Subgroups — C₂²×#10622202 #14906649

Order 2816.

1214 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#66882	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622202	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₁	—	2	1	no	—
6	C₂²×C₁₁×Q₁₆	—	2	3	no	—
7	C₂²×C₁₁×Q₁₆	—	2	4	no	—
8	C₂²×C₁₁×Q₁₆	C₄	1	8	yes	complemented
9	C₂²×C₁₁×Q₁₆	C₂²	1	6	yes	—
10	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₁	—	2	1	no	—
12	C₂×#66882	—	2	3	no	—
13	C₂×#66882	C₂²	1	6	yes	—
14	C₂²×C₁₁⋊C₁₆	—	2	4	no	—
15	C₂²×C₁₁⋊C₁₆	C₂²	1	6	yes	—
16	#10622202	C₂²	1	16	yes	direct product factor
17	C₂²×Q₈×C₁₁	—	2	8	no	—
18	C₂²×Q₈×C₁₁	—	2	6	no	—
19	C₂²×Q₈×C₁₁	—	4	3	no	—
20	C₂²×Q₈×C₁₁	—	4	4	no	—
21	C₂³×C₄×C₁₁	—	4	1	no	—
22	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂³×Dic₁₁	—	4	1	no	—
24	C₂×C₁₁×Q₁₆	—	2	24	no	—
25	C₂×C₁₁×Q₁₆	—	2	12	no	—
26	C₂×C₁₁×Q₁₆	—	2	6	no	—
27	C₂×C₁₁×Q₁₆	C₂×C₄	1	24	yes	complemented
28	C₂×C₁₁×Q₁₆	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₁	D₄	1	3	yes	—
31	C₂²×C₈×C₁₁	D₄	1	4	yes	—
32	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
33	C₂×C₄⋊Dic₁₁	—	2	6	no	—
34	C₂×C₄⋊Dic₁₁	—	4	3	no	—
35	#66882	—	2	6	no	—
36	#66882	C₂³	1	4	yes	—
37	C₂×C₁₁⋊C₁₆	—	2	12	no	—
38	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
39	C₂²×#358	—	11	1	no	maximal, Sylow
40	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
41	C₂×Q₈×C₁₁	—	2	24	no	—
42	C₂×Q₈×C₁₁	—	2	4	no	—
43	C₂×Q₈×C₁₁	—	4	24	no	—
44	C₂×Q₈×C₁₁	—	4	6	no	—
45	C₂×Q₈×C₁₁	—	4	12	no	—
46	C₂²×C₄×C₁₁	—	4	8	no	—
47	C₂²×C₄×C₁₁	—	4	6	no	—
48	C₂²×C₄×C₁₁	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₁	D₈	1	3	yes	—
50	C₂²×C₄×C₁₁	SD₁₆	1	4	yes	—
51	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
52	C₂²×Dic₁₁	—	4	6	no	—
53	C₂²×Dic₁₁	—	8	4	no	—
54	C₁₁×Q₁₆	—	2	24	no	—
55	C₁₁×Q₁₆	—	2	4	no	—
56	C₁₁×Q₁₆	C₂²×C₄	1	8	yes	complemented
57	C₂×C₈×C₁₁	C₂²⋊C₄	1	12	yes	—
58	C₂×C₈×C₁₁	C₂²×C₄	1	6	yes	—
59	C₂×C₈×C₁₁	C₂×D₄	1	6	yes	—
60	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
62	C₄⋊Dic₁₁	—	2	4	no	—
63	C₄⋊Dic₁₁	—	4	6	no	—
64	C₁₁⋊C₁₆	—	2	4	no	—
65	C₂³×Q₁₆	D₁₁	1	1	yes	characteristic
66	C₂³×C₁₆	—	11	1	no	—
67	C₂²×#107	—	11	1	no	—
68	C₂×#358	—	11	12	no	—
69	C₂³×C₁₁	#102	1	4	yes	—
70	C₂³×C₁₁	SD₃₂	1	4	yes	—
71	C₂³×C₁₁	Q₃₂	1	4	yes	—
72	C₂³×C₁₁	C₂×D₈	1	3	yes	—
73	Q₈×C₁₁	—	2	8	no	—
74	Q₈×C₁₁	—	4	4	no	—
75	Q₈×C₁₁	—	4	24	no	—
