Subgroups — C₂²×C₄⋊#6090 #16015894

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6090	C₂	1	2	yes	maximal
2	C₂×C₄⋊#6090	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	6	yes	—
9	C₂²×#6090	—	2	8	no	—
10	C₂²×#6090	C₂²	1	12	yes	—
11	C₄⋊#6090	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₁₃	D₄	1	4	yes	—
19	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
21	C₂×C₄²×C₁₃	C₈	1	6	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
24	C₄×C₈×C₁₃	C₂³	1	4	yes	—
25	C₂×#6090	—	2	24	no	—
26	C₂×#6090	C₂³	1	8	yes	—
27	C₂²×C₄⋊C₁₆	—	13	1	no	maximal, Sylow
28	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	—	2	4	no	—
30	C₂²×C₄×C₁₃	C₁₆	1	8	yes	complemented
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
32	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
33	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
35	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
36	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
37	C₂×C₈×C₁₃	—	2	12	no	—
38	C₂×C₈×C₁₃	C₄⋊C₄	1	12	yes	—
39	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
40	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
41	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
42	C₂×C₈×C₁₃	C₂×Q₈	1	6	yes	—
43	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
44	C₄²×C₁₃	C₂×C₈	1	12	yes	—
45	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
46	#6090	—	2	8	no	—
47	C₂³×C₁₆	—	13	2	no	—
48	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
49	C₂×C₄⋊C₁₆	—	13	12	no	—
50	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
51	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
52	C₂³×C₁₃	M₃₂	1	4	yes	—
53	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
54	C₂×C₄×C₁₃	—	2	12	no	—
55	C₂×C₄×C₁₃	C₄⋊C₈	1	12	yes	—
56	C₂×C₄×C₁₃	C₂×C₁₆	1	24	yes	complemented
57	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
58	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
59	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
60	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
61	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
62	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₁₃	—	2	4	no	—
64	C₈×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
66	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
68	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
69	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
70	C₂²×C₁₆	—	13	12	no	—
71	C₂²×C₁₆	—	26	8	no	—
72	C₂×C₄×C₈	Dic₁₃	1	6	yes	—
73	C₂×C₄×C₈	C₂×D₁₃	1	6	yes	—
74	C₄⋊C₁₆	—	13	16	no	—
75	C₂²×C₁₃	C₄⋊C₁₆	1	16	yes	complemented
76	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
77	C₂²×C₁₃	C₂²×C₁₆	1	6	yes	—
78	C₂²×C₁₃	C₂×M₃₂	1	6	yes	—
79	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₁₃	—	2	4	no	—
81	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
82	C₄×C₁₃	C₂²×C₁₆	1	8	yes	complemented
83	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
85	C₂³×C₄	#1055	1	1	yes	characteristic
86	C₂³×C₄	#1055	1	1	yes	characteristic
87	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	Dic₂₆	1	4	yes	—
90	C₂²×C₈	D₅₂	1	4	yes	—
91	C₂²×C₈	C₂×Dic₁₃	1	6	yes	—
92	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
93	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
94	C₂×C₄²	#1055	1	6	yes	—
95	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
96	C₂×C₁₆	—	13	8	no	—
97	C₂×C₁₆	—	26	24	no	—
98	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
99	C₄×C₈	C₂²×D₁₃	1	4	yes	—
100	C₂×C₁₃	C₂×C₄⋊C₁₆	1	12	yes	complemented
101	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
103	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
104	C₂⁴	C₂×#1055	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	#6090	1	8	yes	complemented
107	C₂²×C₄	C₂×#1055	1	6	yes	—
108	C₂²×C₄	C₂×#1055	1	3	yes	—
109	C₂²×C₄	C₂×#1055	1	6	yes	—
110	C₂²×C₄	#6099	1	4	yes	—
111	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
112	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
113	C₂×C₈	—	2	12	no	—
114	C₂×C₈	C₄⋊Dic₁₃	1	12	yes	—
115	C₂×C₈	C₂×Dic₂₆	1	6	yes	—
116	C₂×C₈	C₂×D₅₂	1	6	yes	—
117	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
118	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
119	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
120	C₄²	C₂×#1055	1	12	yes	—
121	C₄²	C₂²×Dic₁₃	1	4	yes	—
122	C₁₆	—	26	8	no	—
123	C₁₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
124	C₂³	C₄⋊#1055	1	4	yes	—
125	C₂³	C₂×#6090	1	4	yes	—
126	C₂³	#88052	1	4	yes	—
127	C₂³	C₂²×#1055	1	3	yes	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	C₄⋊#1055	1	12	yes	—
130	C₂×C₄	C₂×#6090	1	24	yes	complemented
131	C₂×C₄	C₂²×#1055	1	4	yes	—
132	C₂×C₄	C₂²×#1055	1	4	yes	—
133	C₂×C₄	C₂²×#1055	1	3	yes	—
134	C₂×C₄	C₂×#6099	1	6	yes	—
135	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
136	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	C₂×C₄⋊Dic₁₃	1	6	yes	—
139	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
140	C₈	C₂²×D₅₂	1	1	yes	characteristic
141	C₂²	C₄⋊#6090	1	16	yes	direct product factor
142	C₂²	C₂×C₄⋊#1055	1	6	yes	—
143	C₂²	C₂²×#6090	1	6	yes	—
144	C₂²	C₂×#88052	1	6	yes	—
145	C₂²	C₂³×#1055	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	C₂×C₄⋊#1055	1	6	yes	—
148	C₄	C₂²×#6090	1	8	yes	complemented
149	C₄	C₂²×#6099	1	1	yes	characteristic
150	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
151	C₂	C₂×C₄⋊#6090	1	12	yes	direct product factor
152	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
153	C₂	C₂³×#6090	1	1	yes	characteristic
154	C₂	C₂²×#88052	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#6090	C₂	direct (N × H)
11. C₄⋊#6090	C₂²	direct (N × H)
30. C₂²×C₄×C₁₃	C₁₆	semidirect
56. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
75. C₂²×C₁₃	C₄⋊C₁₆	semidirect
82. C₄×C₁₃	C₂²×C₁₆	semidirect
100. C₂×C₁₃	C₂×C₄⋊C₁₆	semidirect
106. C₂²×C₄	#6090	semidirect
123. C₁₃	C₂²×C₄⋊C₁₆	semidirect
130. C₂×C₄	C₂×#6090	semidirect
141. C₂²	C₄⋊#6090	direct (N × H)
148. C₄	C₂²×#6090	semidirect
151. C₂	C₂×C₄⋊#6090	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×#6090	2
C₂×C₄⋊#6090	2
C₂²×C₄×C₈×C₁₃	2
C₂²×C₄⋊C₁₆	13

Subgroups — C22×C4⋊#6090 #16015894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊#6090 #16015894