Subgroups — C₃₁×#2563 #16842496

Order 3968.

564 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#393	C₂	1	4	yes	maximal, complemented
4	C₃₁×#390	C₂	1	2	yes	maximal
5	C₃₁×#382	C₂	1	2	yes	maximal, complemented
6	C₃₁×#380	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
9	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
11	C₃₁×C₄⋊Q₈	—	2	4	no	—
12	C₄×D₄×C₃₁	—	2	4	no	—
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
19	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#95	C₂²	1	2	yes	—
22	C₃₁×#95	C₂²	1	8	yes	—
23	C₂⁴×C₃₁	C₂³	1	2	yes	—
24	C₂×Q₈×C₃₁	—	2	4	no	—
25	C₂×Q₈×C₃₁	—	4	4	no	—
26	C₂×D₄×C₃₁	—	2	2	no	—
27	C₂×D₄×C₃₁	—	2	2	no	—
28	C₂×D₄×C₃₁	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₃₁	—	2	4	no	—
30	C₂²×C₄×C₃₁	—	2	8	no	—
31	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
33	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂²⋊C₄×C₃₁	—	2	4	no	—
42	C₂²⋊C₄×C₃₁	—	4	8	no	—
43	C₄²×C₃₁	—	2	4	no	—
44	C₄²×C₃₁	D₄	1	4	yes	—
45	C₂³×C₃₁	—	2	2	no	—
46	C₂³×C₃₁	—	2	4	no	—
47	C₂³×C₃₁	—	2	2	no	—
48	C₂³×C₃₁	—	2	4	no	—
49	C₂³×C₃₁	C₂×Q₈	1	4	yes	—
50	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
51	Q₈×C₃₁	—	4	8	no	—
52	D₄×C₃₁	—	2	8	no	—
53	C₂×C₄×C₃₁	—	2	2	no	—
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	—	2	4	no	—
57	C₂×C₄×C₃₁	—	2	4	no	—
58	C₂×C₄×C₃₁	—	2	4	no	—
59	C₂×C₄×C₃₁	—	2	4	no	—
60	C₂×C₄×C₃₁	—	2	8	no	—
61	C₂×C₄×C₃₁	—	4	8	no	—
62	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
66	C₂×C₄×C₃₁	#41	1	2	yes	—
67	C₂×C₄×C₃₁	#41	1	4	yes	—
68	#2563	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₃₁	—	2	4	no	—
70	C₂²×C₃₁	—	2	8	no	—
71	C₂²×C₃₁	—	4	2	no	—
72	C₂²×C₃₁	—	4	2	no	—
73	C₂²×C₃₁	—	4	4	no	—
74	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
77	C₂²×C₃₁	C₂×#41	1	2	yes	—
78	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
79	C₄×C₃₁	—	2	2	no	—
80	C₄×C₃₁	—	4	2	no	—
81	C₄×C₃₁	—	4	4	no	—
82	C₄×C₃₁	—	4	4	no	—
83	C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
84	C₄×C₃₁	#124	1	4	yes	—
85	C₂×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
86	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
87	#393	C₂×C₃₁	1	4	yes	complemented
88	#390	C₂×C₃₁	1	2	yes	—
89	#382	C₂×C₃₁	1	2	yes	complemented
90	#380	C₂×C₃₁	1	4	yes	complemented
91	C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
92	C₂×C₃₁	—	4	4	no	—
93	C₂×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
95	C₂×C₃₁	#535	1	1	yes	characteristic
96	C₂×C₃₁	#554	1	2	yes	—
97	C₂×C₃₁	#555	1	2	yes	—
98	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
99	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
100	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
101	C₄⋊Q₈	—	2	4	no	—
102	C₄×D₄	—	2	4	no	—
103	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
104	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
109	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
110	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
111	#95	C₂²×C₃₁	1	2	yes	—
112	#95	C₂²×C₃₁	1	8	yes	—
113	C₃₁	#2563	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂³×C₃₁	1	2	yes	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	Q₈×C₃₁	1	4	yes	complemented
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
123	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
124	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
125	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	4	8	no	—
134	C₄²	—	2	4	no	—
135	C₄²	D₄×C₃₁	1	4	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	4	no	—
140	C₂³	C₂×Q₈×C₃₁	1	4	yes	—
141	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
142	Q₈	—	4	8	no	—
143	D₄	—	2	8	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
154	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
156	C₂×C₄	C₂×Q₈×C₃₁	1	2	yes	—
157	C₂×C₄	#41×C₃₁	1	2	yes	—
158	C₂×C₄	#41×C₃₁	1	4	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	8	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
165	C₂²	C₂²×Q₈×C₃₁	1	1	yes	characteristic
166	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
167	C₂²	C₂×#41×C₃₁	1	2	yes	—
168	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	4	no	—
172	C₄	—	4	4	no	—
173	C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
174	C₄	C₃₁×#124	1	4	yes	—
175	C₂	—	4	4	no	—
176	C₂	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
177	C₂	C₂×C₃₁×#124	1	1	yes	characteristic
178	C₂	C₃₁×#535	1	1	yes	characteristic
179	C₂	C₃₁×#554	1	2	yes	—
180	C₂	C₃₁×#555	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×D₄×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×C₄⋊Q₈	C₂	semidirect
3. C₃₁×#393	C₂	semidirect
5. C₃₁×#382	C₂	semidirect
6. C₃₁×#380	C₂	semidirect
7. C₄×C₄⋊C₄×C₃₁	C₂	semidirect
28. C₂×D₄×C₃₁	Q₈	semidirect
68. #2563	C₃₁	direct (N × H)
85. C₂×C₄⋊Q₈	C₂×C₃₁	semidirect
87. #393	C₂×C₃₁	semidirect
89. #382	C₂×C₃₁	semidirect
90. #380	C₂×C₃₁	semidirect
91. C₄×C₄⋊C₄	C₂×C₃₁	semidirect
113. C₃₁	#2563	direct (N × H)
119. C₂×D₄	Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×C₄⋊Q₈	2
C₂×C₄×D₄×C₃₁	2
C₃₁×#393	2
C₃₁×#390	2
C₃₁×#382	2
C₃₁×#380	2
C₄×C₄⋊C₄×C₃₁	2
#2563	31

Subgroups — C31×#2563 #16842496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2563 #16842496