Subgroups — C₃₁×#33398 #21490157

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#504	C₂	1	2	yes	maximal
2	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2151	C₂	1	6	yes	maximal, complemented
4	C₃₁×#2150	C₂	1	3	yes	maximal, complemented
5	C₂×C₃₁×C₄⋊C₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
8	C₄×#41×C₃₁	C₄	1	1	yes	characteristic
9	C₃₁×#504	—	2	8	no	—
10	C₂×C₃₁×M₃₂	C₂²	1	6	yes	—
11	C₂²×C₁₆×C₃₁	C₂²	1	6	yes	—
12	C₃₁×#405	C₄	1	3	yes	—
13	C₃₁×#405	C₂²	1	3	yes	—
14	C₄×M₁₆×C₃₁	C₄	1	3	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
16	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊C₁₆	C₂²	1	3	yes	—
18	C₃₁×C₄⋊C₁₆	C₂²	1	6	yes	—
19	C₃₁×C₄⋊C₁₆	C₂²	1	6	yes	—
20	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₃₁×#131	—	2	4	no	—
22	C₃₁×#131	D₄	1	4	yes	—
23	C₃₁×#131	Q₈	1	4	yes	—
24	C₂×M₁₆×C₃₁	C₂×C₄	1	3	yes	—
25	C₂×M₁₆×C₃₁	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
28	C₄×Q₈×C₃₁	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₃₁	C₈	1	1	yes	characteristic
30	C₄×D₄×C₃₁	C₈	1	3	yes	—
31	C₄×D₄×C₃₁	C₈	1	3	yes	—
32	C₃₁×#117	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
34	C₃₁×M₃₂	—	2	12	no	—
35	C₂×C₁₆×C₃₁	—	2	12	no	—
36	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂×C₁₆×C₃₁	C₂³	1	6	yes	—
38	C₃₁×#97	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₃₁	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₃₁	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₄×C₈×C₃₁	C₂³	1	3	yes	—
43	#41×C₃₁	C₄⋊C₄	1	4	yes	—
44	#41×C₃₁	M₁₆	1	4	yes	—
45	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₃₁	C₂×C₈	1	3	yes	—
47	C₂²×C₄×C₃₁	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₃₁	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
50	M₁₆×C₃₁	—	2	6	no	—
51	M₁₆×C₃₁	C₄⋊C₄	1	12	yes	—
52	C₂×C₈×C₃₁	—	2	6	no	—
53	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
57	C₂×C₈×C₃₁	C₂×Q₈	1	6	yes	—
58	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₃₁	C₂×C₈	1	3	yes	—
60	C₄⋊C₄×C₃₁	C₂×C₈	1	3	yes	—
61	C₂²⋊C₄×C₃₁	C₂×C₈	1	6	yes	—
62	C₄²×C₃₁	C₂×C₈	1	3	yes	—
63	C₄²×C₃₁	C₂×C₈	1	3	yes	—
64	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
66	C₁₆×C₃₁	—	2	2	no	—
67	C₁₆×C₃₁	—	2	6	no	—
68	#33398	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	C₂²×C₈	1	3	yes	—
70	Q₈×C₃₁	C₄⋊C₈	1	4	yes	—
71	D₄×C₃₁	C₄⋊C₈	1	12	yes	—
72	C₂×C₄×C₃₁	—	2	6	no	—
73	C₂×C₄×C₃₁	—	4	3	no	—
74	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
75	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
78	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
79	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×M₁₆	1	6	yes	—
81	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₃₁	—	2	1	no	—
83	C₈×C₃₁	—	2	3	no	—
84	C₈×C₃₁	C₂×C₄⋊C₄	1	6	yes	—
85	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
87	C₂×#504	C₂×C₃₁	1	2	yes	—
88	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
89	#2151	C₂×C₃₁	1	6	yes	complemented
90	#2150	C₂×C₃₁	1	3	yes	complemented
91	C₂×C₄⋊C₁₆	C₂×C₃₁	1	3	yes	complemented
92	C₂²×C₃₁	—	2	3	no	—
93	C₂²×C₃₁	—	2	3	no	—
94	C₂²×C₃₁	C₂×C₄⋊C₈	1	6	yes	—
95	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₃₁	—	2	3	no	—
97	C₄×C₃₁	—	2	1	no	—
98	C₄×C₃₁	—	2	3	no	—
99	C₄×C₃₁	C₂×C₄⋊C₈	1	6	yes	—
100	C₄×C₃₁	#504	1	2	yes	complemented
101	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
104	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
105	C₄×#41	C₄×C₃₁	1	1	yes	characteristic
106	#504	—	2	8	no	—
107	C₂×M₃₂	C₂²×C₃₁	1	6	yes	—
108	C₂²×C₁₆	C₂²×C₃₁	1	6	yes	—
109	#405	C₄×C₃₁	1	3	yes	—
110	#405	C₂²×C₃₁	1	3	yes	—
111	C₄×M₁₆	C₄×C₃₁	1	3	yes	—
112	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
113	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₃₁	1	3	yes	—
