Subgroups — C₃₁×#14713 #21471472

Order 7936.

3342 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#390	C₂	1	12	yes	maximal, complemented
3	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₄×C₃₁	C₂²	1	6	yes	—
6	C₂×C₃₁×#127	—	2	4	no	—
7	C₂×C₃₁×#127	—	2	4	no	—
8	C₂×C₃₁×#127	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
10	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	6	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#390	—	2	48	no	—
14	C₃₁×#384	—	2	4	no	—
15	C₄³×C₃₁	—	2	4	no	—
16	C₂⁵×C₃₁	C₂³	1	4	yes	—
17	C₂²×D₄×C₃₁	—	2	12	no	—
18	C₂²×D₄×C₃₁	—	2	12	no	—
19	C₂²×D₄×C₃₁	—	2	24	no	—
20	C₂²×D₄×C₃₁	—	2	24	no	—
21	C₂²×D₄×C₃₁	C₂×C₄	1	24	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
24	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
25	C₃₁×#127	—	2	32	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
28	C₂×C₂²⋊C₄×C₃₁	—	2	96	no	—
29	C₂×C₄²×C₃₁	—	2	12	no	—
30	C₂×C₄²×C₃₁	—	2	12	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₃₁	D₄	1	4	yes	—
33	C₂×C₄²×C₃₁	D₄	1	4	yes	—
34	C₂×C₄²×C₃₁	D₄	1	24	yes	complemented
35	C₂⁴×C₃₁	—	2	24	no	—
36	C₂⁴×C₃₁	—	2	4	no	—
37	C₂⁴×C₃₁	—	2	4	no	—
38	C₂⁴×C₃₁	—	2	24	no	—
39	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
40	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
41	C₂×D₄×C₃₁	—	2	48	no	—
42	C₂×D₄×C₃₁	—	2	96	no	—
43	C₂×D₄×C₃₁	—	4	48	no	—
44	C₂×D₄×C₃₁	—	4	24	no	—
45	C₂×D₄×C₃₁	—	4	24	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	—	2	12	no	—
48	C₂²×C₄×C₃₁	—	2	24	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₃₁	C₂×D₄	1	24	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₃₁	#41	1	12	yes	—
56	C₄⋊C₄×C₃₁	—	4	24	no	—
57	C₂²⋊C₄×C₃₁	—	4	64	no	—
58	C₄²×C₃₁	—	2	48	no	—
59	C₄²×C₃₁	C₂²⋊C₄	1	16	yes	complemented
60	#14713	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	—	2	8	no	—
62	C₂³×C₃₁	—	2	48	no	—
63	C₂³×C₃₁	—	4	24	no	—
64	C₂³×C₃₁	—	4	12	no	—
65	C₂³×C₃₁	—	4	16	no	—
66	C₂³×C₃₁	—	4	12	no	—
67	C₂³×C₃₁	—	4	48	no	—
68	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
69	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
70	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
71	C₂³×C₃₁	C₂²×D₄	1	6	yes	—
72	C₂³×C₃₁	C₂×#41	1	6	yes	—
73	D₄×C₃₁	—	4	96	no	—
74	C₂×C₄×C₃₁	—	2	24	no	—
75	C₂×C₄×C₃₁	—	2	24	no	—
76	C₂×C₄×C₃₁	—	4	16	no	—
77	C₂×C₄×C₃₁	—	4	12	no	—
78	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
79	C₂×C₄×C₃₁	C₄×D₄	1	24	yes	—
80	C₂×C₄×C₃₁	#121	1	24	yes	—
81	C₂×C₄×C₃₁	#124	1	12	yes	—
82	C₂×C₄×C₃₁	#127	1	12	yes	—
83	C₂×C₄×C₃₁	#127	1	16	yes	complemented
84	C₂²×#127	C₂×C₃₁	1	1	yes	characteristic
85	C₂×#390	C₂×C₃₁	1	12	yes	complemented
86	C₂×#384	C₂×C₃₁	1	1	yes	characteristic, complemented
87	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic, complemented
88	C₂²×C₃₁	—	4	32	no	—
89	C₂²×C₃₁	—	4	24	no	—
90	C₂²×C₃₁	—	8	16	no	—
91	C₂²×C₃₁	—	8	12	no	—
92	C₂²×C₃₁	—	8	4	no	—
93	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₃₁	C₂×C₄×D₄	1	6	yes	—
95	C₂²×C₃₁	C₂×#121	1	6	yes	—
96	C₂²×C₃₁	C₂×#124	1	3	yes	—
97	C₂²×C₃₁	C₂×#127	1	4	yes	—
98	C₂²×C₃₁	C₂×#127	1	3	yes	—
99	C₂²×C₃₁	#532	1	12	yes	—
100	C₄×C₃₁	—	2	16	no	—
101	C₄×C₃₁	—	8	4	no	—
102	C₄×C₃₁	#390	1	24	yes	complemented
103	C₂³×D₄	C₂²×C₃₁	1	6	yes	—
104	C₂×#127	—	2	4	no	—
105	C₂×#127	—	2	4	no	—
106	C₂×#127	C₄×C₃₁	1	8	yes	complemented
107	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
108	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	—
109	C₂²×C₄²	C₂²×C₃₁	1	6	yes	—
110	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
111	#390	—	2	48	no	—
112	#384	—	2	4	no	—
113	C₄³	—	2	4	no	—
114	C₂×C₃₁	—	8	8	no	—
115	C₂×C₃₁	C₂×#390	1	3	yes	—
116	C₂×C₃₁	C₄×#127	1	4	yes	—
117	C₂×C₃₁	#2856	1	4	yes	—
118	C₂×C₃₁	#2858	1	3	yes	—
