Subgroups — #14668781

Order 2816.

338 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#66876	C₂	1	1	yes	maximal, characteristic
2	#13068302	C₂	1	1	yes	maximal, characteristic
3	#13066695	C₂	1	4	yes	maximal
4	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×#66879	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₁	C₄	1	2	yes	—
9	C₂×#66876	C₂²	1	2	yes	—
10	#10622545	C₄	1	1	yes	characteristic
11	C₂×#66867	C₄	1	1	yes	characteristic
12	C₁₁×#346	C₄	1	2	yes	complemented
13	C₄×C₁₁×C₁₆	C₄	1	2	yes	complemented
14	#10622099	C₄	1	2	yes	complemented
15	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	complemented
16	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₁₁×M₃₂	—	2	4	no	—
22	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₁₁⋊#95	—	2	2	no	—
28	#66879	—	2	4	no	—
29	C₈×Dic₁₁	C₈	1	8	yes	—
30	#66876	—	2	4	no	—
31	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
32	#66867	—	2	2	no	—
33	#12593	—	11	1	no	maximal, Sylow
34	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
35	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₁₁	—	2	2	no	—
37	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₁	#41	1	4	yes	—
43	C₄²×C₁₁	C₄²	1	1	yes	characteristic
44	C₄²×C₁₁	C₄²	1	1	yes	characteristic
45	C₄²×C₁₁	M₁₆	1	2	yes	—
46	C₄×Dic₁₁	—	2	2	no	—
47	C₄×Dic₁₁	C₂×C₈	1	4	yes	—
48	C₂×C₁₁⋊C₈	—	2	2	no	—
49	C₂×C₁₁⋊C₈	—	2	2	no	—
50	C₂×C₁₁⋊C₈	C₂×C₈	1	4	yes	—
51	C₁₁×C₁₆	—	2	4	no	—
52	C₁₁⋊C₁₆	—	2	4	no	—
53	C₄×M₃₂	—	11	1	no	—
54	C₄×M₃₂	D₁₁	1	1	yes	characteristic
55	#1726	—	11	1	no	—
56	#1375	—	11	4	no	—
57	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₁	—	4	2	no	—
59	C₂×C₄×C₁₁	#97	1	2	yes	—
60	C₂×C₄×C₁₁	#97	1	2	yes	—
61	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₁	#117	1	2	yes	—
63	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₁₁	—	2	4	no	—
66	C₂×Dic₁₁	—	4	4	no	—
67	C₈×C₁₁	C₄×C₈	1	2	yes	—
68	C₈×C₁₁	C₄×C₈	1	2	yes	—
69	C₈×C₁₁	#117	1	2	yes	—
70	C₈×C₁₁	#117	1	2	yes	—
71	C₁₁⋊C₈	—	2	4	no	—
72	C₂×M₃₂	—	11	2	no	—
73	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
74	C₂×#97	—	11	2	no	—
75	C₂×#97	—	11	1	no	—
76	C₂×C₄×C₈	—	11	2	no	—
77	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
78	#376	—	11	1	no	—
79	#346	—	11	2	no	—
80	#346	Dic₁₁	1	2	yes	complemented
81	C₄×C₁₆	—	11	2	no	—
82	C₄×C₁₆	Dic₁₁	1	2	yes	complemented
83	C₂²×C₁₁	—	2	2	no	—
84	C₂²×C₁₁	C₂×#97	1	1	yes	characteristic
85	C₂²×C₁₁	#431	1	2	yes	—
86	C₄×C₁₁	—	2	2	no	—
87	C₄×C₁₁	—	2	2	no	—
88	C₄×C₁₁	#376	1	1	yes	characteristic
89	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₁₁	#431	1	2	yes	—
91	Dic₁₁	—	4	4	no	—
92	C₂²×C₈	—	11	2	no	—
93	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
95	C₂×C₄²	—	11	2	no	—
96	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
97	M₃₂	—	2	4	no	—
98	M₃₂	—	22	4	no	—
99	C₂×C₁₆	—	11	4	no	—
100	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	—
101	#97	—	22	4	no	—
102	#97	—	22	2	no	—
103	C₄×C₈	—	11	8	no	—
104	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
105	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
108	#95	—	22	2	no	—
109	C₂×C₁₁	—	2	2	no	—
110	C₂×C₁₁	#1744	1	1	yes	characteristic
111	C₂×C₁₁	#2110	1	2	yes	—
112	C₂²×C₄	—	22	2	no	—
113	C₂²×C₄	#4313	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
115	C₂×C₈	—	11	4	no	—
116	C₂×C₈	—	22	2	no	—
117	C₂×C₈	—	22	2	no	—
118	C₂×C₈	C₈×D₁₁	1	4	yes	—
119	C₂×C₈	C₄×Dic₁₁	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₁₁	1	1	yes	characteristic
121	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
122	C₂×C₈	#4339	1	4	yes	—
123	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
124	C₄²	—	11	4	no	—
125	C₄²	—	22	2	no	—
126	C₄²	#4313	1	2	yes	—
127	C₄²	C₄×Dic₁₁	1	1	yes	characteristic
128	C₄²	C₄×Dic₁₁	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	22	4	no	—
131	C₁₁	#12593	1	1	yes	characteristic, complemented, Sylow
132	C₂³	C₂×#4313	1	1	yes	characteristic
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	22	4	no	—
135	C₂×C₄	—	44	4	no	—
136	C₂×C₄	#66879	1	2	yes	—
137	C₂×C₄	#66879	1	2	yes	—
138	C₂×C₄	#66925	1	2	yes	—
139	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
140	C₂×C₄	C₂×#4313	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
142	C₈	—	22	4	no	—
143	C₈	C₈×Dic₁₁	1	2	yes	—
144	C₈	C₈×Dic₁₁	1	2	yes	—
145	C₈	#66925	1	2	yes	—
146	C₈	#66979	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#10622298	1	2	yes	—
149	C₂²	C₂×#66879	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	44	4	no	—
153	C₄	#10622298	1	2	yes	—
154	C₄	#10622545	1	1	yes	characteristic
155	C₄	C₂×C₈×Dic₁₁	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#13068361	1	1	yes	characteristic
158	C₂	#13070220	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₁
commutator	C₄×C₁₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₄×C₁₁×M₃₂	C₂
socle	C₂²×C₁₁	C₂×#97
prime radical	C₂³×C₁₁	C₂×M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#12593
LCS	G ▷ C₄×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
12. C₁₁×#346	C₄	semidirect
13. C₄×C₁₁×C₁₆	C₄	semidirect
14. #10622099	C₄	semidirect
15. C₄×C₁₁⋊C₁₆	C₄	semidirect
80. #346	Dic₁₁	semidirect
82. C₄×C₁₆	Dic₁₁	semidirect
131. C₁₁	#12593	semidirect

Maximal subgroups

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

subgroup	index
C₄×#66876	2
#13068302	2
#13066695	2
C₄×C₁₁×M₃₂	2
#12593	11