Subgroups — C₄²×C₄⋊C₄×C₃₁ #21470507

Order 7936.

2478 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	6	yes	maximal
2	C₂×C₄³×C₃₁	C₂	1	2	yes	maximal
3	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#95	C₂	1	6	yes	maximal
5	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	12	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊C₄×C₃₁	C₄	1	48	yes	direct product factor
11	C₂×C₃₁×#95	C₂²	1	12	yes	—
12	C₄³×C₃₁	—	2	8	no	—
13	C₄³×C₃₁	C₄	1	8	yes	complemented
14	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
15	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
16	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
18	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	48	yes	—
19	C₂×C₄²×C₃₁	—	2	24	no	—
20	C₂×C₄²×C₃₁	—	2	12	no	—
21	C₂×C₄²×C₃₁	—	2	24	no	—
22	C₂×C₄²×C₃₁	C₂×C₄	1	12	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	48	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₃₁	C₂×C₄	1	24	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₃₁	D₄	1	4	yes	—
28	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
29	C₃₁×#95	C₂×C₄	1	48	yes	—
30	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	—	2	8	no	—
32	C₂²×C₄×C₃₁	—	2	12	no	—
33	C₂²×C₄×C₃₁	—	2	24	no	—
34	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
35	C₂²×C₄×C₃₁	C₂²×C₄	1	48	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
40	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
41	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₃₁	C₂×Q₈	1	6	yes	—
43	C₂²×C₄×C₃₁	#41	1	12	yes	—
44	C₄⋊C₄×C₃₁	C₄²	1	64	yes	direct product factor
45	C₄²×C₃₁	—	2	24	no	—
46	C₄²×C₃₁	—	2	96	no	—
47	C₄²×C₃₁	C₄²	1	48	yes	complemented
48	C₄²×C₃₁	C₄⋊C₄	1	16	yes	direct product factor
49	C₄²×C₄⋊C₄	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
51	C₂³×C₃₁	C₂³×C₄	1	6	yes	—
52	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
53	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
54	C₂³×C₃₁	C₂×#41	1	6	yes	—
55	C₂×C₄×C₃₁	—	2	12	no	—
56	C₂×C₄×C₃₁	—	2	48	no	—
57	C₂×C₄×C₃₁	—	2	24	no	—
58	C₂×C₄×C₃₁	—	2	48	no	—
59	C₂×C₄×C₃₁	C₂×C₄²	1	12	yes	—
60	C₂×C₄×C₃₁	C₂×C₄²	1	32	yes	—
61	C₂×C₄×C₃₁	C₂×C₄²	1	12	yes	—
62	C₂×C₄×C₃₁	C₂×C₄²	1	16	yes	—
63	C₂×C₄×C₃₁	C₂×C₄²	1	12	yes	—
64	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	12	yes	—
65	C₂×C₄×C₃₁	#117	1	12	yes	—
66	C₂×C₄×C₃₁	C₄×D₄	1	24	yes	—
67	C₂×C₄×C₃₁	C₄×Q₈	1	24	yes	—
68	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	6	yes	—
69	C₂×C₄³	C₂×C₃₁	1	2	yes	—
70	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
71	C₄×#95	C₂×C₃₁	1	6	yes	—
72	C₂²×C₃₁	C₂²×C₄²	1	3	yes	—
73	C₂²×C₃₁	C₂²×C₄²	1	4	yes	—
74	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₃₁	C₂×#117	1	3	yes	—
76	C₂²×C₃₁	C₂×C₄×D₄	1	6	yes	—
77	C₂²×C₃₁	C₂×C₄×Q₈	1	6	yes	—
78	C₂²×C₃₁	C₄×#41	1	12	yes	—
79	C₄×C₃₁	—	2	12	no	—
80	C₄×C₃₁	—	2	32	no	—
81	C₄×C₃₁	C₄³	1	8	yes	complemented
82	C₄×C₃₁	C₄×C₄⋊C₄	1	24	yes	direct product factor
83	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
84	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
85	C₂²×C₄²	C₂²×C₃₁	1	12	yes	—
86	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
87	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
88	C₄×C₄⋊C₄	C₄×C₃₁	1	48	yes	direct product factor
89	C₂×#95	C₂²×C₃₁	1	12	yes	—
90	C₄³	—	2	8	no	—
91	C₄³	C₄×C₃₁	1	8	yes	complemented
92	C₂×C₃₁	C₂×C₄³	1	1	yes	characteristic
93	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	3	yes	—
94	C₂×C₃₁	C₄×#117	1	3	yes	—
95	C₂×C₃₁	C₄²×D₄	1	4	yes	—
