristic
117	#340	C₂²×C₃₁	1	4	yes	—
118	#340	C₂²×C₃₁	1	2	yes	—
119	#340	C₂²×C₃₁	1	4	yes	—
120	#340	C₂²×C₃₁	1	2	yes	—
121	C₂×C₃₁	#2582	1	1	yes	characteristic
122	C₂×C₃₁	#3083	1	4	yes	—
123	C₂×C₃₁	#3259	1	1	yes	characteristic
124	C₂×C₃₁	#3261	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
126	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
127	#125	—	2	4	no	—
128	#125	Q₈×C₃₁	1	8	yes	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
133	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
134	C₂×C₄²	—	2	4	no	—
135	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
137	#107	—	4	2	no	—
138	#107	—	4	2	no	—
139	C₄⋊C₈	—	2	8	no	—
140	C₄⋊C₈	—	4	1	no	—
141	C₄⋊C₈	—	4	1	no	—
142	#95	—	2	4	no	—
143	#95	—	2	4	no	—
144	#95	D₄×C₃₁	1	2	yes	—
145	#95	D₄×C₃₁	1	2	yes	—
146	C₃₁	#29294	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
152	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₄⋊C₄	—	2	16	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	8	4	no	—
162	C₄⋊C₄	—	8	4	no	—
163	C₄⋊C₄	C₂×Q₈×C₃₁	1	8	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	8	4	no	—
167	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
168	C₄²	#41×C₃₁	1	4	yes	—
169	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
183	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
184	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
185	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	—	4	2	no	—
189	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
190	C₂²	C₃₁×#537	1	1	yes	characteristic
191	C₂²	C₃₁×#576	1	2	yes	—
192	C₂²	C₃₁×#578	1	2	yes	—
193	C₄	—	2	8	no	—
194	C₄	—	4	8	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₃₁×#557	1	2	yes	—
199	C₄	Q₈²×C₃₁	1	1	yes	characteristic
200	C₄	C₃₁×#559	1	1	yes	characteristic
201	C₂	C₃₁×#2582	1	1	yes	characteristic
202	C₂	C₃₁×#3083	1	4	yes	—
203	C₂	C₃₁×#3259	1	1	yes	characteristic
204	C₂	C₃₁×#3261	1	1	yes	characteristic

	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 ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
66. #29294	C₃₁	direct (N × H)
146. C₃₁	#29294	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2330	2
C₃₁×#2087	2
C₃₁×#2086	2
C₃₁×#1989	2
C₃₁×#1947	2
C₃₁×#1843	2
#29294	31

Subgroups — C31×#29294 #21486053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29294 #21486053