Subgroups — C₃₁×#13173 #21469932

Order 7936.

262 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1953	C₂	1	2	yes	maximal
5	C₃₁×#1388	C₂	1	2	yes	maximal
6	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#430	—	2	2	no	—
9	C₂×C₃₁×#108	C₂²	1	2	yes	—
10	C₂×C₃₁×#108	C₂²	1	2	yes	—
11	C₃₁×#405	C₂²	1	1	yes	characteristic
12	C₃₁×#368	—	2	1	no	—
13	C₃₁×#368	—	2	1	no	—
14	C₃₁×#368	C₄	1	4	yes	complemented
15	C₃₁×#346	—	2	1	no	—
16	C₄×C₁₆×C₃₁	—	2	1	no	—
17	C₃₁×#344	—	2	2	no	—
18	C₂×M₁₆×C₃₁	—	2	2	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×#117	D₄	1	1	yes	characteristic
23	C₃₁×M₃₂	—	2	1	no	—
24	C₃₁×M₃₂	Q₈	1	2	yes	—
25	C₂×C₁₆×C₃₁	—	2	1	no	—
26	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
30	C₃₁×#108	—	2	2	no	—
31	C₃₁×#108	—	2	2	no	—
32	C₃₁×#108	—	2	2	no	—
33	C₃₁×#108	—	4	4	no	—
34	C₃₁×#108	C₂×C₄	1	4	yes	—
35	C₃₁×#97	—	2	1	no	—
36	C₄×C₈×C₃₁	—	2	1	no	—
37	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₃₁	—	2	1	no	—
39	M₁₆×C₃₁	—	2	4	no	—
40	M₁₆×C₃₁	—	4	2	no	—
41	M₁₆×C₃₁	—	4	2	no	—
42	M₁₆×C₃₁	—	4	2	no	—
43	M₁₆×C₃₁	D₈	1	2	yes	—
44	C₂×C₈×C₃₁	—	2	1	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂²⋊C₄×C₃₁	Q₁₆	1	2	yes	—
55	C₄²×C₃₁	—	2	1	no	—
56	C₁₆×C₃₁	—	2	2	no	—
57	C₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
58	C₁₆×C₃₁	C₄⋊C₄	1	2	yes	—
59	#13173	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
67	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
68	C₈×C₃₁	—	2	2	no	—
69	C₈×C₃₁	—	4	4	no	—
70	C₈×C₃₁	—	8	2	no	—
71	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₈×C₃₁	#125	1	1	yes	characteristic
75	C₂×#368	C₂×C₃₁	1	1	yes	characteristic
76	C₂×#368	C₂×C₃₁	1	1	yes	characteristic
77	#2108	C₂×C₃₁	1	1	yes	characteristic
78	#1953	C₂×C₃₁	1	2	yes	—
79	#1388	C₂×C₃₁	1	2	yes	—
80	C₂²×C₃₁	—	4	1	no	—
81	C₂²×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
82	C₂²×C₃₁	#467	1	1	yes	characteristic
83	C₂²×C₃₁	#500	1	1	yes	characteristic
84	C₄×C₃₁	—	4	2	no	—
85	C₄×C₃₁	#389	1	1	yes	characteristic
86	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
87	C₄×C₃₁	#466	1	1	yes	characteristic
88	C₄×C₃₁	#499	1	1	yes	characteristic
89	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
90	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
91	#430	—	2	2	no	—
92	C₂×#108	C₂²×C₃₁	1	2	yes	—
93	C₂×#108	C₂²×C₃₁	1	2	yes	—
94	#405	C₂²×C₃₁	1	1	yes	characteristic
95	#368	—	2	1	no	—
96	#368	—	2	1	no	—
97	#368	C₄×C₃₁	1	4	yes	complemented
98	#346	—	2	1	no	—
99	C₄×C₁₆	—	2	1	no	—
100	#344	—	2	2	no	—
101	C₂×C₃₁	—	2	1	no	—
102	C₂×C₃₁	—	2	1	no	—
103	C₂×C₃₁	—	2	1	no	—
104	C₂×C₃₁	#1942	1	1	yes	characteristic
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
109	#117	D₄×C₃₁	1	1	yes	characteristic
110	M₃₂	—	2	1	no	—
111	M₃₂	Q₈×C₃₁	1	2	yes	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
114	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
115	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
116	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#108	—	4	4	no	—
121	#108	C₂×C₄×C₃₁	1	4	yes	—
122	#97	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₃₁	#13173	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
126	M₁₆	—	2	1	no	—
127	M₁₆	—	2	4	no	—
128	M₁₆	—	4	2	no	—
129	M₁₆	—	4	2	no	—
130	M₁₆	—	4	2	no	—
131	M₁₆	D₈×C₃₁	1	2	yes	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
136	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
137	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
138	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
139	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
140	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
141	C₄⋊C₄	—	2	1	no	—
142	C₂²⋊C₄	Q₁₆×C₃₁	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
146	C₁₆	C₄⋊C₄×C₃₁	1	2	yes	—
147	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
151	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
153	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
154	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
155	C₈	—	2	2	no	—
156	C₈	—	4	4	no	—
157	C₈	—	8	2	no	—
158	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
159	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
160	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
161	C₈	C₃₁×#125	1	1	yes	characteristic
162	C₂²	—	4	1	no	—
163	C₂²	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
164	C₂²	C₃₁×#467	1	1	yes	characteristic
165	C₂²	C₃₁×#500	1	1	yes	characteristic
166	C₄	—	4	2	no	—
167	C₄	C₃₁×#389	1	1	yes	characteristic
168	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
169	C₄	C₃₁×#466	1	1	yes	characteristic
170	C₄	C₃₁×#499	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	C₃₁×#1942	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#466
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#1942
prime radical	C₂³×C₃₁	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
14. C₃₁×#368	C₄	semidirect
59. #13173	C₃₁	direct (N × H)
97. #368	C₄×C₃₁	semidirect
124. C₃₁	#13173	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#368	2
C₃₁×#2108	2
C₃₁×#1953	2
C₃₁×#1388	2
#13173	31

Subgroups — C31×#13173 #21469932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13173 #21469932