Subgroups — C₂²×Q₈×Dic₂₂ #15660840

Order 2816.

7846 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₂	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#66922	C₂	1	3	yes	maximal
3	C₂²×#66941	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₁₁	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₂₂	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₂	C₂²	1	6	yes	—
9	C₂³×Dic₂₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₁	C₂²	1	24	yes	—
11	C₂²×C₁₁×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂²×C₄²×C₁₁	C₂²	1	3	yes	—
13	C₂×Q₈×Dic₁₁	C₂²	1	48	yes	—
14	C₂²×C₄⋊Dic₁₁	C₂²	1	6	yes	—
15	C₂²×C₄⋊Dic₁₁	C₂²	1	3	yes	—
16	C₂²×Dic₁₁⋊C₄	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₁₁	C₂²	1	6	yes	—
18	Q₈×Dic₂₂	C₂²	1	256	yes	direct product factor
19	C₂×#66941	C₂²	1	144	yes	—
20	C₂×#66922	C₂²	1	72	yes	—
21	C₂×C₄×Dic₂₂	C₂²	1	72	yes	—
22	C₂²×Q₈×C₁₁	Q₈	1	16	yes	complemented
23	C₂²×Q₈×C₁₁	C₂³	1	12	yes	—
24	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
25	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
26	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₁	C₂³	1	6	yes	—
28	C₂³×Dic₁₁	C₂³	1	2	yes	—
29	C₂²×Dic₂₂	—	4	24	no	—
30	C₂²×Dic₂₂	Q₈	1	16	yes	direct product factor
31	C₂²×Dic₂₂	C₂³	1	72	yes	—
32	C₂²×Dic₂₂	C₂³	1	12	yes	—
33	C₄×Q₈×C₁₁	C₂³	1	64	yes	—
34	C₂×C₁₁×C₄⋊C₄	C₂³	1	36	yes	—
35	C₂×C₄²×C₁₁	C₂³	1	36	yes	—
36	Q₈×Dic₁₁	C₂³	1	128	yes	—
37	C₂×C₄⋊Dic₁₁	C₂³	1	36	yes	—
38	C₂×C₄⋊Dic₁₁	C₂³	1	72	yes	—
39	C₂×Dic₁₁⋊C₄	C₂³	1	72	yes	—
40	C₂×C₄×Dic₁₁	C₂³	1	72	yes	—
41	#66941	C₂³	1	384	yes	—
42	#66922	C₂³	1	192	yes	—
43	C₄×Dic₂₂	C₂³	1	192	yes	—
44	C₂²×Q₈²	—	11	1	no	maximal, Sylow
45	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
46	C₂×Q₈×C₁₁	C₂×Q₈	1	96	yes	complemented
47	C₂×Q₈×C₁₁	C₂⁴	1	16	yes	—
48	C₂²×C₄×C₁₁	—	2	12	no	—
49	C₂²×C₄×C₁₁	C₂×Q₈	1	24	yes	—
50	C₂²×C₄×C₁₁	C₂×Q₈	1	8	yes	—
51	C₂²×C₄×C₁₁	C₂⁴	1	18	yes	—
52	C₂²×C₄×C₁₁	C₂⁴	1	18	yes	—
53	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
54	C₂²×Dic₁₁	—	2	24	no	—
55	C₂²×Dic₁₁	C₂×Q₈	1	16	yes	—
56	C₂²×Dic₁₁	C₂⁴	1	36	yes	—
57	C₂²×Dic₁₁	C₂⁴	1	12	yes	—
58	C₂×Dic₂₂	—	4	144	no	—
59	C₂×Dic₂₂	C₂×Q₈	1	96	yes	direct product factor
60	C₂×Dic₂₂	C₂⁴	1	96	yes	—
61	C₂×Dic₂₂	C₂⁴	1	16	yes	—
62	C₁₁×C₄⋊C₄	C₂⁴	1	48	yes	—
63	C₄²×C₁₁	C₂⁴	1	48	yes	—
64	C₄⋊Dic₁₁	C₂⁴	1	48	yes	—
65	C₄⋊Dic₁₁	C₂⁴	1	96	yes	—
66	Dic₁₁⋊C₄	C₂⁴	1	96	yes	—
67	C₄×Dic₁₁	C₂⁴	1	96	yes	—
68	C₂×Q₈²	—	11	48	no	—
69	C₂²×C₄⋊Q₈	—	11	3	no	—
