Subgroups — C₃₁×#12668 #21469427

Order 7936.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2155	C₂	1	2	yes	maximal
2	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1772	C₂	1	2	yes	maximal
4	C₃₁×#1385	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	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×#365	C₄	1	8	yes	complemented
13	C₃₁×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₃₁	C₄	1	2	yes	complemented
15	C₃₁×#339	—	2	2	no	—
16	C₂×M₁₆×C₃₁	—	2	2	no	—
17	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁×#117	—	2	2	no	—
20	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
21	C₃₁×M₃₂	—	2	2	no	—
22	C₃₁×M₃₂	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₃₁×#108	—	2	2	no	—
26	C₃₁×#108	C₂×C₄	1	4	yes	—
27	C₃₁×#107	—	2	2	no	—
28	C₃₁×#106	C₂×C₄	1	4	yes	—
29	C₃₁×#97	—	4	2	no	—
30	C₄×C₈×C₃₁	—	4	2	no	—
31	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	—	2	1	no	—
36	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	M₁₆×C₃₁	—	2	4	no	—
38	M₁₆×C₃₁	—	4	2	no	—
39	C₂×C₈×C₃₁	—	4	2	no	—
40	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
43	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
44	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₃₁	#41	1	2	yes	—
48	C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₂²⋊C₄×C₃₁	—	4	2	no	—
50	C₄²×C₃₁	—	2	1	no	—
51	C₄²×C₃₁	—	4	2	no	—
52	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₃₁	—	2	2	no	—
55	C₁₆×C₃₁	C₄²	1	4	yes	—
56	#12668	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₃₁	#135	1	1	yes	characteristic
58	C₂×C₄×C₃₁	—	2	1	no	—
59	C₂×C₄×C₃₁	—	2	1	no	—
60	C₂×C₄×C₃₁	—	4	2	no	—
61	C₂×C₄×C₃₁	—	4	4	no	—
62	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
65	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
66	C₈×C₃₁	—	4	4	no	—
67	C₈×C₃₁	#106	1	2	yes	—
68	C₈×C₃₁	#106	1	1	yes	characteristic
69	C₈×C₃₁	#106	1	1	yes	characteristic
70	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₃₁	#117	1	1	yes	characteristic
72	C₈×C₃₁	C₄×Q₈	1	2	yes	—
73	#2155	C₂×C₃₁	1	2	yes	—
74	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
75	#1772	C₂×C₃₁	1	2	yes	—
76	#1385	C₂×C₃₁	1	1	yes	characteristic
77	#1384	C₂×C₃₁	1	1	yes	characteristic
78	C₂²×C₃₁	—	4	1	no	—
79	C₂²×C₃₁	#427	1	1	yes	characteristic
80	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
81	C₄×C₃₁	—	2	2	no	—
82	C₄×C₃₁	—	4	1	no	—
83	C₄×C₃₁	—	8	4	no	—
84	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
86	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
87	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
88	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
89	C₂×#108	C₂²×C₃₁	1	2	yes	—
90	#427	C₂²×C₃₁	1	2	yes	—
91	#405	—	2	2	no	—
92	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
93	#365	C₄×C₃₁	1	8	yes	complemented
94	#346	C₄×C₃₁	1	2	yes	complemented
95	C₄×C₁₆	C₄×C₃₁	1	2	yes	complemented
96	#339	—	2	2	no	—
97	C₂×C₃₁	—	2	1	no	—
98	C₂×C₃₁	—	2	2	no	—
99	C₂×C₃₁	C₄×#106	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
103	#117	—	2	2	no	—
104	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
105	M₃₂	—	2	2	no	—
106	M₃₂	C₂×C₄×C₃₁	1	4	yes	—
107	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
108	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
109	#108	—	2	2	no	—
110	#108	C₂×C₄×C₃₁	1	4	yes	—
111	#107	—	2	2	no	—
112	#106	C₂×C₄×C₃₁	1	4	yes	—
113	#97	—	4	2	no	—
114	C₄×C₈	—	4	2	no	—
115	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
116	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
119	C₃₁	#12668	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
126	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
127	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
128	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
130	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
131	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
132	C₂×C₈	#41×C₃₁	1	2	yes	—
133	C₄⋊C₄	—	2	4	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	4	2	no	—
137	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄²×C₃₁	1	4	yes	—
141	C₂³	C₃₁×#135	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
148	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
149	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	C₃₁×#106	1	2	yes	—
152	C₈	C₃₁×#106	1	1	yes	characteristic
153	C₈	C₃₁×#106	1	1	yes	characteristic
154	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
155	C₈	C₃₁×#117	1	1	yes	characteristic
156	C₈	C₄×Q₈×C₃₁	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	C₃₁×#427	1	1	yes	characteristic
159	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	8	4	no	—
163	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
164	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
165	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×C₃₁×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₃₁	#106
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	C₄×#106
prime radical	C₂³×C₃₁	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ 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×#365	C₄	semidirect
13. C₃₁×#346	C₄	semidirect
14. C₄×C₁₆×C₃₁	C₄	semidirect
56. #12668	C₃₁	direct (N × H)
93. #365	C₄×C₃₁	semidirect
94. #346	C₄×C₃₁	semidirect
95. C₄×C₁₆	C₄×C₃₁	semidirect
119. C₃₁	#12668	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×#2155	2
C₄×C₃₁×M₃₂	2
C₃₁×#1772	2
C₃₁×#1385	2
C₃₁×#1384	2
#12668	31

Subgroups — C31×#12668 #21469427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12668 #21469427