Subgroups — C₃₁×#9634 #21466393

Order 7936.

268 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1574	C₂	1	2	yes	maximal
2	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1299	C₂	1	2	yes	maximal
6	C₃₁×#431	C₂²	1	2	yes	—
7	C₃₁×#431	C₂²	1	1	yes	characteristic
8	C₃₁×#423	C₂²	1	2	yes	—
9	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#335	—	2	2	no	—
12	C₃₁×#335	—	2	1	no	—
13	C₃₁×#335	—	2	1	no	—
14	C₃₁×#335	C₄	1	2	yes	—
15	C₃₁×#334	—	2	2	no	—
16	C₃₁×#334	C₄	1	2	yes	—
17	C₃₁×#322	—	2	1	no	—
18	C₈²×C₃₁	—	2	1	no	—
19	C₂×M₁₆×C₃₁	—	2	1	no	—
20	C₂×M₁₆×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
24	C₃₁×C₄⋊C₈	—	2	2	no	—
25	C₃₁×C₄⋊C₈	—	2	2	no	—
26	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
27	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
28	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
29	C₃₁×C₂²⋊C₈	—	2	2	no	—
30	C₃₁×C₂²⋊C₈	—	2	1	no	—
31	C₃₁×#97	Q₈	1	2	yes	—
32	C₄×C₈×C₃₁	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₄×C₈×C₃₁	—	2	2	no	—
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
42	M₁₆×C₃₁	—	2	4	no	—
43	M₁₆×C₃₁	—	8	2	no	—
44	C₂×C₈×C₃₁	—	2	2	no	—
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	2	4	no	—
47	C₂×C₈×C₃₁	—	2	1	no	—
48	C₂×C₈×C₃₁	—	2	1	no	—
49	C₂×C₈×C₃₁	—	4	1	no	—
50	C₂×C₈×C₃₁	—	4	2	no	—
51	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	#9634	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₃₁	#125	1	1	yes	characteristic
61	C₂×C₄×C₃₁	—	2	1	no	—
62	C₂×C₄×C₃₁	—	2	1	no	—
63	C₂×C₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	1	no	—
66	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₈×C₃₁	—	2	4	no	—
73	C₈×C₃₁	—	2	1	no	—
74	C₈×C₃₁	—	2	1	no	—
75	C₈×C₃₁	—	2	1	no	—
76	C₈×C₃₁	—	4	4	no	—
77	C₈×C₃₁	—	4	1	no	—
78	C₈×C₃₁	—	4	1	no	—
79	C₈×C₃₁	—	8	2	no	—
80	C₈×C₃₁	#108	1	2	yes	—
81	#1574	C₂×C₃₁	1	2	yes	—
82	#1567	C₂×C₃₁	1	1	yes	characteristic
83	#1564	C₂×C₃₁	1	1	yes	characteristic
84	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic
85	#1299	C₂×C₃₁	1	2	yes	—
86	C₂²×C₃₁	—	2	1	no	—
87	C₂²×C₃₁	—	2	1	no	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	#389	1	1	yes	characteristic
90	C₄×C₃₁	—	2	1	no	—
91	C₄×C₃₁	—	2	1	no	—
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
94	C₄×C₃₁	#430	1	1	yes	characteristic
95	C₄×C₃₁	#443	1	2	yes	—
96	C₄×C₃₁	#472	1	1	yes	characteristic
97	C₄×C₃₁	#473	1	1	yes	characteristic
98	#431	C₂²×C₃₁	1	2	yes	—
99	#431	C₂²×C₃₁	1	1	yes	characteristic
100	#423	C₂²×C₃₁	1	2	yes	—
101	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
103	#335	—	2	2	no	—
104	#335	—	2	1	no	—
105	#335	—	2	1	no	—
106	#335	C₄×C₃₁	1	2	yes	—
107	#334	—	2	2	no	—
108	#334	C₄×C₃₁	1	2	yes	—
109	#322	—	2	1	no	—
110	C₈²	—	2	1	no	—
111	C₂×C₃₁	—	4	1	no	—
112	C₂×C₃₁	#1945	1	1	yes	characteristic
113	C₂×C₃₁	#1950	1	1	yes	characteristic
114	C₂×C₃₁	#1952	1	1	yes	characteristic
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
118	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
123	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
124	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
125	C₂²⋊C₈	—	2	2	no	—
126	C₂²⋊C₈	—	2	1	no	—
127	#97	Q₈×C₃₁	1	2	yes	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
132	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
133	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
134	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
135	C₃₁	#9634	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	M₁₆	—	8	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
149	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
150	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
152	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
153	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
154	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
155	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
156	C₂³	C₃₁×#125	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
166	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	4	4	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	8	2	no	—
176	C₈	C₃₁×#108	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	C₃₁×#389	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
185	C₄	C₃₁×#430	1	1	yes	characteristic
186	C₄	C₃₁×#443	1	2	yes	—
187	C₄	C₃₁×#472	1	1	yes	characteristic
188	C₄	C₃₁×#473	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	C₃₁×#1945	1	1	yes	characteristic
191	C₂	C₃₁×#1950	1	1	yes	characteristic
192	C₂	C₃₁×#1952	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#389
prime radical	C₂³×C₃₁	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
59. #9634	C₃₁	direct (N × H)
135. C₃₁	#9634	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×#1574	2
C₃₁×#1567	2
C₃₁×#1564	2
C₈×M₁₆×C₃₁	2
C₃₁×#1299	2
#9634	31

Subgroups — C31×#9634 #21466393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9634 #21466393