119	C₂×C₃₁	#2867	1	1	yes	characteristic
120	C₂⁵	C₂³×C₃₁	1	4	yes	—
121	C₂²×D₄	—	2	12	no	—
122	C₂²×D₄	—	2	12	no	—
123	C₂²×D₄	—	2	24	no	—
124	C₂²×D₄	—	2	24	no	—
125	C₂²×D₄	C₂×C₄×C₃₁	1	24	yes	—
126	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
127	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
128	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
129	#127	—	2	32	no	—
130	C₂×C₄⋊C₄	—	2	12	no	—
131	C₂×C₄⋊C₄	—	2	24	no	—
132	C₂×C₂²⋊C₄	—	2	96	no	—
133	C₂×C₄²	—	2	12	no	—
134	C₂×C₄²	—	2	12	no	—
135	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
136	C₂×C₄²	D₄×C₃₁	1	4	yes	—
137	C₂×C₄²	D₄×C₃₁	1	4	yes	—
138	C₂×C₄²	D₄×C₃₁	1	24	yes	complemented
139	C₃₁	#14713	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁴	—	2	24	no	—
141	C₂⁴	—	2	4	no	—
142	C₂⁴	—	2	4	no	—
143	C₂⁴	—	2	24	no	—
144	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
145	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
146	C₂×D₄	—	2	48	no	—
147	C₂×D₄	—	2	96	no	—
148	C₂×D₄	—	4	48	no	—
149	C₂×D₄	—	4	24	no	—
150	C₂×D₄	—	4	24	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	12	no	—
153	C₂²×C₄	—	2	24	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
156	C₂²×C₄	C₂×D₄×C₃₁	1	24	yes	—
157	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
158	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
159	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
160	C₂²×C₄	#41×C₃₁	1	12	yes	—
161	C₄⋊C₄	—	4	24	no	—
162	C₂²⋊C₄	—	4	64	no	—
163	C₄²	—	2	48	no	—
164	C₄²	C₂²⋊C₄×C₃₁	1	16	yes	complemented
165	C₂³	—	2	8	no	—
166	C₂³	—	2	48	no	—
167	C₂³	—	4	24	no	—
168	C₂³	—	4	12	no	—
169	C₂³	—	4	16	no	—
170	C₂³	—	4	12	no	—
171	C₂³	—	4	48	no	—
172	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
173	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
174	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
175	C₂³	C₂²×D₄×C₃₁	1	6	yes	—
176	C₂³	C₂×#41×C₃₁	1	6	yes	—
177	D₄	—	4	96	no	—
178	C₂×C₄	—	2	24	no	—
179	C₂×C₄	—	2	24	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	—	4	12	no	—
182	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
183	C₂×C₄	C₄×D₄×C₃₁	1	24	yes	—
184	C₂×C₄	C₃₁×#121	1	24	yes	—
185	C₂×C₄	C₃₁×#124	1	12	yes	—
186	C₂×C₄	C₃₁×#127	1	12	yes	—
187	C₂×C₄	C₃₁×#127	1	16	yes	complemented
188	C₂²	—	4	24	no	—
189	C₂²	—	4	32	no	—
190	C₂²	—	8	16	no	—
191	C₂²	—	8	12	no	—
192	C₂²	—	8	4	no	—
193	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
194	C₂²	C₂×C₄×D₄×C₃₁	1	6	yes	—
195	C₂²	C₂×C₃₁×#121	1	6	yes	—
196	C₂²	C₂×C₃₁×#124	1	3	yes	—
197	C₂²	C₂×C₃₁×#127	1	4	yes	—
198	C₂²	C₂×C₃₁×#127	1	3	yes	—
199	C₂²	C₃₁×#532	1	12	yes	—
200	C₄	—	2	16	no	—
201	C₄	—	8	4	no	—
202	C₄	C₃₁×#390	1	24	yes	complemented
203	C₂	—	8	8	no	—
204	C₂	C₂×C₃₁×#390	1	3	yes	—
205	C₂	C₄×C₃₁×#127	1	4	yes	—
206	C₂	C₃₁×#2856	1	4	yes	—
207	C₂	C₃₁×#2858	1	3	yes	—
208	C₂	C₃₁×#2867	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
prime radical	C₂²×C₃₁×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#390	C₂	semidirect
3. C₂×C₃₁×#384	C₂	semidirect
4. C₂×C₄³×C₃₁	C₂	semidirect
8. C₂×C₃₁×#127	C₄	semidirect
34. C₂×C₄²×C₃₁	D₄	semidirect
59. C₄²×C₃₁	C₂²⋊C₄	semidirect
60. #14713	C₃₁	direct (N × H)
83. C₂×C₄×C₃₁	#127	semidirect
85. C₂×#390	C₂×C₃₁	semidirect
86. C₂×#384	C₂×C₃₁	semidirect
87. C₂×C₄³	C₂×C₃₁	semidirect
102. C₄×C₃₁	#390	semidirect
106. C₂×#127	C₄×C₃₁	semidirect
138. C₂×C₄²	D₄×C₃₁	semidirect
139. C₃₁	#14713	direct (N × H)
164. C₄²	C₂²⋊C₄×C₃₁	semidirect
187. C₂×C₄	C₃₁×#127	semidirect
202. C₄	C₃₁×#390	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#127	2
C₂×C₃₁×#390	2
C₂×C₃₁×#384	2
C₂×C₄³×C₃₁	2
#14713	31

Subgroups — C31×#14713 #21471472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14713 #21471472