76	C₂×C₄×C₁₁	—	4	4	no	—
77	C₂×C₄×C₁₁	—	4	24	no	—
78	C₂×C₄×C₁₁	#102	1	12	yes	—
79	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₁₁	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₁₁	C₂×SD₁₆	1	6	yes	—
82	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₁₁	—	4	4	no	—
84	C₂×Dic₁₁	—	8	12	no	—
85	C₈×C₁₁	C₂×C₂²⋊C₄	1	6	yes	—
86	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
88	C₂³×Q₈	—	2	1	no	—
89	C₂²×Q₁₆	—	2	3	no	—
90	C₂²×Q₁₆	—	2	4	no	—
91	C₂²×Q₁₆	Dic₁₁	1	8	yes	complemented
92	C₂²×Q₁₆	C₂×D₁₁	1	6	yes	—
93	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
94	C₂²×C₄⋊C₄	—	22	1	no	—
95	C₂²×C₁₆	—	11	6	no	—
96	C₂²×C₁₆	—	22	4	no	—
97	C₂×#107	—	11	6	no	—
98	C₂×#107	—	22	3	no	—
99	#358	—	11	16	no	—
100	C₂²×C₁₁	#358	1	16	yes	complemented
101	C₂²×C₁₁	C₂×#102	1	6	yes	—
102	C₂²×C₁₁	C₂×SD₃₂	1	6	yes	—
103	C₂²×C₁₁	C₂×Q₃₂	1	6	yes	—
104	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
105	C₄×C₁₁	—	4	8	no	—
106	C₄×C₁₁	C₂×#102	1	6	yes	—
107	C₄×C₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
109	Dic₁₁	—	8	4	no	—
110	C₂²×Q₈	—	2	8	no	—
111	C₂²×Q₈	—	2	6	no	—
112	C₂²×Q₈	—	4	3	no	—
113	C₂²×Q₈	—	4	4	no	—
114	C₂³×C₄	—	4	1	no	—
115	C₂³×C₄	—	44	1	no	—
116	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
117	C₂×Q₁₆	—	2	24	no	—
118	C₂×Q₁₆	—	2	6	no	—
119	C₂×Q₁₆	—	2	12	no	—
120	C₂×Q₁₆	C₂×Dic₁₁	1	24	yes	complemented
121	C₂×Q₁₆	C₂²×D₁₁	1	4	yes	—
122	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
123	C₂²×C₈	C₂²⋊D₁₁	1	3	yes	—
124	C₂²×C₈	C₂²⋊D₁₁	1	4	yes	—
125	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
126	C₂×C₄⋊C₄	—	22	6	no	—
127	C₂×C₄⋊C₄	—	44	3	no	—
128	C₂×C₁₆	—	11	4	no	—
129	C₂×C₁₆	—	22	12	no	—
130	#107	—	11	4	no	—
131	#107	—	22	6	no	—
132	C₂×C₁₁	C₂×#358	1	12	yes	complemented
133	C₂×C₁₁	C₂²×#102	1	1	yes	characteristic
134	C₂×C₁₁	C₂²×SD₃₂	1	1	yes	characteristic
135	C₂×C₁₁	C₂²×Q₃₂	1	1	yes	characteristic
136	C₂⁴	#4323	1	1	yes	characteristic
137	C₂×Q₈	—	2	24	no	—
138	C₂×Q₈	—	2	4	no	—
139	C₂×Q₈	—	4	24	no	—
140	C₂×Q₈	—	4	6	no	—
141	C₂×Q₈	—	4	12	no	—
142	C₂²×C₄	—	4	8	no	—
143	C₂²×C₄	—	4	6	no	—
144	C₂²×C₄	—	44	6	no	—
145	C₂²×C₄	—	88	4	no	—
146	C₂²×C₄	#4323	1	3	yes	—
147	C₂²×C₄	#4324	1	4	yes	—
148	C₂²×C₄	C₂²⋊Dic₁₁	1	4	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₁₁	1	3	yes	—
150	Q₁₆	—	2	24	no	—
151	Q₁₆	—	2	4	no	—
152	Q₁₆	C₂²×Dic₁₁	1	8	yes	complemented
