Subgroups — C₂×D₄×Dic₂₂ #13082090

Order 1408.

1568 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	C₂³×Dic₂₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×D₄×Dic₁₁	C₂	1	2	yes	maximal
4	C₂×#66977	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₂₂	C₂	1	16	yes	maximal, direct product factor
6	C₂×#66941	C₂	1	2	yes	maximal, complemented
7	C₂×#66931	C₂	1	4	yes	maximal, complemented
8	C₂×#66922	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₁	C₂²	1	2	yes	—
12	C₂³×Dic₁₁	C₂²	1	4	yes	—
13	C₂²×Dic₂₂	—	2	16	no	—
14	C₂²×Dic₂₂	C₂²	1	2	yes	—
15	C₂²×Dic₂₂	C₂²	1	4	yes	—
16	C₂²×Dic₂₂	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₂	C₂²	1	16	yes	complemented
18	C₄×D₄×C₁₁	C₂²	1	8	yes	—
19	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₁×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₁	C₂²	1	4	yes	—
23	D₄×Dic₁₁	C₂²	1	16	yes	—
24	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
26	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
27	#66977	C₂²	1	16	yes	complemented
28	C₂×Dic₁₁⋊C₄	C₂²	1	2	yes	—
29	C₂×Dic₁₁⋊C₄	C₂²	1	4	yes	—
30	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
31	#66941	C₂²	1	16	yes	complemented
32	#66931	C₂²	1	32	yes	complemented
33	#66922	C₂²	1	8	yes	complemented
34	C₄×Dic₂₂	C₂²	1	8	yes	complemented
35	C₂⁴×C₁₁	C₂³	1	2	yes	—
36	C₂×D₄×C₁₁	Q₈	1	8	yes	complemented
37	C₂×D₄×C₁₁	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₁	—	2	8	no	—
39	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂³	1	8	yes	—
42	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₁	—	2	16	no	—
45	C₂²×Dic₁₁	C₂³	1	16	yes	—
46	C₂²×Dic₁₁	C₂³	1	2	yes	—
47	C₂²×Dic₁₁	C₂³	1	4	yes	—
48	C₂²×Dic₁₁	C₂³	1	2	yes	—
49	C₂×Dic₂₂	—	2	16	no	—
50	C₂×Dic₂₂	—	2	64	no	—
51	C₂×Dic₂₂	—	2	8	no	—
52	C₂×Dic₂₂	D₄	1	8	yes	direct product factor
53	C₂×Dic₂₂	C₂³	1	8	yes	—
54	C₂×Dic₂₂	C₂³	1	16	yes	—
55	C₂×Dic₂₂	C₂³	1	4	yes	—
56	C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
57	C₁₁×C₂²⋊C₄	C₂³	1	8	yes	—
58	C₄²×C₁₁	C₂³	1	4	yes	—
59	C₂²⋊Dic₁₁	C₂³	1	16	yes	—
60	C₄⋊Dic₁₁	C₂³	1	4	yes	—
61	C₄⋊Dic₁₁	C₂³	1	8	yes	—
62	C₄⋊Dic₁₁	C₂³	1	8	yes	—
63	Dic₁₁⋊C₄	C₂³	1	8	yes	—
64	Dic₁₁⋊C₄	C₂³	1	16	yes	—
65	C₄×Dic₁₁	C₂³	1	8	yes	—
66	C₂×D₄×Q₈	—	11	1	no	maximal, Sylow
67	C₂³×C₁₁	—	2	4	no	—
68	C₂³×C₁₁	—	2	4	no	—
69	C₂³×C₁₁	C₂×Q₈	1	8	yes	complemented
70	C₂³×C₁₁	C₂⁴	1	4	yes	—
71	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
72	D₄×C₁₁	C₂×Q₈	1	16	yes	complemented
73	C₂×C₄×C₁₁	—	2	4	no	—
74	C₂×C₄×C₁₁	—	2	2	no	—
75	C₂×C₄×C₁₁	—	2	16	no	—
76	C₂×C₄×C₁₁	C₂×D₄	1	4	yes	—
77	C₂×C₄×C₁₁	C₂×Q₈	1	4	yes	complemented
78	C₂×C₄×C₁₁	C₂⁴	1	2	yes	—
79	C₂×C₄×C₁₁	C₂⁴	1	4	yes	—
