Subgroups — C₂²×#10622537 #15617385

Order 2816.

3438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#4318	C₂	1	2	yes	maximal
2	C₂×#10622537	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₁	C₄	1	2	yes	—
6	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₁	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₁₁	C₄	1	12	yes	—
9	C₂²×C₄²×C₁₁	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₁₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₁	C₂²	1	12	yes	—
12	C₂²×#4318	—	2	16	no	—
13	C₂²×#4318	C₂²	1	24	yes	—
14	C₂³×C₁₁⋊C₈	C₂²	1	4	yes	—
15	#10622537	C₂²	1	64	yes	direct product factor
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	48	yes	complemented
17	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₁	—	2	16	no	—
19	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₁	C₂×C₄	1	24	yes	—
21	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
22	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₁	D₄	1	8	yes	—
25	C₂³×C₄×C₁₁	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
27	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	C₂×C₄	1	12	yes	—
29	C₂×C₄²×C₁₁	C₂×C₄	1	48	yes	—
30	C₂×C₄²×C₁₁	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	12	yes	—
32	C₂×C₄²×C₁₁	C₂³	1	12	yes	—
33	C₂×C₄²×C₁₁	C₂³	1	16	yes	—
34	C₂×#4318	—	2	96	no	—
35	C₂×#4318	C₂³	1	32	yes	—
36	C₂²×C₁₁⋊C₈	—	2	16	no	—
37	C₂²×C₁₁⋊C₈	C₂³	1	24	yes	—
38	C₄⋊(C₁₁⋊C₈)	C₂³	1	64	yes	complemented
39	C₂²×#423	—	11	1	no	maximal, Sylow
40	C₂⁴×C₁₁	—	2	8	no	—
41	C₂⁴×C₁₁	C₄⋊C₄	1	8	yes	—
42	C₂⁴×C₁₁	C₂²×C₄	1	6	yes	—
43	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	—	2	8	no	—
45	C₂²×C₄×C₁₁	—	2	96	no	—
46	C₂²×C₄×C₁₁	C₄⋊C₄	1	48	yes	—
47	C₂²×C₄×C₁₁	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₁₁	M₁₆	1	16	yes	complemented
49	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₁₁	C₂²×C₄	1	32	yes	—
51	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₁₁	C₂×Q₈	1	24	yes	—
58	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
61	C₄²×C₁₁	C₂²×C₄	1	16	yes	—
62	C₄²×C₁₁	C₂²×C₄	1	24	yes	—
63	C₄²×C₁₁	C₂²×C₄	1	16	yes	—
64	C₄²×C₁₁	C₂⁴	1	8	yes	—
65	#4318	—	2	64	no	—
66	C₂×C₁₁⋊C₈	—	2	48	no	—
67	C₂×C₁₁⋊C₈	C₂⁴	1	16	yes	—
68	C₂³×M₁₆	—	11	2	no	—
69	C₂³×C₄²	D₁₁	1	1	yes	characteristic
70	C₂×#423	—	11	24	no	—
71	C₂²×C₄⋊C₈	—	11	4	no	—
72	C₂³×C₁₁	—	2	24	no	—
73	C₂³×C₁₁	—	2	32	no	—
74	C₂³×C₁₁	C₂×C₄⋊C₄	1	24	yes	—
75	C₂³×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
76	C₂³×C₁₁	C₂×M₁₆	1	8	yes	—
77	C₂³×C₁₁	C₂³×C₄	1	4	yes	—
78	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
79	C₂×C₄×C₁₁	—	2	24	no	—
80	C₂×C₄×C₁₁	—	2	64	no	—
81	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	48	yes	—
82	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	12	yes	—
83	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	24	yes	—
