Subgroups — C₃₁×#20768 #21477527

Order 7936.

778 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2905	C₂	1	2	yes	maximal
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1838	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1837	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1738	C₂	1	2	yes	maximal
6	C₂×C₃₁×#341	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₃₁×#424	—	2	8	no	—
12	C₃₁×#423	C₂²	1	8	yes	—
13	C₃₁×#405	C₄	1	8	yes	—
14	C₂×C₃₁×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#341	C₂²	1	4	yes	—
17	C₃₁×#341	C₂²	1	8	yes	—
18	C₃₁×#341	C₂²	1	4	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	—	2	4	no	—
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	—	2	8	no	—
23	C₂×M₁₆×C₃₁	—	2	8	no	—
24	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
25	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
26	C₂²×C₈×C₃₁	—	2	4	no	—
27	C₂²×C₈×C₃₁	—	2	4	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#117	C₂×C₄	1	4	yes	—
33	C₃₁×#117	D₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₃₁	Q₈	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
37	C₃₁×C₄⋊C₈	—	2	16	no	—
38	C₃₁×#97	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
40	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	#41	1	4	yes	—
46	C₂²×C₄×C₃₁	#41	1	4	yes	—
47	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	M₁₆×C₃₁	—	2	8	no	—
49	M₁₆×C₃₁	—	2	8	no	—
50	M₁₆×C₃₁	—	2	8	no	—
51	M₁₆×C₃₁	—	4	8	no	—
52	C₂×C₈×C₃₁	—	2	8	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	—	2	4	no	—
56	C₂×C₈×C₃₁	—	2	8	no	—
57	C₂×C₈×C₃₁	—	4	8	no	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
61	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
62	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
63	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
64	C₄²×C₃₁	C₂×D₄	1	4	yes	—
65	#20768	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	—	4	2	no	—
67	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₃₁	#125	1	4	yes	—
69	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₃₁	—	4	4	no	—
71	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
72	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₃₁	#117	1	4	yes	—
74	C₂×C₄×C₃₁	#117	1	4	yes	—
75	C₂×C₄×C₃₁	#124	1	8	yes	—
76	C₂×C₄×C₃₁	#125	1	4	yes	—
77	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
81	C₈×C₃₁	—	2	2	no	—
82	C₈×C₃₁	—	2	2	no	—
83	C₈×C₃₁	—	2	4	no	—
84	C₈×C₃₁	—	4	8	no	—
85	#2905	C₂×C₃₁	1	2	yes	—
86	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
87	#1838	C₂×C₃₁	1	4	yes	complemented
88	#1837	C₂×C₃₁	1	4	yes	complemented
89	#1738	C₂×C₃₁	1	2	yes	—
90	C₂×#341	C₂×C₃₁	1	2	yes	—
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	—	2	4	no	—
93	C₂²×C₃₁	—	4	4	no	—
94	C₂²×C₃₁	#382	1	4	yes	—
95	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃₁	C₂×#125	1	2	yes	—
97	C₄×C₃₁	—	2	8	no	—
98	C₄×C₃₁	#382	1	4	yes	—
99	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
100	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
101	C₄×C₃₁	C₂×#124	1	2	yes	—
102	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
103	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
104	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
106	#424	—	2	8	no	—
107	#423	C₂²×C₃₁	1	8	yes	—
108	#405	C₄×C₃₁	1	8	yes	—
109	C₂×#97	C₂²×C₃₁	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
111	#341	C₂²×C₃₁	1	4	yes	—
112	#341	C₂²×C₃₁	1	8	yes	—
113	#341	C₂²×C₃₁	1	4	yes	—
114	C₂×C₃₁	—	2	2	no	—
115	C₂×C₃₁	—	2	4	no	—
116	C₂×C₃₁	C₂×#382	1	1	yes	characteristic
117	C₂×C₃₁	#2916	1	2	yes	—
118	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
119	C₂×M₁₆	—	2	4	no	—
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	2	8	no	—
122	C₂×M₁₆	—	2	8	no	—
123	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
124	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
128	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
131	#117	C₂×C₄×C₃₁	1	4	yes	—
132	#117	D₄×C₃₁	1	4	yes	—
133	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
134	C₂×C₂²⋊C₄	Q₈×C₃₁	1	2	yes	—
135	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
136	C₄⋊C₈	—	2	16	no	—
137	#97	C₂×C₄×C₃₁	1	8	yes	—
138	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
139	C₃₁	#20768	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
143	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
144	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
145	C₂²×C₄	#41×C₃₁	1	4	yes	—
146	C₂²×C₄	#41×C₃₁	1	4	yes	—
147	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
148	M₁₆	—	2	8	no	—
149	M₁₆	—	2	8	no	—
150	M₁₆	—	2	8	no	—
151	M₁₆	—	4	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	4	8	no	—
158	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
160	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
161	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
162	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
163	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
164	C₄²	C₂×D₄×C₃₁	1	4	yes	—
165	C₂³	—	4	2	no	—
166	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
167	C₂³	C₃₁×#125	1	4	yes	—
168	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
172	C₂×C₄	C₃₁×#117	1	4	yes	—
173	C₂×C₄	C₃₁×#117	1	4	yes	—
174	C₂×C₄	C₃₁×#124	1	8	yes	—
175	C₂×C₄	C₃₁×#125	1	4	yes	—
176	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
179	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	4	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₃₁×#382	1	4	yes	—
188	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂²	C₂×C₃₁×#125	1	2	yes	—
190	C₄	—	2	8	no	—
191	C₄	C₃₁×#382	1	4	yes	—
192	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
193	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
194	C₄	C₂×C₃₁×#124	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂×C₃₁×#382	1	1	yes	characteristic
198	C₂	C₃₁×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1838	C₂	semidirect
4. C₃₁×#1837	C₂	semidirect
65. #20768	C₃₁	direct (N × H)
87. #1838	C₂×C₃₁	semidirect
88. #1837	C₂×C₃₁	semidirect
139. C₃₁	#20768	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×#2905	2
C₂×C₃₁×#405	2
C₃₁×#1838	2
C₃₁×#1837	2
C₃₁×#1738	2
C₂×C₃₁×#341	2
#20768	31

Subgroups — C31×#20768 #21477527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20768 #21477527