Subgroups — C₂²×C₂²⋊#6092 #16806762

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×#6092	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#6092	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×#1055	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×#1055	—	2	8	no	—
6	C₂³×#1055	C₄	1	6	yes	—
7	C₂³×#1055	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	6	yes	—
9	C₂³×#1055	C₂²	1	1	yes	characteristic
10	C₂²×#6092	—	2	8	no	—
11	C₂²×#6092	C₂²	1	12	yes	complemented
12	C₂²⋊#6092	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×#1055	—	2	4	no	—
21	C₂²×#1055	—	2	24	no	—
22	C₂²×#1055	—	2	24	no	—
23	C₂²×#1055	C₂×C₄	1	6	yes	—
24	C₂²×#1055	C₂×C₄	1	12	yes	—
25	C₂²×#1055	C₂×C₄	1	3	yes	—
26	C₂²×#1055	D₄	1	8	yes	—
27	C₂²×#1055	C₂³	1	4	yes	—
28	C₂²×#1055	C₂³	1	3	yes	—
29	C₂×#6092	—	2	24	no	—
30	C₂×#6092	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×#1055	—	2	12	no	—
49	C₂×#1055	—	2	24	no	—
50	C₂×#1055	—	2	8	no	—
51	C₂×#1055	C₂²⋊C₄	1	12	yes	—
52	C₂×#1055	C₂²×C₄	1	4	yes	—
53	C₂×#1055	C₂²×C₄	1	3	yes	—
54	C₂×#1055	C₂×D₄	1	12	yes	—
55	C₂×#1055	C₂⁴	1	1	yes	characteristic
56	#6092	—	2	8	no	—
57	C₂⁴×C₈	—	13	1	no	—
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	#1055	—	2	4	no	—
80	#1055	C₂×C₂²⋊C₄	1	6	yes	—
81	#1055	C₂²×D₄	1	2	yes	—
82	C₂⁴×C₄	#261	1	1	yes	characteristic
83	C₂³×C₈	—	13	1	no	—
84	C₂³×C₈	—	13	6	no	—
85	C₂³×C₈	—	13	6	no	—
86	C₂³×C₈	—	13	1	no	—
87	C₂³×C₈	—	26	8	no	—
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	#1057	1	1	yes	characteristic
105	C₂³×C₄	—	2	4	no	—
106	C₂³×C₄	—	2	4	no	—
107	C₂³×C₄	#1057	1	1	yes	characteristic
108	C₂³×C₄	#1057	1	6	yes	—
109	C₂³×C₄	C₂×#261	1	6	yes	—
110	C₂³×C₄	C₂×#261	1	1	yes	characteristic
111	C₂²×C₈	—	13	8	no	—
112	C₂²×C₈	—	13	3	no	—
113	C₂²×C₈	—	13	3	no	—
114	C₂²×C₈	—	13	4	no	—
115	C₂²×C₈	—	13	12	no	—
116	C₂²×C₈	—	13	6	no	—
117	C₂²×C₈	—	26	24	no	—
118	C₂²×C₈	—	26	24	no	—
119	C₂²×C₈	—	26	4	no	—
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	#6092	1	8	yes	complemented
130	C₂⁴	C₂×#1057	1	6	yes	—
131	C₂⁴	C₂×#1057	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×#1057	1	6	yes	—
137	C₂²×C₄	C₂×#1057	1	12	yes	—
138	C₂²×C₄	C₂×#1057	1	3	yes	—
139	C₂²×C₄	#6122	1	4	yes	—
140	C₂²×C₄	C₂²⋊#261	1	4	yes	—
141	C₂²×C₄	C₂²×#261	1	4	yes	—
142	C₂²×C₄	C₂²×#261	1	3	yes	—
143	C₂×C₈	—	13	12	no	—
144	C₂×C₈	—	13	12	no	—
145	C₂×C₈	—	13	1	no	—
146	C₂×C₈	—	13	3	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	—	26	8	no	—
149	C₂×C₈	—	26	24	no	—
150	C₂×C₈	—	26	12	no	—
151	C₁₆	—	26	8	no	—
152	C₁₃	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
153	C₂³	—	2	12	no	—
154	C₂³	—	2	32	no	—
155	C₂³	—	2	12	no	—
156	C₂³	C₂×#6092	1	4	yes	—
157	C₂³	C₂×#6092	1	24	yes	complemented
158	C₂³	#88106	1	4	yes	—
159	C₂³	C₂²⋊#1057	1	4	yes	—
160	C₂³	C₂²×#1057	1	4	yes	—
161	C₂³	C₂²×#1057	1	3	yes	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₂²⋊#1057	1	12	yes	—
167	C₂×C₄	C₂²×#1057	1	4	yes	—
168	C₂×C₄	C₂²×#1057	1	3	yes	—
169	C₂×C₄	C₂×#6122	1	6	yes	—
170	C₂×C₄	C₂×C₂²⋊#261	1	6	yes	—
171	C₂×C₄	C₂³×#261	1	1	yes	characteristic
172	C₈	—	13	2	no	—
173	C₈	—	13	6	no	—
174	C₈	—	26	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	48	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂²⋊#6092	1	16	yes	direct product factor
179	C₂²	C₂²×#6092	1	6	yes	—
180	C₂²	C₂²×#6092	1	8	yes	complemented
181	C₂²	C₂×#88106	1	6	yes	—
182	C₂²	C₂×C₂²⋊#1057	1	6	yes	—
183	C₂²	C₂³×#1057	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	C₂×C₂²⋊#1057	1	6	yes	—
186	C₄	C₂²×#6122	1	1	yes	characteristic
187	C₄	C₂²×C₂²⋊#261	1	1	yes	characteristic
188	C₂	—	2	8	no	—
189	C₂	C₂×C₂²⋊#6092	1	12	yes	direct product factor
190	C₂	C₂²×C₂²⋊#1057	1	1	yes	characteristic
191	C₂	C₂³×#6092	1	1	yes	characteristic
192	C₂	C₂²×#88106	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂⁴×C₄×C₁₃	C₄
socle	C₂⁴×C₁₃	C₂×C₈
prime radical	C₂⁵×C₁₃	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂⁴×C₄	#261
hypocenter	C₁₃	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#6092	C₂	semidirect
2. C₂×C₂²⋊#6092	C₂	direct (N × H)
11. C₂²×#6092	C₂²	semidirect
12. C₂²⋊#6092	C₂²	direct (N × H)
30. C₂×#6092	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	#6092	semidirect
152. C₁₃	C₂²×C₂²⋊C₁₆	semidirect
157. C₂³	C₂×#6092	semidirect
178. C₂²	C₂²⋊#6092	direct (N × H)
180. C₂²	C₂²×#6092	semidirect
189. C₂	C₂×C₂²⋊#6092	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×#6092	2
C₂×C₂²⋊#6092	2
C₂⁴×#1055	2
C₂²×C₂²⋊C₁₆	13

Subgroups — C22×C22⋊#6092 #16806762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊#6092 #16806762