96	C₂×C₃₁	C₄²×Q₈	1	4	yes	—
97	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
98	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
99	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
100	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
101	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	48	yes	—
102	C₂×C₄²	—	2	24	no	—
103	C₂×C₄²	—	2	12	no	—
104	C₂×C₄²	—	2	24	no	—
105	C₂×C₄²	C₂×C₄×C₃₁	1	12	yes	—
106	C₂×C₄²	C₂×C₄×C₃₁	1	48	yes	—
107	C₂×C₄²	C₂×C₄×C₃₁	1	12	yes	—
108	C₂×C₄²	C₂×C₄×C₃₁	1	24	yes	—
109	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
110	C₂×C₄²	D₄×C₃₁	1	4	yes	—
111	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
112	#95	C₂×C₄×C₃₁	1	48	yes	—
113	C₃₁	C₄²×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
119	C₂²×C₄	C₂²×C₄×C₃₁	1	48	yes	—
120	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
121	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
122	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
123	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
124	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
125	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
126	C₂²×C₄	C₂×Q₈×C₃₁	1	6	yes	—
127	C₂²×C₄	#41×C₃₁	1	12	yes	—
128	C₄⋊C₄	C₄²×C₃₁	1	64	yes	direct product factor
129	C₄²	—	2	24	no	—
130	C₄²	—	2	96	no	—
131	C₄²	C₄²×C₃₁	1	48	yes	complemented
132	C₄²	C₄⋊C₄×C₃₁	1	16	yes	direct product factor
133	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
134	C₂³	C₂³×C₄×C₃₁	1	6	yes	—
135	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
136	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
137	C₂³	C₂×#41×C₃₁	1	6	yes	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	48	no	—
140	C₂×C₄	—	2	24	no	—
141	C₂×C₄	—	2	48	no	—
142	C₂×C₄	C₂×C₄²×C₃₁	1	12	yes	—
143	C₂×C₄	C₂×C₄²×C₃₁	1	32	yes	—
144	C₂×C₄	C₂×C₄²×C₃₁	1	12	yes	—
145	C₂×C₄	C₂×C₄²×C₃₁	1	16	yes	—
146	C₂×C₄	C₂×C₄²×C₃₁	1	12	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	12	yes	—
148	C₂×C₄	C₃₁×#117	1	12	yes	—
149	C₂×C₄	C₄×D₄×C₃₁	1	24	yes	—
150	C₂×C₄	C₄×Q₈×C₃₁	1	24	yes	—
151	C₂²	C₂²×C₄²×C₃₁	1	3	yes	—
152	C₂²	C₂²×C₄²×C₃₁	1	4	yes	—
153	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
154	C₂²	C₂×C₃₁×#117	1	3	yes	—
155	C₂²	C₂×C₄×D₄×C₃₁	1	6	yes	—
156	C₂²	C₂×C₄×Q₈×C₃₁	1	6	yes	—
157	C₂²	C₄×#41×C₃₁	1	12	yes	—
158	C₄	—	2	12	no	—
159	C₄	—	2	32	no	—
160	C₄	C₄³×C₃₁	1	8	yes	complemented
161	C₄	C₄×C₄⋊C₄×C₃₁	1	24	yes	direct product factor
162	C₂	C₂×C₄³×C₃₁	1	1	yes	characteristic
163	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	3	yes	—
164	C₂	C₄×C₃₁×#117	1	3	yes	—
165	C₂	C₄²×D₄×C₃₁	1	4	yes	—
166	C₂	C₄²×Q₈×C₃₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₃₁	C₂²
commutator	C₂	C₂×C₄³×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₄⋊C₄×C₃₁	C₄	direct (N × H)
13. C₄³×C₃₁	C₄	semidirect
44. C₄⋊C₄×C₃₁	C₄²	direct (N × H)
47. C₄²×C₃₁	C₄²	semidirect
48. C₄²×C₃₁	C₄⋊C₄	direct (N × H)
49. C₄²×C₄⋊C₄	C₃₁	direct (N × H)
81. C₄×C₃₁	C₄³	semidirect
82. C₄×C₃₁	C₄×C₄⋊C₄	direct (N × H)
88. C₄×C₄⋊C₄	C₄×C₃₁	direct (N × H)
91. C₄³	C₄×C₃₁	semidirect
113. C₃₁	C₄²×C₄⋊C₄	direct (N × H)
128. C₄⋊C₄	C₄²×C₃₁	direct (N × H)
131. C₄²	C₄²×C₃₁	semidirect
132. C₄²	C₄⋊C₄×C₃₁	direct (N × H)
160. C₄	C₄³×C₃₁	semidirect
161. C₄	C₄×C₄⋊C₄×C₃₁	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×C₄×C₄⋊C₄×C₃₁	2
C₂×C₄³×C₃₁	2
C₄×C₃₁×#95	2
C₄²×C₄⋊C₄	31

Subgroups — C42×C4⋊C4×C31 #21470507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊C₄×C₃₁ #21470507