80	C₂×C₄×C₁₁	C₂⁴	1	2	yes	—
81	C₂×C₄×C₁₁	C₂⁴	1	2	yes	—
82	C₂×Dic₁₁	—	2	8	no	—
83	C₂×Dic₁₁	—	2	4	no	—
84	C₂×Dic₁₁	—	2	32	no	—
85	C₂×Dic₁₁	C₂×D₄	1	8	yes	—
86	C₂×Dic₁₁	C₂⁴	1	4	yes	—
87	C₂×Dic₁₁	C₂⁴	1	8	yes	—
88	C₂×Dic₁₁	C₂⁴	1	4	yes	—
89	Dic₂₂	—	2	16	no	—
90	Dic₂₂	—	2	32	no	—
91	Dic₂₂	C₂×D₄	1	16	yes	direct product factor
92	C₂³×Q₈	—	11	2	no	—
93	D₄×Q₈	—	11	16	no	—
94	C₂×C₄⋊Q₈	—	11	2	no	—
95	C₂×C₄⋊Q₈	—	11	1	no	—
96	C₂×C₂²⋊Q₈	—	11	2	no	—
97	C₂×C₂²⋊Q₈	—	11	4	no	—
98	C₂×C₄×Q₈	—	11	1	no	—
99	C₂×C₄×D₄	—	11	2	no	—
100	C₂×C₄×D₄	D₁₁	1	1	yes	characteristic
101	C₂²×C₁₁	—	2	16	no	—
102	C₂²×C₁₁	—	2	4	no	—
103	C₂²×C₁₁	—	2	4	no	—
104	C₂²×C₁₁	C₂²×D₄	1	2	yes	—
105	C₂²×C₁₁	C₂²×Q₈	1	2	yes	—
106	C₂²×C₁₁	C₂²×Q₈	1	8	yes	complemented
107	C₂²×C₁₁	2₋¹⁺⁴	1	2	yes	—
108	C₂²×C₁₁	C₂⁵	1	1	yes	characteristic
109	C₄×C₁₁	—	2	4	no	—
110	C₄×C₁₁	—	2	2	no	—
111	C₄×C₁₁	C₂²×D₄	1	4	yes	—
112	C₄×C₁₁	C₂²×Q₈	1	4	yes	complemented
113	Dic₁₁	—	2	8	no	—
114	Dic₁₁	—	2	4	no	—
115	Dic₁₁	C₂²×D₄	1	8	yes	—
116	C₂²×Q₈	—	11	1	no	—
117	C₂²×Q₈	—	11	4	no	—
118	C₂²×Q₈	—	11	2	no	—
119	C₂²×Q₈	—	11	16	no	—
120	C₂²×Q₈	—	22	16	no	—
121	C₂²×D₄	C₂×D₁₁	1	1	yes	characteristic
122	C₂³×C₄	—	11	4	no	—
123	C₂³×C₄	C₂×D₁₁	1	2	yes	—
124	C₄⋊Q₈	—	11	16	no	—
125	C₄⋊Q₈	—	11	8	no	—
126	C₂²⋊Q₈	—	11	16	no	—
127	C₂²⋊Q₈	—	11	32	no	—
128	C₄×Q₈	—	11	8	no	—
129	C₄×D₄	—	11	16	no	—
130	C₄×D₄	C₂×D₁₁	1	8	yes	—
131	C₂×C₄⋊C₄	—	11	2	no	—
132	C₂×C₄⋊C₄	—	11	2	no	—
133	C₂×C₄⋊C₄	—	11	1	no	—
134	C₂×C₄⋊C₄	—	11	4	no	—
135	C₂×C₄⋊C₄	—	11	2	no	—
136	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	—	11	4	no	—
138	C₂×C₂²⋊C₄	C₂×D₁₁	1	2	yes	—
139	C₂×C₄²	—	11	2	no	—
140	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
141	C₂×C₁₁	—	2	8	no	—
142	C₂×C₁₁	D₄×Q₈	1	4	yes	complemented
143	C₂×C₁₁	C₂³×D₄	1	1	yes	characteristic
144	C₂×C₁₁	C₂³×Q₈	1	1	yes	characteristic
145	C₂×C₁₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	C₂⁴	C₂²×D₁₁	1	2	yes	—
147	C₂×Q₈	—	11	8	no	—
148	C₂×Q₈	—	11	4	no	—
149	C₂×Q₈	—	11	16	no	—
150	C₂×Q₈	—	11	8	no	—
151	C₂×Q₈	—	22	16	no	—
152	C₂×Q₈	—	22	64	no	—
153	C₂×Q₈	—	22	8	no	—
154	C₂×D₄	Dic₂₂	1	8	yes	direct product factor
155	C₂×D₄	C₂²×D₁₁	1	4	yes	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	11	2	no	—
158	C₂²×C₄	—	11	16	no	—
159	C₂²×C₄	—	11	2	no	—
160	C₂²×C₄	—	11	4	no	—
161	C₂²×C₄	—	22	16	no	—
162	C₂²×C₄	C₂²×D₁₁	1	8	yes	—
163	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
167	C₄⋊C₄	—	11	8	no	—
168	C₄⋊C₄	—	11	8	no	—
169	C₄⋊C₄	—	11	4	no	—
170	C₄⋊C₄	—	11	16	no	—
171	C₄⋊C₄	—	11	8	no	—
172	C₄⋊C₄	C₂²×D₁₁	1	4	yes	—
173	C₂²⋊C₄	—	11	16	no	—
174	C₂²⋊C₄	C₂²×D₁₁	1	8	yes	—
175	C₄²	—	11	8	no	—
176	C₄²	C₂²×D₁₁	1	4	yes	—
