Subgroups — C₃₁×#7849 #21464608

Order 7936.

734 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1273	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₃₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#431	—	2	4	no	—
7	C₃₁×#431	—	2	4	no	—
8	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	—
9	C₂×C₃₁×C₂²⋊C₈	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₃₁	—	2	4	no	—
11	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
14	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	—	2	4	no	—
17	C₂²×C₈×C₃₁	—	2	4	no	—
18	C₂²×C₈×C₃₁	C₈	1	8	yes	complemented
19	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₃₁	D₄	1	4	yes	—
24	C₂×C₄²×C₃₁	D₄	1	2	yes	—
25	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
26	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	—	2	8	no	—
28	C₃₁×C₂²⋊C₈	—	2	8	no	—
29	C₄×C₈×C₃₁	—	2	4	no	—
30	C₄×C₈×C₃₁	—	2	4	no	—
31	C₄×C₈×C₃₁	—	2	4	no	—
32	C₄×C₈×C₃₁	—	2	4	no	—
33	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	—	2	8	no	—
35	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
36	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
43	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₃₁	—	2	16	no	—
46	C₂×C₈×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	—	2	8	no	—
49	C₂×C₈×C₃₁	—	2	16	no	—
50	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	—
51	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
55	C₄²×C₃₁	C₂×D₄	1	2	yes	—
56	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	#7849	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	—	2	4	no	—
60	C₂³×C₃₁	#95	1	2	yes	—
61	C₂³×C₃₁	C₄×C₈	1	2	yes	—
62	C₂³×C₃₁	#97	1	2	yes	—
63	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₃₁	—	2	4	no	—
65	C₂×C₄×C₃₁	—	2	8	no	—
66	C₂×C₄×C₃₁	—	2	8	no	—
67	C₂×C₄×C₃₁	#95	1	4	yes	—
68	C₂×C₄×C₃₁	#95	1	2	yes	—
69	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₃₁	#97	1	2	yes	—
71	C₂×C₄×C₃₁	C₂²⋊C₈	1	8	yes	—
72	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
73	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₃₁	#131	1	8	yes	—
80	C₈×C₃₁	—	2	8	no	—
81	C₈×C₃₁	—	4	8	no	—
82	C₂×#431	C₂×C₃₁	1	2	yes	—
83	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
84	#1273	C₂×C₃₁	1	4	yes	complemented
85	C₂²×C₃₁	—	2	4	no	—
86	C₂²×C₃₁	—	2	4	no	—
87	C₂²×C₃₁	—	2	4	no	—
88	C₂²×C₃₁	#336	1	4	yes	—
89	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#97	1	1	yes	characteristic
92	C₂²×C₃₁	#405	1	4	yes	—
93	C₄×C₃₁	—	2	4	no	—
94	C₄×C₃₁	#336	1	4	yes	—
95	C₄×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
96	C₄×C₃₁	#408	1	4	yes	—
97	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
98	C₄×C₃₁	#424	1	4	yes	—
99	C₂³×C₈	C₄×C₃₁	1	2	yes	—
100	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
101	#431	—	2	4	no	—
102	#431	—	2	4	no	—
103	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	—
104	C₂×C₂²⋊C₈	C₄×C₃₁	1	4	yes	—
105	C₂×C₄×C₈	—	2	4	no	—
106	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
108	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
109	C₂×C₃₁	—	2	4	no	—
110	C₂×C₃₁	#1450	1	2	yes	—
111	C₂×C₃₁	#1452	1	2	yes	—
112	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
113	C₂×C₃₁	#1729	1	2	yes	—
114	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
115	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	C₈×C₃₁	1	8	yes	complemented
119	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
120	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
121	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
123	C₂×C₄²	D₄×C₃₁	1	4	yes	—
124	C₂×C₄²	D₄×C₃₁	1	2	yes	—
125	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
126	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
127	C₄⋊C₈	—	2	8	no	—
128	C₂²⋊C₈	—	2	8	no	—
129	C₄×C₈	—	2	4	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	2	4	no	—
133	C₃₁	#7849	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
137	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
139	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
140	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
141	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
143	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
146	C₂×C₈	—	2	16	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	16	no	—
151	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	—
152	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
153	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
154	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
155	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
156	C₄²	C₂×D₄×C₃₁	1	2	yes	—
157	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
158	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
159	C₂³	—	2	4	no	—
160	C₂³	C₃₁×#95	1	2	yes	—
161	C₂³	C₄×C₈×C₃₁	1	2	yes	—
162	C₂³	C₃₁×#97	1	2	yes	—
163	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	C₃₁×#95	1	4	yes	—
168	C₂×C₄	C₃₁×#95	1	2	yes	—
169	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
170	C₂×C₄	C₃₁×#97	1	2	yes	—
171	C₂×C₄	C₃₁×C₂²⋊C₈	1	8	yes	—
172	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
177	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
178	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
179	C₂×C₄	C₃₁×#131	1	8	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	4	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	C₃₁×#336	1	4	yes	—
186	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
187	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
188	C₂²	C₂×C₃₁×#97	1	1	yes	characteristic
189	C₂²	C₃₁×#405	1	4	yes	—
190	C₄	—	2	4	no	—
191	C₄	C₃₁×#336	1	4	yes	—
192	C₄	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
193	C₄	C₃₁×#408	1	4	yes	—
194	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
195	C₄	C₃₁×#424	1	4	yes	—
196	C₂	—	2	4	no	—
197	C₂	C₃₁×#1450	1	2	yes	—
198	C₂	C₃₁×#1452	1	2	yes	—
199	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
200	C₂	C₃₁×#1729	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1273	C₂	semidirect
18. C₂²×C₈×C₃₁	C₈	semidirect
58. #7849	C₃₁	direct (N × H)
84. #1273	C₂×C₃₁	semidirect
118. C₂²×C₈	C₈×C₃₁	semidirect
133. C₃₁	#7849	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×#431	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1273	2
#7849	31

Subgroups — C31×#7849 #21464608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7849 #21464608