Subgroups — C₂×#22553067 #30665652

Order 16768.

528 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃₁⋊C₈	C₂	1	2	yes	maximal
2	#22553067	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄²×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃₁⋊C₈	—	2	4	no	—
5	C₂²×C₁₃₁⋊C₈	—	2	4	no	—
6	C₂²×C₁₃₁⋊C₈	C₄	1	8	yes	complemented
7	C₂²×C₁₃₁⋊C₈	C₂²	1	4	yes	—
8	C₂³×C₄×C₁₃₁	C₄	1	2	yes	—
9	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₁₃₁	—	2	4	no	—
11	C₂×C₄²×C₁₃₁	C₄	1	2	yes	—
12	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
13	C₂×C₁₃₁⋊C₈	—	2	16	no	—
14	C₂×C₁₃₁⋊C₈	—	2	4	no	—
15	C₂×C₁₃₁⋊C₈	—	2	4	no	—
16	C₂×C₁₃₁⋊C₈	C₂×C₄	1	8	yes	complemented
17	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₁	—	2	4	no	—
19	C₂²×C₄×C₁₃₁	—	2	4	no	—
20	C₂²×C₄×C₁₃₁	C₈	1	8	yes	complemented
21	C₂²×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₃₁	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₁	D₄	1	4	yes	—
25	C₂²×C₄×C₁₃₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₁	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₄²×C₁₃₁	—	2	4	no	—
29	C₄²×C₁₃₁	—	2	4	no	—
30	C₁₃₁⋊C₈	—	2	8	no	—
31	C₂³×C₁₃₁	C₄²	1	2	yes	—
32	C₂³×C₁₃₁	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₁₃₁	C₄⋊C₄	1	2	yes	—
34	C₂³×C₁₃₁	C₂×C₈	1	4	yes	—
35	C₂³×C₁₃₁	M₁₆	1	4	yes	—
36	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₁	—	2	16	no	—
38	C₂×C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄×C₁₃₁	—	2	4	no	—
40	C₂×C₄×C₁₃₁	C₄²	1	2	yes	—
41	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	8	yes	—
42	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃₁	C₄⋊C₄	1	8	yes	—
44	C₂×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₃₁	C₂×C₈	1	8	yes	complemented
46	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₁	#95	1	4	yes	—
51	C₂²×C₁₃₁	C₄×C₈	1	4	yes	—
52	C₂²×C₁₃₁	#97	1	4	yes	—
53	C₂²×C₁₃₁	C₂²⋊C₈	1	8	yes	—
54	C₂²×C₁₃₁	C₄⋊C₈	1	8	yes	—
55	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
56	C₂²×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₃₁	C₂²×C₈	1	2	yes	—
59	C₂²×C₁₃₁	C₂×M₁₆	1	2	yes	—
60	C₄×C₁₃₁	—	2	8	no	—
61	C₄×C₁₃₁	#95	1	4	yes	—
62	C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
63	C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₁₃₁	#336	1	8	yes	complemented
65	C₂×C₁₃₁	C₂×#95	1	1	yes	characteristic
66	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂×C₁₃₁	C₂×#97	1	1	yes	characteristic
68	C₂×C₁₃₁	C₂×C₂²⋊C₈	1	2	yes	—
69	C₂×C₁₃₁	C₂×C₄⋊C₈	1	2	yes	—
70	C₁₃₁	C₂×#336	1	1	yes	characteristic, complemented, Sylow
71	C₂×#336	—	131	1	no	maximal, Sylow
72	C₂³×C₈	—	131	2	no	—
73	C₂²×C₄²	D₁₃₁	1	1	yes	characteristic
74	#336	—	131	4	no	—
75	C₂³×C₄	Dic₁₃₁	1	2	yes	—
76	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