84	C₂×C₄×C₁₁	C₂×M₁₆	1	48	yes	complemented
85	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
87	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
89	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
90	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
91	C₂×C₄×C₁₁	C₂²×Q₈	1	8	yes	—
92	C₂×C₄×C₁₁	C₂²×Q₈	1	6	yes	—
93	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
94	C₁₁⋊C₈	—	2	16	no	—
95	C₂⁴×C₄	Dic₁₁	1	2	yes	—
96	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
97	C₂²×M₁₆	—	11	24	no	—
98	C₂²×M₁₆	—	22	16	no	—
99	C₂³×C₈	—	11	4	no	—
100	C₂²×C₄²	Dic₁₁	1	1	yes	characteristic
101	C₂²×C₄²	Dic₁₁	1	12	yes	—
102	C₂²×C₄²	Dic₁₁	1	1	yes	characteristic
103	C₂²×C₄²	C₂×D₁₁	1	12	yes	—
104	C₂²×C₄²	C₂×D₁₁	1	2	yes	—
105	#423	—	11	64	no	—
106	C₂×C₄⋊C₈	—	11	48	no	—
107	C₂²×C₁₁	—	2	48	no	—
108	C₂²×C₁₁	—	2	8	no	—
109	C₂²×C₁₁	#423	1	16	yes	complemented
110	C₂²×C₁₁	C₂²×C₄⋊C₄	1	8	yes	—
111	C₂²×C₁₁	C₂²×C₄⋊C₄	1	6	yes	—
112	C₂²×C₁₁	C₂²×M₁₆	1	12	yes	—
113	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
114	C₄×C₁₁	—	2	8	no	—
115	C₄×C₁₁	C₂²×C₄⋊C₄	1	6	yes	—
116	C₄×C₁₁	C₂²×C₄⋊C₄	1	8	yes	—
117	C₄×C₁₁	C₂²×M₁₆	1	16	yes	complemented
118	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
119	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
120	C₂⁵	C₂×Dic₁₁	1	1	yes	characteristic
121	C₂³×C₄	—	2	16	no	—
122	C₂³×C₄	Dic₂₂	1	8	yes	—
123	C₂³×C₄	D₄₄	1	8	yes	—
124	C₂³×C₄	C₂×Dic₁₁	1	2	yes	—
125	C₂³×C₄	C₂×Dic₁₁	1	24	yes	—
126	C₂³×C₄	C₂×Dic₁₁	1	2	yes	—
127	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
128	C₂³×C₄	C₂×Dic₁₁	1	6	yes	—
129	C₂³×C₄	C₂²×D₁₁	1	6	yes	—
130	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
131	C₂×M₁₆	—	11	32	no	—
132	C₂×M₁₆	—	22	96	no	—
133	C₂²×C₈	—	11	24	no	—
134	C₂²×C₈	—	22	16	no	—
135	C₂×C₄²	C₂×Dic₁₁	1	12	yes	—
136	C₂×C₄²	C₂×Dic₁₁	1	48	yes	—
137	C₂×C₄²	C₂×Dic₁₁	1	12	yes	—
138	C₂×C₄²	C₂×Dic₁₁	1	12	yes	—
139	C₂×C₄²	C₂²×D₁₁	1	12	yes	—
140	C₂×C₄²	C₂²×D₁₁	1	16	yes	—
141	C₄⋊C₈	—	11	64	no	—
142	C₂×C₁₁	—	2	8	no	—
143	C₂×C₁₁	C₂×#423	1	12	yes	complemented
144	C₂×C₁₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₁₁	C₂³×M₁₆	1	2	yes	—
146	C₂⁴	—	2	8	no	—
147	C₂⁴	C₄⋊Dic₁₁	1	8	yes	—
148	C₂⁴	C₂²×Dic₁₁	1	6	yes	—
149	C₂⁴	C₂²×Dic₁₁	1	1	yes	characteristic
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	96	no	—
152	C₂²×C₄	#4318	1	16	yes	complemented
153	C₂²×C₄	C₄⋊Dic₁₁	1	48	yes	—
154	C₂²×C₄	C₄⋊Dic₁₁	1	8	yes	—
155	C₂²×C₄	C₂×Dic₂₂	1	24	yes	—
156	C₂²×C₄	C₂×Dic₂₂	1	4	yes	—
157	C₂²×C₄	C₂×D₄₄	1	4	yes	—
158	C₂²×C₄	C₂×D₄₄	1	24	yes	—
159	C₂²×C₄	C₂²×Dic₁₁	1	12	yes	—
160	C₂²×C₄	C₂²×Dic₁₁	1	32	yes	—
161	C₂²×C₄	C₂²×Dic₁₁	1	12	yes	—
162	C₂²×C₄	C₂²×Dic₁₁	1	12	yes	—
163	C₂²×C₄	C₂²×Dic₁₁	1	6	yes	—
164	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
