Subgroups — #14670184

Order 2816.

414 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×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁⋊C₁₆	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₁	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
9	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
10	C₄²×Dic₁₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
12	C₁₁×C₄⋊C₁₆	—	2	2	no	—
13	Dic₁₁⋊C₁₆	—	2	8	no	—
14	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
15	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₁	C₈	1	2	yes	—
19	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₁	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₁₁×C₁₆	—	2	2	no	—
24	C₂×C₁₁×C₁₆	—	2	4	no	—
25	C₄×C₈×C₁₁	D₄	1	2	yes	—
26	C₄×C₈×C₁₁	Q₈	1	2	yes	—
27	C₈×Dic₁₁	—	2	4	no	—
28	C₈×Dic₁₁	D₄	1	4	yes	—
29	C₈×Dic₁₁	Q₈	1	4	yes	—
30	C₂×C₁₁⋊C₁₆	—	2	2	no	—
31	C₂×C₁₁⋊C₁₆	—	2	4	no	—
32	C₄×C₁₁⋊C₈	—	2	2	no	—
33	#12898	—	11	1	no	maximal, Sylow
34	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
38	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₁	—	2	2	no	—
40	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
42	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
45	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
46	C₄²×C₁₁	M₁₆	1	2	yes	—
47	C₄×Dic₁₁	—	2	4	no	—
48	C₄×Dic₁₁	—	2	2	no	—
49	C₄×Dic₁₁	C₁₆	1	4	yes	complemented
50	C₄×Dic₁₁	C₄⋊C₄	1	4	yes	—
51	C₄×Dic₁₁	M₁₆	1	4	yes	—
52	C₂×C₁₁⋊C₈	—	2	2	no	—
53	C₂×C₁₁⋊C₈	—	2	2	no	—
54	C₂×C₁₁⋊C₈	C₄⋊C₄	1	4	yes	—
55	C₁₁×C₁₆	—	4	2	no	—
56	C₁₁⋊C₁₆	—	4	2	no	—
57	C₂×C₄⋊C₁₆	—	11	4	no	—
58	C₂×C₄⋊C₁₆	—	11	1	no	—
59	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic
60	C₄²×C₈	—	11	1	no	—
61	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₁	—	2	2	no	—
63	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
64	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
65	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₁₁	M₃₂	1	2	yes	—
67	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
71	C₂×Dic₁₁	—	2	2	no	—
72	C₂×Dic₁₁	—	2	2	no	—
73	C₂×Dic₁₁	C₄⋊C₈	1	4	yes	—
74	C₂×Dic₁₁	C₄⋊C₈	1	4	yes	—
75	C₂×Dic₁₁	C₂×C₁₆	1	4	yes	—
76	C₂×Dic₁₁	M₃₂	1	4	yes	—
77	C₈×C₁₁	—	2	2	no	—
78	C₈×C₁₁	#127	1	1	yes	characteristic
79	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
80	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₁₁⋊C₈	—	2	4	no	—
82	C₂²×C₁₆	—	11	2	no	—
83	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
84	C₂×C₄×C₈	—	11	2	no	—
85	C₂×C₄×C₈	—	11	2	no	—
86	C₂×C₄×C₈	—	11	1	no	—
87	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
88	C₄³	—	11	1	no	—
89	C₄⋊C₁₆	—	2	2	no	—
90	C₄⋊C₁₆	—	22	8	no	—
91	C₄⋊C₁₆	—	22	2	no	—
92	C₂²×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
93	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
95	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₁₁	C₂×M₃₂	1	2	yes	—
97	C₄×C₁₁	—	2	2	no	—
98	C₄×C₁₁	C₄⋊C₁₆	1	4	yes	complemented
99	C₄×C₁₁	#384	1	1	yes	characteristic