70	C₂²×C₄⋊Q₈	—	11	6	no	—
71	C₂²×C₄×Q₈	—	11	3	no	—
72	C₂²×C₄×Q₈	—	11	2	no	—
73	C₂²×C₄×Q₈	D₁₁	1	1	yes	characteristic
74	C₂³×C₁₁	C₂²×Q₈	1	4	yes	—
75	C₂³×C₁₁	C₂²×Q₈	1	4	yes	—
76	C₂³×C₁₁	H(5,𝔽₂)	1	4	yes	—
77	C₂³×C₁₁	C₂⁵	1	3	yes	—
78	Q₈×C₁₁	C₂²×Q₈	1	64	yes	complemented
79	C₂×C₄×C₁₁	—	2	36	no	—
80	C₂×C₄×C₁₁	C₂²×Q₈	1	72	yes	—
81	C₂×C₄×C₁₁	C₂²×Q₈	1	24	yes	—
82	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
83	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
84	C₂×C₄×C₁₁	C₂⁵	1	4	yes	—
85	C₂×Dic₁₁	—	2	72	no	—
86	C₂×Dic₁₁	C₂²×Q₈	1	48	yes	—
87	C₂×Dic₁₁	C₂⁵	1	24	yes	—
88	C₂×Dic₁₁	C₂⁵	1	8	yes	—
89	Dic₂₂	—	4	96	no	—
90	Dic₂₂	C₂²×Q₈	1	64	yes	direct product factor
91	C₂³×Q₈	—	11	1	no	—
92	C₂³×Q₈	—	11	6	no	—
93	C₂³×Q₈	C₂×D₁₁	1	1	yes	characteristic
94	Q₈²	—	11	256	no	—
95	C₂×C₄⋊Q₈	—	11	144	no	—
96	C₂×C₄⋊Q₈	—	11	72	no	—
97	C₂×C₄×Q₈	—	11	48	no	—
98	C₂×C₄×Q₈	—	11	72	no	—
99	C₂×C₄×Q₈	C₂×D₁₁	1	24	yes	—
100	C₂²×C₄⋊C₄	—	11	6	no	—
101	C₂²×C₄⋊C₄	—	11	6	no	—
102	C₂²×C₄⋊C₄	—	11	3	no	—
103	C₂²×C₄⋊C₄	C₂×D₁₁	1	3	yes	—
104	C₂²×C₄²	—	11	6	no	—
105	C₂²×C₄²	C₂×D₁₁	1	3	yes	—
106	C₂²×C₁₁	Q₈²	1	16	yes	complemented
107	C₂²×C₁₁	C₂³×Q₈	1	6	yes	—
108	C₂²×C₁₁	C₂³×Q₈	1	6	yes	—
109	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	6	yes	—
110	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
111	C₄×C₁₁	—	2	12	no	—
112	C₄×C₁₁	C₂³×Q₈	1	24	yes	—
113	C₄×C₁₁	C₂³×Q₈	1	8	yes	—
114	Dic₁₁	—	2	24	no	—
115	Dic₁₁	C₂³×Q₈	1	16	yes	—
116	C₂²×Q₈	—	11	16	no	—
117	C₂²×Q₈	—	11	12	no	—
118	C₂²×Q₈	—	11	72	no	—
119	C₂²×Q₈	—	44	24	no	—
120	C₂²×Q₈	Dic₂₂	1	16	yes	direct product factor
121	C₂²×Q₈	C₂²×D₁₁	1	12	yes	—
122	C₂³×C₄	—	11	2	no	—
123	C₂³×C₄	—	11	6	no	—
124	C₂³×C₄	C₂²×D₁₁	1	3	yes	—
125	C₂³×C₄	C₂²×D₁₁	1	3	yes	—
126	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	11	384	no	—
128	C₄⋊Q₈	—	11	192	no	—
129	C₄×Q₈	—	11	128	no	—
130	C₄×Q₈	—	11	192	no	—
131	C₄×Q₈	C₂²×D₁₁	1	64	yes	—
132	C₂×C₄⋊C₄	—	11	72	no	—
133	C₂×C₄⋊C₄	—	11	72	no	—
134	C₂×C₄⋊C₄	—	11	36	no	—
135	C₂×C₄⋊C₄	C₂²×D₁₁	1	36	yes	—
136	C₂×C₄²	—	11	72	no	—
137	C₂×C₄²	C₂²×D₁₁	1	36	yes	—
138	C₂×C₁₁	C₂×Q₈²	1	12	yes	complemented
139	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
140	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
141	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
142	C₂⁴	C₂³×D₁₁	1	1	yes	characteristic
143	C₂×Q₈	—	11	96	no	—
144	C₂×Q₈	—	11	16	no	—
145	C₂×Q₈	—	11	96	no	—
146	C₂×Q₈	—	44	144	no	—
