Subgroups — C₃₁×#12911 #21469670

Order 7936.

266 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2155	C₂	1	2	yes	maximal
4	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1833	C₂	1	2	yes	maximal
6	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#108	C₂²	1	2	yes	—
9	C₃₁×#427	C₂²	1	2	yes	—
10	C₃₁×#405	C₂²	1	1	yes	characteristic
11	C₃₁×#368	—	2	1	no	—
12	C₃₁×#368	—	2	1	no	—
13	C₃₁×#367	—	2	1	no	—
14	C₃₁×#366	—	2	1	no	—
15	C₃₁×#365	—	2	4	no	—
16	C₃₁×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₃₁	C₄	1	2	yes	complemented
18	C₃₁×#344	—	2	2	no	—
19	C₃₁×H(3,ℤ/4)	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×#117	—	2	2	no	—
24	C₃₁×#117	D₄	1	1	yes	characteristic
25	C₃₁×M₃₂	D₄	1	2	yes	—
26	C₃₁×M₃₂	Q₈	1	2	yes	—
27	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₃₁	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₃₁	Q₈	1	1	yes	characteristic
34	C₃₁×#108	—	2	2	no	—
35	C₃₁×#108	—	2	2	no	—
36	C₃₁×#107	—	2	2	no	—
37	C₃₁×#106	—	2	2	no	—
38	C₃₁×#97	Q₈	1	2	yes	—
39	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₃₁	—	2	1	no	—
42	M₁₆×C₃₁	—	2	1	no	—
43	M₁₆×C₃₁	—	4	2	no	—
44	M₁₆×C₃₁	—	4	2	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
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×D₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₄⋊C₄×C₃₁	—	4	2	no	—
55	C₂²⋊C₄×C₃₁	—	2	1	no	—
56	C₂²⋊C₄×C₃₁	—	4	2	no	—
57	C₄²×C₃₁	—	4	2	no	—
58	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₃₁	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₃₁	C₄⋊C₄	1	1	yes	characteristic
63	C₁₆×C₃₁	C₄⋊C₄	1	4	yes	—
64	#12911	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
72	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
73	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
74	C₈×C₃₁	—	2	2	no	—
75	C₈×C₃₁	—	8	2	no	—
76	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×#368	C₂×C₃₁	1	1	yes	characteristic
81	#2158	C₂×C₃₁	1	1	yes	characteristic
82	#2155	C₂×C₃₁	1	2	yes	—
83	#2108	C₂×C₃₁	1	1	yes	characteristic
84	#1833	C₂×C₃₁	1	2	yes	—
85	C₂²×C₃₁	—	4	1	no	—
86	C₂²×C₃₁	#427	1	1	yes	characteristic
87	C₂²×C₃₁	#495	1	1	yes	characteristic
88	C₂²×C₃₁	#499	1	1	yes	characteristic
89	C₄×C₃₁	—	4	2	no	—
90	C₄×C₃₁	—	16	2	no	—
91	C₄×C₃₁	#384	1	1	yes	characteristic
92	C₄×C₃₁	#427	1	1	yes	characteristic
93	C₄×C₃₁	#495	1	1	yes	characteristic
94	C₄×C₃₁	#499	1	1	yes	characteristic
95	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
96	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
97	C₂×#108	C₂²×C₃₁	1	2	yes	—
98	#427	C₂²×C₃₁	1	2	yes	—
99	#405	C₂²×C₃₁	1	1	yes	characteristic
100	#368	—	2	1	no	—
101	#368	—	2	1	no	—
102	#367	—	2	1	no	—
103	#366	—	2	1	no	—
104	#365	—	2	4	no	—
105	#346	C₄×C₃₁	1	2	yes	complemented
106	C₄×C₁₆	C₄×C₃₁	1	2	yes	complemented
107	#344	—	2	2	no	—
108	H(3,ℤ/4)	—	2	2	no	—
109	C₂×C₃₁	—	2	1	no	—
110	C₂×C₃₁	—	2	1	no	—
111	C₂×C₃₁	—	2	1	no	—
112	C₂×C₃₁	#1846	1	1	yes	characteristic
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
116	#117	—	2	2	no	—
117	#117	D₄×C₃₁	1	1	yes	characteristic
118	M₃₂	D₄×C₃₁	1	2	yes	—
119	M₃₂	Q₈×C₃₁	1	2	yes	—
120	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
121	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
122	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
123	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
124	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
125	C₂×C₁₆	Q₈×C₃₁	1	1	yes	characteristic
126	C₂×C₁₆	Q₈×C₃₁	1	1	yes	characteristic
127	#108	—	2	2	no	—
128	#108	—	2	2	no	—
129	#107	—	2	2	no	—
130	#106	—	2	2	no	—
131	#97	Q₈×C₃₁	1	2	yes	—
132	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
133	C₃₁	#12911	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	4	2	no	—
138	M₁₆	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₂²⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	4	2	no	—
151	C₄²	—	4	2	no	—
152	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
153	C₁₆	C₄⋊C₄×C₃₁	1	1	yes	characteristic
154	C₁₆	C₄⋊C₄×C₃₁	1	1	yes	characteristic
155	C₁₆	C₄⋊C₄×C₃₁	1	1	yes	characteristic
156	C₁₆	C₄⋊C₄×C₃₁	1	1	yes	characteristic
157	C₁₆	C₄⋊C₄×C₃₁	1	4	yes	—
158	C₂³	C₃₁×#135	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
163	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
164	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
166	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	8	2	no	—
169	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
170	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
171	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
172	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	C₃₁×#427	1	1	yes	characteristic
175	C₂²	C₃₁×#495	1	1	yes	characteristic
176	C₂²	C₃₁×#499	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	16	2	no	—
179	C₄	C₃₁×#384	1	1	yes	characteristic
180	C₄	C₃₁×#427	1	1	yes	characteristic
181	C₄	C₃₁×#495	1	1	yes	characteristic
182	C₄	C₃₁×#499	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	C₃₁×#1846	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#427
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#1846
prime radical	C₂³×C₃₁	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
16. C₃₁×#346	C₄	semidirect
17. C₄×C₁₆×C₃₁	C₄	semidirect
64. #12911	C₃₁	direct (N × H)
105. #346	C₄×C₃₁	semidirect
106. C₄×C₁₆	C₄×C₃₁	semidirect
133. C₃₁	#12911	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×#2158	2
C₃₁×#2155	2
C₃₁×#2108	2
C₃₁×#1833	2
#12911	31

Subgroups — C31×#12911 #21469670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12911 #21469670