Subgroups — C₂²×C₂²⋊#6090 #16015584

Order 3328.

1470 conjugacy classes of subgroups.

The subgroup lattice has 194 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, complemented
2	C₂×C₂²⋊#6090	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	—	2	8	no	—
6	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₄	1	6	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	6	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6090	—	2	8	no	—
11	C₂²×#6090	C₂²	1	12	yes	complemented
12	C₂²⋊#6090	C₂²	1	16	yes	direct product factor
13	C₂⁵×C₁₃	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	—	2	4	no	—
15	C₂³×C₄×C₁₃	—	2	4	no	—
16	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₈	1	6	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	6	yes	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	—	2	4	no	—
21	C₂²×C₈×C₁₃	—	2	24	no	—
22	C₂²×C₈×C₁₃	—	2	24	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	12	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₁₃	D₄	1	8	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
29	C₂×#6090	—	2	24	no	—
30	C₂×#6090	C₂³	1	8	yes	complemented
31	C₂²×C₂²⋊C₁₆	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	—	2	4	no	—
33	C₂⁴×C₁₃	—	2	4	no	—
34	C₂⁴×C₁₃	C₁₆	1	8	yes	complemented
35	C₂⁴×C₁₃	C₂×C₈	1	6	yes	—
36	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	—	2	4	no	—
38	C₂²×C₄×C₁₃	—	2	24	no	—
39	C₂²×C₄×C₁₃	—	2	12	no	—
40	C₂²×C₄×C₁₃	—	2	12	no	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₃	—	2	12	no	—
49	C₂×C₈×C₁₃	—	2	24	no	—
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	C₂²⋊C₄	1	12	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₁₃	C₂×D₄	1	12	yes	—
55	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
56	#6090	—	2	8	no	—
57	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
58	C₂³×C₁₆	—	13	2	no	—
59	C₂×C₂²⋊C₁₆	—	13	12	no	—
60	C₂³×C₁₃	—	2	32	no	—
61	C₂³×C₁₃	—	2	12	no	—
62	C₂³×C₁₃	—	2	12	no	—
63	C₂³×C₁₃	C₂²⋊C₈	1	4	yes	—
64	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
65	C₂³×C₁₃	C₂×C₁₆	1	24	yes	complemented
66	C₂³×C₁₃	M₃₂	1	4	yes	—
67	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
68	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₁₃	—	2	12	no	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	24	no	—
73	C₂×C₄×C₁₃	C₂²⋊C₈	1	12	yes	—
74	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
77	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
78	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₁₃	—	2	4	no	—
80	C₈×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
81	C₈×C₁₃	C₂²×D₄	1	2	yes	—
82	C₂⁴×C₄	Dic₁₃	1	1	yes	characteristic
83	C₂³×C₈	—	2	8	no	—
84	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
85	C₂³×C₈	Dic₁₃	1	6	yes	—
86	C₂³×C₈	C₂×D₁₃	1	6	yes	—
87	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₂²×C₁₆	—	13	12	no	—
89	C₂²×C₁₆	—	26	8	no	—
90	C₂²⋊C₁₆	—	13	16	no	—
91	C₂²×C₁₃	—	2	4	no	—
92	C₂²×C₁₃	—	2	48	no	—
93	C₂²×C₁₃	—	2	4	no	—
94	C₂²×C₁₃	C₂²⋊C₁₆	1	16	yes	complemented
95	C₂²×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
96	C₂²×C₁₃	C₂²×C₁₆	1	6	yes	—
97	C₂²×C₁₃	C₂²×C₁₆	1	8	yes	complemented
98	C₂²×C₁₃	C₂×M₃₂	1	6	yes	—
99	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
100	C₄×C₁₃	—	2	4	no	—
101	C₄×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
102	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
104	C₂⁵	#1055	1	1	yes	characteristic
105	C₂³×C₄	—	2	4	no	—
106	C₂³×C₄	—	2	4	no	—
107	C₂³×C₄	#1055	1	1	yes	characteristic
108	C₂³×C₄	#1055	1	6	yes	—
109	C₂³×C₄	C₂×Dic₁₃	1	6	yes	—
110	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	—	2	24	no	—
