Subgroups — C₃₁×#33248 #21490007

Order 7936.

1006 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	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#490	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2641	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1972	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1970	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
9	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#127	C₂²	1	2	yes	—
11	C₂×C₃₁×#125	C₂²	1	2	yes	—
12	C₃₁×#499	—	2	4	no	—
13	C₃₁×#493	—	2	4	no	—
14	C₃₁×#490	—	2	8	no	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	8	yes	—
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#394	—	2	8	no	—
23	C₃₁×#390	C₂²	1	8	yes	—
24	C₃₁×#376	C₂²	1	1	yes	characteristic
25	C₂²×D₄×C₃₁	—	2	4	no	—
26	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
27	C₂×D₈×C₃₁	—	2	4	no	—
28	C₂×D₈×C₃₁	—	2	4	no	—
29	C₂×D₈×C₃₁	—	2	8	no	—
30	C₂×D₈×C₃₁	—	2	8	no	—
31	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×#127	—	2	4	no	—
35	C₃₁×#127	—	2	4	no	—
36	C₃₁×#125	—	2	4	no	—
37	C₃₁×#125	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
41	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#107	—	2	4	no	—
44	C₃₁×#106	—	2	4	no	—
45	C₃₁×#102	—	2	16	no	—
46	C₃₁×#97	—	2	4	no	—
47	C₃₁×#97	D₄	1	4	yes	—
48	C₄×C₈×C₃₁	—	2	4	no	—
49	C₄×C₈×C₃₁	D₄	1	4	yes	—
50	C₃₁×#95	D₄	1	4	yes	—
51	C₂⁴×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	2	8	no	—
55	C₂×D₄×C₃₁	—	2	8	no	—
56	C₂×D₄×C₃₁	—	4	8	no	—
57	C₂×D₄×C₃₁	—	8	8	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	D₈×C₃₁	—	4	8	no	—
63	D₈×C₃₁	—	4	8	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₃₁	#41	1	4	yes	—
70	C₂×C₈×C₃₁	#41	1	4	yes	—
71	C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₄⋊C₄×C₃₁	—	2	4	no	—
73	C₄⋊C₄×C₃₁	—	4	8	no	—
74	C₂²⋊C₄×C₃₁	—	8	16	no	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	—	2	2	no	—
77	C₄²×C₃₁	C₂×D₄	1	2	yes	—
78	C₄²×C₃₁	C₂×D₄	1	2	yes	—
79	#33248	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	4	4	no	—
81	C₂³×C₃₁	—	4	8	no	—
82	C₂³×C₃₁	—	4	16	no	—
83	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	D₄×C₃₁	—	4	4	no	—
85	D₄×C₃₁	—	4	4	no	—
86	D₄×C₃₁	—	4	8	no	—
87	D₄×C₃₁	—	8	16	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	8	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	4	8	no	—
93	C₂×C₄×C₃₁	#127	1	4	yes	—
94	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
98	C₈×C₃₁	—	2	4	no	—
99	C₈×C₃₁	#124	1	4	yes	—
100	C₈×C₃₁	#124	1	4	yes	—
101	C₂×#499	C₂×C₃₁	1	1	yes	characteristic, complemented
102	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
103	C₂×#490	C₂×C₃₁	1	2	yes	complemented
104	#2641	C₂×C₃₁	1	2	yes	complemented
105	#1972	C₂×C₃₁	1	4	yes	complemented
106	#1970	C₂×C₃₁	1	4	yes	complemented
107	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂²×C₃₁	—	8	4	no	—
109	C₂²×C₃₁	—	8	8	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	#576	1	4	yes	—
114	C₄×C₃₁	—	2	4	no	—
115	C₄×C₃₁	—	4	4	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
118	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
119	C₄×C₃₁	#532	1	2	yes	—
120	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
121	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
122	C₂×#127	C₂²×C₃₁	1	2	yes	—
123	C₂×#125	C₂²×C₃₁	1	2	yes	—
124	#499	—	2	4	no	—
125	#493	—	2	4	no	—
126	#490	—	2	8	no	—
127	C₂×#107	C₂²×C₃₁	1	2	yes	—
128	C₂×#106	C₂²×C₃₁	1	2	yes	—
129	C₂×#102	C₂²×C₃₁	1	8	yes	—
130	C₂×#97	C₂²×C₃₁	1	2	yes	—
131	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
133	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
134	#394	—	2	8	no	—
135	#390	C₂²×C₃₁	1	8	yes	—
136	#376	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₃₁	—	16	4	no	—
138	C₂×C₃₁	#2858	1	1	yes	characteristic
139	C₂×C₃₁	#3155	1	2	yes	—
140	C₂×C₃₁	#3176	1	4	yes	—
141	C₂²×D₄	—	2	4	no	—
142	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
143	C₂×D₈	—	2	4	no	—
144	C₂×D₈	—	2	4	no	—
145	C₂×D₈	—	2	8	no	—
146	C₂×D₈	—	2	8	no	—
147	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
148	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
150	#127	—	2	4	no	—
151	#127	—	2	4	no	—
152	#125	—	2	4	no	—
153	#125	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	16	no	—
157	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
158	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
159	#107	—	2	4	no	—
160	#106	—	2	4	no	—
161	#102	—	2	16	no	—
162	#97	—	2	4	no	—
163	#97	D₄×C₃₁	1	4	yes	—
164	C₄×C₈	—	2	4	no	—
165	C₄×C₈	D₄×C₃₁	1	4	yes	—
166	#95	D₄×C₃₁	1	4	yes	—
167	C₃₁	#33248	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁴	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	2	8	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	8	8	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
178	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
179	D₈	—	4	8	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
184	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
186	C₂×C₈	#41×C₃₁	1	4	yes	—
187	C₂×C₈	#41×C₃₁	1	4	yes	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	8	16	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
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	16	no	—
199	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	8	no	—
203	D₄	—	8	16	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	C₃₁×#127	1	4	yes	—
210	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
213	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	C₃₁×#124	1	4	yes	—
216	C₈	C₃₁×#124	1	4	yes	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	8	no	—
219	C₂²	—	8	16	no	—
220	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
221	C₂²	C₃₁×#532	1	2	yes	—
222	C₂²	C₃₁×#576	1	4	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	4	no	—
226	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
227	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
228	C₄	C₃₁×#532	1	2	yes	—
229	C₂	—	16	4	no	—
230	C₂	C₃₁×#2858	1	1	yes	characteristic
231	C₂	C₃₁×#3155	1	2	yes	—
232	C₂	C₃₁×#3176	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 ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#499	C₂	semidirect
3. C₂×C₃₁×#490	C₂	semidirect
4. C₃₁×#2641	C₂	semidirect
5. C₃₁×#1972	C₂	semidirect
6. C₃₁×#1970	C₂	semidirect
7. C₃₁×#1726	C₂	semidirect
79. #33248	C₃₁	direct (N × H)
101. C₂×#499	C₂×C₃₁	semidirect
103. C₂×#490	C₂×C₃₁	semidirect
104. #2641	C₂×C₃₁	semidirect
105. #1972	C₂×C₃₁	semidirect
106. #1970	C₂×C₃₁	semidirect
107. #1726	C₂×C₃₁	semidirect
167. C₃₁	#33248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#499	2
C₂×C₃₁×#493	2
C₂×C₃₁×#490	2
C₃₁×#2641	2
C₃₁×#1972	2
C₃₁×#1970	2
C₃₁×#1726	2
#33248	31

Subgroups — C31×#33248 #21490007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33248 #21490007