Subgroups — C₃₁×#9897 #21466656

Order 7936.

430 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1296	C₂	1	4	yes	maximal
5	C₂×C₃₁×#125	—	2	1	no	—
6	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	—	2	1	no	—
9	C₃₁×#389	—	2	2	no	—
10	C₃₁×#389	C₄	1	4	yes	—
11	C₃₁×#384	C₂²	1	2	yes	—
12	C₃₁×#382	—	2	1	no	—
13	C₄³×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#336	—	2	1	no	—
16	C₂²×C₈×C₃₁	—	2	2	no	—
17	C₂²×C₈×C₃₁	—	2	2	no	—
18	C₃₁×#125	—	4	4	no	—
19	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	D₄	1	2	yes	—
26	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×C₄⋊C₈	—	2	2	no	—
29	C₃₁×C₄⋊C₈	—	2	4	no	—
30	C₃₁×C₄⋊C₈	Q₈	1	4	yes	—
31	C₄×C₈×C₃₁	—	4	2	no	—
32	C₃₁×#95	—	2	4	no	—
33	C₂²×C₄×C₃₁	—	2	4	no	—
34	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	—	2	4	no	—
40	C₂×C₈×C₃₁	—	4	2	no	—
41	C₂×C₈×C₃₁	—	4	2	no	—
42	C₂×C₈×C₃₁	—	4	2	no	—
43	C₂×C₈×C₃₁	—	4	4	no	—
44	C₄⋊C₄×C₃₁	—	4	8	no	—
45	C₄⋊C₄×C₃₁	—	4	4	no	—
46	C₄⋊C₄×C₃₁	—	4	4	no	—
47	C₄⋊C₄×C₃₁	—	4	4	no	—
48	C₄²×C₃₁	—	2	4	no	—
49	C₄²×C₃₁	—	2	2	no	—
50	C₄²×C₃₁	—	4	2	no	—
51	C₄²×C₃₁	SD₁₆	1	2	yes	—
52	C₄²×C₃₁	Q₁₆	1	2	yes	—
53	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
54	C₄²×C₃₁	#41	1	2	yes	—
55	#9897	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	—	2	2	no	—
58	C₂×C₄×C₃₁	—	2	2	no	—
59	C₂×C₄×C₃₁	—	2	2	no	—
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	4	4	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	#103	1	4	yes	—
67	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₃₁	#135	1	4	yes	—
74	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
75	C₈×C₃₁	—	4	4	no	—
76	C₈×C₃₁	—	8	2	no	—
77	#2859	C₂×C₃₁	1	1	yes	characteristic
78	#1831	C₂×C₃₁	1	1	yes	characteristic
79	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
80	#1296	C₂×C₃₁	1	4	yes	—
81	C₂²×C₃₁	#391	1	1	yes	characteristic
82	C₂²×C₃₁	#413	1	2	yes	—
83	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
84	C₂²×C₃₁	#416	1	2	yes	—
85	C₂²×C₃₁	#418	1	1	yes	characteristic
86	C₄×C₃₁	—	2	4	no	—
87	C₄×C₃₁	—	2	2	no	—
88	C₄×C₃₁	—	4	2	no	—
89	C₄×C₃₁	—	8	4	no	—
90	C₄×C₃₁	#476	1	2	yes	—
91	C₄×C₃₁	#477	1	2	yes	—
92	C₄×C₃₁	#488	1	2	yes	—
93	C₄×C₃₁	#499	1	2	yes	—
94	C₂×#125	—	2	1	no	—
95	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
96	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
97	C₂×C₄×C₈	—	2	1	no	—
98	#389	—	2	2	no	—
99	#389	C₄×C₃₁	1	4	yes	—
100	#384	C₂²×C₃₁	1	2	yes	—
101	#382	—	2	1	no	—
102	C₄³	C₂²×C₃₁	1	1	yes	characteristic
103	#336	—	2	2	no	—
104	#336	—	2	1	no	—
105	C₂×C₃₁	#1542	1	2	yes	—
106	C₂×C₃₁	#1553	1	2	yes	—
107	C₂×C₃₁	#1987	1	1	yes	characteristic
108	C₂×C₃₁	#1991	1	1	yes	characteristic
109	C₂×C₃₁	#1998	1	1	yes	characteristic
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	2	no	—
112	#125	—	4	4	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
117	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
118	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
119	C₂×C₄²	D₄×C₃₁	1	2	yes	—
120	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
121	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	Q₈×C₃₁	1	4	yes	—
125	C₄×C₈	—	4	2	no	—
126	#95	—	2	4	no	—
127	C₃₁	#9897	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
130	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	SD₁₆×C₃₁	1	2	yes	—
147	C₄²	Q₁₆×C₃₁	1	2	yes	—
148	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
149	C₄²	#41×C₃₁	1	2	yes	—
150	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	C₃₁×#103	1	4	yes	—
161	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
162	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
163	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
164	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₃₁×#135	1	4	yes	—
168	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	8	2	no	—
171	C₂²	C₃₁×#391	1	1	yes	characteristic
172	C₂²	C₃₁×#413	1	2	yes	—
173	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
174	C₂²	C₃₁×#416	1	2	yes	—
175	C₂²	C₃₁×#418	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	8	4	no	—
180	C₄	C₃₁×#476	1	2	yes	—
181	C₄	C₃₁×#477	1	2	yes	—
182	C₄	C₃₁×#488	1	2	yes	—
183	C₄	C₃₁×#499	1	2	yes	—
184	C₂	C₃₁×#1542	1	2	yes	—
185	C₂	C₃₁×#1553	1	2	yes	—
186	C₂	C₃₁×#1987	1	1	yes	characteristic
187	C₂	C₃₁×#1991	1	1	yes	characteristic
188	C₂	C₃₁×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
55. #9897	C₃₁	direct (N × H)
127. C₃₁	#9897	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2859	2
C₃₁×#1831	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1296	2
#9897	31

Subgroups — C31×#9897 #21466656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9897 #21466656