177	C₁₁	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
178	C₂³	—	2	4	no	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×Dic₂₂	1	8	yes	complemented
181	C₂³	C₂³×D₁₁	1	4	yes	—
182	C₂³	C₂³×D₁₁	1	1	yes	characteristic
183	Q₈	—	11	16	no	—
184	Q₈	—	22	32	no	—
185	Q₈	—	22	16	no	—
186	D₄	C₂×Dic₂₂	1	16	yes	direct product factor
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	11	8	no	—
191	C₂×C₄	—	11	4	no	—
192	C₂×C₄	—	11	4	no	—
193	C₂×C₄	—	11	8	no	—
194	C₂×C₄	—	22	32	no	—
195	C₂×C₄	—	22	8	no	—
196	C₂×C₄	—	22	4	no	—
197	C₂×C₄	C₂×Dic₂₂	1	4	yes	complemented
198	C₂×C₄	D₄×D₁₁	1	4	yes	—
199	C₂×C₄	C₂³×D₁₁	1	2	yes	—
200	C₂×C₄	C₂³×D₁₁	1	4	yes	—
201	C₂×C₄	C₂³×D₁₁	1	2	yes	—
202	C₂×C₄	C₂³×D₁₁	1	2	yes	—
203	C₂²	—	2	16	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₂²×Dic₂₂	1	2	yes	—
207	C₂²	C₂²×Dic₂₂	1	8	yes	complemented
208	C₂²	C₂×D₄×D₁₁	1	2	yes	—
209	C₂²	#67043	1	2	yes	—
210	C₂²	C₂⁴×D₁₁	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	2	no	—
213	C₄	—	11	8	no	—
214	C₄	—	22	8	no	—
215	C₄	—	22	4	no	—
216	C₄	C₂²×Dic₂₂	1	4	yes	complemented
217	C₄	C₂×D₄×D₁₁	1	4	yes	—
218	C₂	—	2	8	no	—
219	C₂	D₄×Dic₂₂	1	4	yes	direct product factor
220	C₂	C₂³×Dic₂₂	1	1	yes	characteristic
221	C₂	C₂²×D₄×D₁₁	1	1	yes	characteristic
222	C₂	C₂×#67043	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁
Fitting	C₂×C₄×D₄×C₁₁	C₂
socle	C₂³×C₁₁	C₂⁴
prime radical	C₂²×D₄×C₁₁	C₂²
hypercenter	C₂×C₄×D₄	D₁₁
hypocenter	C₁₁	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₂₂	C₂	semidirect
4. C₂×#66977	C₂	semidirect
5. D₄×Dic₂₂	C₂	direct (N × H)
6. C₂×#66941	C₂	semidirect
7. C₂×#66931	C₂	semidirect
8. C₂×#66922	C₂	semidirect
9. C₂×C₄×Dic₂₂	C₂	semidirect
17. C₂²×Dic₂₂	C₂²	semidirect
27. #66977	C₂²	semidirect
31. #66941	C₂²	semidirect
32. #66931	C₂²	semidirect
33. #66922	C₂²	semidirect
34. C₄×Dic₂₂	C₂²	semidirect
36. C₂×D₄×C₁₁	Q₈	semidirect
52. C₂×Dic₂₂	D₄	direct (N × H)
69. C₂³×C₁₁	C₂×Q₈	semidirect
72. D₄×C₁₁	C₂×Q₈	semidirect
77. C₂×C₄×C₁₁	C₂×Q₈	semidirect
91. Dic₂₂	C₂×D₄	direct (N × H)
106. C₂²×C₁₁	C₂²×Q₈	semidirect
112. C₄×C₁₁	C₂²×Q₈	semidirect
142. C₂×C₁₁	D₄×Q₈	semidirect
154. C₂×D₄	Dic₂₂	direct (N × H)
177. C₁₁	C₂×D₄×Q₈	semidirect
180. C₂³	C₂×Dic₂₂	semidirect
186. D₄	C₂×Dic₂₂	direct (N × H)
197. C₂×C₄	C₂×Dic₂₂	semidirect
207. C₂²	C₂²×Dic₂₂	semidirect
216. C₄	C₂²×Dic₂₂	semidirect
219. C₂	D₄×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
C₂³×Dic₂₂	2
C₂×C₄×D₄×C₁₁	2
C₂×D₄×Dic₁₁	2
C₂×#66977	2
D₄×Dic₂₂	2
C₂×#66941	2
C₂×#66931	2
C₂×#66922	2
C₂×C₄×Dic₂₂	2
C₂×D₄×Q₈	11

Subgroups — C2×D4×Dic22 #13082090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×Dic₂₂ #13082090