Subgroups — C₃₁×#12670 #21469429

Order 7936.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1772	C₂	1	2	yes	maximal
5	C₃₁×#1382	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₃₁	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	—	2	2	no	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#368	C₄	1	4	yes	complemented
13	C₃₁×#367	C₄	1	2	yes	complemented
14	C₃₁×#366	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₃₁	C₄	1	4	yes	complemented
16	C₃₁×#339	—	2	2	no	—
17	C₂×M₁₆×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₃₁×#117	—	2	2	no	—
21	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₁	—	2	1	no	—
23	C₂×C₁₆×C₃₁	—	2	1	no	—
24	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
29	C₃₁×#108	—	2	2	no	—
30	C₃₁×#108	C₂×C₄	1	4	yes	—
31	C₃₁×#107	C₂×C₄	1	4	yes	—
32	C₃₁×#106	—	2	2	no	—
33	C₃₁×#97	—	4	2	no	—
34	C₄×C₈×C₃₁	—	4	2	no	—
35	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₃₁	Q₈	1	2	yes	—
37	C₂²×C₄×C₃₁	—	2	1	no	—
38	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₃₁	—	2	4	no	—
40	M₁₆×C₃₁	—	4	2	no	—
41	C₂×C₈×C₃₁	—	4	2	no	—
42	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
44	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
45	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
46	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
51	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₂²⋊C₄×C₃₁	—	4	2	no	—
54	C₄²×C₃₁	—	2	1	no	—
55	C₄²×C₃₁	—	4	2	no	—
56	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₃₁	—	2	2	no	—
58	C₁₆×C₃₁	C₄²	1	2	yes	—
59	C₁₆×C₃₁	C₄²	1	2	yes	—
60	#12670	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	#135	1	1	yes	characteristic
62	C₂×C₄×C₃₁	—	2	4	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
71	C₈×C₃₁	—	4	4	no	—
72	C₈×C₃₁	#107	1	2	yes	—
73	C₈×C₃₁	#107	1	2	yes	—
74	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₃₁	#117	1	1	yes	characteristic
76	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
78	C₂×#368	C₂×C₃₁	1	1	yes	characteristic
79	#2158	C₂×C₃₁	1	1	yes	characteristic
80	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
81	#1772	C₂×C₃₁	1	2	yes	—
82	#1382	C₂×C₃₁	1	2	yes	—
83	C₂²×C₃₁	—	2	2	no	—
84	C₂²×C₃₁	#427	1	1	yes	characteristic
85	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
86	C₂²×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
87	C₄×C₃₁	—	2	4	no	—
88	C₄×C₃₁	—	8	4	no	—
89	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₃₁	C₂×#107	1	1	yes	characteristic
91	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
92	C₄×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
93	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
94	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
95	C₂×#108	C₂²×C₃₁	1	2	yes	—
96	#427	C₂²×C₃₁	1	2	yes	—
97	#405	—	2	2	no	—
98	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
99	#368	C₄×C₃₁	1	4	yes	complemented
100	#367	C₄×C₃₁	1	2	yes	complemented
101	#366	C₄×C₃₁	1	2	yes	complemented
102	C₄×C₁₆	C₄×C₃₁	1	4	yes	complemented
103	#339	—	2	2	no	—
104	C₂×C₃₁	—	2	1	no	—
105	C₂×C₃₁	—	2	1	no	—
106	C₂×C₃₁	C₄×#107	1	1	yes	characteristic
107	C₂×C₃₁	#2178	1	2	yes	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
111	#117	—	2	2	no	—
112	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
116	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
117	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
118	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
119	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
120	#108	—	2	2	no	—
121	#108	C₂×C₄×C₃₁	1	4	yes	—
122	#107	C₂×C₄×C₃₁	1	4	yes	—
123	#106	—	2	2	no	—
124	#97	—	4	2	no	—
125	C₄×C₈	—	4	2	no	—
126	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
127	C₄×C₈	Q₈×C₃₁	1	2	yes	—
128	C₃₁	#12670	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
135	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
136	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
137	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
142	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
143	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	4	2	no	—
148	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
149	C₁₆	—	2	2	no	—
150	C₁₆	C₄²×C₃₁	1	2	yes	—
151	C₁₆	C₄²×C₃₁	1	2	yes	—
152	C₂³	C₃₁×#135	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
160	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
161	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
162	C₈	—	4	4	no	—
163	C₈	C₃₁×#107	1	2	yes	—
164	C₈	C₃₁×#107	1	2	yes	—
165	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
166	C₈	C₃₁×#117	1	1	yes	characteristic
167	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
168	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	C₃₁×#427	1	1	yes	characteristic
171	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
172	C₂²	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	8	4	no	—
175	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
176	C₄	C₂×C₃₁×#107	1	1	yes	characteristic
177	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
178	C₄	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₃₁×#107	1	1	yes	characteristic
182	C₂	C₃₁×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	D₈
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×D₈
prime radical	C₂³×C₃₁	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#368	C₄	semidirect
13. C₃₁×#367	C₄	semidirect
14. C₃₁×#366	C₄	semidirect
15. C₄×C₁₆×C₃₁	C₄	semidirect
60. #12670	C₃₁	direct (N × H)
99. #368	C₄×C₃₁	semidirect
100. #367	C₄×C₃₁	semidirect
101. #366	C₄×C₃₁	semidirect
102. C₄×C₁₆	C₄×C₃₁	semidirect
128. C₃₁	#12670	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
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1772	2
C₃₁×#1382	2
#12670	31

Subgroups — C31×#12670 #21469429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12670 #21469429