Subgroups — C₃₁×#13286 #21470045

Order 7936.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1382	C₂	1	4	yes	maximal
5	C₃₁×#533	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	2	yes	—
8	C₃₁×#499	—	2	1	no	—
9	C₃₁×#499	C₄	1	2	yes	—
10	C₃₁×#498	—	2	1	no	—
11	C₂×C₃₁×#108	—	2	1	no	—
12	C₂×C₃₁×#108	C₂²	1	2	yes	—
13	C₃₁×#427	—	2	1	no	—
14	C₃₁×#427	C₂²	1	2	yes	—
15	C₃₁×#423	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#346	—	2	1	no	—
18	C₃₁×#346	—	2	1	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	—	2	2	no	—
23	C₃₁×#125	—	2	2	no	—
24	C₃₁×C₂²⋊Q₈	—	4	2	no	—
25	C₄×Q₈×C₃₁	—	4	2	no	—
26	C₃₁×#117	—	2	2	no	—
27	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₃₁	—	2	2	no	—
29	C₂×C₁₆×C₃₁	—	2	2	no	—
30	C₂×C₁₆×C₃₁	Q₈	1	4	yes	complemented
31	C₃₁×#108	—	2	4	no	—
32	C₃₁×#108	—	4	2	no	—
33	C₃₁×#107	—	2	2	no	—
34	C₃₁×#107	C₂×C₄	1	4	yes	—
35	C₃₁×#106	—	2	2	no	—
36	C₃₁×#106	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	4	2	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	—	4	2	no	—
43	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₃₁	—	4	2	no	—
46	M₁₆×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
51	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
52	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₃₁	#41	1	2	yes	—
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	4	4	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₂²⋊C₄×C₃₁	—	4	2	no	—
60	C₄²×C₃₁	—	4	2	no	—
61	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₃₁	D₈	1	2	yes	—
64	C₁₆×C₃₁	—	2	4	no	—
65	#13286	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₃₁	—	8	4	no	—
68	C₂×C₄×C₃₁	—	4	2	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	#102	1	2	yes	—
71	C₂×C₄×C₃₁	#102	1	2	yes	—
72	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
78	C₈×C₃₁	—	8	2	no	—
79	C₈×C₃₁	#103	1	2	yes	—
80	C₈×C₃₁	#103	1	2	yes	—
81	C₈×C₃₁	C₄×Q₈	1	2	yes	—
82	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
83	#3160	C₂×C₃₁	1	1	yes	characteristic
84	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
85	#1848	C₂×C₃₁	1	1	yes	characteristic
86	#1382	C₂×C₃₁	1	4	yes	—
87	C₂²×C₃₁	—	2	2	no	—
88	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
89	C₂²×C₃₁	#487	1	1	yes	characteristic
90	C₂²×C₃₁	#498	1	1	yes	characteristic
91	C₄×C₃₁	—	2	2	no	—
92	C₄×C₃₁	—	2	2	no	—
93	C₄×C₃₁	—	8	4	no	—
94	C₄×C₃₁	#391	1	1	yes	characteristic
95	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
96	C₄×C₃₁	#486	1	1	yes	characteristic
97	C₄×C₃₁	#500	1	1	yes	characteristic
98	#533	—	2	1	no	—
99	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
100	#500	C₄×C₃₁	1	2	yes	—
101	#499	—	2	1	no	—
102	#499	C₄×C₃₁	1	2	yes	—
103	#498	—	2	1	no	—
104	C₂×#108	—	2	1	no	—
105	C₂×#108	C₂²×C₃₁	1	2	yes	—
106	#427	—	2	1	no	—
107	#427	C₂²×C₃₁	1	2	yes	—
108	#423	—	2	1	no	—
109	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
110	#346	—	2	1	no	—
111	#346	—	2	1	no	—
112	C₂×C₃₁	—	2	1	no	—
113	C₂×C₃₁	—	2	1	no	—
114	C₂×C₃₁	#1990	1	1	yes	characteristic
115	C₂×C₃₁	#2143	1	2	yes	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
119	C₄⋊Q₈	—	2	2	no	—
120	#125	—	2	2	no	—
121	C₂²⋊Q₈	—	4	2	no	—
122	C₄×Q₈	—	4	2	no	—
123	#117	—	2	2	no	—
124	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	Q₈×C₃₁	1	4	yes	complemented
128	#108	—	2	4	no	—
129	#108	—	4	2	no	—
130	#107	—	2	2	no	—
131	#107	C₂×C₄×C₃₁	1	4	yes	—
132	#106	—	2	2	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
138	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
139	C₃₁	#13286	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×Q₈	—	4	2	no	—
141	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
148	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
149	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
153	C₂×C₈	#41×C₃₁	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
160	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
161	C₄²	D₈×C₃₁	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	C₃₁×#102	1	2	yes	—
168	C₂×C₄	C₃₁×#102	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
171	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
175	C₈	—	8	2	no	—
176	C₈	C₃₁×#103	1	2	yes	—
177	C₈	C₃₁×#103	1	2	yes	—
178	C₈	C₄×Q₈×C₃₁	1	2	yes	—
179	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
182	C₂²	C₃₁×#487	1	1	yes	characteristic
183	C₂²	C₃₁×#498	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	8	4	no	—
187	C₄	C₃₁×#391	1	1	yes	characteristic
188	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
189	C₄	C₃₁×#486	1	1	yes	characteristic
190	C₄	C₃₁×#500	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	C₃₁×#1990	1	1	yes	characteristic
194	C₂	C₃₁×#2143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#102
prime radical	C₂³×C₃₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₆×C₃₁	Q₈	semidirect
65. #13286	C₃₁	direct (N × H)
127. C₂×C₁₆	Q₈×C₃₁	semidirect
139. C₃₁	#13286	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×#3160	2
C₂×C₃₁×#346	2
C₃₁×#1848	2
C₃₁×#1382	2
#13286	31

Subgroups — C31×#13286 #21470045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13286 #21470045