Subgroups — C₃₁×#22258 #21479017

Order 7936.

578 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2961	C₂	1	2	yes	maximal
3	C₃₁×#1995	C₂	1	2	yes	maximal
4	C₃₁×#1922	C₂	1	8	yes	maximal
5	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂²	1	2	yes	—
8	C₃₁×#500	C₄	1	2	yes	complemented
9	C₃₁×#499	C₄	1	4	yes	complemented
10	C₃₁×#498	C₄	1	2	yes	complemented
11	C₃₁×#446	—	2	8	no	—
12	C₃₁×#427	C₂²	1	4	yes	—
13	C₃₁×#424	C₂²	1	4	yes	—
14	C₃₁×#423	C₂²	1	2	yes	—
15	C₃₁×#405	C₂²	1	4	yes	—
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#376	C₂²	1	1	yes	characteristic
19	C₃₁×#341	C₂²	1	4	yes	—
20	C₃₁×#339	C₂²	1	8	yes	—
21	C₃₁×#336	C₂²	1	4	yes	—
22	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
23	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
25	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₃₁×#125	C₂×C₄	1	4	yes	—
27	C₃₁×C₂²⋊Q₈	—	2	4	no	—
28	C₄×Q₈×C₃₁	—	2	4	no	—
29	C₃₁×#117	C₂³	1	4	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#107	C₂×C₄	1	8	yes	—
33	C₃₁×#106	C₂×C₄	1	8	yes	—
34	C₃₁×C₄⋊C₈	—	2	8	no	—
35	C₃₁×C₄⋊C₈	—	2	4	no	—
36	C₃₁×#97	—	2	4	no	—
37	C₃₁×#97	D₄	1	4	yes	—
38	C₄×C₈×C₃₁	—	2	4	no	—
39	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₃₁×#95	—	2	2	no	—
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	2	2	no	—
45	C₂²×C₄×C₃₁	#41	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
47	M₁₆×C₃₁	—	2	8	no	—
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	—	2	4	no	—
51	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
54	C₄⋊C₄×C₃₁	—	2	8	no	—
55	C₄⋊C₄×C₃₁	—	2	4	no	—
56	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
57	C₂²⋊C₄×C₃₁	—	2	4	no	—
58	C₄²×C₃₁	—	2	4	no	—
59	C₄²×C₃₁	—	2	4	no	—
60	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂×D₄	1	2	yes	—
63	#22258	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
65	Q₈×C₃₁	—	8	4	no	—
66	C₂×C₄×C₃₁	—	2	8	no	—
67	C₂×C₄×C₃₁	—	4	2	no	—
68	C₂×C₄×C₃₁	—	4	4	no	—
69	C₂×C₄×C₃₁	—	4	2	no	—
70	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
75	C₈×C₃₁	—	2	2	no	—
76	C₈×C₃₁	—	2	2	no	—
77	C₈×C₃₁	—	2	4	no	—
78	C₈×C₃₁	—	4	4	no	—
79	#3160	C₂×C₃₁	1	1	yes	characteristic
80	#2961	C₂×C₃₁	1	2	yes	—
81	#1995	C₂×C₃₁	1	2	yes	—
82	#1922	C₂×C₃₁	1	8	yes	—
83	#1837	C₂×C₃₁	1	1	yes	characteristic
84	#1746	C₂×C₃₁	1	1	yes	characteristic
85	C₂²×C₃₁	—	2	2	no	—
86	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
87	C₂²×C₃₁	#541	1	2	yes	—
88	C₄×C₃₁	—	2	2	no	—
89	C₄×C₃₁	—	2	2	no	—
90	C₄×C₃₁	—	4	8	no	—
91	C₄×C₃₁	—	8	2	no	—
92	C₄×C₃₁	#519	1	2	yes	—
93	C₄×C₃₁	#536	1	2	yes	—
94	#533	C₂²×C₃₁	1	2	yes	—
95	#500	C₄×C₃₁	1	2	yes	complemented
96	#499	C₄×C₃₁	1	4	yes	complemented
97	#498	C₄×C₃₁	1	2	yes	complemented
98	#446	—	2	8	no	—
99	#427	C₂²×C₃₁	1	4	yes	—
100	#424	C₂²×C₃₁	1	4	yes	—
101	#423	C₂²×C₃₁	1	2	yes	—
102	#405	C₂²×C₃₁	1	4	yes	—
103	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
105	#376	C₂²×C₃₁	1	1	yes	characteristic
106	#341	C₂²×C₃₁	1	4	yes	—
107	#339	C₂²×C₃₁	1	8	yes	—
108	#336	C₂²×C₃₁	1	4	yes	—
109	C₂×C₃₁	—	2	2	no	—
110	C₂×C₃₁	#2327	1	1	yes	characteristic
111	C₂×C₃₁	#2957	1	2	yes	—
112	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
113	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
114	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
115	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
116	#125	C₂×C₄×C₃₁	1	4	yes	—
117	C₂²⋊Q₈	—	2	4	no	—
118	C₄×Q₈	—	2	4	no	—
119	#117	C₂³×C₃₁	1	4	yes	—
120	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
121	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
122	#107	C₂×C₄×C₃₁	1	8	yes	—
123	#106	C₂×C₄×C₃₁	1	8	yes	—
124	C₄⋊C₈	—	2	8	no	—
125	C₄⋊C₈	—	2	4	no	—
126	#97	—	2	4	no	—
127	#97	D₄×C₃₁	1	4	yes	—
128	C₄×C₈	—	2	4	no	—
129	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
131	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
132	#95	—	2	2	no	—
133	C₃₁	#22258	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	#41×C₃₁	1	2	yes	—
137	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
138	M₁₆	—	2	8	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₃₁	1	2	yes	—
154	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
155	Q₈	—	8	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
161	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
162	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	2	4	no	—
168	C₈	—	4	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
171	C₂²	C₃₁×#541	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	4	8	no	—
175	C₄	—	8	2	no	—
176	C₄	C₃₁×#519	1	2	yes	—
177	C₄	C₃₁×#536	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	C₃₁×#2327	1	1	yes	characteristic
180	C₂	C₃₁×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
8. C₃₁×#500	C₄	semidirect
9. C₃₁×#499	C₄	semidirect
10. C₃₁×#498	C₄	semidirect
63. #22258	C₃₁	direct (N × H)
95. #500	C₄×C₃₁	semidirect
96. #499	C₄×C₃₁	semidirect
97. #498	C₄×C₃₁	semidirect
133. C₃₁	#22258	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×#3160	2
C₃₁×#2961	2
C₃₁×#1995	2
C₃₁×#1922	2
C₃₁×#1837	2
C₃₁×#1746	2
#22258	31

Subgroups — C31×#22258 #21479017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22258 #21479017