Subgroups — C₃₁×#33734 #21490493

Order 7936.

514 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	2	yes	maximal
2	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2241	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2216	C₂	1	4	yes	maximal
6	C₄×C₃₁×Q₃₂	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	—	2	2	no	—
9	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
10	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#494	C₂²	1	2	yes	—
13	C₃₁×#487	—	2	2	no	—
14	C₃₁×#467	—	2	2	no	—
15	C₃₁×#467	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₃₁	—	2	2	no	—
17	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#366	C₂²	1	4	yes	—
20	C₃₁×#358	C₂²	1	8	yes	—
21	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
22	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×Q₁₆×C₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×C₄⋊Q₈	—	2	2	no	—
29	C₃₁×C₄⋊Q₈	—	2	2	no	—
30	C₃₁×#125	—	4	2	no	—
31	C₃₁×C₂²⋊Q₈	—	2	4	no	—
32	C₃₁×C₂²⋊Q₈	—	4	2	no	—
33	C₄×Q₈×C₃₁	—	2	4	no	—
34	C₄×Q₈×C₃₁	—	4	2	no	—
35	C₃₁×#117	—	4	2	no	—
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×Q₃₂	—	2	8	no	—
38	C₂×C₁₆×C₃₁	—	2	4	no	—
39	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#107	—	2	2	no	—
42	C₃₁×#107	C₂³	1	4	yes	—
43	C₃₁×#103	—	2	4	no	—
44	C₃₁×#103	—	2	4	no	—
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₃₁	—	2	4	no	—
50	C₂×Q₈×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₃₁	—	2	8	no	—
55	Q₁₆×C₃₁	—	4	4	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄⋊C₄×C₃₁	—	4	4	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₃₁	—	4	4	no	—
69	C₄²×C₃₁	—	4	4	no	—
70	C₄²×C₃₁	D₈	1	1	yes	characteristic
71	C₄²×C₃₁	D₈	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₁₆×C₃₁	—	2	2	no	—
75	C₁₆×C₃₁	#41	1	4	yes	—
76	#33734	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₃₁	—	4	4	no	—
79	Q₈×C₃₁	—	4	4	no	—
80	Q₈×C₃₁	—	8	4	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	Q₃₂	1	4	yes	complemented
87	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₃₁	—	2	1	no	—
94	C₈×C₃₁	—	2	1	no	—
95	C₈×C₃₁	#135	1	2	yes	—
96	C₈×C₃₁	C₂×#41	1	2	yes	—
97	#3170	C₂×C₃₁	1	2	yes	—
98	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
99	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
100	#2241	C₂×C₃₁	1	2	yes	complemented
101	#2216	C₂×C₃₁	1	4	yes	—
102	C₄×Q₃₂	C₂×C₃₁	1	4	yes	complemented
103	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	—	2	1	no	—
106	C₂²×C₃₁	C₂×Q₃₂	1	2	yes	—
107	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	2	1	no	—
110	C₄×C₃₁	—	8	4	no	—
111	C₄×C₃₁	—	8	4	no	—
112	C₄×C₃₁	C₂×Q₃₂	1	2	yes	—
113	C₄×C₃₁	C₂×Q₃₂	1	2	yes	—
114	C₄×C₃₁	#508	1	2	yes	—
115	C₄×C₃₁	#532	1	1	yes	characteristic
116	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
117	#533	—	2	2	no	—
118	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
119	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
120	#500	C₂²×C₃₁	1	2	yes	—
121	#494	C₂²×C₃₁	1	2	yes	—
122	#487	—	2	2	no	—
123	#467	—	2	2	no	—
124	#467	C₂²×C₃₁	1	4	yes	—
125	C₄×Q₁₆	—	2	2	no	—
126	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
128	#366	C₂²×C₃₁	1	4	yes	—
129	#358	C₂²×C₃₁	1	8	yes	—
130	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
131	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
132	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×C₃₁	—	2	2	no	—
134	C₂×C₃₁	#3169	1	1	yes	characteristic
135	C₂×C₃₁	C₂²×Q₃₂	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
139	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	#125	—	4	2	no	—
144	C₂²⋊Q₈	—	2	4	no	—
145	C₂²⋊Q₈	—	4	2	no	—
146	C₄×Q₈	—	2	4	no	—
147	C₄×Q₈	—	4	2	no	—
148	#117	—	4	2	no	—
149	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
150	Q₃₂	—	2	8	no	—
151	C₂×C₁₆	—	2	4	no	—
152	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
153	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
154	#107	—	2	2	no	—
155	#107	C₂³×C₃₁	1	4	yes	—
156	#103	—	2	4	no	—
157	#103	—	2	4	no	—
158	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
159	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
161	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
162	C₃₁	#33734	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
168	Q₁₆	—	2	8	no	—
169	Q₁₆	—	4	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
175	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	—	4	4	no	—
184	C₄²	D₈×C₃₁	1	1	yes	characteristic
185	C₄²	D₈×C₃₁	1	1	yes	characteristic
186	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	#41×C₃₁	1	4	yes	—
190	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	8	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₃₁×Q₃₂	1	4	yes	complemented
200	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	C₃₁×#135	1	2	yes	—
209	C₈	C₂×#41×C₃₁	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×C₃₁×Q₃₂	1	2	yes	—
213	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₂×C₃₁×Q₃₂	1	2	yes	—
219	C₄	C₂×C₃₁×Q₃₂	1	2	yes	—
220	C₄	C₃₁×#508	1	2	yes	—
221	C₄	C₃₁×#532	1	1	yes	characteristic
222	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	C₃₁×#3169	1	1	yes	characteristic
225	C₂	C₂²×C₃₁×Q₃₂	1	1	yes	characteristic
226	C₂	C₂×C₃₁×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2252	C₂	semidirect
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
6. C₄×C₃₁×Q₃₂	C₂	semidirect
76. #33734	C₃₁	direct (N × H)
86. C₂×C₄×C₃₁	Q₃₂	semidirect
98. #2252	C₂×C₃₁	semidirect
99. #2247	C₂×C₃₁	semidirect
100. #2241	C₂×C₃₁	semidirect
102. C₄×Q₃₂	C₂×C₃₁	semidirect
162. C₃₁	#33734	direct (N × H)
199. C₂×C₄	C₃₁×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3170	2
C₃₁×#2252	2
C₃₁×#2247	2
C₃₁×#2241	2
C₃₁×#2216	2
C₄×C₃₁×Q₃₂	2
C₂×C₄×C₁₆×C₃₁	2
#33734	31

Subgroups — C31×#33734 #21490493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33734 #21490493