Subgroups — C₃₁×#2561 #16842494

Order 3968.

672 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#397	C₂	1	2	yes	maximal, complemented
4	C₃₁×#393	C₂	1	2	yes	maximal, complemented
5	C₃₁×#388	C₂	1	4	yes	maximal, complemented
6	C₃₁×#381	C₂	1	2	yes	maximal, complemented
7	C₃₁×#379	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×C₂²⋊Q₈	—	2	4	no	—
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
17	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	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	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₃₁	C₂²	1	2	yes	complemented
24	C₃₁×#95	C₂²	1	2	yes	complemented
25	C₃₁×#95	C₂²	1	4	yes	complemented
26	C₃₁×#95	C₂²	1	4	yes	complemented
27	C₂⁴×C₃₁	—	2	2	no	—
28	C₂⁴×C₃₁	—	2	4	no	—
29	C₂⁴×C₃₁	—	2	4	no	—
30	C₂⁴×C₃₁	—	4	2	no	—
31	C₂⁴×C₃₁	C₂³	1	2	yes	—
32	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₃₁	—	2	2	no	—
34	C₂×Q₈×C₃₁	—	4	2	no	—
35	C₂²×C₄×C₃₁	—	2	2	no	—
36	C₂²×C₄×C₃₁	—	2	4	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	2	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂²⋊C₄×C₃₁	—	2	4	no	—
48	C₂²⋊C₄×C₃₁	—	2	8	no	—
49	C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂²⋊C₄×C₃₁	—	2	4	no	—
51	C₂²⋊C₄×C₃₁	—	2	4	no	—
52	C₂²⋊C₄×C₃₁	—	4	4	no	—
53	C₄²×C₃₁	—	4	2	no	—
54	C₂³×C₃₁	—	2	2	no	—
55	C₂³×C₃₁	—	2	2	no	—
56	C₂³×C₃₁	—	2	8	no	—
57	C₂³×C₃₁	—	2	2	no	—
58	C₂³×C₃₁	—	2	2	no	—
59	C₂³×C₃₁	—	2	4	no	—
60	C₂³×C₃₁	—	2	4	no	—
61	C₂³×C₃₁	—	4	16	no	—
62	C₂³×C₃₁	—	4	4	no	—
63	C₂³×C₃₁	—	4	4	no	—
64	C₂³×C₃₁	C₂×D₄	1	2	yes	—
65	C₂³×C₃₁	#41	1	4	yes	—
66	C₂³×C₃₁	#41	1	4	yes	—
67	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
68	Q₈×C₃₁	—	4	4	no	—
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	2	4	no	—
76	C₂×C₄×C₃₁	—	2	4	no	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
84	#2561	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₃₁	—	2	2	no	—
86	C₂²×C₃₁	—	2	2	no	—
87	C₂²×C₃₁	—	2	8	no	—
88	C₂²×C₃₁	—	2	4	no	—
89	C₂²×C₃₁	—	2	8	no	—
90	C₂²×C₃₁	—	4	16	no	—
91	C₂²×C₃₁	—	4	2	no	—
92	C₂²×C₃₁	—	4	2	no	—
93	C₂²×C₃₁	—	4	4	no	—
94	C₂²×C₃₁	#124	1	4	yes	—
95	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₂²×C₃₁	C₂×#41	1	2	yes	—
97	C₂²×C₃₁	C₂×#41	1	2	yes	—
98	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
99	C₄×C₃₁	—	2	2	no	—
100	C₄×C₃₁	—	2	2	no	—
101	C₄×C₃₁	—	4	2	no	—
102	C₄×C₃₁	—	4	2	no	—
103	C₄×C₃₁	—	4	2	no	—
104	C₄×C₃₁	—	4	4	no	—
105	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
106	C₂²×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
107	#397	C₂×C₃₁	1	2	yes	complemented
108	#393	C₂×C₃₁	1	2	yes	complemented
109	#388	C₂×C₃₁	1	4	yes	complemented
110	#381	C₂×C₃₁	1	2	yes	complemented
111	#379	C₂×C₃₁	1	2	yes	—
112	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
113	C₂×C₃₁	—	2	4	no	—
114	C₂×C₃₁	—	4	4	no	—
115	C₂×C₃₁	#525	1	1	yes	characteristic
116	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
117	C₂×C₃₁	#534	1	1	yes	characteristic
118	C₂×C₃₁	#551	1	4	yes	—
119	C₂⁵	C₂²×C₃₁	1	1	yes	characteristic
120	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
121	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
122	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
