Subgroups — C₃₁×#8812 #21465571

Order 7936.

382 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#335	C₂	1	2	yes	maximal
5	C₂×C₃₁×#334	C₂	1	2	yes	maximal
6	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₄³×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#336	—	2	2	no	—
13	C₃₁×#336	C₄	1	2	yes	—
14	C₃₁×#336	C₄	1	2	yes	—
15	C₃₁×#335	—	2	4	no	—
16	C₃₁×#334	—	2	4	no	—
17	C₂²×C₈×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	—	2	1	no	—
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	1	yes	characteristic
23	C₂×C₄²×C₃₁	D₄	1	2	yes	—
24	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	—	2	2	no	—
29	C₃₁×C₄⋊C₈	—	2	2	no	—
30	C₃₁×C₄⋊C₈	—	4	4	no	—
31	C₄×C₈×C₃₁	D₄	1	4	yes	—
32	C₄×C₈×C₃₁	Q₈	1	4	yes	—
33	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₃₁	—	2	4	no	—
39	C₂×C₈×C₃₁	—	2	4	no	—
40	C₂×C₈×C₃₁	—	4	2	no	—
41	C₂×C₈×C₃₁	—	4	1	no	—
42	C₂×C₈×C₃₁	—	4	1	no	—
43	C₂×C₈×C₃₁	—	4	4	no	—
44	C₂×C₈×C₃₁	—	4	2	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
47	C₄²×C₃₁	—	2	4	no	—
48	C₄²×C₃₁	—	2	2	no	—
49	C₄²×C₃₁	—	2	2	no	—
50	C₄²×C₃₁	—	2	2	no	—
51	C₄²×C₃₁	—	2	2	no	—
52	C₄²×C₃₁	C₂×D₄	1	2	yes	—
53	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
54	#8812	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	—	2	2	no	—
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	#108	1	4	yes	—
63	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
65	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₃₁	#131	1	2	yes	—
68	C₂×C₄×C₃₁	#131	1	2	yes	—
69	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×C₄×C₃₁	#137	1	2	yes	—
71	C₈×C₃₁	—	2	8	no	—
72	C₈×C₃₁	—	8	2	no	—
73	C₈×C₃₁	—	8	1	no	—
74	C₈×C₃₁	—	8	1	no	—
75	#1842	C₂×C₃₁	1	1	yes	characteristic
76	#1831	C₂×C₃₁	1	1	yes	characteristic
77	#1744	C₂×C₃₁	1	1	yes	characteristic
78	C₂×#335	C₂×C₃₁	1	2	yes	—
79	C₂×#334	C₂×C₃₁	1	2	yes	—
80	C₂²×C₃₁	#384	1	1	yes	characteristic
81	C₂²×C₃₁	#424	1	1	yes	characteristic
82	C₂²×C₃₁	#424	1	1	yes	characteristic
83	C₂²×C₃₁	#428	1	2	yes	—
84	C₂²×C₃₁	C₂×#108	1	2	yes	—
85	C₄×C₃₁	—	2	4	no	—
86	C₄×C₃₁	—	2	2	no	—
87	C₄×C₃₁	—	4	2	no	—
88	C₄×C₃₁	#424	1	4	yes	—
89	C₄×C₃₁	#496	1	1	yes	characteristic
90	C₄×C₃₁	#497	1	1	yes	characteristic
91	C₄×C₃₁	#501	1	2	yes	—
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
94	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
97	C₄³	C₂²×C₃₁	1	1	yes	characteristic
98	#336	—	2	2	no	—
99	#336	C₄×C₃₁	1	2	yes	—
100	#336	C₄×C₃₁	1	2	yes	—
101	#335	—	2	4	no	—
102	#334	—	2	4	no	—
103	C₂×C₃₁	#1573	1	2	yes	—
104	C₂×C₃₁	#1574	1	2	yes	—
105	C₂×C₃₁	#1843	1	1	yes	characteristic
106	C₂×C₃₁	#1847	1	1	yes	characteristic
107	C₂×C₃₁	#1848	1	1	yes	characteristic
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
112	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
114	C₂×C₄²	D₄×C₃₁	1	2	yes	—
115	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
116	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄×C₈	D₄×C₃₁	1	4	yes	—
123	C₄×C₈	Q₈×C₃₁	1	4	yes	—
124	C₃₁	#8812	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
126	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
129	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×D₄×C₃₁	1	2	yes	—
145	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	C₃₁×#108	1	4	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
155	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
156	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
157	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	C₃₁×#131	1	2	yes	—
159	C₂×C₄	C₃₁×#131	1	2	yes	—
160	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
161	C₂×C₄	C₃₁×#137	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	8	2	no	—
164	C₈	—	8	1	no	—
165	C₈	—	8	1	no	—
166	C₂²	C₃₁×#384	1	1	yes	characteristic
167	C₂²	C₃₁×#424	1	1	yes	characteristic
168	C₂²	C₃₁×#424	1	1	yes	characteristic
169	C₂²	C₃₁×#428	1	2	yes	—
170	C₂²	C₂×C₃₁×#108	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	C₃₁×#424	1	4	yes	—
175	C₄	C₃₁×#496	1	1	yes	characteristic
176	C₄	C₃₁×#497	1	1	yes	characteristic
177	C₄	C₃₁×#501	1	2	yes	—
178	C₂	C₃₁×#1573	1	2	yes	—
179	C₂	C₃₁×#1574	1	2	yes	—
180	C₂	C₃₁×#1843	1	1	yes	characteristic
181	C₂	C₃₁×#1847	1	1	yes	characteristic
182	C₂	C₃₁×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
54. #8812	C₃₁	direct (N × H)
124. C₃₁	#8812	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×#1842	2
C₃₁×#1831	2
C₃₁×#1744	2
C₂×C₃₁×#335	2
C₂×C₃₁×#334	2
#8812	31

Subgroups — C31×#8812 #21465571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8812 #21465571