Subgroups — C₃₁×#33963 #21490722

Order 7936.

532 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3396	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2249	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2210	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2207	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₃₁×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₃₁×#559	—	2	2	no	—
10	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
12	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
13	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
14	C₃₁×#495	C₂²	1	4	yes	—
15	C₃₁×#493	C₂²	1	2	yes	—
16	C₃₁×#463	—	2	4	no	—
17	C₃₁×#460	—	2	2	no	—
18	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
20	C₄×D₈×C₃₁	C₂²	1	4	yes	—
21	C₃₁×#366	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₃₁×#358	C₂²	1	2	yes	—
25	C₃₁×#357	C₂²	1	4	yes	—
26	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
27	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
29	C₂×SD₁₆×C₃₁	—	2	4	no	—
30	C₂×D₈×C₃₁	—	2	2	no	—
31	C₂×D₈×C₃₁	C₂³	1	4	yes	—
32	C₃₁×#127	—	2	2	no	—
33	C₃₁×#124	—	2	4	no	—
34	C₃₁×#124	—	4	2	no	—
35	C₃₁×#123	—	4	4	no	—
36	C₄×Q₈×C₃₁	—	2	2	no	—
37	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₄×D₄×C₃₁	—	2	4	no	—
39	C₄×D₄×C₃₁	—	4	2	no	—
40	C₃₁×Q₃₂	—	2	2	no	—
41	C₃₁×SD₃₂	—	4	4	no	—
42	C₃₁×D₁₆	—	2	4	no	—
43	C₃₁×D₁₆	—	4	2	no	—
44	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#107	C₂³	1	2	yes	—
48	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
50	C₃₁×#103	—	2	2	no	—
51	C₃₁×#102	—	2	4	no	—
52	C₄×C₈×C₃₁	C₂³	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₃₁	—	2	2	no	—
55	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
56	C₂×D₄×C₃₁	—	2	4	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	Q₁₆×C₃₁	—	2	2	no	—
61	Q₁₆×C₃₁	—	2	2	no	—
62	SD₁₆×C₃₁	—	8	4	no	—
63	D₈×C₃₁	—	2	4	no	—
64	D₈×C₃₁	—	2	4	no	—
65	D₈×C₃₁	—	8	2	no	—
66	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₄⋊C₄×C₃₁	—	4	2	no	—
71	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
73	C₂²⋊C₄×C₃₁	—	4	4	no	—
74	C₂²⋊C₄×C₃₁	—	4	4	no	—
75	C₂²⋊C₄×C₃₁	—	4	4	no	—
76	C₄²×C₃₁	—	4	2	no	—
77	C₄²×C₃₁	C₂×D₄	1	2	yes	—
78	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₁₆×C₃₁	—	2	1	no	—
80	C₁₆×C₃₁	—	2	2	no	—
81	C₁₆×C₃₁	#41	1	2	yes	—
82	#33963	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	4	no	—
84	Q₈×C₃₁	—	2	2	no	—
85	Q₈×C₃₁	—	4	2	no	—
86	Q₈×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	4	4	no	—
88	D₄×C₃₁	—	4	4	no	—
89	D₄×C₃₁	—	8	4	no	—
90	D₄×C₃₁	—	16	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	16	4	no	—
94	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	2	no	—
101	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
102	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
103	#3396	C₂×C₃₁	1	2	yes	complemented
104	#2249	C₂×C₃₁	1	2	yes	complemented
105	#2246	C₂×C₃₁	1	1	yes	characteristic
106	#2210	C₂×C₃₁	1	4	yes	complemented
107	#2207	C₂×C₃₁	1	2	yes	complemented
108	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
110	C₄×D₁₆	C₂×C₃₁	1	2	yes	complemented
111	C₂²×C₃₁	—	8	4	no	—
112	C₂²×C₃₁	—	8	4	no	—
113	C₂²×C₃₁	—	8	4	no	—
114	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	—	2	2	no	—
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	8	2	no	—
119	C₄×C₃₁	—	8	2	no	—
120	C₄×C₃₁	#508	1	2	yes	—
121	C₄×C₃₁	#550	1	1	yes	characteristic
122	C₄×C₃₁	#576	1	1	yes	characteristic
123	#559	—	2	2	no	—
124	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
126	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
127	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
