Subgroups — C₃₁×#33475 #21490234

Order 7936.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2183	C₂	1	8	yes	maximal, complemented
3	C₃₁×#2151	C₂	1	2	yes	maximal
4	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2108	C₂	1	2	yes	maximal
6	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₄	1	1	yes	characteristic
8	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#446	C₄	1	4	yes	—
12	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
13	C₃₁×#424	C₄	1	2	yes	—
14	C₃₁×#423	C₄	1	1	yes	characteristic
15	C₃₁×#405	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	—
18	C₃₁×C₄⋊C₁₆	C₂²	1	8	yes	—
19	C₃₁×#346	C₂²	1	4	yes	—
20	C₃₁×#346	C₂²	1	4	yes	—
21	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
22	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	C₈	1	2	yes	—
26	C₃₁×#125	C₈	1	2	yes	—
27	C₃₁×C₂²⋊Q₈	C₈	1	4	yes	—
28	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
29	C₃₁×#117	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₃₁×M₃₂	—	2	4	no	—
32	C₂×C₁₆×C₃₁	—	2	2	no	—
33	C₂×C₁₆×C₃₁	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
35	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
37	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
38	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
39	C₃₁×#97	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₃₁	C₂³	1	2	yes	—
42	C₄×C₈×C₃₁	C₂³	1	4	yes	—
43	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	M₁₆×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₃₁	#41	1	2	yes	—
53	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
55	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
56	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
58	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
61	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
62	C₁₆×C₃₁	—	2	2	no	—
63	C₁₆×C₃₁	—	2	4	no	—
64	C₁₆×C₃₁	C₂×Q₈	1	4	yes	—
65	#33475	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
67	Q₈×C₃₁	—	4	4	no	—
68	C₂×C₄×C₃₁	—	4	2	no	—
69	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₃₁	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	#131	1	2	yes	—
75	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
76	C₈×C₃₁	—	2	2	no	—
77	C₈×C₃₁	—	2	4	no	—
78	C₈×C₃₁	C₄×Q₈	1	2	yes	—
79	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
81	#2960	C₂×C₃₁	1	1	yes	characteristic
82	#2183	C₂×C₃₁	1	8	yes	complemented
83	#2151	C₂×C₃₁	1	2	yes	—
84	#2150	C₂×C₃₁	1	1	yes	characteristic
85	#2108	C₂×C₃₁	1	2	yes	—
86	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	C₈×Q₈	1	2	yes	—
89	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	2	2	no	—
92	C₄×C₃₁	—	2	8	no	—
93	C₄×C₃₁	C₈×Q₈	1	2	yes	—
94	C₄×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
96	#533	C₄×C₃₁	1	1	yes	characteristic
97	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
98	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
100	#446	C₄×C₃₁	1	4	yes	—
101	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
102	#424	C₄×C₃₁	1	2	yes	—
103	#423	C₄×C₃₁	1	1	yes	characteristic
104	#405	C₂²×C₃₁	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
106	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	—
107	C₄⋊C₁₆	C₂²×C₃₁	1	8	yes	—
108	#346	C₂²×C₃₁	1	4	yes	—
109	#346	C₂²×C₃₁	1	4	yes	—
110	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
111	C₂×C₃₁	—	2	2	no	—
112	C₂×C₃₁	C₂×C₈×Q₈	1	1	yes	characteristic
113	C₂×C₃₁	#3407	1	2	yes	—
114	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
115	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
116	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
117	C₄⋊Q₈	C₈×C₃₁	1	2	yes	—
118	#125	C₈×C₃₁	1	2	yes	—
119	C₂²⋊Q₈	C₈×C₃₁	1	4	yes	—
120	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
121	#117	C₂×C₄×C₃₁	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
123	M₃₂	—	2	4	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	Q₈×C₃₁	1	4	yes	complemented
126	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
127	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
128	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
129	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
130	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
131	#97	C₂×C₄×C₃₁	1	4	yes	—
132	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
133	C₄×C₈	C₂³×C₃₁	1	4	yes	—
134	C₄×C₈	C₂³×C₃₁	1	2	yes	—
135	C₃₁	#33475	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
137	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
138	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
142	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
143	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
144	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
145	C₂×C₈	#41×C₃₁	1	2	yes	—
146	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
148	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
149	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
150	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
151	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
152	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
153	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
154	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	2	4	no	—
157	C₁₆	C₂×Q₈×C₃₁	1	4	yes	—
158	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
162	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
163	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
164	C₂×C₄	C₂²×C₈×C₃₁	1	8	yes	—
165	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
166	C₂×C₄	C₃₁×#131	1	2	yes	—
167	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	C₄×Q₈×C₃₁	1	2	yes	—
171	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
172	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	C₈×Q₈×C₃₁	1	2	yes	—
175	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	C₈×Q₈×C₃₁	1	2	yes	—
180	C₄	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
181	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×C₈×Q₈×C₃₁	1	1	yes	characteristic
184	C₂	C₃₁×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2183	C₂	semidirect
33. C₂×C₁₆×C₃₁	Q₈	semidirect
65. #33475	C₃₁	direct (N × H)
82. #2183	C₂×C₃₁	semidirect
125. C₂×C₁₆	Q₈×C₃₁	semidirect
135. C₃₁	#33475	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×#2960	2
C₃₁×#2183	2
C₃₁×#2151	2
C₃₁×#2150	2
C₃₁×#2108	2
C₂×C₃₁×#346	2
#33475	31

Subgroups — C31×#33475 #21490234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33475 #21490234