Subgroups — #14668602

Order 2816.

402 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622104	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622150	C₂	1	2	yes	maximal
3	M₃₂×Dic₁₁	C₂	1	2	yes	maximal
4	#13068304	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#347	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
8	C₂×#66879	C₄	1	1	yes	characteristic
9	C₂×#66879	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66876	C₂²	1	2	yes	—
13	#10622545	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	C₁₁×#347	C₄	1	4	yes	complemented
16	#10622150	C₄	1	8	yes	complemented
17	#10622140	C₄	1	4	yes	complemented
18	C₁₆×Dic₁₁	C₄	1	4	yes	complemented
19	#10622104	C₄	1	4	yes	complemented
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₁₁×M₃₂	C₂×C₄	1	4	yes	—
28	C₁₁×M₃₂	C₂×C₄	1	4	yes	—
29	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
30	C₁₁×#97	C₂×C₄	1	2	yes	—
31	C₁₁×#97	C₂×C₄	1	2	yes	—
32	C₁₁⋊#95	—	2	2	no	—
33	#66879	—	2	2	no	—
34	#66879	C₂×C₄	1	4	yes	—
35	C₈×Dic₁₁	—	2	2	no	—
36	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
37	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
38	#66876	C₂×C₄	1	4	yes	—
39	#66876	C₂×C₄	1	4	yes	—
40	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
41	C₄×C₁₁⋊C₈	—	2	2	no	—
42	#12541	—	11	1	no	maximal, Sylow
43	C₂²×C₄×C₁₁	—	2	1	no	—
44	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₁	—	2	1	no	—
46	C₂²×Dic₁₁	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₁	—	2	1	no	—
48	C₂×C₈×C₁₁	—	2	1	no	—
49	C₂×C₈×C₁₁	C₄²	1	2	yes	—
50	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₁₁	—	2	1	no	—
55	C₄²×C₁₁	C₄²	1	2	yes	—
56	C₄×Dic₁₁	—	2	1	no	—
57	C₄×Dic₁₁	C₄²	1	2	yes	—
58	C₄×Dic₁₁	C₄²	1	2	yes	—
59	C₄×Dic₁₁	M₁₆	1	2	yes	—
60	C₂×C₁₁⋊C₈	—	2	1	no	—
61	C₂×C₁₁⋊C₈	—	2	1	no	—
62	C₂×C₁₁⋊C₈	C₄²	1	2	yes	—
63	C₂×C₁₁⋊C₈	C₄²	1	1	yes	characteristic
64	C₂×C₁₁⋊C₈	C₄²	1	1	yes	characteristic
65	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
66	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
67	C₁₁×C₁₆	C₄²	1	8	yes	—
68	C₁₁⋊C₁₆	C₄²	1	8	yes	—
69	C₂×#347	—	11	2	no	—
70	C₂×#347	—	11	1	no	—
71	C₂×#347	D₁₁	1	1	yes	characteristic
72	C₄×M₃₂	—	11	2	no	—
73	#1726	—	11	1	no	—
74	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₁	—	2	2	no	—
76	C₂×C₄×C₁₁	—	4	2	no	—
77	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₁₁	—	2	2	no	—
84	C₂×Dic₁₁	—	4	2	no	—
85	C₂×Dic₁₁	—	4	2	no	—
86	C₂×Dic₁₁	#97	1	4	yes	—
87	C₈×C₁₁	—	2	2	no	—
88	C₈×C₁₁	C₂×C₄²	1	2	yes	—
89	C₈×C₁₁	C₂×C₄²	1	2	yes	—
90	C₁₁⋊C₈	—	2	2	no	—
91	C₁₁⋊C₈	#97	1	2	yes	—
92	C₁₁⋊C₈	#97	1	2	yes	—
93	C₂×M₃₂	—	11	2	no	—
94	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
95	C₂×#97	—	11	1	no	—
96	C₂×#97	—	11	1	no	—
97	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
98	C₂×C₄×C₈	—	11	1	no	—
99	C₂×C₄×C₈	—	11	1	no	—
100	C₂×C₄×C₈	—	11	1	no	—
101	#376	—	11	1	no	—
102	#347	—	11	8	no	—
103	#347	—	11	4	no	—
104	#347	Dic₁₁	1	4	yes	complemented
105	#346	—	11	4	no	—
106	C₄×C₁₆	—	11	4	no	—
107	C₂²×C₁₁	—	2	2	no	—
108	C₂²×C₁₁	C₂×#97	1	1	yes	characteristic
109	C₂²×C₁₁	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₁₁	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₁₁	—	4	2	no	—
