Subgroups — C₃₁×#33231 #21489990

Order 7936.

1206 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#490	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2867	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1961	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1957	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#127	—	2	4	no	—
8	C₂×C₃₁×#127	C₂²	1	2	yes	—
9	C₂×C₃₁×#127	C₂²	1	2	yes	—
10	C₂×C₃₁×#127	C₂²	1	2	yes	—
11	C₂×C₃₁×#125	C₂²	1	2	yes	—
12	C₃₁×#490	—	2	16	no	—
13	C₂×C₃₁×#102	C₂²	1	16	yes	—
14	C₂×C₃₁×#97	C₂²	1	2	yes	—
15	C₂×C₃₁×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#389	—	2	4	no	—
18	C₃₁×#384	C₂²	1	2	yes	—
19	C₃₁×#382	C₂²	1	2	yes	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₃₁	—	2	4	no	—
22	C₂²×D₄×C₃₁	—	2	4	no	—
23	C₂²×D₄×C₃₁	—	2	4	no	—
24	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
26	C₃₁×#127	—	2	8	no	—
27	C₃₁×#127	—	2	8	no	—
28	C₃₁×#127	—	2	8	no	—
29	C₃₁×#127	—	4	16	no	—
30	C₃₁×#125	—	2	8	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄²×C₃₁	D₄	1	4	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#102	—	2	32	no	—
39	C₃₁×#97	—	2	4	no	—
40	C₃₁×#97	D₄	1	8	yes	complemented
41	C₄×C₈×C₃₁	—	2	4	no	—
42	C₃₁×#95	—	2	4	no	—
43	C₂⁴×C₃₁	—	2	4	no	—
44	C₂×D₄×C₃₁	—	2	8	no	—
45	C₂×D₄×C₃₁	—	2	8	no	—
46	C₂×D₄×C₃₁	—	2	8	no	—
47	C₂×D₄×C₃₁	—	4	16	no	—
48	C₂×D₄×C₃₁	—	4	16	no	—
49	C₂×D₄×C₃₁	—	4	8	no	—
50	C₂×D₄×C₃₁	—	4	8	no	—
51	C₂×D₄×C₃₁	—	4	8	no	—
52	C₂×D₄×C₃₁	—	4	16	no	—
53	C₂²×C₄×C₃₁	—	2	4	no	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
61	C₄⋊C₄×C₃₁	—	2	8	no	—
62	C₄⋊C₄×C₃₁	—	4	8	no	—
63	C₄⋊C₄×C₃₁	—	4	8	no	—
64	C₄²×C₃₁	—	2	8	no	—
65	C₄²×C₃₁	C₂×D₄	1	4	yes	—
66	C₄²×C₃₁	#41	1	4	yes	—
67	C₄²×C₃₁	#41	1	4	yes	—
68	#33231	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	—	4	4	no	—
70	C₂³×C₃₁	—	4	4	no	—
71	C₂³×C₃₁	—	4	4	no	—
72	C₂³×C₃₁	—	4	16	no	—
73	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
74	D₄×C₃₁	—	4	16	no	—
75	D₄×C₃₁	—	8	16	no	—
76	D₄×C₃₁	—	8	16	no	—
77	D₄×C₃₁	—	8	16	no	—
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	8	no	—
82	C₂×C₄×C₃₁	#124	1	4	yes	—
83	C₂×C₄×C₃₁	#124	1	4	yes	—
84	C₂×C₄×C₃₁	#127	1	4	yes	—
85	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	8	yes	—
86	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
88	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
89	C₈×C₃₁	—	2	8	no	—
90	C₂×#490	C₂×C₃₁	1	4	yes	complemented
91	#2867	C₂×C₃₁	1	1	yes	characteristic
92	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
93	#1961	C₂×C₃₁	1	4	yes	complemented
94	#1957	C₂×C₃₁	1	4	yes	complemented
95	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
96	C₂²×C₃₁	—	8	4	no	—
97	C₂²×C₃₁	—	8	8	no	—
98	C₂²×C₃₁	—	8	8	no	—
99	C₂²×C₃₁	—	8	8	no	—
100	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
103	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
104	C₄×C₃₁	—	2	4	no	—
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	8	4	no	—
107	C₄×C₃₁	#490	1	8	yes	complemented
108	C₄×C₃₁	#532	1	4	yes	—
109	C₂×#127	—	2	4	no	—
110	C₂×#127	C₂²×C₃₁	1	2	yes	—
111	C₂×#127	C₂²×C₃₁	1	2	yes	—
112	C₂×#127	C₂²×C₃₁	1	2	yes	—
113	C₂×#125	C₂²×C₃₁	1	2	yes	—
114	#490	—	2	16	no	—
115	C₂×#102	C₂²×C₃₁	1	16	yes	—
116	C₂×#97	C₂²×C₃₁	1	2	yes	—
117	C₂×#97	C₂²×C₃₁	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
119	#389	—	2	4	no	—
120	#384	C₂²×C₃₁	1	2	yes	—
