Subgroups — #14672301

Order 2816.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 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	C₄×#66876	C₂	1	1	yes	maximal, characteristic
3	#13066705	C₂	1	4	yes	maximal
4	C₁₁×#2960	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#533	C₄	1	1	yes	characteristic
6	C₁₁×#446	—	2	1	no	—
7	C₁₁×#446	—	2	1	no	—
8	C₈×Q₈×C₁₁	—	2	1	no	—
9	C₈×Q₈×C₁₁	—	2	1	no	—
10	C₁₁×#424	C₄	1	2	yes	—
11	C₁₁×#423	C₄	1	1	yes	characteristic
12	C₁₁×#405	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×#66876	C₂²	1	2	yes	—
15	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
17	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
18	#10622099	—	2	1	no	—
19	C₄×C₁₁⋊C₁₆	—	2	1	no	—
20	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
23	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
24	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₁₁×#125	C₈	1	2	yes	—
26	C₁₁×C₂²⋊Q₈	—	2	1	no	—
27	C₁₁×C₂²⋊Q₈	—	2	1	no	—
28	C₄×Q₈×C₁₁	—	2	1	no	—
29	C₄×Q₈×C₁₁	—	2	1	no	—
30	C₁₁×#117	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₁₁×C₄⋊C₈	—	2	2	no	—
33	C₁₁×C₄⋊C₈	—	2	2	no	—
34	C₁₁×C₄⋊C₈	—	2	1	no	—
35	C₁₁×C₄⋊C₈	—	2	1	no	—
36	C₁₁×#97	—	2	2	no	—
37	C₄×C₈×C₁₁	—	2	2	no	—
38	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
42	#66876	—	2	2	no	—
43	#66876	Q₈	1	4	yes	complemented
44	C₂×C₁₁⋊C₁₆	—	2	2	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	C₂×C₁₁⋊C₁₆	—	2	4	no	—
47	#13265	—	11	1	no	maximal, Sylow
48	C₂×Q₈×C₁₁	—	2	1	no	—
49	C₂×Q₈×C₁₁	—	2	1	no	—
50	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₁₁×M₁₆	—	2	4	no	—
54	C₂×C₈×C₁₁	—	2	2	no	—
55	C₂×C₈×C₁₁	—	2	2	no	—
56	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₁	#41	1	2	yes	—
62	C₁₁×C₄⋊C₄	—	2	2	no	—
63	C₁₁×C₄⋊C₄	—	2	2	no	—
64	C₁₁×C₄⋊C₄	—	2	1	no	—
65	C₁₁×C₄⋊C₄	—	2	1	no	—
66	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
67	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
68	C₁₁×C₂²⋊C₄	—	2	2	no	—
69	C₄²×C₁₁	—	2	2	no	—
70	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
74	C₁₁⋊C₁₆	—	2	4	no	—
75	C₁₁⋊C₁₆	—	4	2	no	—
76	#2960	D₁₁	1	1	yes	characteristic
77	C₂×C₄⋊C₁₆	—	11	1	no	—
78	C₄×M₃₂	—	11	1	no	—
79	#1379	—	11	4	no	—
80	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
81	Q₈×C₁₁	—	4	2	no	—
82	Q₈×C₁₁	—	4	2	no	—
83	C₂×C₄×C₁₁	—	2	2	no	—
84	C₂×C₄×C₁₁	—	2	2	no	—
85	C₂×C₄×C₁₁	—	2	2	no	—
86	C₂×C₄×C₁₁	—	4	1	no	—
87	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₁	#131	1	2	yes	—
95	C₈×C₁₁	—	2	1	no	—
96	C₈×C₁₁	—	2	1	no	—
97	C₈×C₁₁	—	4	2	no	—
98	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
99	C₈×C₁₁	#124	1	1	yes	characteristic
100	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
101	#533	Dic₁₁	1	1	yes	characteristic
102	C₂×M₃₂	—	11	2	no	—
103	C₂²×C₁₆	—	11	2	no	—
104	#446	—	2	1	no	—
105	#446	—	2	1	no	—
106	C₈×Q₈	—	2	1	no	—
107	C₈×Q₈	—	2	1	no	—
108	#424	Dic₁₁	1	2	yes	—
109	#423	Dic₁₁	1	1	yes	characteristic
110	#405	C₂×D₁₁	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
112	C₄⋊C₁₆	—	22	1	no	—
113	C₄⋊C₁₆	—	22	1	no	—
114	#346	—	22	1	no	—
115	C₄×C₁₆	—	22	1	no	—
116	C₂²×C₁₁	—	2	1	no	—
117	C₂²×C₁₁	—	2	1	no	—
118	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₁	#446	1	2	yes	—
120	C₄×C₁₁	—	2	1	no	—
121	C₄×C₁₁	—	2	1	no	—
122	C₄×C₁₁	—	2	1	no	—
123	C₄×C₁₁	—	4	2	no	—
124	C₄×C₁₁	—	4	2	no	—
125	C₄×C₁₁	#350	1	2	yes	—
126	C₄×C₁₁	#391	1	1	yes	characteristic
127	C₄×C₁₁	#408	1	1	yes	characteristic
128	C₄×C₁₁	C₈×Q₈	1	2	yes	—
129	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
130	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
132	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
133	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
134	#125	C₁₁⋊C₈	1	2	yes	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	#117	C₂×Dic₁₁	1	2	yes	—
140	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
141	M₃₂	—	11	4	no	—
142	M₃₂	—	22	2	no	—
143	C₂×C₁₆	—	22	2	no	—
144	C₂×C₁₆	—	22	4	no	—
145	C₂×C₁₆	—	22	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	—	2	1	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
156	C₂×C₁₁	—	2	2	no	—
157	C₂×C₁₁	#1984	1	1	yes	characteristic
158	C₂×C₁₁	C₂×#350	1	1	yes	characteristic
159	C₂×C₁₁	#2117	1	1	yes	characteristic
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂²×C₄	#4318	1	1	yes	characteristic
163	C₂²×C₄	#4318	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
165	M₁₆	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₁₁	1	2	yes	—
171	C₂×C₈	#4343	1	2	yes	—
172	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
179	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
183	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
185	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
186	C₁₆	—	22	4	no	—
187	C₁₆	—	44	2	no	—
188	C₁₁	#13265	1	1	yes	characteristic, complemented, Sylow
189	C₂³	C₂×#4318	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
199	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
201	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
202	C₂×C₄	#67001	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	2	no	—
207	C₈	#66997	1	1	yes	characteristic
208	C₈	#66999	1	2	yes	—
209	C₈	#67000	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#10622637	1	2	yes	—
213	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	4	2	no	—
219	C₄	#10622199	1	2	yes	—
220	C₄	Q₈×C₁₁⋊C₈	1	2	yes	—
221	C₄	#10622843	1	1	yes	characteristic
222	C₄	#10622847	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	#13069553	1	1	yes	characteristic
225	C₂	C₂×#10622199	1	1	yes	characteristic
226	C₂	#13070244	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2960	C₂
socle	C₂²×C₁₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₁	C₂×M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2960	D₁₁
hypocenter	C₁₁	#13265
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
43. #66876	Q₈	semidirect
188. C₁₁	#13265	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
C₄×#66876	2
#13066705	2
C₁₁×#2960	2
#13265	11