123	C₂²⋊Q₈	—	2	4	no	—
124	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
125	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
126	C₂×C₂²⋊C₄	—	2	4	no	—
127	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
133	C₂×C₄²	C₂²×C₃₁	1	2	yes	complemented
134	#95	C₂²×C₃₁	1	2	yes	complemented
135	#95	C₂²×C₃₁	1	4	yes	complemented
136	#95	C₂²×C₃₁	1	4	yes	complemented
137	C₃₁	#2561	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴	—	2	2	no	—
139	C₂⁴	—	2	4	no	—
140	C₂⁴	—	2	4	no	—
141	C₂⁴	—	4	2	no	—
142	C₂⁴	C₂³×C₃₁	1	2	yes	—
143	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	D₄×C₃₁	1	4	yes	—
150	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
151	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
152	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
153	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
154	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	8	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	4	4	no	—
164	C₄²	—	4	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	8	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	—	2	4	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	4	16	no	—
173	C₂³	—	4	4	no	—
174	C₂³	—	4	4	no	—
175	C₂³	C₂×D₄×C₃₁	1	2	yes	—
176	C₂³	#41×C₃₁	1	4	yes	—
177	C₂³	#41×C₃₁	1	4	yes	—
178	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	4	16	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	C₃₁×#124	1	4	yes	—
205	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
206	C₂²	C₂×#41×C₃₁	1	2	yes	—
207	C₂²	C₂×#41×C₃₁	1	2	yes	—
208	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	4	no	—
215	C₂	—	2	4	no	—
216	C₂	—	4	4	no	—
217	C₂	C₃₁×#525	1	1	yes	characteristic
218	C₂	C₂×C₃₁×#124	1	1	yes	characteristic
219	C₂	C₃₁×#534	1	1	yes	characteristic
220	C₂	C₃₁×#551	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	C₂²
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
1. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#397	C₂	semidirect
4. C₃₁×#393	C₂	semidirect
5. C₃₁×#388	C₂	semidirect
6. C₃₁×#381	C₂	semidirect
8. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
10. C₂²×Q₈×C₃₁	C₂²	semidirect
14. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
15. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
23. C₂×C₄²×C₃₁	C₂²	semidirect
24. C₃₁×#95	C₂²	semidirect
25. C₃₁×#95	C₂²	semidirect
26. C₃₁×#95	C₂²	semidirect
84. #2561	C₃₁	direct (N × H)
105. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
107. #397	C₂×C₃₁	semidirect
108. #393	C₂×C₃₁	semidirect
109. #388	C₂×C₃₁	semidirect
110. #381	C₂×C₃₁	semidirect
112. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
120. C₂²×Q₈	C₂²×C₃₁	semidirect
124. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
125. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
133. C₂×C₄²	C₂²×C₃₁	semidirect
134. #95	C₂²×C₃₁	semidirect
135. #95	C₂²×C₃₁	semidirect
136. #95	C₂²×C₃₁	semidirect
137. C₃₁	#2561	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×C₂²⋊Q₈	2
C₂²×C₂²⋊C₄×C₃₁	2
C₃₁×#397	2
C₃₁×#393	2
C₃₁×#388	2
C₃₁×#381	2
C₃₁×#379	2
C₄×C₂²⋊C₄×C₃₁	2
#2561	31

Subgroups — C31×#2561 #16842494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2561 #16842494