Subgroups — C₃₁×#22259 #21479018

Order 7936.

594 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2961	C₂	1	2	yes	maximal
3	C₃₁×#1994	C₂	1	2	yes	maximal
4	C₃₁×#1922	C₂	1	8	yes	maximal
5	C₃₁×#1838	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1745	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂²	1	2	yes	—
8	C₃₁×#501	C₄	1	8	yes	complemented
9	C₃₁×#446	—	2	8	no	—
10	C₃₁×#427	C₂²	1	4	yes	—
11	C₃₁×#424	C₂²	1	4	yes	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₃₁×#405	C₂²	1	4	yes	—
14	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
16	C₄³×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#341	C₂²	1	4	yes	—
18	C₃₁×#339	C₂²	1	8	yes	—
19	C₃₁×#336	C₂²	1	4	yes	—
20	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
21	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
22	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
23	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₃₁×#125	C₂×C₄	1	4	yes	—
25	C₃₁×C₂²⋊Q₈	—	2	4	no	—
26	C₄×Q₈×C₃₁	—	2	4	no	—
27	C₃₁×#117	C₂³	1	4	yes	—
28	C₂×C₄²×C₃₁	—	2	2	no	—
29	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×#107	C₂×C₄	1	8	yes	—
32	C₃₁×#106	C₂×C₄	1	8	yes	—
33	C₃₁×C₄⋊C₈	—	2	8	no	—
34	C₃₁×C₄⋊C₈	—	2	4	no	—
35	C₃₁×#97	—	2	4	no	—
36	C₃₁×#97	C₂×C₄	1	4	yes	—
37	C₃₁×#97	D₄	1	4	yes	—
38	C₄×C₈×C₃₁	—	2	4	no	—
39	C₂×Q₈×C₃₁	—	2	4	no	—
40	C₂²×C₄×C₃₁	—	2	2	no	—
41	C₂²×C₄×C₃₁	#41	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
43	M₁₆×C₃₁	—	2	8	no	—
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	2	4	no	—
46	C₂×C₈×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
49	C₄⋊C₄×C₃₁	—	2	8	no	—
50	C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
52	C₂²⋊C₄×C₃₁	—	2	4	no	—
53	C₄²×C₃₁	—	2	4	no	—
54	C₄²×C₃₁	—	2	4	no	—
55	C₄²×C₃₁	—	4	4	no	—
56	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
57	C₄²×C₃₁	C₂×D₄	1	2	yes	—
58	#22259	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
60	Q₈×C₃₁	—	8	4	no	—
61	C₂×C₄×C₃₁	—	2	4	no	—
62	C₂×C₄×C₃₁	—	2	8	no	—
63	C₂×C₄×C₃₁	—	4	2	no	—
64	C₂×C₄×C₃₁	—	4	2	no	—
65	C₂×C₄×C₃₁	—	4	2	no	—
66	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
67	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
70	C₈×C₃₁	—	2	4	no	—
71	C₈×C₃₁	—	2	4	no	—
72	C₈×C₃₁	—	4	4	no	—
73	#3162	C₂×C₃₁	1	1	yes	characteristic
74	#2961	C₂×C₃₁	1	2	yes	—
75	#1994	C₂×C₃₁	1	2	yes	—
76	#1922	C₂×C₃₁	1	8	yes	—
77	#1838	C₂×C₃₁	1	1	yes	characteristic
78	#1745	C₂×C₃₁	1	1	yes	characteristic
79	C₂²×C₃₁	—	2	2	no	—
80	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂²×C₃₁	#541	1	2	yes	—
82	C₄×C₃₁	—	2	4	no	—
83	C₄×C₃₁	—	4	4	no	—
84	C₄×C₃₁	—	4	8	no	—
85	C₄×C₃₁	#519	1	2	yes	—
86	C₄×C₃₁	#536	1	2	yes	—
87	#533	C₂²×C₃₁	1	2	yes	—
88	#501	C₄×C₃₁	1	8	yes	complemented
89	#446	—	2	8	no	—
90	#427	C₂²×C₃₁	1	4	yes	—
91	#424	C₂²×C₃₁	1	4	yes	—
92	#423	C₂²×C₃₁	1	2	yes	—
93	#405	C₂²×C₃₁	1	4	yes	—
94	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
95	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
96	C₄³	C₂²×C₃₁	1	1	yes	characteristic
97	#341	C₂²×C₃₁	1	4	yes	—
98	#339	C₂²×C₃₁	1	8	yes	—
99	#336	C₂²×C₃₁	1	4	yes	—
100	C₂×C₃₁	—	2	2	no	—
101	C₂×C₃₁	#2327	1	1	yes	characteristic
102	C₂×C₃₁	#2956	1	2	yes	—
103	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
104	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
105	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
106	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
107	#125	C₂×C₄×C₃₁	1	4	yes	—
108	C₂²⋊Q₈	—	2	4	no	—
109	C₄×Q₈	—	2	4	no	—
110	#117	C₂³×C₃₁	1	4	yes	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
113	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
114	#107	C₂×C₄×C₃₁	1	8	yes	—
115	#106	C₂×C₄×C₃₁	1	8	yes	—
116	C₄⋊C₈	—	2	8	no	—
117	C₄⋊C₈	—	2	4	no	—
118	#97	—	2	4	no	—
119	#97	C₂×C₄×C₃₁	1	4	yes	—
120	#97	D₄×C₃₁	1	4	yes	—
121	C₄×C₈	—	2	4	no	—
122	C₃₁	#22259	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×Q₈	—	2	4	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	#41×C₃₁	1	2	yes	—
126	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
132	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
141	C₄²	C₂×D₄×C₃₁	1	2	yes	—
142	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
143	Q₈	—	8	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
150	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
158	C₂²	C₃₁×#541	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	8	no	—
162	C₄	C₃₁×#519	1	2	yes	—
163	C₄	C₃₁×#536	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	C₃₁×#2327	1	1	yes	characteristic
166	C₂	C₃₁×#2956	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×#501	C₄	semidirect
58. #22259	C₃₁	direct (N × H)
88. #501	C₄×C₃₁	semidirect
122. C₃₁	#22259	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×#3162	2
C₃₁×#2961	2
C₃₁×#1994	2
C₃₁×#1922	2
C₃₁×#1838	2
C₃₁×#1745	2
#22259	31

Subgroups — C31×#22259 #21479018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22259 #21479018