Subgroups — C₂²×C₈×Dic₂₂ #15618124

Order 2816.

2478 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₂	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₁⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₁₁	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₂	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₁	C₄	1	1	yes	characteristic
12	C₂²×Dic₁₁⋊C₄	C₄	1	2	yes	—
13	C₂²×C₄×Dic₁₁	C₂²	1	2	yes	—
14	C₂³×C₁₁⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×Dic₂₂	C₄	1	12	yes	—
16	C₂×C₄×Dic₂₂	C₂²	1	12	yes	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	12	yes	—
18	C₂×Dic₁₁⋊C₈	C₂²	1	24	yes	—
19	C₂×C₈×Dic₁₁	C₂²	1	24	yes	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	12	yes	—
21	C₈×Dic₂₂	C₂²	1	64	yes	direct product factor
22	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₁	C₂×C₄	1	2	yes	—
26	C₂³×Dic₁₁	C₂×C₄	1	2	yes	—
27	C₂²×Dic₂₂	C₈	1	16	yes	direct product factor
28	C₂²×Dic₂₂	C₂×C₄	1	12	yes	—
29	C₂²×C₈×C₁₁	—	2	4	no	—
30	C₂²×C₈×C₁₁	Q₈	1	8	yes	complemented
31	C₂²×C₈×C₁₁	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₁	C₂³	1	6	yes	—
33	C₂×C₄²×C₁₁	C₂×C₄	1	6	yes	—
34	C₂×C₄²×C₁₁	C₂³	1	6	yes	—
35	C₂×C₄⋊Dic₁₁	C₂×C₄	1	12	yes	—
36	C₂×Dic₁₁⋊C₄	C₂×C₄	1	24	yes	—
37	C₂×C₄×Dic₁₁	C₂×C₄	1	12	yes	—
38	C₂×C₄×Dic₁₁	C₂³	1	12	yes	—
39	C₂²×C₁₁⋊C₈	—	2	8	no	—
40	C₂²×C₁₁⋊C₈	C₂³	1	12	yes	—
41	C₄×Dic₂₂	C₂×C₄	1	48	yes	—
42	C₄×Dic₂₂	C₂³	1	16	yes	—
43	C₄×C₈×C₁₁	C₂³	1	16	yes	—
44	Dic₁₁⋊C₈	C₂³	1	32	yes	—
45	C₈×Dic₁₁	C₂³	1	32	yes	—
46	C₄⋊(C₁₁⋊C₈)	C₂³	1	16	yes	—
47	C₂²×C₈×Q₈	—	11	1	no	maximal, Sylow
48	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	—	2	4	no	—
50	C₂²×C₄×C₁₁	C₂×C₈	1	8	yes	—
51	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
54	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₁₁	#41	1	4	yes	—
56	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
57	C₂²×Dic₁₁	—	2	8	no	—
58	C₂²×Dic₁₁	C₂×C₈	1	16	yes	—
59	C₂²×Dic₁₁	C₂²×C₄	1	12	yes	—
60	C₂²×Dic₁₁	C₂²×C₄	1	12	yes	—
61	C₂×Dic₂₂	C₂×C₈	1	96	yes	direct product factor
62	C₂×Dic₂₂	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₁₁	—	2	12	no	—
64	C₂×C₈×C₁₁	C₂×Q₈	1	24	yes	complemented
65	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
66	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
67	C₄²×C₁₁	C₂²×C₄	1	12	yes	—
68	C₄²×C₁₁	C₂⁴	1	4	yes	—
69	C₄⋊Dic₁₁	C₂²×C₄	1	16	yes	—
70	Dic₁₁⋊C₄	C₂²×C₄	1	32	yes	—
71	C₄×Dic₁₁	C₂²×C₄	1	24	yes	—
72	C₄×Dic₁₁	C₂⁴	1	8	yes	—
73	C₂×C₁₁⋊C₈	—	2	24	no	—
74	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
75	C₂²×C₄×Q₈	—	11	1	no	—
76	C₂×C₈×Q₈	—	11	24	no	—
77	C₂²×C₄⋊C₈	—	11	1	no	—
78	C₂²×C₄⋊C₈	—	11	2	no	—
79	C₂²×C₄×C₈	—	11	2	no	—
80	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
81	C₂³×C₁₁	C₄×Q₈	1	4	yes	—
