Subgroups — C₃₁×#19667 #21476426

Order 7936.

780 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3207	C₂	1	4	yes	maximal, complemented
2	C₃₁×#3169	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1717	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1706	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#532	C₂²	1	4	yes	—
9	C₃₁×#500	C₂²	1	2	yes	—
10	C₃₁×#493	C₂²	1	2	yes	—
11	C₃₁×#493	C₂²	1	4	yes	—
12	C₃₁×#492	C₂²	1	4	yes	—
13	C₃₁×#487	C₂²	1	4	yes	—
14	C₃₁×#486	C₂²	1	4	yes	—
15	C₃₁×#486	C₂²	1	4	yes	—
16	C₃₁×#466	—	2	4	no	—
17	C₃₁×#463	—	2	8	no	—
18	C₃₁×#449	—	2	8	no	—
19	C₄×D₈×C₃₁	—	2	4	no	—
20	C₃₁×#431	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#322	C₂²	1	2	yes	—
23	C₈²×C₃₁	C₂²	1	1	yes	characteristic
24	C₈²×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×#41×C₃₁	—	2	4	no	—
26	C₂×SD₁₆×C₃₁	—	2	8	no	—
27	C₂×D₈×C₃₁	—	2	4	no	—
28	C₂×D₈×C₃₁	—	2	8	no	—
29	C₂²×C₈×C₃₁	D₄	1	2	yes	—
30	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
31	C₃₁×#127	C₂³	1	4	yes	—
32	C₃₁×#124	—	2	4	no	—
33	C₃₁×#121	—	2	4	no	—
34	C₃₁×#121	—	2	4	no	—
35	C₄×D₄×C₃₁	—	2	4	no	—
36	C₄×D₄×C₃₁	—	2	4	no	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#107	—	2	4	no	—
39	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
40	C₃₁×#103	—	2	8	no	—
41	C₃₁×#102	—	2	8	no	—
42	C₃₁×#102	—	2	8	no	—
43	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
44	C₄×C₈×C₃₁	D₄	1	2	yes	—
45	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂³	1	2	yes	—
48	C₄×C₈×C₃₁	C₂³	1	2	yes	—
49	#41×C₃₁	—	8	8	no	—
50	C₂×Q₈×C₃₁	—	2	4	no	—
51	C₂×D₄×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	4	4	no	—
55	C₂²×C₄×C₃₁	—	4	4	no	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	SD₁₆×C₃₁	—	8	8	no	—
59	D₈×C₃₁	—	4	8	no	—
60	D₈×C₃₁	—	8	8	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₂²⋊C₄×C₃₁	—	4	4	no	—
69	C₂²⋊C₄×C₃₁	—	4	4	no	—
70	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×D₄	1	2	yes	—
72	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
73	#19667	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	—	4	4	no	—
75	C₂³×C₃₁	#127	1	1	yes	characteristic
76	Q₈×C₃₁	—	4	8	no	—
77	D₄×C₃₁	—	4	8	no	—
78	D₄×C₃₁	—	4	8	no	—
79	D₄×C₃₁	—	4	8	no	—
80	D₄×C₃₁	—	8	8	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	8	8	no	—
85	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#127	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
89	C₈×C₃₁	—	2	2	no	—
90	C₈×C₃₁	—	2	4	no	—
91	C₈×C₃₁	—	2	4	no	—
92	C₈×C₃₁	#135	1	4	yes	—
93	#3207	C₂×C₃₁	1	4	yes	complemented
94	#3169	C₂×C₃₁	1	2	yes	complemented
95	#1717	C₂×C₃₁	1	4	yes	complemented
96	#1712	C₂×C₃₁	1	1	yes	characteristic, complemented
97	#1710	C₂×C₃₁	1	1	yes	characteristic, complemented
98	#1706	C₂×C₃₁	1	2	yes	complemented
99	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	8	4	no	—
103	C₂²×C₃₁	—	8	4	no	—
104	C₂²×C₃₁	—	8	4	no	—
105	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	8	4	no	—
109	C₄×C₃₁	C₂×#135	1	2	yes	—
110	C₄×C₃₁	#576	1	4	yes	—
111	#532	C₂²×C₃₁	1	4	yes	—
112	#500	C₂²×C₃₁	1	2	yes	—
113	#493	C₂²×C₃₁	1	2	yes	—
114	#493	C₂²×C₃₁	1	4	yes	—
115	#492	C₂²×C₃₁	1	4	yes	—
116	#487	C₂²×C₃₁	1	4	yes	—
117	#486	C₂²×C₃₁	1	4	yes	—