153	C₂×C₈	C₂²⋊Dic₁₁	1	12	yes	—
154	C₂×C₈	C₂²×Dic₁₁	1	6	yes	—
155	C₂×C₈	C₂×C₂²⋊D₁₁	1	3	yes	—
156	C₂×C₈	C₂×C₂²⋊D₁₁	1	6	yes	—
157	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
158	C₄⋊C₄	—	22	4	no	—
159	C₄⋊C₄	—	44	6	no	—
160	C₁₆	—	22	4	no	—
161	C₁₁	C₂²×#358	1	1	yes	characteristic, complemented, Sylow
162	C₂³	#66892	1	4	yes	—
163	C₂³	#66893	1	4	yes	—
164	C₂³	#66896	1	4	yes	—
165	C₂³	C₂×#4323	1	3	yes	—
166	Q₈	—	2	8	no	—
167	Q₈	—	4	4	no	—
168	Q₈	—	4	24	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	24	no	—
171	C₂×C₄	—	44	4	no	—
172	C₂×C₄	—	88	12	no	—
173	C₂×C₄	#66896	1	12	yes	—
174	C₂×C₄	C₂×#4323	1	3	yes	—
175	C₂×C₄	C₂×#4324	1	6	yes	—
176	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	6	yes	—
177	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
178	C₈	C₂×C₂²⋊Dic₁₁	1	6	yes	—
179	C₈	C₂³×Dic₁₁	1	1	yes	characteristic
180	C₈	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
181	C₂²	#10622202	1	16	yes	direct product factor
182	C₂²	C₂×#66892	1	6	yes	—
183	C₂²	C₂×#66893	1	6	yes	—
184	C₂²	C₂×#66896	1	6	yes	—
185	C₂²	C₂²×#4323	1	1	yes	characteristic
186	C₄	—	4	8	no	—
187	C₄	—	88	4	no	—
188	C₄	C₂×#66896	1	6	yes	—
189	C₄	C₂²×#4324	1	1	yes	characteristic
190	C₄	C₂²×C₂²⋊Dic₁₁	1	1	yes	characteristic
191	C₂	C₂×#10622202	1	12	yes	direct product factor
192	C₂	C₂²×#66896	1	1	yes	characteristic
193	C₂	C₂²×#66892	1	1	yes	characteristic
194	C₂	C₂²×#66893	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#4323
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂³×C₁₁×Q₁₆	C₂
socle, prime radical	C₂⁴×C₁₁	D₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂³×Q₁₆	D₁₁
hypocenter	C₁₁	C₂²×#358
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#10622202	C₂	direct (N × H)
8. C₂²×C₁₁×Q₁₆	C₄	semidirect
16. #10622202	C₂²	direct (N × H)
27. C₂×C₁₁×Q₁₆	C₂×C₄	semidirect
56. C₁₁×Q₁₆	C₂²×C₄	semidirect
91. C₂²×Q₁₆	Dic₁₁	semidirect
100. C₂²×C₁₁	#358	semidirect
120. C₂×Q₁₆	C₂×Dic₁₁	semidirect
132. C₂×C₁₁	C₂×#358	semidirect
152. Q₁₆	C₂²×Dic₁₁	semidirect
161. C₁₁	C₂²×#358	semidirect
181. C₂²	#10622202	direct (N × H)
191. C₂	C₂×#10622202	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×C₁₁⋊C₁₆	2
C₂²×#66882	2
C₂×#10622202	2
C₂³×C₁₁×Q₁₆	2
C₂²×#358	11

Subgroups — C22×#10622202 #14906649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622202 #14906649