Subgroups — C₃₁×#2591 #16842524

Order 3968.

596 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#400	C₂	1	2	yes	maximal, complemented
4	C₃₁×#394	C₂	1	2	yes	maximal, complemented
5	C₃₁×#388	C₂	1	4	yes	maximal, complemented
6	C₃₁×#386	C₂	1	2	yes	maximal, complemented
7	C₃₁×#386	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
10	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#121	—	2	4	no	—
15	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#95	C₂²	1	2	yes	—
25	C₃₁×#95	C₂²	1	4	yes	—
26	C₃₁×#95	C₂²	1	4	yes	—
27	C₂⁴×C₃₁	C₂³	1	2	yes	—
28	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₃₁	—	2	4	no	—
30	C₂×D₄×C₃₁	—	2	8	no	—
31	C₂×D₄×C₃₁	—	2	2	no	—
32	C₂×D₄×C₃₁	—	4	2	no	—
33	C₂²×C₄×C₃₁	—	2	2	no	—
34	C₂²×C₄×C₃₁	—	2	4	no	—
35	C₂²×C₄×C₃₁	—	2	2	no	—
36	C₂²×C₄×C₃₁	—	4	2	no	—
37	C₂²×C₄×C₃₁	—	4	2	no	—
38	C₂²×C₄×C₃₁	D₄	1	4	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂²⋊C₄×C₃₁	—	2	4	no	—
49	C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂²⋊C₄×C₃₁	—	2	8	no	—
51	C₂²⋊C₄×C₃₁	—	4	4	no	—
52	C₄²×C₃₁	—	4	2	no	—
53	C₂³×C₃₁	—	2	2	no	—
54	C₂³×C₃₁	—	2	2	no	—
55	C₂³×C₃₁	—	2	2	no	—
56	C₂³×C₃₁	—	2	4	no	—
57	C₂³×C₃₁	—	2	4	no	—
58	C₂³×C₃₁	—	2	4	no	—
59	C₂³×C₃₁	C₂×D₄	1	2	yes	—
60	C₂³×C₃₁	#41	1	4	yes	—
61	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
62	D₄×C₃₁	—	4	4	no	—
63	D₄×C₃₁	—	4	8	no	—
64	C₂×C₄×C₃₁	—	2	8	no	—
65	C₂×C₄×C₃₁	—	2	2	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	—	2	4	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	2	4	no	—
76	C₂×C₄×C₃₁	—	4	4	no	—
77	C₂×C₄×C₃₁	—	4	4	no	—
78	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₃₁	#41	1	4	yes	—
81	#2591	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₃₁	—	2	2	no	—
83	C₂²×C₃₁	—	2	2	no	—
84	C₂²×C₃₁	—	2	8	no	—
85	C₂²×C₃₁	—	4	2	no	—
86	C₂²×C₃₁	—	4	2	no	—
87	C₂²×C₃₁	—	4	2	no	—
88	C₂²×C₃₁	—	4	8	no	—
89	C₂²×C₃₁	#124	1	4	yes	—
90	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#41	1	2	yes	—
92	C₂²×C₃₁	C₂×#41	1	2	yes	—
93	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
94	C₄×C₃₁	—	2	4	no	—
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	—	2	2	no	—
97	C₄×C₃₁	—	4	2	no	—
98	C₄×C₃₁	—	4	2	no	—
99	C₄×C₃₁	—	4	4	no	—
100	C₂×#121	C₂×C₃₁	1	1	yes	characteristic
101	C₂²×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
102	#400	C₂×C₃₁	1	2	yes	complemented
103	#394	C₂×C₃₁	1	2	yes	complemented
104	#388	C₂×C₃₁	1	4	yes	complemented
105	#386	C₂×C₃₁	1	2	yes	complemented
106	#386	C₂×C₃₁	1	2	yes	complemented
107	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×C₃₁	—	2	4	no	—
109	C₂×C₃₁	—	8	2	no	—
110	C₂×C₃₁	#525	1	1	yes	characteristic
111	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
112	C₂×C₃₁	#534	1	1	yes	characteristic
113	C₂×C₃₁	#553	1	4	yes	—
114	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
115	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
116	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
117	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
118	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
119	#121	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
128	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
129	#95	C₂²×C₃₁	1	2	yes	—
130	#95	C₂²×C₃₁	1	4	yes	—
131	#95	C₂²×C₃₁	1	4	yes	—
132	C₃₁	#2591	1	1	yes	characteristic, direct product factor, Sylow
133	C₂⁴	C₂³×C₃₁	1	2	yes	—
134	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	2	8	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	D₄×C₃₁	1	4	yes	—
145	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
146	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
147	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
148	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₂²⋊C₄	—	4	4	no	—
158	C₄²	—	4	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	4	no	—
163	C₂³	—	2	4	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×D₄×C₃₁	1	2	yes	—
166	C₂³	#41×C₃₁	1	4	yes	—
167	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
168	D₄	—	4	4	no	—
169	D₄	—	4	8	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
186	C₂×C₄	#41×C₃₁	1	4	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	8	no	—
194	C₂²	C₃₁×#124	1	4	yes	—
195	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
196	C₂²	C₂×#41×C₃₁	1	2	yes	—
197	C₂²	C₂×#41×C₃₁	1	2	yes	—
198	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₂	—	2	4	no	—
206	C₂	—	8	2	no	—
207	C₂	C₃₁×#525	1	1	yes	characteristic
208	C₂	C₂×C₃₁×#124	1	1	yes	characteristic
209	C₂	C₃₁×#534	1	1	yes	characteristic
210	C₂	C₃₁×#553	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₃₁	C₂	semidirect
3. C₃₁×#400	C₂	semidirect
4. C₃₁×#394	C₂	semidirect
5. C₃₁×#388	C₂	semidirect
6. C₃₁×#386	C₂	semidirect
7. C₃₁×#386	C₂	semidirect
8. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
81. #2591	C₃₁	direct (N × H)
101. C₂²×C₄⋊C₄	C₂×C₃₁	semidirect
102. #400	C₂×C₃₁	semidirect
103. #394	C₂×C₃₁	semidirect
104. #388	C₂×C₃₁	semidirect
105. #386	C₂×C₃₁	semidirect
106. #386	C₂×C₃₁	semidirect
107. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
132. C₃₁	#2591	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#121	2
C₂²×C₄⋊C₄×C₃₁	2
C₃₁×#400	2
C₃₁×#394	2
C₃₁×#388	2
C₃₁×#386	2
C₄×C₂²⋊C₄×C₃₁	2
#2591	31

Subgroups — C31×#2591 #16842524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2591 #16842524