121	#382	C₂²×C₃₁	1	2	yes	—
122	C₄³	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×C₃₁	—	16	4	no	—
124	C₂×C₃₁	#2858	1	1	yes	characteristic
125	C₂×C₃₁	C₂×#490	1	2	yes	—
126	C₂×C₃₁	#3138	1	2	yes	—
127	C₂×C₃₁	#3151	1	2	yes	—
128	C₂²×D₄	—	2	4	no	—
129	C₂²×D₄	—	2	4	no	—
130	C₂²×D₄	—	2	4	no	—
131	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
132	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
133	#127	—	2	8	no	—
134	#127	—	2	8	no	—
135	#127	—	2	8	no	—
136	#127	—	4	16	no	—
137	#125	—	2	8	no	—
138	C₂×C₄⋊C₄	—	2	4	no	—
139	C₂×C₄⋊C₄	—	2	4	no	—
140	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
141	C₂×C₄²	D₄×C₃₁	1	4	yes	—
142	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
145	#102	—	2	32	no	—
146	#97	—	2	4	no	—
147	#97	D₄×C₃₁	1	8	yes	complemented
148	C₄×C₈	—	2	4	no	—
149	#95	—	2	4	no	—
150	C₃₁	#33231	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁴	—	2	4	no	—
152	C₂×D₄	—	2	8	no	—
153	C₂×D₄	—	2	8	no	—
154	C₂×D₄	—	2	8	no	—
155	C₂×D₄	—	4	16	no	—
156	C₂×D₄	—	4	16	no	—
157	C₂×D₄	—	4	8	no	—
158	C₂×D₄	—	4	8	no	—
159	C₂×D₄	—	4	8	no	—
160	C₂×D₄	—	4	16	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
165	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
169	C₄⋊C₄	—	2	8	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄⋊C₄	—	4	8	no	—
172	C₄²	—	2	8	no	—
173	C₄²	C₂×D₄×C₃₁	1	4	yes	—
174	C₄²	#41×C₃₁	1	4	yes	—
175	C₄²	#41×C₃₁	1	4	yes	—
176	C₂³	—	4	4	no	—
177	C₂³	—	4	4	no	—
178	C₂³	—	4	4	no	—
179	C₂³	—	4	16	no	—
180	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
181	D₄	—	4	16	no	—
182	D₄	—	8	16	no	—
183	D₄	—	8	16	no	—
184	D₄	—	8	16	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	8	no	—
189	C₂×C₄	C₃₁×#124	1	4	yes	—
190	C₂×C₄	C₃₁×#124	1	4	yes	—
191	C₂×C₄	C₃₁×#127	1	4	yes	—
192	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	8	yes	—
193	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
194	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
195	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
196	C₈	—	2	8	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	8	no	—
199	C₂²	—	8	8	no	—
200	C₂²	—	8	8	no	—
201	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
202	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
203	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
204	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	8	4	no	—
208	C₄	C₃₁×#490	1	8	yes	complemented
209	C₄	C₃₁×#532	1	4	yes	—
210	C₂	—	16	4	no	—
211	C₂	C₃₁×#2858	1	1	yes	characteristic
212	C₂	C₂×C₃₁×#490	1	2	yes	—
213	C₂	C₃₁×#3138	1	2	yes	—
214	C₂	C₃₁×#3151	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2867	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×#490	C₂	semidirect
3. C₃₁×#2859	C₂	semidirect
4. C₃₁×#1961	C₂	semidirect
5. C₃₁×#1957	C₂	semidirect
6. C₄×C₃₁×#97	C₂	semidirect
40. C₃₁×#97	D₄	semidirect
68. #33231	C₃₁	direct (N × H)
90. C₂×#490	C₂×C₃₁	semidirect
92. #2859	C₂×C₃₁	semidirect
93. #1961	C₂×C₃₁	semidirect
94. #1957	C₂×C₃₁	semidirect
95. C₄×#97	C₂×C₃₁	semidirect
107. C₄×C₃₁	#490	semidirect
147. #97	D₄×C₃₁	semidirect
150. C₃₁	#33231	direct (N × H)
208. C₄	C₃₁×#490	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#490	2
C₃₁×#2867	2
C₃₁×#2859	2
C₃₁×#1961	2
C₃₁×#1957	2
C₄×C₃₁×#97	2
#33231	31

Subgroups — C31×#33231 #21489990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33231 #21489990