82	C₂³×C₁₁	C₂²×C₈	1	4	yes	—
83	C₂³×C₁₁	#131	1	4	yes	—
84	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
85	C₂×C₄×C₁₁	—	2	12	no	—
86	C₂×C₄×C₁₁	C₄×Q₈	1	12	yes	—
87	C₂×C₄×C₁₁	C₂²×C₈	1	24	yes	—
88	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
89	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₁₁	C₂²×Q₈	1	6	yes	—
92	C₂×C₄×C₁₁	C₂×#41	1	6	yes	—
93	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
94	C₂×Dic₁₁	—	2	24	no	—
95	C₂×Dic₁₁	C₂²×C₈	1	48	yes	—
96	C₂×Dic₁₁	C₂³×C₄	1	8	yes	—
97	C₂×Dic₁₁	C₂³×C₄	1	8	yes	—
98	Dic₂₂	C₂²×C₈	1	64	yes	direct product factor
99	C₈×C₁₁	—	2	4	no	—
100	C₈×C₁₁	C₂²×Q₈	1	8	yes	complemented
101	C₁₁⋊C₈	—	2	8	no	—
102	C₂³×Q₈	—	11	1	no	—
103	C₂³×C₈	—	11	2	no	—
104	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
106	C₂×C₄×Q₈	—	11	12	no	—
107	C₂×C₄×Q₈	—	11	12	no	—
108	C₂²×C₄⋊C₄	—	11	2	no	—
109	C₂²×C₄⋊C₄	—	11	1	no	—
110	C₂²×C₄²	—	11	2	no	—
111	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
112	C₈×Q₈	—	11	64	no	—
113	C₂×C₄⋊C₈	—	11	24	no	—
114	C₂×C₄⋊C₈	—	11	12	no	—
115	C₂×C₄×C₈	—	11	24	no	—
116	C₂×C₄×C₈	C₂×D₁₁	1	12	yes	—
117	C₂²×C₁₁	C₈×Q₈	1	16	yes	complemented
118	C₂²×C₁₁	C₂×C₄×Q₈	1	6	yes	—
119	C₂²×C₁₁	C₂³×C₈	1	6	yes	—
120	C₂²×C₁₁	C₂×#131	1	6	yes	—
121	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
122	C₄×C₁₁	—	2	4	no	—
123	C₄×C₁₁	C₂×C₄×Q₈	1	6	yes	—
124	C₄×C₁₁	C₂³×C₈	1	8	yes	—
125	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
126	C₄×C₁₁	C₂²×#41	1	1	yes	characteristic
127	Dic₁₁	—	2	8	no	—
128	Dic₁₁	C₂³×C₈	1	16	yes	—
129	C₂²×Q₈	—	11	16	no	—
130	C₂²×Q₈	—	11	12	no	—
131	C₂³×C₄	—	11	2	no	—
132	C₂³×C₄	—	11	2	no	—
133	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
134	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	—	11	12	no	—
138	C₂²×C₈	—	22	8	no	—
139	C₂²×C₈	Dic₂₂	1	8	yes	direct product factor
140	C₂²×C₈	C₂²×D₁₁	1	6	yes	—
141	C₂²×C₈	C₂²×D₁₁	1	6	yes	—
142	C₄×Q₈	—	11	48	no	—
143	C₄×Q₈	—	11	16	no	—
144	C₂×C₄⋊C₄	—	11	24	no	—
145	C₂×C₄⋊C₄	—	11	12	no	—
146	C₂×C₄²	—	11	12	no	—
147	C₂×C₄²	—	11	12	no	—
148	C₂×C₄²	C₄×D₁₁	1	6	yes	—
149	C₂×C₄²	C₂²×D₁₁	1	6	yes	—
150	C₄⋊C₈	—	11	32	no	—
151	C₄⋊C₈	—	11	16	no	—
152	C₄×C₈	—	11	32	no	—
153	C₄×C₈	C₂²×D₁₁	1	16	yes	—
154	C₂×C₁₁	C₂×C₈×Q₈	1	12	yes	complemented
155	C₂×C₁₁	C₂²×C₄×Q₈	1	1	yes	characteristic
156	C₂×C₁₁	C₂⁴×C₈	1	1	yes	characteristic
157	C₂×C₁₁	C₂²×#131	1	1	yes	characteristic
158	C₂⁴	C₂×C₄×D₁₁	1	1	yes	characteristic
159	C₂×Q₈	—	11	96	no	—
160	C₂×Q₈	—	11	16	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	11	16	no	—
163	C₂²×C₄	—	11	12	no	—
164	C₂²×C₄	—	11	12	no	—
165	C₂²×C₄	—	22	8	no	—
166	C₂²×C₄	C₈×D₁₁	1	8	yes	—
167	C₂²×C₄	C₂×Dic₂₂	1	4	yes	—
168	C₂²×C₄	C₂×C₄×D₁₁	1	6	yes	—
169	C₂²×C₄	C₂×C₄×D₁₁	1	3	yes	—