165	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
166	C₂²×C₄	C₂³×D₁₁	1	4	yes	—
167	M₁₆	—	22	64	no	—
168	C₂×C₈	—	11	16	no	—
169	C₂×C₈	—	22	48	no	—
170	C₄²	C₂²×Dic₁₁	1	16	yes	—
171	C₄²	C₂²×Dic₁₁	1	24	yes	—
172	C₄²	C₂²×Dic₁₁	1	16	yes	—
173	C₄²	C₂³×D₁₁	1	8	yes	—
174	C₁₁	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	2	32	no	—
176	C₂³	—	2	24	no	—
177	C₂³	C₂×#4318	1	8	yes	—
178	C₂³	C₂×C₄⋊Dic₁₁	1	24	yes	—
179	C₂³	C₂×C₄⋊Dic₁₁	1	4	yes	—
180	C₂³	C₂³×Dic₁₁	1	4	yes	—
181	C₂³	C₂³×Dic₁₁	1	3	yes	—
182	C₂×C₄	—	2	24	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	C₂×#4318	1	48	yes	complemented
185	C₂×C₄	C₂×C₄⋊Dic₁₁	1	48	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₁₁	1	24	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₁₁	1	12	yes	—
188	C₂×C₄	C₂²×Dic₂₂	1	8	yes	—
189	C₂×C₄	C₂²×Dic₂₂	1	6	yes	—
190	C₂×C₄	C₂²×D₄₄	1	6	yes	—
191	C₂×C₄	C₂²×D₄₄	1	8	yes	—
192	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
193	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
194	C₂×C₄	C₂³×Dic₁₁	1	4	yes	—
195	C₂×C₄	C₂³×Dic₁₁	1	3	yes	—
196	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
197	C₈	—	22	16	no	—
198	C₂²	—	2	48	no	—
199	C₂²	—	2	8	no	—
200	C₂²	#10622537	1	16	yes	direct product factor
201	C₂²	C₂²×#4318	1	12	yes	—
202	C₂²	C₂²×C₄⋊Dic₁₁	1	8	yes	—
203	C₂²	C₂²×C₄⋊Dic₁₁	1	6	yes	—
204	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
205	C₄	—	2	8	no	—
206	C₄	C₂²×#4318	1	16	yes	complemented
207	C₄	C₂²×C₄⋊Dic₁₁	1	6	yes	—
208	C₄	C₂²×C₄⋊Dic₁₁	1	8	yes	—
209	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
210	C₄	C₂³×D₄₄	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#10622537	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊Dic₁₁	1	1	yes	characteristic
214	C₂	C₂³×#4318	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂³×C₄²×C₁₁	C₂
socle	C₂⁴×C₁₁	C₂²×C₄
prime radical	C₂⁵×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂³×C₄²	D₁₁
hypocenter	C₁₁	C₂²×#423
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#10622537	C₂	direct (N × H)
3. C₂²×C₄⋊(C₁₁⋊C₈)	C₂	semidirect
15. #10622537	C₂²	direct (N × H)
16. C₂×C₄⋊(C₁₁⋊C₈)	C₂²	semidirect
38. C₄⋊(C₁₁⋊C₈)	C₂³	semidirect
48. C₂²×C₄×C₁₁	M₁₆	semidirect
84. C₂×C₄×C₁₁	C₂×M₁₆	semidirect
109. C₂²×C₁₁	#423	semidirect
117. C₄×C₁₁	C₂²×M₁₆	semidirect
143. C₂×C₁₁	C₂×#423	semidirect
152. C₂²×C₄	#4318	semidirect
174. C₁₁	C₂²×#423	semidirect
184. C₂×C₄	C₂×#4318	semidirect
200. C₂²	#10622537	direct (N × H)
206. C₄	C₂²×#4318	semidirect
212. C₂	C₂×#10622537	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×#4318	2
C₂×#10622537	2
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂³×C₄²×C₁₁	2
C₂²×#423	11

Subgroups — C22×#10622537 #15617385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622537 #15617385