Subgroups — #14668773

Order 2816.

390 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622140	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	#13066694	C₂	1	1	yes	maximal, characteristic
4	#13066693	C₂	1	2	yes	maximal
5	C₁₁×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
13	C₄²×Dic₁₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	#10622140	C₄	1	8	yes	complemented
16	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₁	C₈	1	2	yes	—
20	C₂×C₄×Dic₁₁	C₈	1	2	yes	—
21	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₁×C₁₆	—	2	2	no	—
26	C₂×C₁₁×C₁₆	—	2	2	no	—
27	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₁	—	2	2	no	—
29	C₈×Dic₁₁	—	2	1	no	—
30	C₈×Dic₁₁	—	2	1	no	—
31	C₈×Dic₁₁	—	2	2	no	—
32	C₈×Dic₁₁	C₈	1	4	yes	—
33	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
34	C₂×C₁₁⋊C₁₆	—	2	2	no	—
35	C₂×C₁₁⋊C₁₆	—	2	2	no	—
36	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
37	C₄×C₁₁⋊C₈	—	2	2	no	—
38	#12591	—	11	1	no	maximal, Sylow
39	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₁	—	2	1	no	—
45	C₂×C₈×C₁₁	—	2	1	no	—
46	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₁₁	#41	1	2	yes	—
49	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
50	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
51	C₄²×C₁₁	—	2	1	no	—
52	C₄²×C₁₁	—	2	1	no	—
53	C₄×Dic₁₁	—	2	4	no	—
54	C₄×Dic₁₁	—	2	4	no	—
55	C₄×Dic₁₁	—	2	1	no	—
56	C₄×Dic₁₁	—	2	1	no	—
57	C₄×Dic₁₁	C₄²	1	2	yes	—
58	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
59	C₂×C₁₁⋊C₈	—	2	1	no	—
60	C₂×C₁₁⋊C₈	—	2	1	no	—
61	C₂×C₁₁⋊C₈	C₄²	1	2	yes	—
62	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
63	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
64	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
65	C₁₁×C₁₆	—	2	4	no	—
66	C₁₁⋊C₁₆	—	2	4	no	—
67	C₂×#346	—	11	2	no	—
68	C₄²×C₈	—	11	1	no	—
69	#1374	—	11	2	no	—
70	#1374	—	11	1	no	—
71	#1374	D₁₁	1	1	yes	characteristic
72	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	2	no	—
75	C₂×C₄×C₁₁	M₃₂	1	4	yes	—
76	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
78	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×Dic₁₁	—	2	2	no	—
83	C₂×Dic₁₁	—	2	2	no	—
84	C₂×Dic₁₁	—	2	2	no	—
85	C₂×Dic₁₁	—	2	2	no	—
86	C₂×Dic₁₁	C₄×C₈	1	2	yes	—
87	C₂×Dic₁₁	C₄×C₈	1	2	yes	—
88	C₂×Dic₁₁	M₃₂	1	4	yes	—
89	C₈×C₁₁	—	2	2	no	—
90	C₈×C₁₁	#117	1	2	yes	—
91	C₈×C₁₁	#117	1	2	yes	—
92	C₁₁⋊C₈	—	2	2	no	—
93	C₁₁⋊C₈	#97	1	4	yes	—
94	C₂²×C₁₆	—	11	2	no	—
95	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
96	C₂×C₄×C₈	—	11	1	no	—
97	C₂×C₄×C₈	—	11	1	no	—
98	C₂×C₄×C₈	—	11	1	no	—
99	C₂×C₄×C₈	—	11	1	no	—
100	C₂×C₄×C₈	—	11	1	no	—
101	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
102	C₄³	—	11	1	no	—
103	#346	—	11	8	no	—
104	C₂²×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
105	C₂²×C₁₁	#431	1	1	yes	characteristic
106	C₂²×C₁₁	#431	1	1	yes	characteristic
107	C₂²×C₁₁	C₂×M₃₂	1	2	yes	—
108	C₂²×C₁₁	C₂×M₃₂	1	2	yes	—
109	C₄×C₁₁	—	2	4	no	—
110	C₄×C₁₁	#376	1	1	yes	characteristic
111	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
112	C₄×C₁₁	#431	1	1	yes	characteristic
