Subgroups — C₃₁×#33241 #21490000

Order 7936.

1174 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#486	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2858	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1970	C₂	1	8	yes	maximal, complemented
6	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₂²	1	4	yes	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃₁×#127	C₂²	1	2	yes	—
10	C₂×C₃₁×#124	—	2	4	no	—
11	C₃₁×#500	—	2	4	no	—
12	C₃₁×#493	—	2	4	no	—
13	C₃₁×#486	—	2	8	no	—
14	C₂×C₃₁×#107	C₂²	1	4	yes	—
15	C₂×C₃₁×#102	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#390	—	2	4	no	—
20	C₃₁×#390	C₂²	1	8	yes	—
21	C₄³×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₃₁	—	2	2	no	—
23	C₂²×D₄×C₃₁	—	2	2	no	—
24	C₂²×D₄×C₃₁	—	2	2	no	—
25	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
26	C₂×D₈×C₃₁	—	2	8	no	—
27	C₂×D₈×C₃₁	—	2	16	no	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
30	C₃₁×C₄⋊Q₈	—	2	4	no	—
31	C₃₁×C₄⋊Q₈	—	2	4	no	—
32	C₃₁×#127	—	2	4	no	—
33	C₃₁×#127	—	2	4	no	—
34	C₃₁×#124	—	4	16	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
38	C₂×C₄²×C₃₁	D₄	1	2	yes	—
39	C₂×C₄²×C₃₁	D₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#107	—	2	8	no	—
43	C₃₁×#102	—	2	16	no	—
44	C₄×C₈×C₃₁	—	2	8	no	—
45	C₄×C₈×C₃₁	D₄	1	4	yes	—
46	C₄×C₈×C₃₁	D₄	1	4	yes	—
47	C₂⁴×C₃₁	—	2	4	no	—
48	C₂×Q₈×C₃₁	—	4	4	no	—
49	C₂×Q₈×C₃₁	—	4	8	no	—
50	C₂×D₄×C₃₁	—	2	4	no	—
51	C₂×D₄×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	2	16	no	—
53	C₂×D₄×C₃₁	—	4	4	no	—
54	C₂×D₄×C₃₁	—	4	4	no	—
55	C₂×D₄×C₃₁	—	4	8	no	—
56	C₂×D₄×C₃₁	—	4	8	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	D₈×C₃₁	—	4	16	no	—
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₃₁	#41	1	8	yes	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	2	4	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₂²⋊C₄×C₃₁	—	4	32	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	2	4	no	—
73	C₄²×C₃₁	D₈	1	8	yes	complemented
74	C₄²×C₃₁	C₂×D₄	1	2	yes	—
75	C₄²×C₃₁	C₂×D₄	1	2	yes	—
76	#33241	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	—	4	4	no	—
78	C₂³×C₃₁	—	4	8	no	—
79	C₂³×C₃₁	—	4	8	no	—
80	C₂³×C₃₁	—	4	8	no	—
81	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₃₁	—	8	8	no	—
83	D₄×C₃₁	—	4	8	no	—
84	D₄×C₃₁	—	4	8	no	—
85	D₄×C₃₁	—	8	8	no	—
86	D₄×C₃₁	—	8	8	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	4	8	no	—
91	C₂×C₄×C₃₁	#127	1	2	yes	—
92	C₂×C₄×C₃₁	#127	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×D₈	1	8	yes	—
94	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
95	C₂×C₄×C₃₁	#135	1	4	yes	—
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
99	C₈×C₃₁	—	2	4	no	—
100	C₈×C₃₁	#124	1	8	yes	complemented
101	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
102	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
103	C₂×#486	C₂×C₃₁	1	2	yes	complemented
104	#2858	C₂×C₃₁	1	2	yes	complemented
105	#1970	C₂×C₃₁	1	8	yes	complemented
106	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
107	C₂²×C₃₁	—	8	4	no	—
108	C₂²×C₃₁	—	8	4	no	—
109	C₂²×C₃₁	—	8	4	no	—
110	C₂²×C₃₁	—	8	16	no	—
111	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
112	C₂²×C₃₁	#532	1	2	yes	—
113	C₂²×C₃₁	C₂²×D₈	1	2	yes	—
114	C₂²×C₃₁	C₂×#135	1	2	yes	—
115	C₄×C₃₁	—	2	8	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	#493	1	4	yes	—
118	C₄×C₃₁	#495	1	4	yes	—
119	C₄×C₃₁	C₂×#124	1	2	yes	—
120	C₄×C₃₁	#532	1	2	yes	—
121	C₂²×D₈	C₂²×C₃₁	1	4	yes	—
122	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
123	C₂×#127	C₂²×C₃₁	1	2	yes	—
124	C₂×#124	—	2	4	no	—
125	#500	—	2	4	no	—
126	#493	—	2	4	no	—
127	#486	—	2	8	no	—
128	C₂×#107	C₂²×C₃₁	1	4	yes	—
