Subgroups — #14640977

Order 2816.

226 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068788	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622095	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#2150	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×M₃₂	—	2	1	no	—
5	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×#66883	—	2	1	no	—
7	C₂×#66883	C₄	1	2	yes	—
8	#10622537	—	2	1	no	—
9	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
10	C₁₁×C₄⋊C₁₆	—	2	1	no	—
11	C₁₁×C₄⋊C₁₆	C₄	1	2	yes	—
12	#10622095	—	2	1	no	—
13	#10622095	C₄	1	2	yes	—
14	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
17	C₂×#4318	—	2	2	no	—
18	C₂²×C₁₁⋊C₈	—	4	1	no	—
19	C₁₁×M₃₂	—	4	1	no	—
20	C₁₁×M₃₂	—	4	1	no	—
21	C₂×C₁₁×C₁₆	—	2	2	no	—
22	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
26	#66883	—	4	1	no	—
27	#66883	—	4	1	no	—
28	#66883	—	4	2	no	—
29	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
31	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
32	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
33	#7280	—	11	1	no	maximal, Sylow
34	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₁	Q₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₁	—	2	1	no	—
38	C₂×C₈×C₁₁	—	2	1	no	—
39	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	D₈	1	1	yes	characteristic
44	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₁	Q₁₆	1	1	yes	characteristic
47	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₁	D₈	1	1	yes	characteristic
50	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
51	#4318	—	4	2	no	—
52	#4318	—	4	2	no	—
53	C₂×C₁₁⋊C₈	—	4	2	no	—
54	C₂×C₁₁⋊C₈	—	4	2	no	—
55	C₂×C₁₁⋊C₈	—	8	2	no	—
56	C₁₁×C₁₆	—	4	2	no	—
57	#2150	D₁₁	1	1	yes	characteristic
58	#1848	—	11	1	no	—
59	C₂×#334	—	11	1	no	—
60	C₂³×C₁₁	#103	1	1	yes	characteristic
61	C₂×C₄×C₁₁	—	2	2	no	—
62	C₂×C₄×C₁₁	—	4	1	no	—
63	C₂×C₄×C₁₁	#95	1	1	yes	characteristic
64	C₂×C₄×C₁₁	#100	1	1	yes	characteristic
65	C₂×C₄×C₁₁	#101	1	1	yes	characteristic
66	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
67	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
68	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
69	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
71	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
72	C₂×C₄×C₁₁	#107	1	1	yes	characteristic
73	C₈×C₁₁	—	2	1	no	—
74	C₈×C₁₁	—	2	1	no	—
75	C₈×C₁₁	#102	1	1	yes	characteristic
76	C₈×C₁₁	#103	1	1	yes	characteristic
77	C₈×C₁₁	#106	1	1	yes	characteristic
78	C₈×C₁₁	#107	1	1	yes	characteristic
79	C₁₁⋊C₈	—	8	2	no	—
80	C₁₁⋊C₈	—	8	2	no	—
81	C₂×M₃₂	—	2	1	no	—
82	C₂×#108	—	11	2	no	—
83	C₂×#108	—	22	1	no	—
84	#423	—	22	1	no	—
85	C₂×C₄⋊C₈	—	22	1	no	—
86	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
87	C₄⋊C₁₆	—	2	1	no	—
88	C₄⋊C₁₆	Dic₁₁	1	2	yes	—
89	#334	—	11	2	no	—
90	#334	—	22	1	no	—
91	C₂²×C₁₁	—	2	1	no	—
92	C₂²×C₁₁	—	2	1	no	—
93	C₂²×C₁₁	#332	1	1	yes	characteristic
94	C₂²×C₁₁	#333	1	1	yes	characteristic
95	C₂²×C₁₁	#340	1	1	yes	characteristic
96	C₄×C₁₁	—	2	1	no	—
97	C₄×C₁₁	—	2	1	no	—
98	C₄×C₁₁	—	2	1	no	—
99	C₄×C₁₁	#331	1	1	yes	characteristic
100	C₄×C₁₁	#332	1	1	yes	characteristic
101	C₄×C₁₁	#340	1	1	yes	characteristic
102	C₄×C₁₁	#343	1	1	yes	characteristic
103	C₄×C₁₁	#365	1	2	yes	—
