Subgroups — #14665331

Order 2816.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13067364	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622095	C₂	1	1	yes	maximal, characteristic
3	#13066682	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×M₃₂	C₄	1	1	yes	characteristic
6	C₂×C₁₁×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	#10622538	—	2	1	no	—
9	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
10	C₁₁×#346	C₂²	1	2	yes	—
11	C₄×C₁₁×C₁₆	C₄	1	2	yes	—
12	#10622096	—	2	1	no	—
13	#10622095	—	2	1	no	—
14	#10622095	C₂²	1	2	yes	—
15	#10622095	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₁₁⋊C₈	—	4	1	no	—
20	C₁₁×M₃₂	—	2	2	no	—
21	C₁₁×M₃₂	C₈	1	4	yes	complemented
22	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₂²⋊(C₁₁⋊C₈)	—	4	1	no	—
29	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
31	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
32	C₄×C₁₁⋊C₈	—	4	1	no	—
33	#12000	—	11	1	no	maximal, Sylow
34	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₁	—	2	2	no	—
38	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
42	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
46	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
47	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₁⋊C₈	—	4	2	no	—
49	C₂×C₁₁⋊C₈	—	4	2	no	—
50	C₂×C₁₁⋊C₈	—	8	2	no	—
51	C₁₁×C₁₆	—	2	2	no	—
52	C₁₁×C₁₆	C₂×C₈	1	4	yes	complemented
53	C₄×M₃₂	D₁₁	1	1	yes	characteristic
54	#1564	—	11	1	no	—
55	C₂×#334	—	11	1	no	—
56	#1368	—	11	4	no	—
57	C₂³×C₁₁	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₁	—	2	2	no	—
59	C₂×C₄×C₁₁	—	4	1	no	—
60	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
62	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
64	C₂×C₄×C₁₁	#108	1	2	yes	—
65	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₁₁	—	2	1	no	—
69	C₈×C₁₁	—	2	1	no	—
70	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
71	C₈×C₁₁	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₁₁	C₂×M₁₆	1	1	yes	characteristic
73	C₁₁⋊C₈	—	8	2	no	—
74	C₁₁⋊C₈	—	8	2	no	—
75	C₂×M₃₂	Dic₁₁	1	1	yes	characteristic
76	C₂×M₃₂	Dic₁₁	1	1	yes	characteristic
77	#431	—	22	1	no	—
78	C₂×C₄⋊C₈	—	22	1	no	—
79	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
80	#346	C₂×D₁₁	1	2	yes	—
81	C₄×C₁₆	Dic₁₁	1	2	yes	—
82	#335	—	22	1	no	—
83	#334	—	11	2	no	—
84	#334	—	11	2	no	—
85	#334	—	22	1	no	—
86	C₂²×C₁₁	—	2	1	no	—
87	C₂²×C₁₁	—	2	1	no	—
88	C₂²×C₁₁	#334	1	2	yes	—
89	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
90	C₄×C₁₁	—	2	1	no	—
91	C₄×C₁₁	—	2	1	no	—
92	C₄×C₁₁	—	2	1	no	—
93	C₄×C₁₁	#334	1	2	yes	—
94	C₄×C₁₁	#365	1	2	yes	—
95	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₁₁	C₂×#108	1	1	yes	characteristic
97	C₂²×C₈	—	44	1	no	—
98	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
100	C₂×C₄²	D₄₄	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	C₁₁⋊C₈	1	4	yes	complemented
103	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
104	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
105	C₄⋊C₈	—	22	2	no	—
106	C₄⋊C₈	—	22	2	no	—
107	C₄⋊C₈	—	44	1	no	—
108	C₂²⋊C₈	—	44	1	no	—
109	C₄×C₈	—	44	1	no	—
110	C₄×C₈	Dic₂₂	1	1	yes	characteristic
111	C₄×C₈	D₄₄	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
114	C₂×C₁₁	—	2	2	no	—
115	C₂×C₁₁	C₂×#334	1	1	yes	characteristic
116	C₂×C₁₁	C₂×#365	1	1	yes	characteristic
117	C₂×C₁₁	#2155	1	1	yes	characteristic
118	C₂²×C₄	#4314	1	1	yes	characteristic
119	C₂²×C₄	#4314	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	44	2	no	—
123	C₂×C₈	—	44	2	no	—
124	C₂×C₈	—	88	2	no	—
125	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
126	C₂×C₈	#4318	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
130	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
131	C₄²	#4314	1	1	yes	characteristic
132	C₄²	#4314	1	1	yes	characteristic
133	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
134	C₄²	C₂×D₄₄	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	C₂×C₁₁⋊C₈	1	4	yes	complemented
137	C₁₁	#12000	1	1	yes	characteristic, complemented, Sylow
138	C₂³	#66881	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
142	C₂×C₄	#66881	1	1	yes	characteristic
143	C₂×C₄	#66881	1	1	yes	characteristic
144	C₂×C₄	#66881	1	1	yes	characteristic
145	C₂×C₄	#66883	1	2	yes	—
146	C₂×C₄	C₂×#4314	1	1	yes	characteristic
147	C₂×C₄	C₂×#4314	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	88	2	no	—
152	C₈	—	88	2	no	—
153	C₈	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
154	C₈	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
155	C₈	C₂×#4318	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#10622095	1	2	yes	—
159	C₂²	C₂×#66881	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#10622095	1	2	yes	—
164	C₄	#10622151	1	2	yes	—
165	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
166	C₄	C₂×#66883	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#10622095	1	1	yes	characteristic
169	C₂	C₂×#10622151	1	1	yes	characteristic
170	C₂	#13070389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#66881
commutator	C₈×C₁₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₄×C₁₁×M₃₂	C₂
socle	C₂²×C₁₁	C₂×#106
prime radical	C₂³×C₁₁	#106
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#12000
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13066682	C₂	semidirect
21. C₁₁×M₃₂	C₈	semidirect
52. C₁₁×C₁₆	C₂×C₈	semidirect
102. M₃₂	C₁₁⋊C₈	semidirect
136. C₁₆	C₂×C₁₁⋊C₈	semidirect
137. C₁₁	#12000	semidirect

Maximal subgroups

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

subgroup	index
#13067364	2
C₂×#10622095	2
#13066682	2
C₄×C₁₁×M₃₂	2
#12000	11