Subgroups — #14669169

Order 2816.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622144	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622140	C₂	1	1	yes	maximal, characteristic
3	#13068497	C₂	1	2	yes	maximal
4	#13066711	C₂	1	2	yes	maximal
5	C₁₁×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₁₁×#427	C₂²	1	2	yes	—
8	C₂×#66883	C₂²	1	2	yes	—
9	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#10622610	—	2	2	no	—
12	C₁₁×#367	C₄	1	2	yes	complemented
13	C₁₁×#366	C₄	1	2	yes	complemented
14	#10622170	—	2	2	no	—
15	#10622144	C₄	1	4	yes	complemented
16	#10622140	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₁₁×#117	—	2	2	no	—
19	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
20	C₂×#4318	—	2	2	no	—
21	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₁₁×#107	C₂×C₄	1	4	yes	—
27	C₁₁×#106	—	2	2	no	—
28	#66883	—	2	2	no	—
29	#66883	C₂×C₄	1	4	yes	—
30	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₁	Q₈	1	2	yes	—
32	C₂×C₁₁⋊C₁₆	—	2	1	no	—
33	C₂×C₁₁⋊C₁₆	—	2	1	no	—
34	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
35	#66867	—	4	2	no	—
36	C₄×C₁₁⋊C₈	—	4	2	no	—
37	#12671	—	11	1	no	maximal, Sylow
38	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₁	—	2	1	no	—
40	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
46	C₁₁×C₄⋊C₄	—	2	4	no	—
47	C₁₁×C₂²⋊C₄	—	4	2	no	—
48	C₄²×C₁₁	—	4	2	no	—
49	C₄×Dic₁₁	—	2	1	no	—
50	C₄×Dic₁₁	C₄⋊C₄	1	2	yes	—
51	#4318	—	2	4	no	—
52	#4318	—	4	2	no	—
53	C₂×C₁₁⋊C₈	—	4	2	no	—
54	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₁₁⋊C₈	D₈	1	2	yes	—
56	C₂×C₁₁⋊C₈	Q₁₆	1	2	yes	—
57	C₁₁×C₁₆	C₄²	1	2	yes	—
58	C₁₁×C₁₆	C₄²	1	2	yes	—
59	C₁₁⋊C₁₆	—	2	2	no	—
60	C₂×#368	—	11	1	no	—
61	#2158	D₁₁	1	1	yes	characteristic
62	C₂×#346	—	11	1	no	—
63	#1772	—	11	2	no	—
64	#1382	—	11	2	no	—
65	C₂³×C₁₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₁₁	—	4	4	no	—
67	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
73	C₂×Dic₁₁	—	2	2	no	—
74	C₂×Dic₁₁	—	4	2	no	—
75	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₁₁	#117	1	1	yes	characteristic
77	C₈×C₁₁	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₁₁	C₄×Q₈	1	1	yes	characteristic
79	C₁₁⋊C₈	—	4	4	no	—
80	C₁₁⋊C₈	#107	1	4	yes	—
81	C₂²×C₁₆	—	11	1	no	—
82	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
83	C₂×#108	—	11	2	no	—
84	#427	C₂×D₁₁	1	2	yes	—
85	#405	—	22	2	no	—
86	C₂×C₄×C₈	—	11	1	no	—
87	#368	—	11	4	no	—
88	#367	Dic₁₁	1	2	yes	complemented
89	#366	Dic₁₁	1	2	yes	complemented
90	#346	—	11	4	no	—
91	#339	—	22	2	no	—
92	C₂²×C₁₁	—	2	2	no	—
93	C₂²×C₁₁	#427	1	1	yes	characteristic
94	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
95	C₂²×C₁₁	C₄×Q₁₆	1	1	yes	characteristic
96	C₄×C₁₁	—	8	4	no	—
97	C₄×C₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
99	C₄×C₁₁	C₄×D₈	1	1	yes	characteristic
100	C₄×C₁₁	C₄×Q₁₆	1	1	yes	characteristic
101	Dic₁₁	—	4	2	no	—
102	C₂×M₁₆	—	22	2	no	—
103	C₂²×C₈	—	11	1	no	—
104	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	C₂×C₄²	—	11	1	no	—
