Subgroups — #14647455

Order 2816.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068780	C₂	1	1	yes	maximal, characteristic
2	#13066899	C₂	1	1	yes	maximal, characteristic
3	#13066476	C₂	1	4	yes	maximal
4	C₁₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#108	—	2	1	no	—
6	C₂×C₁₁×#108	—	2	1	no	—
7	C₁₁×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×#66882	—	2	1	no	—
10	C₂×#66881	—	2	1	no	—
11	#10622549	C₂²	1	2	yes	—
12	#10622538	C₂²	1	2	yes	—
13	#10622093	C₄	1	2	yes	complemented
14	C₈×C₁₁⋊C₈	C₄	1	2	yes	complemented
15	C₂×C₁₁×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₁	—	2	2	no	—
20	C₁₁×#108	—	4	2	no	—
21	C₁₁×#108	—	4	2	no	—
22	C₁₁×C₄⋊C₈	—	2	4	no	—
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₁₁⋊#95	—	2	4	no	—
28	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
29	#66882	—	4	2	no	—
30	#66881	—	4	2	no	—
31	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
32	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
33	#8790	—	11	1	no	maximal, Sylow
34	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	#41	1	2	yes	—
36	C₂²×Dic₁₁	—	4	2	no	—
37	C₁₁×M₁₆	—	4	2	no	—
38	C₁₁×M₁₆	—	4	2	no	—
39	C₂×C₈×C₁₁	—	4	2	no	—
40	C₂×C₈×C₁₁	C₄⋊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	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	D₈	1	2	yes	—
45	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
48	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₁	#41	1	2	yes	—
51	C₄⋊Dic₁₁	—	4	2	no	—
52	C₄⋊Dic₁₁	—	4	2	no	—
53	C₂×C₁₁⋊C₈	—	2	4	no	—
54	#1848	D₁₁	1	1	yes	characteristic
55	#1846	—	11	1	no	—
56	#1450	—	11	1	no	—
57	#1286	—	11	4	no	—
58	C₂³×C₁₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₁	—	4	2	no	—
60	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₁	#117	1	2	yes	—
62	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
63	C₂×C₄×C₁₁	#125	1	2	yes	—
64	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₁₁	—	8	4	no	—
69	C₂×Dic₁₁	—	8	2	no	—
70	C₈×C₁₁	—	8	2	no	—
71	C₈×C₁₁	#106	1	2	yes	—
72	C₈×C₁₁	#106	1	2	yes	—
73	C₈×C₁₁	#107	1	2	yes	—
74	C₈×C₁₁	#107	1	2	yes	—
75	C₁₁⋊C₈	—	4	4	no	—
76	#431	—	11	2	no	—
77	C₂×#108	—	2	1	no	—
78	C₂×#108	—	2	1	no	—
79	C₂×#107	—	22	1	no	—
80	C₂×#106	—	22	1	no	—
81	#423	C₂×D₁₁	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
83	#382	—	11	2	no	—
84	#322	—	11	2	no	—
85	C₈²	—	11	2	no	—
86	C₂²×C₁₁	—	2	2	no	—
87	C₂²×C₁₁	#382	1	1	yes	characteristic
88	C₂²×C₁₁	#486	1	1	yes	characteristic
89	C₂²×C₁₁	#487	1	1	yes	characteristic
90	C₄×C₁₁	—	2	2	no	—
91	C₄×C₁₁	—	2	2	no	—
92	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
93	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
94	C₄×C₁₁	#486	1	1	yes	characteristic
95	C₄×C₁₁	#487	1	1	yes	characteristic
96	Dic₁₁	—	16	2	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	22	2	no	—
101	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
102	#108	—	4	2	no	—
103	#108	—	4	2	no	—
104	#107	—	44	2	no	—
105	#106	—	44	2	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	22	2	no	—
108	C₂²⋊C₈	—	22	2	no	—
109	C₄×C₈	—	11	4	no	—
110	C₄×C₈	Dic₂₂	1	1	yes	characteristic
111	C₄×C₈	Dic₂₂	1	1	yes	characteristic
112	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
113	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
114	#95	—	22	4	no	—
115	C₂×C₁₁	—	2	1	no	—
116	C₂×C₁₁	—	2	1	no	—
117	C₂×C₁₁	#1833	1	2	yes	—
118	C₂×C₁₁	#1834	1	1	yes	characteristic
119	C₂²×C₄	—	44	2	no	—
120	C₂²×C₄	#4343	1	2	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
122	M₁₆	—	4	2	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	22	4	no	—
126	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
127	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
129	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#4323	1	2	yes	—
131	C₂×C₈	#4324	1	2	yes	—
132	C₂×C₈	#4325	1	2	yes	—
133	C₂×C₈	#4326	1	2	yes	—
134	C₄⋊C₄	—	44	2	no	—
135	C₄⋊C₄	—	44	2	no	—
136	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
138	C₄²	#4341	1	2	yes	—
139	C₁₁	#8790	1	1	yes	characteristic, complemented, Sylow
140	C₂³	#67000	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	88	4	no	—
143	C₂×C₄	—	88	2	no	—
144	C₂×C₄	#66943	1	2	yes	—
145	C₂×C₄	#66945	1	2	yes	—
146	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×#4323	1	1	yes	characteristic
148	C₂×C₄	C₂×#4324	1	1	yes	characteristic
149	C₂×C₄	#66989	1	1	yes	characteristic
150	C₂×C₄	C₂×#4325	1	1	yes	characteristic
151	C₂×C₄	C₂×#4326	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	44	4	no	—
154	C₈	#66871	1	2	yes	—
155	C₈	#66871	1	2	yes	—
156	C₈	#66872	1	2	yes	—
157	C₈	#66872	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#10622590	1	1	yes	characteristic
160	C₂²	#10622701	1	1	yes	characteristic
161	C₂²	#10622706	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	176	2	no	—
165	C₄	C₂×#66871	1	1	yes	characteristic
166	C₄	C₂×#66872	1	1	yes	characteristic
167	C₄	#10622683	1	1	yes	characteristic
168	C₄	#10622694	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#13068741	1	2	yes	—
172	C₂	#13068743	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622590
commutator	C₄²×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#1848	C₂
socle	C₂²×C₁₁	#382
prime radical	C₂³×C₁₁	#124
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#1848	D₁₁
hypocenter	C₁₁	#8790
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
13. #10622093	C₄	semidirect
14. C₈×C₁₁⋊C₈	C₄	semidirect
139. C₁₁	#8790	semidirect

Maximal subgroups

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

subgroup	index
#13068780	2
#13066899	2
#13066476	2
C₁₁×#1848	2
#8790	11