Subgroups — C₃₁×#3263 #16843196

Order 3968.

346 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	C₃₁×#542	C₂	1	2	yes	maximal, complemented
2	C₃₁×#476	C₂	1	2	yes	maximal, complemented
3	C₃₁×#475	C₂	1	2	yes	maximal, complemented
4	C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#469	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#465	C₂	1	2	yes	maximal, complemented
7	C₃₁×#464	C₂	1	2	yes	maximal, complemented
8	C₃₁×#463	C₂	1	2	yes	maximal, complemented
9	C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
12	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₃₁×#126	—	2	2	no	—
17	C₃₁×#124	—	2	2	no	—
18	C₃₁×#124	C₂²	1	2	yes	—
19	C₃₁×#123	—	2	2	no	—
20	C₃₁×C₂²⋊Q₈	—	2	2	no	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃₁×#121	C₂²	1	2	yes	—
23	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₃₁×#106	C₂²	1	2	yes	—
27	C₃₁×#106	C₂²	1	1	yes	characteristic
28	C₃₁×#106	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
31	C₃₁×#103	C₂²	1	2	yes	—
32	C₃₁×#103	C₂²	1	2	yes	—
33	C₃₁×#102	C₂²	1	2	yes	—
34	C₃₁×#102	C₂²	1	2	yes	—
35	C₂×Q₈×C₃₁	—	2	2	no	—
36	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂×D₄×C₃₁	—	2	2	no	—
38	C₂×D₄×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₁	—	2	2	no	—
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	Q₁₆×C₃₁	—	4	1	no	—
42	SD₁₆×C₃₁	—	4	2	no	—
43	D₈×C₃₁	—	4	1	no	—
44	M₁₆×C₃₁	—	4	1	no	—
45	C₂×C₈×C₃₁	—	4	1	no	—
46	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
53	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₄²×C₃₁	—	2	2	no	—
63	C₄²×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₃₁	—	2	2	no	—
65	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₃₁	—	2	2	no	—
67	Q₈×C₃₁	—	2	2	no	—
68	Q₈×C₃₁	—	8	2	no	—
69	D₄×C₃₁	—	2	2	no	—
70	D₄×C₃₁	—	2	2	no	—
71	D₄×C₃₁	—	8	2	no	—
72	C₂×C₄×C₃₁	—	2	1	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	8	2	no	—
78	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₈×C₃₁	—	2	2	no	—
83	C₈×C₃₁	—	2	1	no	—
84	C₈×C₃₁	—	2	1	no	—
85	#3263	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₃₁	—	2	1	no	—
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	—	4	2	no	—
89	C₂²×C₃₁	—	4	2	no	—
90	C₂²×C₃₁	—	4	2	no	—
91	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	—	2	2	no	—
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	4	2	no	—
96	C₄×C₃₁	—	4	2	no	—
97	C₄×C₃₁	—	4	2	no	—
98	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
99	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
100	#542	C₂×C₃₁	1	2	yes	complemented
101	#476	C₂×C₃₁	1	2	yes	complemented
102	#475	C₂×C₃₁	1	2	yes	complemented
103	#470	C₂×C₃₁	1	1	yes	characteristic, complemented
104	#469	C₂×C₃₁	1	1	yes	characteristic
105	#465	C₂×C₃₁	1	2	yes	complemented
106	#464	C₂×C₃₁	1	2	yes	complemented
107	#463	C₂×C₃₁	1	2	yes	complemented
108	#424	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂×C₃₁	—	4	1	no	—
110	C₂×C₃₁	—	8	2	no	—
111	C₂×C₃₁	#537	1	1	yes	characteristic
112	C₂×C₃₁	#577	1	2	yes	—
113	C₂×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
114	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
115	C₂×D₈	C₂²×C₃₁	1	1	yes	characteristic
116	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
118	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
119	#126	—	2	2	no	—
120	#124	—	2	2	no	—
121	#124	C₂²×C₃₁	1	2	yes	—
122	#123	—	2	2	no	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
125	#121	C₂²×C₃₁	1	2	yes	—
126	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
127	C₄×D₄	C₂²×C₃₁	1	2	yes	—
128	#117	C₂²×C₃₁	1	1	yes	characteristic
129	#106	C₂²×C₃₁	1	2	yes	—
130	#106	C₂²×C₃₁	1	1	yes	characteristic
131	#106	C₂²×C₃₁	1	1	yes	characteristic
132	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
133	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
134	#103	C₂²×C₃₁	1	2	yes	—
135	#103	C₂²×C₃₁	1	2	yes	—
136	#102	C₂²×C₃₁	1	2	yes	—
137	#102	C₂²×C₃₁	1	2	yes	—
138	C₃₁	#3263	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	C₂³×C₃₁	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	4	2	no	—
147	D₈	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂³×C₃₁	1	2	yes	—
152	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	D₄×C₃₁	1	2	yes	—
157	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
158	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
165	C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
166	C₄²	—	2	2	no	—
167	C₄²	C₂³×C₃₁	1	2	yes	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	8	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	8	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
184	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
202	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	C₃₁×#537	1	1	yes	characteristic
206	C₂	C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#542	C₂	semidirect
2. C₃₁×#476	C₂	semidirect
3. C₃₁×#475	C₂	semidirect
4. C₃₁×#470	C₂	semidirect
6. C₃₁×#465	C₂	semidirect
7. C₃₁×#464	C₂	semidirect
8. C₃₁×#463	C₂	semidirect
9. C₃₁×#424	C₂	semidirect
85. #3263	C₃₁	direct (N × H)
100. #542	C₂×C₃₁	semidirect
101. #476	C₂×C₃₁	semidirect
102. #475	C₂×C₃₁	semidirect
103. #470	C₂×C₃₁	semidirect
105. #465	C₂×C₃₁	semidirect
106. #464	C₂×C₃₁	semidirect
107. #463	C₂×C₃₁	semidirect
108. #424	C₂×C₃₁	semidirect
138. C₃₁	#3263	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×#542	2
C₃₁×#476	2
C₃₁×#475	2
C₃₁×#470	2
C₃₁×#469	2
C₃₁×#465	2
C₃₁×#464	2
C₃₁×#463	2
C₃₁×#424	2
#3263	31

Subgroups — C31×#3263 #16843196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3263 #16843196