104	C₂×M₁₆	—	22	2	no	—
105	C₂²×C₈	—	44	1	no	—
106	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
108	C₂×C₄²	D₄₄	1	1	yes	characteristic
109	M₃₂	—	4	1	no	—
110	M₃₂	—	4	1	no	—
111	C₂×C₁₆	—	2	2	no	—
112	#108	—	44	2	no	—
113	#108	—	44	1	no	—
114	#108	—	44	1	no	—
115	C₄⋊C₈	—	22	2	no	—
116	C₄⋊C₈	—	44	1	no	—
117	C₄⋊C₈	—	44	1	no	—
118	C₄⋊C₈	—	44	1	no	—
119	C₄×C₈	Dic₂₂	1	1	yes	characteristic
120	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
123	C₂×C₁₁	—	2	1	no	—
124	C₂×C₁₁	—	2	1	no	—
125	C₂×C₁₁	#1296	1	1	yes	characteristic
126	C₂×C₁₁	#1383	1	1	yes	characteristic
127	C₂×C₁₁	#1388	1	1	yes	characteristic
128	C₂²×C₄	#4314	1	1	yes	characteristic
129	C₂²×C₄	Dic₄₄	1	1	yes	characteristic
130	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
131	M₁₆	—	44	2	no	—
132	M₁₆	—	44	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	44	2	no	—
136	C₂×C₈	—	44	2	no	—
137	C₂×C₈	—	88	2	no	—
138	C₂×C₈	C₄×Dic₁₁	1	1	yes	characteristic
139	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
140	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
141	C₂×C₈	#4323	1	1	yes	characteristic
142	C₂×C₈	#4324	1	1	yes	characteristic
143	C₂×C₈	#4325	1	1	yes	characteristic
144	C₂×C₈	#4326	1	1	yes	characteristic
145	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
146	C₄²	#4314	1	1	yes	characteristic
147	C₄²	D₈₈	1	1	yes	characteristic
148	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
149	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
150	C₁₆	—	4	2	no	—
151	C₁₁	#7280	1	1	yes	characteristic, complemented, Sylow
152	C₂³	#66880	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	#66873	1	1	yes	characteristic
156	C₂×C₄	#66874	1	1	yes	characteristic
157	C₂×C₄	#66880	1	1	yes	characteristic
158	C₂×C₄	#66881	1	1	yes	characteristic
159	C₂×C₄	#66882	1	1	yes	characteristic
160	C₂×C₄	#66885	1	1	yes	characteristic
161	C₂×C₄	#66885	1	1	yes	characteristic
162	C₂×C₄	#66887	1	1	yes	characteristic
163	C₂×C₄	#66888	1	1	yes	characteristic
164	C₂×C₄	C₁₁⋊#95	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	88	2	no	—
168	C₈	—	88	2	no	—
169	C₈	#66871	1	1	yes	characteristic
170	C₈	#66872	1	1	yes	characteristic
171	C₈	#66896	1	1	yes	characteristic
172	C₈	#66899	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#10622119	1	1	yes	characteristic
176	C₂²	#10622120	1	1	yes	characteristic
177	C₂²	#10622189	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	#10622123	1	1	yes	characteristic
182	C₄	#10622124	1	1	yes	characteristic
183	C₄	#10622151	1	2	yes	—
184	C₄	#10622167	1	1	yes	characteristic
185	C₄	#10622192	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#13066500	1	1	yes	characteristic
189	C₂	#13066715	1	1	yes	characteristic
190	C₂	#13066728	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622189
commutator	C₂×C₈×C₁₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₁
Fitting	C₁₁×#2150	C₂
socle	C₂²×C₁₁	#340
prime radical	C₂³×C₁₁	#103
cosocle	C₂×C₄×C₈×C₁₁	C₂²
hypercenter	#2150	D₁₁
hypocenter	C₁₁	#7280
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
151. C₁₁	#7280	semidirect

Maximal subgroups

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

subgroup	index
#13068788	2
C₂×#10622095	2
C₁₁×#2150	2
#7280	11