129	C₂×#102	C₂²×C₃₁	1	8	yes	—
130	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
131	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
133	#390	—	2	4	no	—
134	#390	C₂²×C₃₁	1	8	yes	—
135	C₄³	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×C₃₁	—	16	4	no	—
137	C₂×C₃₁	#2858	1	1	yes	characteristic
138	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
139	C₂×C₃₁	C₂×#495	1	1	yes	characteristic
140	C₂×C₃₁	#3169	1	4	yes	—
141	C₂²×Q₈	—	2	2	no	—
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
145	C₂×D₈	—	2	8	no	—
146	C₂×D₈	—	2	16	no	—
147	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
148	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	2	4	no	—
151	#127	—	2	4	no	—
152	#127	—	2	4	no	—
153	#124	—	4	16	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	16	no	—
157	C₂×C₄²	D₄×C₃₁	1	2	yes	—
158	C₂×C₄²	D₄×C₃₁	1	2	yes	—
159	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
160	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
161	#107	—	2	8	no	—
162	#102	—	2	16	no	—
163	C₄×C₈	—	2	8	no	—
164	C₄×C₈	D₄×C₃₁	1	4	yes	—
165	C₄×C₈	D₄×C₃₁	1	4	yes	—
166	C₃₁	#33241	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	—	2	4	no	—
168	C₂×Q₈	—	4	4	no	—
169	C₂×Q₈	—	4	8	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	2	16	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	—	4	8	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
182	D₈	—	4	16	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
185	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
186	C₂×C₈	#41×C₃₁	1	8	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₂²⋊C₄	—	4	32	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	4	no	—
193	C₄²	D₈×C₃₁	1	8	yes	complemented
194	C₄²	C₂×D₄×C₃₁	1	2	yes	—
195	C₄²	C₂×D₄×C₃₁	1	2	yes	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	8	no	—
198	C₂³	—	4	8	no	—
199	C₂³	—	4	8	no	—
200	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
201	Q₈	—	8	8	no	—
202	D₄	—	4	8	no	—
203	D₄	—	4	8	no	—
204	D₄	—	8	8	no	—
205	D₄	—	8	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	C₃₁×#127	1	2	yes	—
211	C₂×C₄	C₃₁×#127	1	2	yes	—
212	C₂×C₄	C₂×D₈×C₃₁	1	8	yes	—
213	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
214	C₂×C₄	C₃₁×#135	1	4	yes	—
215	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
218	C₈	—	2	4	no	—
219	C₈	C₃₁×#124	1	8	yes	complemented
220	C₂²	—	8	4	no	—
221	C₂²	—	8	4	no	—
222	C₂²	—	8	4	no	—
223	C₂²	—	8	16	no	—
224	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
225	C₂²	C₃₁×#532	1	2	yes	—
226	C₂²	C₂²×D₈×C₃₁	1	2	yes	—
227	C₂²	C₂×C₃₁×#135	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	—	8	4	no	—
230	C₄	C₃₁×#493	1	4	yes	—
231	C₄	C₃₁×#495	1	4	yes	—
232	C₄	C₂×C₃₁×#124	1	2	yes	—
233	C₄	C₃₁×#532	1	2	yes	—
234	C₂	—	16	4	no	—
235	C₂	C₃₁×#2858	1	1	yes	characteristic
236	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
237	C₂	C₂×C₃₁×#495	1	1	yes	characteristic
238	C₂	C₃₁×#3169	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#500	C₂	semidirect
3. C₂×C₃₁×#486	C₂	semidirect
4. C₃₁×#2858	C₂	semidirect
5. C₃₁×#1970	C₂	semidirect
6. C₄²×C₈×C₃₁	C₂	semidirect
73. C₄²×C₃₁	D₈	semidirect
76. #33241	C₃₁	direct (N × H)
100. C₈×C₃₁	#124	semidirect
101. C₂×#500	C₂×C₃₁	semidirect
103. C₂×#486	C₂×C₃₁	semidirect
104. #2858	C₂×C₃₁	semidirect
105. #1970	C₂×C₃₁	semidirect
106. C₄²×C₈	C₂×C₃₁	semidirect
166. C₃₁	#33241	direct (N × H)
193. C₄²	D₈×C₃₁	semidirect
219. C₈	C₃₁×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₂×C₃₁×#493	2
C₂×C₃₁×#486	2
C₃₁×#2858	2
C₃₁×#1970	2
C₄²×C₈×C₃₁	2
#33241	31

Subgroups — C31×#33241 #21490000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33241 #21490000