100	C₄×C₁₁	#423	1	1	yes	characteristic
101	C₄×C₁₁	#423	1	2	yes	—
102	Dic₁₁	—	2	4	no	—
103	Dic₁₁	C₄⋊C₁₆	1	8	yes	complemented
104	C₂²×C₈	—	11	2	no	—
105	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
107	C₂×C₄²	—	11	1	no	—
108	C₂×C₄²	—	11	2	no	—
109	C₂×C₄²	—	11	2	no	—
110	C₂×C₄²	—	11	1	no	—
111	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	22	4	no	—
115	C₂×C₁₆	—	22	2	no	—
116	C₄×C₈	—	11	4	no	—
117	C₄×C₈	—	11	4	no	—
118	C₄×C₈	—	22	4	no	—
119	C₄×C₈	—	22	2	no	—
120	C₄×C₈	Dic₂₂	1	2	yes	—
121	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
122	C₂×C₁₁	#1842	1	1	yes	characteristic
123	C₂×C₁₁	C₂×C₄⋊C₁₆	1	2	yes	—
124	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
125	C₂×C₁₁	#2150	1	2	yes	—
126	C₂×C₁₁	#2150	1	1	yes	characteristic
127	C₂²×C₄	—	11	2	no	—
128	C₂²×C₄	—	11	2	no	—
129	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
130	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	11	4	no	—
134	C₂×C₈	—	22	2	no	—
135	C₂×C₈	—	22	2	no	—
136	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
137	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
138	C₂×C₈	D₄×D₁₁	1	2	yes	—
139	C₂×C₈	Q₈×D₁₁	1	2	yes	—
140	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
141	C₄²	—	11	4	no	—
142	C₄²	—	11	4	no	—
143	C₄²	—	11	4	no	—
144	C₄²	—	22	4	no	—
145	C₄²	—	22	2	no	—
146	C₄²	#4313	1	2	yes	—
147	C₄²	Dic₁₁⋊C₄	1	2	yes	—
148	C₁₆	—	4	2	no	—
149	C₁₆	—	44	2	no	—
150	C₁₁	#12898	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	11	4	no	—
154	C₂×C₄	—	11	4	no	—
155	C₂×C₄	—	11	4	no	—
156	C₂×C₄	—	11	4	no	—
157	C₂×C₄	—	22	2	no	—
158	C₂×C₄	—	22	2	no	—
159	C₂×C₄	C₁₆×D₁₁	1	2	yes	—
160	C₂×C₄	#66862	1	2	yes	—
161	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
162	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
163	C₂×C₄	C₄⋊C₄×D₁₁	1	2	yes	—
164	C₂×C₄	C₂×#4313	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₁₁	1	2	yes	—
166	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	22	4	no	—
169	C₈	#66941	1	2	yes	—
170	C₈	#66993	1	1	yes	characteristic
171	C₈	#66997	1	1	yes	characteristic
172	C₂²	D₁₁×C₄⋊C₈	1	2	yes	—
173	C₂²	C₂×C₁₆×D₁₁	1	1	yes	characteristic
174	C₂²	C₂×#66862	1	1	yes	characteristic
175	C₂²	D₁₁×M₃₂	1	2	yes	—
176	C₂²	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	11	8	no	—
179	C₄	—	22	4	no	—
180	C₄	Dic₁₁⋊C₁₆	1	4	yes	complemented
181	C₄	#10622450	1	2	yes	—
182	C₄	#10622589	1	1	yes	characteristic
183	C₄	#10622731	1	1	yes	characteristic
184	C₂	#13068767	1	1	yes	characteristic
185	C₂	D₁₁×C₄⋊C₁₆	1	2	yes	—
186	C₂	C₂×Dic₁₁⋊C₁₆	1	1	yes	characteristic
187	C₂	#13070370	1	2	yes	—
188	C₂	#13070371	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	#12898
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
49. C₄×Dic₁₁	C₁₆	semidirect
98. C₄×C₁₁	C₄⋊C₁₆	semidirect
103. Dic₁₁	C₄⋊C₁₆	semidirect
150. C₁₁	#12898	semidirect
180. C₄	Dic₁₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×Dic₁₁⋊C₁₆	2
C₄×C₈×Dic₁₁	2
C₂×C₁₁×C₄⋊C₁₆	2
#12898	11