115	C₄⋊C₁₆	C₂²×C₃₁	1	6	yes	—
116	C₄⋊C₁₆	C₂²×C₃₁	1	6	yes	—
117	C₂×C₃₁	—	2	6	no	—
118	C₂×C₃₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
119	C₂×C₃₁	C₂×#504	1	1	yes	characteristic
120	C₂×C₃₁	#3407	1	1	yes	characteristic
121	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
122	#131	—	2	4	no	—
123	#131	D₄×C₃₁	1	4	yes	—
124	#131	Q₈×C₃₁	1	4	yes	—
125	C₂×M₁₆	C₂×C₄×C₃₁	1	3	yes	—
126	C₂×M₁₆	C₂×C₄×C₃₁	1	3	yes	—
127	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
128	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
129	C₄×Q₈	C₈×C₃₁	1	1	yes	characteristic
130	C₄×Q₈	C₈×C₃₁	1	1	yes	characteristic
131	C₄×D₄	C₈×C₃₁	1	3	yes	—
132	C₄×D₄	C₈×C₃₁	1	3	yes	—
133	#117	C₂×C₄×C₃₁	1	3	yes	—
134	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
135	M₃₂	—	2	12	no	—
136	C₂×C₁₆	—	2	12	no	—
137	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
138	C₂×C₁₆	C₂³×C₃₁	1	6	yes	—
139	#97	C₂×C₄×C₃₁	1	6	yes	—
140	C₄×C₈	C₂×C₄×C₃₁	1	3	yes	—
141	C₄×C₈	C₂×C₄×C₃₁	1	3	yes	—
142	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
143	C₄×C₈	C₂³×C₃₁	1	3	yes	—
144	C₃₁	#33398	1	1	yes	characteristic, direct product factor, Sylow
145	#41	C₄⋊C₄×C₃₁	1	4	yes	—
146	#41	M₁₆×C₃₁	1	4	yes	—
147	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
148	C₂×D₄	C₂×C₈×C₃₁	1	3	yes	—
149	C₂²×C₄	C₂×C₈×C₃₁	1	3	yes	—
150	C₂²×C₄	C₂×C₈×C₃₁	1	3	yes	—
151	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
152	M₁₆	—	2	6	no	—
153	M₁₆	C₄⋊C₄×C₃₁	1	12	yes	—
154	C₂×C₈	—	2	6	no	—
155	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
157	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
158	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
159	C₂×C₈	C₂×Q₈×C₃₁	1	6	yes	—
160	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₈×C₃₁	1	3	yes	—
162	C₄⋊C₄	C₂×C₈×C₃₁	1	3	yes	—
163	C₂²⋊C₄	C₂×C₈×C₃₁	1	6	yes	—
164	C₄²	C₂×C₈×C₃₁	1	3	yes	—
165	C₄²	C₂×C₈×C₃₁	1	3	yes	—
166	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
167	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
168	C₁₆	—	2	2	no	—
169	C₁₆	—	2	6	no	—
170	C₂³	C₂²×C₈×C₃₁	1	3	yes	—
171	Q₈	C₃₁×C₄⋊C₈	1	4	yes	—
172	D₄	C₃₁×C₄⋊C₈	1	12	yes	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	6	yes	—
176	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
178	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
179	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
180	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
181	C₂×C₄	C₂×M₁₆×C₃₁	1	6	yes	—
182	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	C₂×C₄⋊C₄×C₃₁	1	6	yes	—
186	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
187	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₃₁×C₄⋊C₈	1	6	yes	—
191	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₃₁×C₄⋊C₈	1	6	yes	—
196	C₄	C₃₁×#504	1	2	yes	complemented
197	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
198	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₃₁×C₄⋊C₈	1	1	yes	characteristic
201	C₂	C₂×C₃₁×#504	1	1	yes	characteristic
202	C₂	C₃₁×#3407	1	1	yes	characteristic

	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	C₂×#41×C₃₁	C₂×C₄
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
5. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
68. #33398	C₃₁	direct (N × H)
89. #2151	C₂×C₃₁	semidirect
90. #2150	C₂×C₃₁	semidirect
91. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
100. C₄×C₃₁	#504	semidirect
144. C₃₁	#33398	direct (N × H)
196. C₄	C₃₁×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#504	2
C₄×C₃₁×#131	2
C₃₁×#2151	2
C₃₁×#2150	2
C₂×C₃₁×C₄⋊C₁₆	2
#33398	31

Subgroups — C31×#33398 #21490157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33398 #21490157