170	C₂²×C₄	C₂×C₄×D₁₁	1	6	yes	—
171	C₂²×C₄	#4339	1	4	yes	—
172	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
173	C₂×C₈	—	2	12	no	—
174	C₂×C₈	—	11	8	no	—
175	C₂×C₈	—	22	24	no	—
176	C₂×C₈	C₂×Dic₂₂	1	24	yes	direct product factor
177	C₂×C₈	C₂³×D₁₁	1	4	yes	—
178	C₂×C₈	C₂³×D₁₁	1	4	yes	—
179	C₄⋊C₄	—	11	32	no	—
180	C₄⋊C₄	—	11	16	no	—
181	C₄²	—	11	24	no	—
182	C₄²	—	11	8	no	—
183	C₄²	C₂×C₄×D₁₁	1	12	yes	—
184	C₄²	C₂³×D₁₁	1	4	yes	—
185	C₁₁	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₄×Dic₂₂	1	4	yes	—
187	C₂³	C₂×C₈×D₁₁	1	4	yes	—
188	C₂³	#66954	1	4	yes	—
189	C₂³	C₂²×C₄×D₁₁	1	3	yes	—
190	Q₈	—	11	64	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	11	48	no	—
193	C₂×C₄	—	11	8	no	—
194	C₂×C₄	—	11	8	no	—
195	C₂×C₄	—	22	24	no	—
196	C₂×C₄	C₄×Dic₂₂	1	12	yes	—
197	C₂×C₄	C₂×C₈×D₁₁	1	24	yes	—
198	C₂×C₄	C₂²×Dic₂₂	1	6	yes	—
199	C₂×C₄	C₂²×C₄×D₁₁	1	4	yes	—
200	C₂×C₄	C₂²×C₄×D₁₁	1	3	yes	—
201	C₂×C₄	C₂²×C₄×D₁₁	1	4	yes	—
202	C₂×C₄	C₂×#4339	1	6	yes	—
203	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	—	22	8	no	—
206	C₈	C₂²×Dic₂₂	1	8	yes	direct product factor
207	C₂²	C₈×Dic₂₂	1	16	yes	direct product factor
208	C₂²	C₂×C₄×Dic₂₂	1	6	yes	—
209	C₂²	C₂²×C₈×D₁₁	1	6	yes	—
210	C₂²	C₂×#66954	1	6	yes	—
211	C₂²	C₂³×C₄×D₁₁	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	11	16	no	—
214	C₄	—	22	8	no	—
215	C₄	C₂×C₄×Dic₂₂	1	6	yes	—
216	C₄	C₂²×C₈×D₁₁	1	8	yes	—
217	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
218	C₄	C₂²×#4339	1	1	yes	characteristic
219	C₂	C₂×C₈×Dic₂₂	1	12	yes	direct product factor
220	C₂	C₂²×C₄×Dic₂₂	1	1	yes	characteristic
221	C₂	C₂³×C₈×D₁₁	1	1	yes	characteristic
222	C₂	C₂²×#66954	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₁₁
commutator	C₂×C₁₁	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle, prime radical	C₂⁴×C₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
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×C₈×Dic₂₂	C₂	direct (N × H)
21. C₈×Dic₂₂	C₂²	direct (N × H)
27. C₂²×Dic₂₂	C₈	direct (N × H)
30. C₂²×C₈×C₁₁	Q₈	semidirect
61. C₂×Dic₂₂	C₂×C₈	direct (N × H)
64. C₂×C₈×C₁₁	C₂×Q₈	semidirect
98. Dic₂₂	C₂²×C₈	direct (N × H)
100. C₈×C₁₁	C₂²×Q₈	semidirect
117. C₂²×C₁₁	C₈×Q₈	semidirect
139. C₂²×C₈	Dic₂₂	direct (N × H)
154. C₂×C₁₁	C₂×C₈×Q₈	semidirect
176. C₂×C₈	C₂×Dic₂₂	direct (N × H)
185. C₁₁	C₂²×C₈×Q₈	semidirect
206. C₈	C₂²×Dic₂₂	direct (N × H)
207. C₂²	C₈×Dic₂₂	direct (N × H)
219. C₂	C₂×C₈×Dic₂₂	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×Dic₂₂	2
C₂×C₈×Dic₂₂	2
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂²×Dic₁₁⋊C₈	2
C₂²×C₈×Dic₁₁	2
C₂²×C₄×C₈×C₁₁	2
C₂²×C₈×Q₈	11

Subgroups — C22×C8×Dic22 #15618124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Dic₂₂ #15618124