Subgroups — #14668607

Order 2816.

350 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070220	C₂	1	2	yes	maximal
2	#13070215	C₂	1	2	yes	maximal
3	#13070213	C₂	1	1	yes	maximal, characteristic
4	#13068371	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#2110	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
7	C₁₁×#405	C₂²	1	1	yes	characteristic
8	#10622714	C₂²	1	2	yes	—
9	C₂×#66876	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₁₁×#347	C₄	1	4	yes	complemented
12	C₁₁⋊#338	C₄	1	1	yes	characteristic
13	#10622195	C₄	1	1	yes	characteristic
14	#10622169	C₄	1	2	yes	—
15	#10622150	C₄	1	8	yes	complemented
16	#10622140	C₄	1	4	yes	complemented
17	C₁₆×Dic₁₁	C₄	1	4	yes	complemented
18	#10622099	C₄	1	2	yes	complemented
19	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	complemented
20	C₂×C₁₁×M₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₁₁×#117	C₂×C₄	1	1	yes	characteristic
23	#66979	C₂×C₄	1	2	yes	—
24	C₂×#4318	C₂×C₄	1	2	yes	—
25	C₁₁×M₃₂	C₂×C₄	1	4	yes	—
26	C₁₁×M₃₂	C₂×C₄	1	4	yes	—
27	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
28	C₁₁×#97	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
30	#66879	C₂×C₄	1	4	yes	—
31	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
32	#66876	C₂×C₄	1	4	yes	—
33	C₂×C₁₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
38	#12542	—	11	1	no	maximal, Sylow
39	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₁₁×M₁₆	—	2	2	no	—
41	C₂×C₈×C₁₁	C₄²	1	2	yes	—
42	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
44	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₁₁×C₄⋊C₄	—	2	1	no	—
47	C₁₁×C₂²⋊C₄	—	2	1	no	—
48	C₄²×C₁₁	C₄²	1	2	yes	—
49	C₂²⋊Dic₁₁	—	2	2	no	—
50	C₄⋊Dic₁₁	—	2	2	no	—
51	C₄×Dic₁₁	C₄²	1	4	yes	—
52	#4318	—	2	2	no	—
53	#4318	—	2	2	no	—
54	C₂×C₁₁⋊C₈	C₄²	1	4	yes	—
55	C₁₁×C₁₆	C₄²	1	8	yes	—
56	C₁₁⋊C₁₆	C₄²	1	1	yes	characteristic
57	C₁₁⋊C₁₆	C₄²	1	1	yes	characteristic
58	C₁₁⋊C₁₆	C₄²	1	2	yes	—
59	C₁₁⋊C₁₆	C₄²	1	4	yes	—
60	#2110	—	11	2	no	—
61	#2110	D₁₁	1	1	yes	characteristic
62	#2108	—	11	2	no	—
63	#2108	—	11	1	no	—
64	#1747	—	11	1	no	—
65	C₂³×C₁₁	#131	1	1	yes	characteristic
66	C₂×C₄×C₁₁	—	2	2	no	—
67	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#131	1	2	yes	—
71	C₂×Dic₁₁	—	2	4	no	—
72	C₈×C₁₁	—	2	2	no	—
73	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
74	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₁₁	C₂×C₄²	1	2	yes	—
76	C₁₁⋊C₈	—	2	4	no	—
77	C₂×M₃₂	—	11	1	no	—
78	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
79	C₂²×C₁₆	—	11	1	no	—
80	#405	—	11	2	no	—
81	#405	C₂×D₁₁	1	1	yes	characteristic
82	#347	—	11	8	no	—
83	#347	Dic₁₁	1	4	yes	complemented
84	#346	—	11	4	no	—
85	#346	—	11	2	no	—
86	C₄×C₁₆	—	11	4	no	—
87	C₄×C₁₆	—	11	2	no	—
88	#338	—	11	1	no	—
89	H(3,ℤ/4)	—	11	1	no	—
90	H(3,ℤ/4)	—	11	2	no	—