107	C₂×C₁₆	—	11	2	no	—
108	C₂×C₁₆	—	22	1	no	—
109	C₂×C₁₆	—	22	1	no	—
110	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
111	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
113	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
114	#108	—	11	4	no	—
115	#108	—	22	2	no	—
116	#107	C₂×Dic₁₁	1	4	yes	—
117	#106	—	2	2	no	—
118	#97	—	44	2	no	—
119	C₄×C₈	—	11	2	no	—
120	C₄×C₈	—	11	2	no	—
121	C₄×C₈	—	44	2	no	—
122	C₂×C₁₁	—	2	1	no	—
123	C₂×C₁₁	—	2	1	no	—
124	C₂×C₁₁	C₄×#107	1	1	yes	characteristic
125	C₂×C₁₁	#2179	1	2	yes	—
126	C₂²×C₄	—	22	1	no	—
127	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
128	M₁₆	—	22	4	no	—
129	M₁₆	—	44	2	no	—
130	C₂×C₈	—	11	2	no	—
131	C₂×C₈	—	11	2	no	—
132	C₂×C₈	—	11	2	no	—
133	C₂×C₈	—	44	2	no	—
134	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
136	C₂×C₈	#4341	1	1	yes	characteristic
137	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
138	C₂×C₈	#4343	1	1	yes	characteristic
139	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
140	C₄⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	4	2	no	—
142	C₄²	—	4	2	no	—
143	C₄²	—	11	2	no	—
144	C₄²	—	22	1	no	—
145	C₁₆	—	22	2	no	—
146	C₁₆	C₄×Dic₁₁	1	2	yes	—
147	C₁₆	C₄×Dic₁₁	1	2	yes	—
148	C₁₁	#12671	1	1	yes	characteristic, complemented, Sylow
149	C₂³	#66965	1	1	yes	characteristic
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	22	2	no	—
152	C₂×C₄	—	44	2	no	—
153	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
154	C₂×C₄	#66946	1	1	yes	characteristic
155	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
156	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
157	C₂×C₄	#66972	1	1	yes	characteristic
158	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
159	C₈	—	11	4	no	—
160	C₈	—	44	4	no	—
161	C₈	#66940	1	1	yes	characteristic
162	C₈	#66945	1	1	yes	characteristic
163	C₈	C₂×C₄×Dic₁₁	1	1	yes	characteristic
164	C₈	Q₈×Dic₁₁	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#10622486	1	1	yes	characteristic
167	C₂²	#10622493	1	1	yes	characteristic
168	C₂²	D₈×Dic₁₁	1	1	yes	characteristic
169	C₄	—	8	4	no	—
170	C₄	—	44	2	no	—
171	C₄	#10622485	1	1	yes	characteristic
172	C₄	D₁₁×#107	1	1	yes	characteristic
173	C₄	C₄⋊C₄×Dic₁₁	1	1	yes	characteristic
174	C₄	Q₁₆×Dic₁₁	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#107×Dic₁₁	1	1	yes	characteristic
178	C₂	#13070502	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₁
commutator	C₈×C₁₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2158	C₂
socle	C₂²×C₁₁	C₄×D₈
prime radical	C₂³×C₁₁	#135
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2158	D₁₁
hypocenter	C₁₁	#12671
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
12. C₁₁×#367	C₄	semidirect
13. C₁₁×#366	C₄	semidirect
15. #10622144	C₄	semidirect
16. #10622140	C₄	semidirect
88. #367	Dic₁₁	semidirect
89. #366	Dic₁₁	semidirect
148. C₁₁	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622144	2
C₂×#10622140	2
#13068497	2
#13066711	2
C₁₁×#2158	2
#12671	11