Subgroups — C₃₁×#33397 #21490156

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2151	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2150	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
8	C₂×C₃₁×M₃₂	—	2	8	no	—
9	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
10	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	—
12	C₃₁×#405	C₄	1	4	yes	—
13	C₃₁×#405	C₂²	1	4	yes	—
14	C₂×C₃₁×#97	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×C₄⋊C₁₆	C₂²	1	16	yes	complemented
17	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	—	2	4	no	—
19	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₃₁	D₄	1	4	yes	—
24	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
27	C₃₁×#117	C₈	1	4	yes	—
28	C₃₁×#117	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
30	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₃₁×M₃₂	—	2	8	no	—
33	C₃₁×M₃₂	—	2	8	no	—
34	C₂×C₁₆×C₃₁	—	2	8	no	—
35	C₂×C₁₆×C₃₁	C₂³	1	8	yes	—
36	C₃₁×#97	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₃₁	C₂³	1	4	yes	—
39	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
45	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	M₁₆×C₃₁	—	2	8	no	—
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
51	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
55	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
60	C₂²⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
61	C₄²×C₃₁	C₂×C₈	1	4	yes	—
62	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
63	C₁₆×C₃₁	—	2	8	no	—
64	#33397	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	—	4	2	no	—
66	C₂³×C₃₁	C₄⋊C₈	1	4	yes	—
67	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
68	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
72	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₃₁	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
79	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₃₁	—	2	4	no	—
81	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
83	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×M₃₂	C₂×C₃₁	1	2	yes	—
86	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
87	#2151	C₂×C₃₁	1	8	yes	complemented
88	#2150	C₂×C₃₁	1	4	yes	complemented
89	C₂²×C₃₁	—	2	2	no	—
90	C₂²×C₃₁	—	2	4	no	—
91	C₂²×C₃₁	—	4	4	no	—
92	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
93	C₂²×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
94	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
95	C₄×C₃₁	—	2	8	no	—
96	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
97	C₄×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
98	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
100	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
101	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
102	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
103	C₂×M₃₂	—	2	8	no	—
104	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
105	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
106	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	—
107	#405	C₄×C₃₁	1	4	yes	—
108	#405	C₂²×C₃₁	1	4	yes	—
109	C₂×#97	C₄×C₃₁	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
111	C₄⋊C₁₆	C₂²×C₃₁	1	16	yes	complemented
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	—	2	4	no	—
114	C₂×C₃₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₃₁	#3407	1	2	yes	—
116	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
117	C₂×M₁₆	—	2	4	no	—
118	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
119	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
120	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
121	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
122	C₂²×C₈	D₄×C₃₁	1	4	yes	—
123	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
124	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
126	#117	C₈×C₃₁	1	4	yes	—
127	#117	C₂×C₄×C₃₁	1	4	yes	—
128	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
129	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
130	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
131	M₃₂	—	2	8	no	—
132	M₃₂	—	2	8	no	—
133	C₂×C₁₆	—	2	8	no	—
134	C₂×C₁₆	C₂³×C₃₁	1	8	yes	—
135	#97	C₂×C₄×C₃₁	1	8	yes	—
136	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
137	C₄×C₈	C₂³×C₃₁	1	4	yes	—
138	C₃₁	#33397	1	1	yes	characteristic, direct product factor, Sylow
139	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
141	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
142	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
144	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
147	M₁₆	—	2	8	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
150	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
151	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
152	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
153	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
154	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
155	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
156	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
157	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
158	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
159	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
160	C₂²⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
161	C₄²	C₂×C₈×C₃₁	1	4	yes	—
162	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
163	C₁₆	—	2	8	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₃₁×C₄⋊C₈	1	4	yes	—
166	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
167	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
170	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
172	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
173	C₂×C₄	C₂²×C₈×C₃₁	1	8	yes	—
174	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
176	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
177	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
178	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
181	C₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
182	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
183	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
188	C₂²	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
189	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
192	C₄	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
193	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
194	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂²×C₃₁×C₄⋊C₈	1	1	yes	characteristic
198	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	C₂⁴×C₃₁	C₂×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
3. C₃₁×#2151	C₂	semidirect
4. C₃₁×#2150	C₂	semidirect
16. C₃₁×C₄⋊C₁₆	C₂²	semidirect
64. #33397	C₃₁	direct (N × H)
87. #2151	C₂×C₃₁	semidirect
88. #2150	C₂×C₃₁	semidirect
111. C₄⋊C₁₆	C₂²×C₃₁	semidirect
138. C₃₁	#33397	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×M₃₂	2
C₂×C₃₁×#405	2
C₃₁×#2151	2
C₃₁×#2150	2
#33397	31

Subgroups — C31×#33397 #21490156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33397 #21490156