Subgroups — C₂×C₄×C₄⋊Dic₁₃₁ #30668296

Order 16768.

928 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₃₁	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₃₁	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃₁⋊#95	C₂	1	2	yes	maximal
5	C₂²×C₄²×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₃₁	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₁₃₁	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊Dic₁₃₁	C₂²	1	8	yes	—
9	C₂×C₄×Dic₁₃₁	—	2	8	no	—
10	C₂×C₄×Dic₁₃₁	C₂²	1	8	yes	—
11	C₁₃₁⋊#95	C₂²	1	8	yes	—
12	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₃₁	C₄	1	8	yes	complemented
16	C₂×C₄²×C₁₃₁	C₂²	1	4	yes	—
17	C₂²×Dic₁₃₁	—	2	8	no	—
18	C₂²×Dic₁₃₁	—	2	8	no	—
19	C₂²×Dic₁₃₁	C₂×C₄	1	16	yes	—
20	C₂²×Dic₁₃₁	C₂³	1	8	yes	—
21	C₄⋊Dic₁₃₁	C₂×C₄	1	32	yes	direct product factor
22	C₄×Dic₁₃₁	—	2	16	no	—
23	C₂⁴×C₁₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₁	—	2	4	no	—
25	C₂²×C₄×C₁₃₁	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₃₁	D₄	1	4	yes	—
30	C₂²×C₄×C₁₃₁	Q₈	1	4	yes	—
31	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
34	C₄²×C₁₃₁	C₂×C₄	1	16	yes	complemented
35	C₂×Dic₁₃₁	—	2	32	no	—
36	C₂×Dic₁₃₁	—	2	8	no	—
37	C₂×Dic₁₃₁	—	2	8	no	—
38	C₂×Dic₁₃₁	C₂²×C₄	1	16	yes	—
39	C₂³×C₁₃₁	C₂²×C₄	1	4	yes	—
40	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
41	C₂³×C₁₃₁	C₂×D₄	1	2	yes	—
42	C₂³×C₁₃₁	C₂×Q₈	1	2	yes	—
43	C₂³×C₁₃₁	#41	1	4	yes	—
44	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₁₃₁	—	2	8	no	—
46	C₂×C₄×C₁₃₁	—	2	4	no	—
47	C₂×C₄×C₁₃₁	C₄²	1	16	yes	complemented
48	C₂×C₄×C₁₃₁	C₄⋊C₄	1	16	yes	complemented
49	C₂×C₄×C₁₃₁	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
54	C₂×C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₁₃₁	C₂×C₄²	1	4	yes	—
56	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₁₃₁	#117	1	4	yes	—
58	C₂²×C₁₃₁	C₄×D₄	1	8	yes	—
59	C₂²×C₁₃₁	C₄×Q₈	1	8	yes	—
60	C₂²×C₁₃₁	C₂³×C₄	1	2	yes	—
61	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
62	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
63	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
64	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
65	C₄×C₁₃₁	—	2	4	no	—
66	C₄×C₁₃₁	C₂×C₄²	1	8	yes	complemented
67	C₄×C₁₃₁	C₂×C₄⋊C₄	1	8	yes	complemented
68	Dic₁₃₁	—	2	16	no	—
69	C₂×C₁₃₁	C₄×C₄⋊C₄	1	8	yes	complemented
70	C₂×C₁₃₁	C₂²×C₄²	1	1	yes	characteristic
71	C₂×C₁₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₁₃₁	C₂×#117	1	1	yes	characteristic
73	C₂×C₁₃₁	C₂×C₄×D₄	1	2	yes	—
74	C₂×C₁₃₁	C₂×C₄×Q₈	1	2	yes	—
75	C₁₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄×C₄⋊C₄	—	131	1	no	maximal, Sylow
77	C₂²×C₄⋊C₄	—	131	2	no	—
78	C₂²×C₄²	—	131	2	no	—
79	C₂²×C₄²	D₁₃₁	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	131	8	no	—
81	C₂×#95	—	131	2	no	—
82	C₂³×C₄	—	131	4	no	—
83	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
85	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	131	16	no	—
87	C₂×C₄⋊C₄	—	131	8	no	—
88	C₂×C₄²	—	262	8	no	—
89	C₂×C₄²	—	131	8	no	—
90	C₂×C₄²	Dic₁₃₁	1	8	yes	complemented
91	C₂×C₄²	C₂×D₁₃₁	1	4	yes	—
