Subgroups — #14907263

Order 2816.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077550	C₂	1	1	yes	maximal, characteristic
2	#13070521	C₂	1	4	yes	maximal, complemented
3	#13070518	C₂	1	4	yes	maximal, complemented
4	#13070376	C₂	1	2	yes	maximal
5	#13070215	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₁₁×#2150	C₂	1	1	yes	maximal, characteristic
8	#10623001	C₄	1	1	yes	characteristic
9	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	#10622715	C₄	1	2	yes	—
12	#10622714	C₂²	1	2	yes	—
13	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	#10622537	C₄	1	1	yes	characteristic
16	#10622292	C₄	1	4	yes	—
17	C₈×Dic₂₂	C₂²	1	4	yes	—
18	C₁₁×C₄⋊C₁₆	C₂²	1	4	yes	—
19	#10622140	C₂²	1	4	yes	—
20	Dic₁₁⋊C₁₆	C₂²	1	8	yes	—
21	C₁₆×Dic₁₁	C₂²	1	4	yes	—
22	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
23	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
24	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
27	#66979	C₂×C₄	1	2	yes	—
28	#66977	C₈	1	4	yes	—
29	C₂×#4318	C₂×C₄	1	2	yes	—
30	#66923	C₈	1	2	yes	—
31	#66922	C₈	1	2	yes	—
32	C₄×Dic₂₂	C₂×C₄	1	4	yes	—
33	C₁₁×M₃₂	—	2	4	no	—
34	C₂×C₁₁×C₁₆	C₂³	1	4	yes	—
35	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₁	C₂³	1	2	yes	—
37	#66879	C₂×C₄	1	4	yes	—
38	Dic₁₁⋊C₈	C₂×C₄	1	8	yes	—
39	C₈×Dic₁₁	C₂³	1	4	yes	—
40	C₂×C₁₁⋊C₁₆	—	2	2	no	—
41	C₂×C₁₁⋊C₁₆	Q₈	1	4	yes	complemented
42	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
45	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
46	#33474	—	11	1	no	maximal, Sylow
47	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₂₂	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₁₁	—	2	2	no	—
51	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₁₁	#41	1	2	yes	—
55	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
56	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
59	C₂²⋊Dic₁₁	C₂×C₈	1	4	yes	—
60	C₄⋊Dic₁₁	C₂×C₈	1	4	yes	—
61	C₄⋊Dic₁₁	C₂×C₈	1	4	yes	—
62	Dic₁₁⋊C₄	C₂×C₈	1	8	yes	—
63	C₄×Dic₁₁	C₂²×C₄	1	4	yes	—
64	#4318	—	2	4	no	—
65	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
66	C₁₁×C₁₆	—	2	4	no	—
67	C₁₁⋊C₁₆	—	2	2	no	—
68	C₁₁⋊C₁₆	C₂×Q₈	1	2	yes	—
69	C₁₁⋊C₁₆	C₂×Q₈	1	2	yes	—
70	#2960	—	11	1	no	—
71	#2183	—	11	4	no	—
72	Q₈×C₁₆	—	11	4	no	—
73	#2151	—	11	2	no	—
74	#2150	D₁₁	1	1	yes	characteristic
75	#2108	—	11	2	no	—
76	C₂×C₄×C₁₆	—	11	1	no	—
77	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₁	—	2	4	no	—
79	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
80	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₁	#131	1	2	yes	—
84	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
85	C₂×Dic₁₁	C₂²×C₈	1	8	yes	—
86	Dic₂₂	—	2	8	no	—
87	C₈×C₁₁	—	2	2	no	—
88	C₈×C₁₁	C₄×Q₈	1	2	yes	—
89	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
91	C₁₁⋊C₈	—	2	4	no	—
92	#533	—	11	1	no	—
93	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
94	C₂²×C₁₆	—	11	1	no	—
95	C₂²×C₁₆	—	11	1	no	—
96	#446	—	11	4	no	—
97	C₈×Q₈	—	11	4	no	—
98	#424	—	11	2	no	—
99	#423	—	11	1	no	—
100	#405	—	11	2	no	—
101	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
