Subgroups — #14670220

Order 2816.

314 conjugacy classes of subgroups.

The subgroup lattice has 190 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×#10622131	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#13068786	C₂	1	2	yes	maximal
5	C₂×C₁₁×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#108	C₂²	1	2	yes	—
8	C₂×#66886	—	2	2	no	—
9	#10622731	—	2	2	no	—
10	C₂×#66883	C₂²	1	2	yes	—
11	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₁₁×#368	—	2	2	no	—
14	#10622144	—	2	2	no	—
15	C₁₆×Dic₁₁	C₄	1	2	yes	—
16	C₁₆×Dic₁₁	C₄	1	2	yes	—
17	#10622131	—	2	4	no	—
18	C₂×C₁₁×M₁₆	—	2	2	no	—
19	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
21	C₂×#4318	—	2	2	no	—
22	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
25	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
26	C₁₁×#108	—	2	2	no	—
27	C₁₁×#108	—	2	2	no	—
28	#66886	—	4	4	no	—
29	#66883	—	2	2	no	—
30	#66883	—	2	2	no	—
31	Dic₁₁⋊C₈	—	4	4	no	—
32	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
34	C₈×Dic₁₁	Q₈	1	1	yes	characteristic
35	C₈×Dic₁₁	Q₈	1	1	yes	characteristic
36	C₂×C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
38	C₂×C₁₁⋊C₁₆	Q₈	1	2	yes	—
39	#12909	—	11	1	no	maximal, Sylow
40	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₁	Q₁₆	1	2	yes	—
42	C₁₁×M₁₆	—	4	2	no	—
43	C₁₁×M₁₆	—	4	2	no	—
44	C₂×C₈×C₁₁	—	4	2	no	—
45	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
51	C₄×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₁₁	D₈	1	2	yes	—
54	#4318	—	4	2	no	—
55	#4318	—	4	2	no	—
56	C₂×C₁₁⋊C₈	—	4	2	no	—
57	C₂×C₁₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₁₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₁₁⋊C₈	D₈	1	2	yes	—
60	C₂×C₁₁⋊C₈	Q₁₆	1	2	yes	—
61	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	—
62	C₁₁⋊C₁₆	C₄⋊C₄	1	4	yes	—
63	C₂×#368	—	11	1	no	—
64	C₂×#368	—	11	2	no	—
65	C₂×#368	D₁₁	1	1	yes	characteristic
66	C₂×C₄×C₁₆	—	11	1	no	—
67	#1848	—	11	2	no	—
68	C₂³×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₁₁	—	2	4	no	—
76	C₂×Dic₁₁	#107	1	2	yes	—
77	C₂×Dic₁₁	#107	1	2	yes	—
78	C₈×C₁₁	—	8	2	no	—
79	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₁₁⋊C₈	—	8	2	no	—
84	C₁₁⋊C₈	#107	1	2	yes	—
85	C₁₁⋊C₈	#107	1	2	yes	—
86	C₂²×C₁₆	—	11	1	no	—
87	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
88	C₂×#108	—	11	2	no	—
89	C₂×#108	—	22	2	no	—
90	C₂×#108	C₂×D₁₁	1	2	yes	—
91	#423	—	22	2	no	—
92	C₂×C₄×C₈	—	11	1	no	—
93	#368	—	2	2	no	—
94	#368	—	22	4	no	—
95	#368	—	22	2	no	—
96	C₄×C₁₆	—	11	2	no	—
97	C₄×C₁₆	—	11	2	no	—
98	C₂²×C₁₁	—	2	2	no	—
99	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
100	C₂²×C₁₁	#494	1	1	yes	characteristic
101	C₂²×C₁₁	#500	1	1	yes	characteristic
102	C₄×C₁₁	#384	1	1	yes	characteristic
103	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₁	#493	1	1	yes	characteristic
105	C₄×C₁₁	#500	1	1	yes	characteristic
106	Dic₁₁	—	2	2	no	—
107	Dic₁₁	#368	1	4	yes	complemented
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	22	2	no	—
110	C₂²×C₈	—	11	1	no	—
111	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
112	C₂×C₄²	—	11	1	no	—
113	C₂×C₁₆	—	11	2	no	—
114	C₂×C₁₆	—	11	2	no	—
115	C₂×C₁₆	—	11	2	no	—
116	C₂×C₁₆	Dic₂₂	1	2	yes	—
117	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
118	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
119	#108	—	2	2	no	—
120	#108	—	2	2	no	—
121	#108	—	22	2	no	—
122	#108	—	22	2	no	—
123	#108	—	44	4	no	—
124	C₄⋊C₈	—	44	4	no	—
125	C₄×C₈	—	11	1	no	—
126	C₄×C₈	—	11	1	no	—
127	C₄×C₈	—	11	1	no	—
128	C₄×C₈	—	11	1	no	—
129	C₂×C₁₁	—	2	1	no	—
130	C₂×C₁₁	—	2	1	no	—
131	C₂×C₁₁	#1845	1	1	yes	characteristic
132	C₂×C₁₁	C₂×#368	1	2	yes	—
133	C₂²×C₄	—	11	2	no	—
134	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	44	2	no	—
138	M₁₆	—	44	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	11	1	no	—
141	C₂×C₈	—	11	1	no	—
142	C₂×C₈	—	11	2	no	—
143	C₂×C₈	—	11	2	no	—
144	C₂×C₈	—	44	2	no	—
145	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
146	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
147	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
148	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
150	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
151	C₄²	—	11	1	no	—
152	C₄²	—	11	1	no	—
153	C₄²	—	11	2	no	—
154	C₁₆	—	11	4	no	—
155	C₁₆	Dic₁₁⋊C₄	1	4	yes	—
156	C₁₁	#12909	1	1	yes	characteristic, complemented, Sylow
157	C₂³	Q₁₆×D₁₁	1	1	yes	characteristic
158	C₂×C₄	—	11	2	no	—
159	C₂×C₄	—	11	2	no	—
160	C₂×C₄	—	22	4	no	—
161	C₂×C₄	#66941	1	1	yes	characteristic
162	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
163	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
164	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
166	C₂×C₄	#66993	1	1	yes	characteristic
167	C₈	—	8	2	no	—
168	C₈	—	11	2	no	—
169	C₈	—	11	2	no	—
170	C₈	—	88	2	no	—
171	C₈	#66941	1	1	yes	characteristic
172	C₈	C₄⋊C₄×D₁₁	1	1	yes	characteristic
173	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
174	C₈	#66997	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	#10622487	1	1	yes	characteristic
177	C₂²	D₁₁×#107	1	1	yes	characteristic
178	C₂²	#10622794	1	1	yes	characteristic
179	C₄	—	11	4	no	—
180	C₄	—	22	2	no	—
181	C₄	#10622487	1	1	yes	characteristic
182	C₄	D₁₁×#107	1	1	yes	characteristic
183	C₄	#10622589	1	1	yes	characteristic
184	C₄	#10622764	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#13068774	1	1	yes	characteristic
188	C₂	D₁₁×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₁
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#368	C₂
socle	C₂²×C₁₁	C₂×#107
prime radical	C₂³×C₁₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#368	D₁₁
hypocenter	C₁₁	#12909
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
107. Dic₁₁	#368	semidirect
156. C₁₁	#12909	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×#10622131	2
C₂×C₁₆×Dic₁₁	2
#13068786	2
C₂×C₁₁×#368	2
#12909	11