Subgroups — C₃₁×#33960 #21490719

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3393	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2249	C₂	1	2	yes	maximal
3	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2211	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2208	C₂	1	2	yes	maximal, complemented
6	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₁×Q₃₂	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#556	—	2	2	no	—
10	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
13	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
14	C₃₁×#495	C₂²	1	4	yes	—
15	C₃₁×#494	C₂²	1	2	yes	—
16	C₃₁×#464	—	2	4	no	—
17	C₃₁×#461	—	2	2	no	—
18	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
20	C₄×D₈×C₃₁	C₂²	1	2	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	4	yes	—
25	C₃₁×#357	C₂²	1	2	yes	—
26	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₃₁	—	2	2	no	—
29	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
30	C₂×SD₁₆×C₃₁	—	2	4	no	—
31	C₂×D₈×C₃₁	C₂³	1	2	yes	—
32	C₃₁×C₄⋊Q₈	—	2	2	no	—
33	C₃₁×#126	—	4	4	no	—
34	C₃₁×#124	—	2	4	no	—
35	C₃₁×C₂²⋊Q₈	—	4	2	no	—
36	C₄×Q₈×C₃₁	—	2	4	no	—
37	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
38	C₄×D₄×C₃₁	—	2	2	no	—
39	C₃₁×#117	—	4	2	no	—
40	C₃₁×Q₃₂	—	2	4	no	—
41	C₃₁×Q₃₂	—	4	2	no	—
42	C₃₁×SD₃₂	—	4	4	no	—
43	C₃₁×D₁₆	—	2	2	no	—
44	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
45	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
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	4	no	—
51	C₃₁×#102	—	2	2	no	—
52	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂×Q₈×C₃₁	—	2	4	no	—
55	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	Q₁₆×C₃₁	—	2	4	no	—
59	Q₁₆×C₃₁	—	2	4	no	—
60	Q₁₆×C₃₁	—	8	2	no	—
61	SD₁₆×C₃₁	—	8	4	no	—
62	D₈×C₃₁	—	2	2	no	—
63	D₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄⋊C₄×C₃₁	—	4	2	no	—
69	C₄⋊C₄×C₃₁	—	4	4	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	2	no	—
75	C₂²⋊C₄×C₃₁	—	4	4	no	—
76	C₄²×C₃₁	—	4	4	no	—
77	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂×D₄	1	2	yes	—
79	C₁₆×C₃₁	—	2	2	no	—
80	C₁₆×C₃₁	—	2	1	no	—
81	C₁₆×C₃₁	#41	1	2	yes	—
82	#33960	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	2	no	—
84	Q₈×C₃₁	—	2	2	no	—
85	Q₈×C₃₁	—	4	4	no	—
86	Q₈×C₃₁	—	4	4	no	—
87	D₄×C₃₁	—	4	2	no	—
88	D₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	—	8	4	no	—
92	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
100	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
101	#3393	C₂×C₃₁	1	2	yes	complemented
102	#2249	C₂×C₃₁	1	2	yes	—
103	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
104	#2211	C₂×C₃₁	1	4	yes	complemented
105	#2208	C₂×C₃₁	1	2	yes	complemented
106	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
107	C₄×Q₃₂	C₂×C₃₁	1	2	yes	complemented
108	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic
109	C₂²×C₃₁	—	8	2	no	—
110	C₂²×C₃₁	—	8	2	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	—	8	2	no	—
118	C₄×C₃₁	#508	1	2	yes	—
119	C₄×C₃₁	#550	1	1	yes	characteristic
120	C₄×C₃₁	#576	1	1	yes	characteristic
121	#556	—	2	2	no	—
122	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
123	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
124	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
125	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
126	#495	C₂²×C₃₁	1	4	yes	—
127	#494	C₂²×C₃₁	1	2	yes	—
128	#464	—	2	4	no	—
129	#461	—	2	2	no	—
130	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
131	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
132	C₄×D₈	C₂²×C₃₁	1	2	yes	—
133	#366	C₂²×C₃₁	1	1	yes	characteristic
134	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
135	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
136	#358	C₂²×C₃₁	1	4	yes	—
137	#357	C₂²×C₃₁	1	2	yes	—
138	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
139	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
140	C₂×C₃₁	—	16	2	no	—
141	C₂×C₃₁	#3391	1	1	yes	characteristic
142	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
143	C₂×C₃₁	#3417	1	1	yes	characteristic
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
146	C₂×SD₁₆	—	2	4	no	—
147	C₂×D₈	C₂³×C₃₁	1	2	yes	—
148	C₄⋊Q₈	—	2	2	no	—
149	#126	—	4	4	no	—
150	#124	—	2	4	no	—
151	C₂²⋊Q₈	—	4	2	no	—
152	C₄×Q₈	—	2	4	no	—
153	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
154	C₄×D₄	—	2	2	no	—
155	#117	—	4	2	no	—
156	Q₃₂	—	2	4	no	—
157	Q₃₂	—	4	2	no	—
158	SD₃₂	—	4	4	no	—
159	D₁₆	—	2	2	no	—
160	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
162	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
163	#107	C₂³×C₃₁	1	2	yes	—
164	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
165	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
166	#103	—	2	4	no	—
167	#102	—	2	2	no	—
168	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₃₁	1	2	yes	—
170	C₃₁	#33960	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
173	C₂×D₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	8	2	no	—
178	SD₁₆	—	8	4	no	—
179	D₈	—	2	2	no	—
180	D₈	—	2	2	no	—
181	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
183	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
189	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
190	C₂²⋊C₄	—	4	4	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₄²	—	4	4	no	—
194	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₃₁	1	2	yes	—
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₃₁	1	2	yes	—
199	C₂³	—	4	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
211	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
216	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	—	8	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	8	2	no	—
226	C₄	C₃₁×#508	1	2	yes	—
227	C₄	C₃₁×#550	1	1	yes	characteristic
228	C₄	C₃₁×#576	1	1	yes	characteristic
229	C₂	—	16	2	no	—
230	C₂	C₃₁×#3391	1	1	yes	characteristic
231	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
232	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×D₁₆	C₂²
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×#3393	C₂	semidirect
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2211	C₂	semidirect
5. C₃₁×#2208	C₂	semidirect
6. Q₈×C₁₆×C₃₁	C₂	semidirect
7. C₄×C₃₁×Q₃₂	C₂	semidirect
82. #33960	C₃₁	direct (N × H)
101. #3393	C₂×C₃₁	semidirect
103. #2247	C₂×C₃₁	semidirect
104. #2211	C₂×C₃₁	semidirect
105. #2208	C₂×C₃₁	semidirect
106. Q₈×C₁₆	C₂×C₃₁	semidirect
107. C₄×Q₃₂	C₂×C₃₁	semidirect
170. C₃₁	#33960	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×#3393	2
C₃₁×#2249	2
C₃₁×#2247	2
C₃₁×#2211	2
C₃₁×#2208	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×Q₃₂	2
C₄×C₃₁×D₁₆	2
#33960	31

Subgroups — C31×#33960 #21490719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33960 #21490719