102	C₄⋊C₁₆	—	11	8	no	—
103	C₄⋊C₁₆	C₂×D₁₁	1	4	yes	—
104	#346	—	11	4	no	—
105	C₄×C₁₆	—	11	4	no	—
106	C₄×C₁₆	—	11	2	no	—
107	C₄×C₁₆	—	11	2	no	—
108	C₂²×C₁₁	—	2	2	no	—
109	C₂²×C₁₁	C₈×Q₈	1	2	yes	—
110	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₁	—	2	2	no	—
112	C₄×C₁₁	C₈×Q₈	1	2	yes	—
113	C₄×C₁₁	#504	1	4	yes	—
114	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
116	Dic₁₁	—	2	8	no	—
117	C₂×M₁₆	—	11	2	no	—
118	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
120	C₄⋊Q₈	—	11	2	no	—
121	#125	—	11	2	no	—
122	C₂²⋊Q₈	—	11	4	no	—
123	C₄×Q₈	—	11	4	no	—
124	#117	—	11	2	no	—
125	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
126	M₃₂	—	2	4	no	—
127	C₂×C₁₆	—	11	4	no	—
128	C₂×C₁₆	—	11	1	no	—
129	C₂×C₁₆	—	11	1	no	—
130	C₂×C₁₆	—	11	2	no	—
131	C₂×C₁₆	—	22	2	no	—
132	C₂×C₁₆	C₂²×D₁₁	1	4	yes	—
133	C₄⋊C₈	—	11	8	no	—
134	C₄⋊C₈	—	11	4	no	—
135	#97	—	11	4	no	—
136	C₄×C₈	—	11	4	no	—
137	C₄×C₈	C₄×D₁₁	1	2	yes	—
138	C₄×C₈	C₂²×D₁₁	1	2	yes	—
139	C₂×C₁₁	—	2	2	no	—
140	C₂×C₁₁	C₂×C₈×Q₈	1	1	yes	characteristic
141	C₂×C₁₁	C₂×#504	1	2	yes	—
142	C₂×Q₈	—	11	4	no	—
143	C₂²×C₄	C₈×D₁₁	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
145	M₁₆	—	22	4	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	11	4	no	—
148	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
149	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
150	C₂×C₈	#4341	1	2	yes	—
151	C₂×C₈	Q₈×D₁₁	1	2	yes	—
152	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
153	C₄⋊C₄	—	11	4	no	—
154	C₄⋊C₄	—	11	8	no	—
155	C₄⋊C₄	—	11	4	no	—
156	C₂²⋊C₄	—	11	4	no	—
157	C₄²	—	11	4	no	—
158	C₄²	C₈×D₁₁	1	1	yes	characteristic
159	C₄²	C₈×D₁₁	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₁₁	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₁₆	—	11	2	no	—
163	C₁₆	—	11	2	no	—
164	C₁₆	—	22	2	no	—
165	C₁₁	#33474	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
167	Q₈	—	22	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	11	8	no	—
170	C₂×C₄	#66940	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
173	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
174	C₂×C₄	#66960	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	22	4	no	—
178	C₈	#66940	1	2	yes	—
179	C₈	C₂×#4341	1	1	yes	characteristic
180	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	#10622443	1	2	yes	—
183	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	22	8	no	—
186	C₄	#10622443	1	2	yes	—
187	C₄	#10622520	1	4	yes	—
188	C₄	C₂×#66940	1	1	yes	characteristic
189	C₄	C₂×#66960	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×#10622443	1	1	yes	characteristic
192	C₂	C₂×#10622520	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2150	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2150	D₁₁
hypocenter	C₁₁	#33474
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #13070521	C₂	semidirect
3. #13070518	C₂	semidirect
41. C₂×C₁₁⋊C₁₆	Q₈	semidirect
165. C₁₁	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13077550	2
#13070521	2
#13070518	2
#13070376	2
#13070215	2
C₂×C₄×C₁₁⋊C₁₆	2
C₁₁×#2150	2
#33474	11