Subgroups — C₃₁×#33458 #21490217

Order 7936.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2950	C₂	1	2	yes	maximal
3	C₃₁×#2176	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₁×Q₃₂	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2141	C₂	1	2	yes	maximal
7	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#519	—	2	2	no	—
10	C₂×C₃₁×Q₃₂	C₄	1	8	yes	complemented
11	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#441	—	2	4	no	—
16	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
17	C₃₁×#427	C₂²	1	2	yes	—
18	C₃₁×#419	—	2	2	no	—
19	C₃₁×#405	C₂²	1	1	yes	characteristic
20	C₃₁×#367	C₂²	1	2	yes	—
21	C₃₁×#366	C₂²	1	2	yes	—
22	C₃₁×#358	C₂²	1	8	yes	—
23	C₃₁×#346	C₂²	1	2	yes	—
24	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂×Q₁₆×C₃₁	—	2	2	no	—
27	C₂×Q₁₆×C₃₁	—	2	4	no	—
28	C₂×Q₁₆×C₃₁	C₂×C₄	1	8	yes	—
29	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
30	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₃₁	—	2	8	no	—
33	C₄×Q₈×C₃₁	—	4	4	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	—	4	4	no	—
36	C₃₁×#117	D₄	1	1	yes	characteristic
37	C₃₁×Q₃₂	C₂×C₄	1	8	yes	—
38	C₃₁×Q₃₂	C₂×C₄	1	8	yes	—
39	C₃₁×M₃₂	—	2	2	no	—
40	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#107	C₂³	1	4	yes	—
45	C₃₁×#106	—	2	2	no	—
46	C₃₁×#103	—	2	8	no	—
47	C₃₁×#97	D₄	1	2	yes	—
48	C₄×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂×Q₈×C₃₁	—	2	8	no	—
50	C₂×Q₈×C₃₁	—	2	4	no	—
51	C₂×Q₈×C₃₁	—	4	2	no	—
52	C₂×Q₈×C₃₁	—	4	4	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₃₁	—	2	8	no	—
56	Q₁₆×C₃₁	—	2	4	no	—
57	Q₁₆×C₃₁	C₂²×C₄	1	8	yes	—
58	M₁₆×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	#41	1	2	yes	—
63	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	4	8	no	—
66	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
67	C₂²⋊C₄×C₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
69	C₄²×C₃₁	—	4	2	no	—
70	C₄²×C₃₁	—	4	8	no	—
71	C₄²×C₃₁	C₂×D₄	1	2	yes	—
72	C₁₆×C₃₁	—	2	2	no	—
73	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
74	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
75	#33458	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
77	Q₈×C₃₁	—	2	8	no	—
78	Q₈×C₃₁	—	4	4	no	—
79	Q₈×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	8	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
85	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#135	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	C₄×D₄	1	2	yes	—
90	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
92	C₂²×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
93	#2950	C₂×C₃₁	1	2	yes	—
94	#2176	C₂×C₃₁	1	4	yes	complemented
95	C₄×Q₃₂	C₂×C₃₁	1	4	yes	complemented
96	#2158	C₂×C₃₁	1	1	yes	characteristic
97	#2141	C₂×C₃₁	1	2	yes	—
98	#2108	C₂×C₃₁	1	1	yes	characteristic
99	C₂²×C₃₁	—	2	2	no	—
100	C₂²×C₃₁	C₄×D₈	1	2	yes	—
101	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₃₁	—	2	4	no	—
103	C₄×C₃₁	—	4	8	no	—
104	C₄×C₃₁	—	8	4	no	—
105	C₄×C₃₁	C₄×D₈	1	2	yes	—
106	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
107	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
108	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
109	#519	—	2	2	no	—
110	C₂×Q₃₂	C₄×C₃₁	1	8	yes	complemented
111	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
112	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
113	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
114	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
115	#441	—	2	4	no	—
116	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
117	#427	C₂²×C₃₁	1	2	yes	—
118	#419	—	2	2	no	—
119	#405	C₂²×C₃₁	1	1	yes	characteristic
120	#367	C₂²×C₃₁	1	2	yes	—
121	#366	C₂²×C₃₁	1	2	yes	—
122	#358	C₂²×C₃₁	1	8	yes	—
123	#346	C₂²×C₃₁	1	2	yes	—
124	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
125	C₂×C₃₁	—	2	2	no	—
126	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
127	C₂×C₃₁	#3417	1	2	yes	—
128	C₂²×Q₈	—	2	2	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	2	4	no	—
131	C₂×Q₁₆	C₂×C₄×C₃₁	1	8	yes	—
132	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
133	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
135	C₄×Q₈	—	2	8	no	—
136	C₄×Q₈	—	4	4	no	—
137	#117	—	2	2	no	—
138	#117	—	4	4	no	—
139	#117	D₄×C₃₁	1	1	yes	characteristic
140	Q₃₂	C₂×C₄×C₃₁	1	8	yes	—
141	Q₃₂	C₂×C₄×C₃₁	1	8	yes	—
142	M₃₂	—	2	2	no	—
143	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
144	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
145	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
146	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
147	#107	C₂³×C₃₁	1	4	yes	—
148	#106	—	2	2	no	—
149	#103	—	2	8	no	—
150	#97	D₄×C₃₁	1	2	yes	—
151	C₄×C₈	C₂³×C₃₁	1	2	yes	—
152	C₃₁	#33458	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×C₄×C₃₁	1	8	yes	—
162	M₁₆	—	2	2	no	—
163	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
164	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
166	C₂×C₈	#41×C₃₁	1	2	yes	—
167	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	8	no	—
170	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
173	C₄²	—	4	2	no	—
174	C₄²	—	4	8	no	—
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
178	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
179	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	8	no	—
182	Q₈	—	4	4	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
187	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
188	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₃₁×#135	1	2	yes	—
190	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	C₄×D₄×C₃₁	1	2	yes	—
193	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
194	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	C₄×D₈×C₃₁	1	2	yes	—
197	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	4	8	no	—
200	C₄	—	8	4	no	—
201	C₄	C₄×D₈×C₃₁	1	2	yes	—
202	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
203	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
206	C₂	C₃₁×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2176	C₂	semidirect
4. C₄×C₃₁×Q₃₂	C₂	semidirect
10. C₂×C₃₁×Q₃₂	C₄	semidirect
75. #33458	C₃₁	direct (N × H)
94. #2176	C₂×C₃₁	semidirect
95. C₄×Q₃₂	C₂×C₃₁	semidirect
110. C₂×Q₃₂	C₄×C₃₁	semidirect
152. C₃₁	#33458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×Q₃₂	2
C₃₁×#2950	2
C₃₁×#2176	2
C₄×C₃₁×Q₃₂	2
C₃₁×#2158	2
C₃₁×#2141	2
C₃₁×#2108	2
#33458	31

Subgroups — C31×#33458 #21490217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33458 #21490217