113	C₂²×C₈	—	2	24	no	—
114	C₂²×C₈	C₂×Dic₁₃	1	6	yes	—
115	C₂²×C₈	C₂×Dic₁₃	1	12	yes	—
116	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
117	C₂²×C₈	C₂²⋊D₁₃	1	8	yes	—
118	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
119	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
120	C₂×C₁₆	—	13	8	no	—
121	C₂×C₁₆	—	26	24	no	—
122	C₂×C₁₃	—	2	8	no	—
123	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	12	yes	complemented
124	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
126	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
127	C₂⁴	—	2	4	no	—
128	C₂⁴	—	2	4	no	—
129	C₂⁴	#6090	1	8	yes	complemented
130	C₂⁴	C₂×#1055	1	6	yes	—
131	C₂⁴	C₂×#1055	1	1	yes	characteristic
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	24	no	—
134	C₂²×C₄	—	2	12	no	—
135	C₂²×C₄	—	2	12	no	—
136	C₂²×C₄	C₂×#1055	1	6	yes	—
137	C₂²×C₄	C₂×#1055	1	12	yes	—
138	C₂²×C₄	C₂×#1055	1	3	yes	—
139	C₂²×C₄	#6099	1	4	yes	—
140	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
141	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
142	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
143	C₂×C₈	—	2	12	no	—
144	C₂×C₈	—	2	24	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	C₂²⋊Dic₁₃	1	12	yes	—
147	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
148	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
149	C₂×C₈	C₂×C₂²⋊D₁₃	1	12	yes	—
150	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
151	C₁₆	—	26	8	no	—
152	C₁₃	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
153	C₂³	—	2	12	no	—
154	C₂³	—	2	12	no	—
155	C₂³	—	2	32	no	—
156	C₂³	C₂×#6090	1	4	yes	—
157	C₂³	C₂×#6090	1	24	yes	complemented
158	C₂³	#88052	1	4	yes	—
159	C₂³	C₂²⋊#1055	1	4	yes	—
160	C₂³	C₂²×#1055	1	4	yes	—
161	C₂³	C₂²×#1055	1	3	yes	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	24	no	—
166	C₂×C₄	C₂²⋊#1055	1	12	yes	—
167	C₂×C₄	C₂²×#1055	1	4	yes	—
168	C₂×C₄	C₂²×#1055	1	3	yes	—
169	C₂×C₄	C₂×#6099	1	6	yes	—
170	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	6	yes	—
171	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
172	C₈	—	2	4	no	—
173	C₈	C₂×C₂²⋊Dic₁₃	1	6	yes	—
174	C₈	C₂²×C₂²⋊D₁₃	1	2	yes	—
175	C₂²	—	2	48	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂²⋊#6090	1	16	yes	direct product factor
179	C₂²	C₂²×#6090	1	6	yes	—
180	C₂²	C₂²×#6090	1	8	yes	complemented
181	C₂²	C₂×#88052	1	6	yes	—
182	C₂²	C₂×C₂²⋊#1055	1	6	yes	—
183	C₂²	C₂³×#1055	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	C₂×C₂²⋊#1055	1	6	yes	—
186	C₄	C₂²×#6099	1	1	yes	characteristic
187	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
188	C₂	—	2	8	no	—
189	C₂	C₂×C₂²⋊#6090	1	12	yes	direct product factor
190	C₂	C₂²×C₂²⋊#1055	1	1	yes	characteristic
191	C₂	C₂³×#6090	1	1	yes	characteristic
192	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	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×C₈
prime radical	C₂⁵×C₁₃	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#6090	C₂	semidirect
2. C₂×C₂²⋊#6090	C₂	direct (N × H)
11. C₂²×#6090	C₂²	semidirect
12. C₂²⋊#6090	C₂²	direct (N × H)
30. C₂×#6090	C₂³	semidirect
34. C₂⁴×C₁₃	C₁₆	semidirect
65. C₂³×C₁₃	C₂×C₁₆	semidirect
94. C₂²×C₁₃	C₂²⋊C₁₆	semidirect
97. C₂²×C₁₃	C₂²×C₁₆	semidirect
123. C₂×C₁₃	C₂×C₂²⋊C₁₆	semidirect
129. C₂⁴	#6090	semidirect
152. C₁₃	C₂²×C₂²⋊C₁₆	semidirect
157. C₂³	C₂×#6090	semidirect
178. C₂²	C₂²⋊#6090	direct (N × H)
180. C₂²	C₂²×#6090	semidirect
189. C₂	C₂×C₂²⋊#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×C₂²⋊#6090	2
C₂⁴×C₈×C₁₃	2
C₂²×C₂²⋊C₁₆	13

Subgroups — C22×C22⋊#6090 #16015584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊#6090 #16015584