147	C₂×Q₈	C₂×Dic₂₂	1	96	yes	direct product factor
148	C₂×Q₈	C₂³×D₁₁	1	16	yes	—
149	C₂²×C₄	—	2	12	no	—
150	C₂²×C₄	—	11	16	no	—
151	C₂²×C₄	—	11	12	no	—
152	C₂²×C₄	—	11	36	no	—
153	C₂²×C₄	—	22	24	no	—
154	C₂²×C₄	C₂×Dic₂₂	1	24	yes	—
155	C₂²×C₄	Q₈×D₁₁	1	8	yes	—
156	C₂²×C₄	C₂³×D₁₁	1	18	yes	—
157	C₂²×C₄	C₂³×D₁₁	1	18	yes	—
158	C₂²×C₄	C₂³×D₁₁	1	6	yes	—
159	C₄⋊C₄	—	11	96	no	—
160	C₄⋊C₄	—	11	96	no	—
161	C₄⋊C₄	—	11	48	no	—
162	C₄⋊C₄	C₂³×D₁₁	1	48	yes	—
163	C₄²	—	11	96	no	—
164	C₄²	C₂³×D₁₁	1	48	yes	—
165	C₁₁	C₂²×Q₈²	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂²×Dic₂₂	1	4	yes	—
167	C₂³	C₂×Q₈×D₁₁	1	4	yes	—
168	C₂³	#67042	1	4	yes	—
169	C₂³	C₂⁴×D₁₁	1	3	yes	—
170	Q₈	—	11	64	no	—
171	Q₈	—	44	96	no	—
172	Q₈	C₂²×Dic₂₂	1	64	yes	direct product factor
173	C₂×C₄	—	2	36	no	—
174	C₂×C₄	—	11	48	no	—
175	C₂×C₄	—	11	8	no	—
176	C₂×C₄	—	11	24	no	—
177	C₂×C₄	—	22	72	no	—
178	C₂×C₄	C₂²×Dic₂₂	1	72	yes	—
179	C₂×C₄	C₂×Q₈×D₁₁	1	24	yes	—
180	C₂×C₄	C₂⁴×D₁₁	1	12	yes	—
181	C₂×C₄	C₂⁴×D₁₁	1	12	yes	—
182	C₂×C₄	C₂⁴×D₁₁	1	4	yes	—
183	C₂²	Q₈×Dic₂₂	1	16	yes	direct product factor
184	C₂²	C₂³×Dic₂₂	1	6	yes	—
185	C₂²	C₂²×Q₈×D₁₁	1	6	yes	—
186	C₂²	C₂×#67042	1	6	yes	—
187	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
188	C₄	—	2	12	no	—
189	C₄	—	11	16	no	—
190	C₄	—	22	24	no	—
191	C₄	C₂³×Dic₂₂	1	24	yes	—
192	C₄	C₂²×Q₈×D₁₁	1	8	yes	—
193	C₂	C₂×Q₈×Dic₂₂	1	12	yes	direct product factor
194	C₂	C₂³×Q₈×D₁₁	1	1	yes	characteristic
195	C₂	C₂⁴×Dic₂₂	1	1	yes	characteristic
196	C₂	C₂²×#67042	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂²×C₄×Q₈×C₁₁	C₂
socle, prime radical	C₂⁴×C₁₁	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₁₁
hypocenter	C₁₁	C₂²×Q₈²
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₂₂	C₂	direct (N × H)
18. Q₈×Dic₂₂	C₂²	direct (N × H)
22. C₂²×Q₈×C₁₁	Q₈	semidirect
30. C₂²×Dic₂₂	Q₈	direct (N × H)
46. C₂×Q₈×C₁₁	C₂×Q₈	semidirect
59. C₂×Dic₂₂	C₂×Q₈	direct (N × H)
78. Q₈×C₁₁	C₂²×Q₈	semidirect
90. Dic₂₂	C₂²×Q₈	direct (N × H)
106. C₂²×C₁₁	Q₈²	semidirect
120. C₂²×Q₈	Dic₂₂	direct (N × H)
138. C₂×C₁₁	C₂×Q₈²	semidirect
147. C₂×Q₈	C₂×Dic₂₂	direct (N × H)
165. C₁₁	C₂²×Q₈²	semidirect
172. Q₈	C₂²×Dic₂₂	direct (N × H)
183. C₂²	Q₈×Dic₂₂	direct (N × H)
193. C₂	C₂×Q₈×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
C₂×Q₈×Dic₂₂	2
C₂²×#66922	2
C₂²×#66941	2
C₂²×Q₈×Dic₁₁	2
C₂²×C₄×Dic₂₂	2
C₂²×C₄×Q₈×C₁₁	2
C₂²×Q₈²	11

Subgroups — C22×Q8×Dic22 #15660840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈×Dic₂₂ #15660840