92	#95	—	131	8	no	—
93	C₂⁴	C₂²×D₁₃₁	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	262	8	no	—
96	C₂²×C₄	—	262	8	no	—
97	C₂²×C₄	—	131	16	no	—
98	C₂²×C₄	—	131	8	no	—
99	C₂²×C₄	C₄×D₁₃₁	1	8	yes	—
100	C₂²×C₄	C₂×Dic₁₃₁	1	4	yes	—
101	C₂²×C₄	C₂×Dic₁₃₁	1	4	yes	—
102	C₂²×C₄	C₂×Dic₁₃₁	1	4	yes	—
103	C₂²×C₄	D₅₂₄	1	4	yes	—
104	C₂²×C₄	Dic₂₆₂	1	4	yes	—
105	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
107	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
108	C₄⋊C₄	—	131	32	no	—
109	C₄²	—	262	16	no	—
110	C₄²	C₂×Dic₁₃₁	1	16	yes	complemented
111	C₂³	C₂×C₄×D₁₃₁	1	4	yes	—
112	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
113	C₂³	C₂×D₅₂₄	1	2	yes	—
114	C₂³	C₂×Dic₂₆₂	1	2	yes	—
115	C₂³	#14523513	1	4	yes	—
116	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	8	no	—
119	C₂×C₄	—	262	32	no	—
120	C₂×C₄	—	262	8	no	—
121	C₂×C₄	—	262	8	no	—
122	C₂×C₄	—	131	16	no	—
123	C₂×C₄	C₄×Dic₁₃₁	1	16	yes	complemented
124	C₂×C₄	C₄⋊Dic₁₃₁	1	16	yes	direct product factor
125	C₂×C₄	C₂×C₄×D₁₃₁	1	8	yes	—
126	C₂×C₄	C₂²×Dic₁₃₁	1	4	yes	—
127	C₂×C₄	C₂²×Dic₁₃₁	1	4	yes	—
128	C₂×C₄	C₂²×Dic₁₃₁	1	4	yes	—
129	C₂×C₄	C₂×D₅₂₄	1	4	yes	—
130	C₂×C₄	C₂×Dic₂₆₂	1	4	yes	—
131	C₂²	C₂×C₄×Dic₁₃₁	1	4	yes	—
132	C₂²	C₂×C₄⋊Dic₁₃₁	1	4	yes	—
133	C₂²	#16862205	1	4	yes	—
134	C₂²	C₄×D₅₂₄	1	8	yes	—
135	C₂²	C₄×Dic₂₆₂	1	8	yes	—
136	C₂²	C₂²×C₄×D₁₃₁	1	2	yes	—
137	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
138	C₂²	C₂²×D₅₂₄	1	1	yes	characteristic
139	C₂²	C₂²×Dic₂₆₂	1	1	yes	characteristic
140	C₂²	C₂×#14523513	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	262	16	no	—
143	C₄	C₂×C₄×Dic₁₃₁	1	8	yes	complemented
144	C₄	C₂×C₄⋊Dic₁₃₁	1	8	yes	direct product factor
145	C₂	C₄×C₄⋊Dic₁₃₁	1	8	yes	direct product factor
146	C₂	C₂²×C₄×Dic₁₃₁	1	1	yes	characteristic
147	C₂	C₂²×C₄⋊Dic₁₃₁	1	1	yes	characteristic
148	C₂	C₂×#16862205	1	1	yes	characteristic
149	C₂	C₂×C₄×D₅₂₄	1	2	yes	—
150	C₂	C₂×C₄×Dic₂₆₂	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₃₁
commutator	C₂×C₁₃₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₃₁
Fitting	C₂²×C₄²×C₁₃₁	C₂
socle, prime radical	C₂⁴×C₁₃₁	C₂³
cosocle	C₂³×C₁₃₁	C₂⁴
hypercenter	C₂²×C₄²	D₁₃₁
hypocenter	C₁₃₁	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊Dic₁₃₁	C₂	direct (N × H)
7. C₂×C₄⋊Dic₁₃₁	C₄	direct (N × H)
15. C₂×C₄²×C₁₃₁	C₄	semidirect
21. C₄⋊Dic₁₃₁	C₂×C₄	direct (N × H)
34. C₄²×C₁₃₁	C₂×C₄	semidirect
47. C₂×C₄×C₁₃₁	C₄²	semidirect
48. C₂×C₄×C₁₃₁	C₄⋊C₄	semidirect
66. C₄×C₁₃₁	C₂×C₄²	semidirect
67. C₄×C₁₃₁	C₂×C₄⋊C₄	semidirect
69. C₂×C₁₃₁	C₄×C₄⋊C₄	semidirect
75. C₁₃₁	C₂×C₄×C₄⋊C₄	semidirect
90. C₂×C₄²	Dic₁₃₁	semidirect
110. C₄²	C₂×Dic₁₃₁	semidirect
123. C₂×C₄	C₄×Dic₁₃₁	semidirect
124. C₂×C₄	C₄⋊Dic₁₃₁	direct (N × H)
143. C₄	C₂×C₄×Dic₁₃₁	semidirect
144. C₄	C₂×C₄⋊Dic₁₃₁	direct (N × H)
145. C₂	C₄×C₄⋊Dic₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₃₁	2
C₂²×C₄×Dic₁₃₁	2
C₄×C₄⋊Dic₁₃₁	2
C₂×C₁₃₁⋊#95	2
C₂²×C₄²×C₁₃₁	2
C₂×C₄×C₄⋊C₄	131

Subgroups — C2×C4×C4⋊Dic131 #30668296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊Dic₁₃₁ #30668296