113	C₄×C₁₁	#431	1	1	yes	characteristic
114	Dic₁₁	—	2	2	no	—
115	Dic₁₁	—	2	2	no	—
116	Dic₁₁	—	2	4	no	—
117	C₂²×C₈	—	11	1	no	—
118	C₂²×C₈	—	11	1	no	—
119	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
121	C₂×C₄²	—	11	1	no	—
122	C₂×C₄²	—	11	2	no	—
123	C₂×C₄²	—	11	2	no	—
124	C₂×C₄²	—	11	1	no	—
125	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	11	4	no	—
129	C₂×C₁₆	—	22	2	no	—
130	C₂×C₁₆	—	22	2	no	—
131	C₂×C₁₆	C₄×D₁₁	1	4	yes	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	11	4	no	—
134	C₄×C₈	—	11	4	no	—
135	C₄×C₈	—	22	2	no	—
136	C₄×C₈	—	22	2	no	—
137	C₄×C₈	—	22	1	no	—
138	C₄×C₈	—	22	1	no	—
139	C₂×C₁₁	#1744	1	1	yes	characteristic
140	C₂×C₁₁	C₄×M₃₂	1	1	yes	characteristic
141	C₂×C₁₁	C₄×M₃₂	1	1	yes	characteristic
142	C₂×C₁₁	#2163	1	2	yes	—
143	C₂×C₁₁	#2163	1	2	yes	—
144	C₂²×C₄	—	11	2	no	—
145	C₂²×C₄	—	11	1	no	—
146	C₂²×C₄	—	11	1	no	—
147	C₂²×C₄	C₈×D₁₁	1	2	yes	—
148	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	11	2	no	—
152	C₂×C₈	—	11	2	no	—
153	C₂×C₈	—	11	2	no	—
154	C₂×C₈	—	11	2	no	—
155	C₂×C₈	—	22	1	no	—
156	C₂×C₈	—	22	1	no	—
157	C₂×C₈	C₈×D₁₁	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
159	C₂×C₈	#4341	1	1	yes	characteristic
160	C₂×C₈	#4341	1	2	yes	—
161	C₂×C₈	#4343	1	1	yes	characteristic
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	11	2	no	—
165	C₄²	—	11	2	no	—
166	C₄²	—	22	4	no	—
167	C₄²	—	22	4	no	—
168	C₄²	—	22	1	no	—
169	C₄²	—	22	1	no	—
170	C₁₆	—	2	4	no	—
171	C₁₆	—	22	4	no	—
172	C₁₁	#12591	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	11	4	no	—
177	C₂×C₄	—	11	2	no	—
178	C₂×C₄	—	11	2	no	—
179	C₂×C₄	—	22	2	no	—
180	C₂×C₄	—	22	2	no	—
181	C₂×C₄	—	22	2	no	—
182	C₂×C₄	—	22	2	no	—
183	C₂×C₄	#66862	1	4	yes	—
184	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
185	C₂×C₄	#66930	1	1	yes	characteristic
186	C₂×C₄	#66945	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
188	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
189	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	11	4	no	—
192	C₈	—	22	2	no	—
193	C₈	#66930	1	2	yes	—
194	C₈	#66945	1	2	yes	—
195	C₂²	C₄×C₈×D₁₁	1	1	yes	characteristic
196	C₂²	#10622331	1	1	yes	characteristic
197	C₂²	#10622446	1	1	yes	characteristic
198	C₂²	C₂×#66862	1	2	yes	—
199	C₂²	D₁₁×M₃₂	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	—	22	4	no	—
202	C₄	—	22	2	no	—
203	C₄	—	22	2	no	—
204	C₄	#10622261	1	1	yes	characteristic
205	C₄	D₁₁×#97	1	1	yes	characteristic
206	C₄	#10622331	1	1	yes	characteristic
207	C₄	#10622446	1	1	yes	characteristic
208	C₂	#13068360	1	1	yes	characteristic
209	C₂	C₄×#66862	1	1	yes	characteristic
210	C₂	#13070209	1	1	yes	characteristic
211	C₂	#13070416	1	2	yes	—
212	C₂	#13070418	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂×C₄×C₈
Fitting	C₁₁×#1374	C₂
socle, prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1374	D₁₁
hypocenter	C₁₁	#12591
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
15. #10622140	C₄	semidirect
172. C₁₁	#12591	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622140	2
C₄×C₈×Dic₁₁	2
#13066694	2
#13066693	2
C₁₁×#1374	2
#12591	11