91	C₂²×C₁₁	—	4	1	no	—
92	C₂²×C₁₁	#405	1	1	yes	characteristic
93	C₂²×C₁₁	#405	1	2	yes	—
94	C₄×C₁₁	—	4	2	no	—
95	C₄×C₁₁	C₄³	1	1	yes	characteristic
96	C₄×C₁₁	#405	1	1	yes	characteristic
97	C₄×C₁₁	#405	1	2	yes	—
98	Dic₁₁	—	4	4	no	—
99	C₂×M₁₆	—	11	2	no	—
100	C₂×M₁₆	C₄×D₁₁	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
102	#117	—	11	2	no	—
103	#117	C₄×D₁₁	1	1	yes	characteristic
104	M₃₂	—	11	4	no	—
105	M₃₂	C₄×D₁₁	1	4	yes	—
106	M₃₂	C₂×Dic₁₁	1	4	yes	—
107	C₂×C₁₆	—	11	1	no	—
108	C₂×C₁₆	—	11	2	no	—
109	C₂×C₁₆	—	11	2	no	—
110	C₂×C₁₆	—	11	1	no	—
111	C₂×C₁₆	—	11	2	no	—
112	C₂×C₁₆	C₄×D₁₁	1	4	yes	—
113	#97	—	11	4	no	—
114	#97	C₂×Dic₁₁	1	2	yes	—
115	C₄×C₈	—	11	4	no	—
116	C₄×C₈	C₄×D₁₁	1	2	yes	—
117	C₂×C₁₁	—	2	2	no	—
118	C₂×C₁₁	—	2	1	no	—
119	C₂×C₁₁	#1726	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
121	M₁₆	—	2	2	no	—
122	M₁₆	—	22	2	no	—
123	M₁₆	—	22	2	no	—
124	C₂×C₈	—	11	4	no	—
125	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
126	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
127	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
129	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	22	2	no	—
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	22	2	no	—
134	C₄²	—	11	4	no	—
135	C₄²	C₄×Dic₁₁	1	2	yes	—
136	C₁₆	—	11	4	no	—
137	C₁₆	—	11	1	no	—
138	C₁₆	—	11	1	no	—
139	C₁₆	—	11	2	no	—
140	C₁₆	C₄×Dic₁₁	1	8	yes	—
141	C₁₁	#12542	1	1	yes	characteristic, complemented, Sylow
142	C₂³	#66960	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	22	4	no	—
145	C₂×C₄	C₄²×D₁₁	1	2	yes	—
146	C₂×C₄	#66960	1	2	yes	—
147	C₂×C₄	#66960	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	22	4	no	—
151	C₈	C₄²×D₁₁	1	1	yes	characteristic
152	C₈	C₄²×D₁₁	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₁₁	1	2	yes	—
154	C₂²	—	4	1	no	—
155	C₂²	#10622321	1	2	yes	—
156	C₂²	#10622735	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	44	4	no	—
159	C₄	#10622321	1	2	yes	—
160	C₄	C₄²×Dic₁₁	1	1	yes	characteristic
161	C₄	#10622735	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#13068304	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₁₁
commutator	C₄×C₁₁	C₄³
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2110	C₂
socle	C₂×C₁₁	#1726
prime radical	C₂³×C₁₁	#131
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2110	D₁₁
hypocenter	C₁₁	#12542
LCS	G ▷ C₄×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
11. C₁₁×#347	C₄	semidirect
15. #10622150	C₄	semidirect
16. #10622140	C₄	semidirect
17. C₁₆×Dic₁₁	C₄	semidirect
18. #10622099	C₄	semidirect
19. C₄×C₁₁⋊C₁₆	C₄	semidirect
83. #347	Dic₁₁	semidirect
141. C₁₁	#12542	semidirect

Maximal subgroups

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

subgroup	index
#13070220	2
#13070215	2
#13070213	2
#13068371	2
C₁₁×#2110	2
#12542	11