128	#495	C₂²×C₃₁	1	4	yes	—
129	#493	C₂²×C₃₁	1	2	yes	—
130	#463	—	2	4	no	—
131	#460	—	2	2	no	—
132	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
133	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
134	C₄×D₈	C₂²×C₃₁	1	4	yes	—
135	#366	C₂²×C₃₁	1	1	yes	characteristic
136	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
137	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
138	#358	C₂²×C₃₁	1	2	yes	—
139	#357	C₂²×C₃₁	1	4	yes	—
140	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
141	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₃₁	—	16	4	no	—
143	C₂×C₃₁	#3391	1	1	yes	characteristic
144	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
145	C₂×C₃₁	#3415	1	1	yes	characteristic
146	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	C₂³×C₃₁	1	4	yes	—
150	#127	—	2	2	no	—
151	#124	—	2	4	no	—
152	#124	—	4	2	no	—
153	#123	—	4	4	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
156	C₄×D₄	—	2	4	no	—
157	C₄×D₄	—	4	2	no	—
158	Q₃₂	—	2	2	no	—
159	SD₃₂	—	4	4	no	—
160	D₁₆	—	2	4	no	—
161	D₁₆	—	4	2	no	—
162	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
163	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
164	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
165	#107	C₂³×C₃₁	1	2	yes	—
166	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
168	#103	—	2	2	no	—
169	#102	—	2	4	no	—
170	C₄×C₈	C₂³×C₃₁	1	2	yes	—
171	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	C₃₁	#33963	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂²×C₄	—	4	4	no	—
179	Q₁₆	—	2	2	no	—
180	Q₁₆	—	2	2	no	—
181	SD₁₆	—	8	4	no	—
182	D₈	—	2	4	no	—
183	D₈	—	2	4	no	—
184	D₈	—	8	2	no	—
185	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
186	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
187	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
191	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	—	4	4	no	—
195	C₄²	—	4	2	no	—
196	C₄²	C₂×D₄×C₃₁	1	2	yes	—
197	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₁₆	—	2	1	no	—
199	C₁₆	—	2	2	no	—
200	C₁₆	#41×C₃₁	1	2	yes	—
201	C₂³	—	4	4	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	4	no	—
208	D₄	—	16	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	16	4	no	—
212	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
213	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
214	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
220	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	4	no	—
224	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	8	2	no	—
229	C₄	—	8	2	no	—
230	C₄	C₃₁×#508	1	2	yes	—
231	C₄	C₃₁×#550	1	1	yes	characteristic
232	C₄	C₃₁×#576	1	1	yes	characteristic
233	C₂	—	16	4	no	—
234	C₂	C₃₁×#3391	1	1	yes	characteristic
235	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
236	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#2246	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3396	C₂	semidirect
2. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2210	C₂	semidirect
5. C₃₁×#2207	C₂	semidirect
6. Q₈×C₁₆×C₃₁	C₂	semidirect
7. C₄×C₃₁×Q₃₂	C₂	semidirect
8. C₄×C₃₁×D₁₆	C₂	semidirect
82. #33963	C₃₁	direct (N × H)
103. #3396	C₂×C₃₁	semidirect
104. #2249	C₂×C₃₁	semidirect
106. #2210	C₂×C₃₁	semidirect
107. #2207	C₂×C₃₁	semidirect
108. Q₈×C₁₆	C₂×C₃₁	semidirect
109. C₄×Q₃₂	C₂×C₃₁	semidirect
110. C₄×D₁₆	C₂×C₃₁	semidirect
172. C₃₁	#33963	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×#3396	2
C₃₁×#2249	2
C₃₁×#2246	2
C₃₁×#2210	2
C₃₁×#2207	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×Q₃₂	2
C₄×C₃₁×D₁₆	2
#33963	31

Subgroups — C31×#33963 #21490722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33963 #21490722