112	C₄×C₁₁	C₄³	1	1	yes	characteristic
113	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
114	C₄×C₁₁	C₄×M₁₆	1	1	yes	characteristic
115	C₄×C₁₁	C₄×M₁₆	1	1	yes	characteristic
116	Dic₁₁	—	2	4	no	—
117	Dic₁₁	—	4	2	no	—
118	C₂²×C₈	—	11	1	no	—
119	C₂²×C₈	—	11	1	no	—
120	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
122	C₂×C₄²	—	11	1	no	—
123	C₂×C₄²	—	11	1	no	—
124	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
125	M₃₂	—	11	4	no	—
126	M₃₂	—	11	4	no	—
127	M₃₂	C₄×D₁₁	1	4	yes	—
128	M₃₂	C₂×Dic₁₁	1	4	yes	—
129	C₂×C₁₆	—	11	4	no	—
130	C₂×C₁₆	C₄×D₁₁	1	4	yes	—
131	#97	—	11	4	no	—
132	#97	—	22	2	no	—
133	#97	C₄×D₁₁	1	2	yes	—
134	#97	C₂×Dic₁₁	1	2	yes	—
135	C₄×C₈	—	11	2	no	—
136	C₄×C₈	—	11	2	no	—
137	C₄×C₈	—	22	2	no	—
138	C₄×C₈	—	22	2	no	—
139	#95	—	22	2	no	—
140	C₂×C₁₁	—	2	1	no	—
141	C₂×C₁₁	—	2	1	no	—
142	C₂×C₁₁	C₄×#97	1	1	yes	characteristic
143	C₂×C₁₁	#2110	1	2	yes	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	11	2	no	—
146	C₂²×C₄	—	22	1	no	—
147	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	11	1	no	—
151	C₂×C₈	—	11	1	no	—
152	C₂×C₈	—	11	2	no	—
153	C₂×C₈	—	11	2	no	—
154	C₂×C₈	—	11	2	no	—
155	C₂×C₈	—	22	1	no	—
156	C₂×C₈	—	22	1	no	—
157	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
161	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
162	C₄²	—	2	1	no	—
163	C₄²	—	11	2	no	—
164	C₄²	—	11	2	no	—
165	C₄²	—	11	2	no	—
166	C₄²	—	22	1	no	—
167	C₄²	C₄×Dic₁₁	1	2	yes	—
168	C₁₆	—	11	8	no	—
169	C₁₆	C₄×Dic₁₁	1	8	yes	—
170	C₁₁	#12541	1	1	yes	characteristic, complemented, Sylow
171	C₂³	M₁₆×D₁₁	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	11	4	no	—
175	C₂×C₄	—	22	2	no	—
176	C₂×C₄	—	44	2	no	—
177	C₂×C₄	—	44	2	no	—
178	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
179	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
182	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	11	2	no	—
186	C₈	—	11	2	no	—
187	C₈	—	22	2	no	—
188	C₈	C₄²×D₁₁	1	2	yes	—
189	C₈	C₂×C₄×Dic₁₁	1	2	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	D₁₁×#97	1	1	yes	characteristic
192	C₂²	#10622320	1	1	yes	characteristic
193	C₂²	M₁₆×Dic₁₁	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	22	4	no	—
196	C₄	—	44	2	no	—
197	C₄	D₁₁×#97	1	1	yes	characteristic
198	C₄	#10622320	1	1	yes	characteristic
199	C₄	C₄²×Dic₁₁	1	1	yes	characteristic
200	C₄	M₁₆×Dic₁₁	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#97×Dic₁₁	1	1	yes	characteristic
204	C₂	#13070219	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₁
commutator	C₄×C₁₁	C₄³
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#347	C₂
socle	C₂²×C₁₁	C₂×#97
prime radical	C₂³×C₁₁	C₂×M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#347	D₁₁
hypocenter	C₁₁	#12541
LCS	G ▷ C₄×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
15. C₁₁×#347	C₄	semidirect
16. #10622150	C₄	semidirect
17. #10622140	C₄	semidirect
18. C₁₆×Dic₁₁	C₄	semidirect
19. #10622104	C₄	semidirect
104. #347	Dic₁₁	semidirect
170. C₁₁	#12541	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#10622104	2
C₂×#10622150	2
M₃₂×Dic₁₁	2
#13068304	2
C₂×C₁₁×#347	2
#12541	11