77	C₂²×C₈	—	262	4	no	—
78	C₂²×C₈	—	262	4	no	—
79	C₂²×C₈	—	131	8	no	—
80	C₂²×C₈	—	131	4	no	—
81	C₂×C₄²	—	2	4	no	—
82	C₂×C₄²	Dic₁₃₁	1	2	yes	—
83	C₂×C₄²	C₂×D₁₃₁	1	2	yes	—
84	C₂⁴	C₂×Dic₁₃₁	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	2	4	no	—
87	C₂²×C₄	C₁₃₁⋊C₈	1	8	yes	complemented
88	C₂²×C₄	C₄×D₁₃₁	1	4	yes	—
89	C₂²×C₄	C₂×Dic₁₃₁	1	4	yes	—
90	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
91	C₂²×C₄	C₂²⋊D₁₃₁	1	4	yes	—
92	C₂²×C₄	D₅₂₄	1	2	yes	—
93	C₂²×C₄	Dic₂₆₂	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
95	C₂×C₈	—	262	16	no	—
96	C₂×C₈	—	262	4	no	—
97	C₂×C₈	—	262	4	no	—
98	C₂×C₈	—	131	8	no	—
99	C₄²	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₂³	C₄×Dic₁₃₁	1	2	yes	—
102	C₂³	C₂²⋊Dic₁₃₁	1	2	yes	—
103	C₂³	C₄⋊Dic₁₃₁	1	2	yes	—
104	C₂³	C₂×C₁₃₁⋊C₈	1	4	yes	—
105	C₂³	#14523497	1	4	yes	—
106	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	16	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	C₄×Dic₁₃₁	1	2	yes	—
111	C₂×C₄	(C₂×D₁₃₁)⋊C₄	1	8	yes	—
112	C₂×C₄	C₂²⋊Dic₁₃₁	1	2	yes	—
113	C₂×C₄	Dic₁₃₁⋊C₄	1	8	yes	—
114	C₂×C₄	C₄⋊Dic₁₃₁	1	2	yes	—
115	C₂×C₄	C₂×C₁₃₁⋊C₈	1	8	yes	complemented
116	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
118	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₂₆₂	1	1	yes	characteristic
120	C₈	—	262	8	no	—
121	C₂²	C₁₃₁⋊#95	1	4	yes	—
122	C₂²	C₄×C₁₃₁⋊C₈	1	4	yes	—
123	C₂²	#16862157	1	4	yes	—
124	C₂²	C₂²⋊(C₁₃₁⋊C₈)	1	8	yes	—
125	C₂²	C₄⋊(C₁₃₁⋊C₈)	1	8	yes	—
126	C₂²	C₂×C₄×Dic₁₃₁	1	1	yes	characteristic
127	C₂²	C₂×C₂²⋊Dic₁₃₁	1	1	yes	characteristic
128	C₂²	C₂×C₄⋊Dic₁₃₁	1	1	yes	characteristic
129	C₂²	C₂²×C₁₃₁⋊C₈	1	2	yes	—
130	C₂²	C₂×#14523497	1	2	yes	—
131	C₄	—	2	8	no	—
132	C₄	C₁₃₁⋊#95	1	4	yes	—
133	C₄	C₂×(C₂×D₁₃₁)⋊C₄	1	2	yes	—
134	C₄	C₂×Dic₁₃₁⋊C₄	1	2	yes	—
135	C₂	#22553067	1	8	yes	direct product factor
136	C₂	C₂×C₁₃₁⋊#95	1	1	yes	characteristic
137	C₂	C₂×C₄×C₁₃₁⋊C₈	1	1	yes	characteristic
138	C₂	C₂×#16862157	1	1	yes	characteristic
139	C₂	C₂×C₂²⋊(C₁₃₁⋊C₈)	1	2	yes	—
140	C₂	C₂×C₄⋊(C₁₃₁⋊C₈)	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₃₁
Fitting	C₂²×C₄²×C₁₃₁	C₂
socle, prime radical	C₂⁴×C₁₃₁	C₂×C₄
cosocle	C₂²×C₄×C₁₃₁	C₂³
hypercenter	C₂²×C₄²	D₁₃₁
hypocenter	C₁₃₁	C₂×#336
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
2. #22553067	C₂	direct (N × H)
6. C₂²×C₁₃₁⋊C₈	C₄	semidirect
16. C₂×C₁₃₁⋊C₈	C₂×C₄	semidirect
20. C₂²×C₄×C₁₃₁	C₈	semidirect
45. C₂×C₄×C₁₃₁	C₂×C₈	semidirect
64. C₂×C₁₃₁	#336	semidirect
70. C₁₃₁	C₂×#336	semidirect
87. C₂²×C₄	C₁₃₁⋊C₈	semidirect
115. C₂×C₄	C₂×C₁₃₁⋊C₈	semidirect
135. C₂	#22553067	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃₁⋊C₈	2
#22553067	2
C₂²×C₄²×C₁₃₁	2
C₂×#336	131

Subgroups — C2×#22553067 #30665652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553067 #30665652