118	#486	C₂²×C₃₁	1	4	yes	—
119	#466	—	2	4	no	—
120	#463	—	2	8	no	—
121	#449	—	2	8	no	—
122	C₄×D₈	—	2	4	no	—
123	#431	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	#322	C₂²×C₃₁	1	2	yes	—
126	C₈²	C₂²×C₃₁	1	1	yes	characteristic
127	C₈²	C₂²×C₃₁	1	1	yes	characteristic
128	C₂×C₃₁	—	4	1	no	—
129	C₂×C₃₁	—	16	4	no	—
130	C₂×C₃₁	#3147	1	1	yes	characteristic
131	C₂×C₃₁	#3155	1	2	yes	—
132	C₂×#41	—	2	4	no	—
133	C₂×SD₁₆	—	2	8	no	—
134	C₂×D₈	—	2	4	no	—
135	C₂×D₈	—	2	8	no	—
136	C₂²×C₈	D₄×C₃₁	1	2	yes	—
137	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
138	#127	C₂³×C₃₁	1	4	yes	—
139	#124	—	2	4	no	—
140	#121	—	2	4	no	—
141	#121	—	2	4	no	—
142	C₄×D₄	—	2	4	no	—
143	C₄×D₄	—	2	4	no	—
144	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
145	#107	—	2	4	no	—
146	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
147	#103	—	2	8	no	—
148	#102	—	2	8	no	—
149	#102	—	2	8	no	—
150	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
151	C₄×C₈	D₄×C₃₁	1	2	yes	—
152	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₃₁	1	2	yes	—
154	C₄×C₈	C₂³×C₃₁	1	2	yes	—
155	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
156	C₃₁	#19667	1	1	yes	characteristic, direct product factor, Sylow
157	#41	—	8	8	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
165	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
166	SD₁₆	—	8	8	no	—
167	D₈	—	4	8	no	—
168	D₈	—	8	8	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
172	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
173	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	4	4	no	—
177	C₂²⋊C₄	—	4	4	no	—
178	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₃₁	1	2	yes	—
180	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
181	C₂³	—	4	4	no	—
182	C₂³	C₃₁×#127	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	D₄	—	4	8	no	—
185	D₄	—	4	8	no	—
186	D₄	—	4	8	no	—
187	D₄	—	8	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	8	no	—
192	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
193	C₂×C₄	C₃₁×#127	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	4	no	—
198	C₈	—	2	4	no	—
199	C₈	C₃₁×#135	1	4	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	2	no	—
202	C₂²	—	8	4	no	—
203	C₂²	—	8	4	no	—
204	C₂²	—	8	4	no	—
205	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	C₂×C₃₁×#135	1	2	yes	—
210	C₄	C₃₁×#576	1	4	yes	—
211	C₂	—	4	1	no	—
212	C₂	—	16	4	no	—
213	C₂	C₃₁×#3147	1	1	yes	characteristic
214	C₂	C₃₁×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3207	C₂	semidirect
2. C₃₁×#3169	C₂	semidirect
3. C₃₁×#1717	C₂	semidirect
4. C₃₁×#1712	C₂	semidirect
5. C₃₁×#1710	C₂	semidirect
6. C₃₁×#1706	C₂	semidirect
7. C₃₁×#1450	C₂	semidirect
73. #19667	C₃₁	direct (N × H)
93. #3207	C₂×C₃₁	semidirect
94. #3169	C₂×C₃₁	semidirect
95. #1717	C₂×C₃₁	semidirect
96. #1712	C₂×C₃₁	semidirect
97. #1710	C₂×C₃₁	semidirect
98. #1706	C₂×C₃₁	semidirect
99. #1450	C₂×C₃₁	semidirect
156. C₃₁	#19667	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×#3207	2
C₃₁×#3169	2
C₃₁×#1717	2
C₃₁×#1712	2
C₃₁×#1710	2
C₃₁×#1706	2
C₃₁×#1450	2
#19667	31

Subgroups — C31×#19667 #21476426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19667 #21476426