Subgroups — #14907409

Order 2816.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070448	C₂	1	2	yes	maximal, complemented
2	#13070442	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#13070419	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070415	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070230	C₂	1	4	yes	maximal, complemented
7	C₁₁×#2931	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#131	C₂²	1	2	yes	—
9	C₁₁×#532	C₄	1	1	yes	characteristic
10	C₁₁×#436	C₄	1	2	yes	—
11	C₁₁×#436	C₄	1	2	yes	—
12	C₈×D₄×C₁₁	C₂²	1	2	yes	—
13	C₈×D₄×C₁₁	C₂²	1	2	yes	—
14	C₁₁×#423	C₄	1	1	yes	characteristic
15	C₁₁×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
17	C₂×#66876	C₂²	1	4	yes	—
18	C₂²×C₁₁⋊C₁₆	C₂²	1	4	yes	—
19	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
20	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
21	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
22	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
23	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
24	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
25	#10622099	C₂²	1	2	yes	—
26	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
29	C₁₁×#131	—	4	4	no	—
30	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
35	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
36	C₁₁×#127	C₈	1	1	yes	characteristic
37	C₁₁×#124	C₈	1	2	yes	—
38	C₁₁×#121	C₈	1	2	yes	—
39	C₁₁×#121	C₈	1	2	yes	—
40	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
43	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	#66876	—	4	4	no	—
52	C₂×C₁₁⋊C₁₆	—	4	4	no	—
53	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
56	#33495	—	11	1	no	maximal, Sylow
57	C₁₁×#41	—	4	2	no	—
58	C₁₁×#41	—	4	2	no	—
59	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
62	C₂×D₄×C₁₁	C₂×C₈	1	4	yes	—
63	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₁₁×M₁₆	—	4	2	no	—
69	C₁₁×M₁₆	—	4	4	no	—
70	C₂×C₈×C₁₁	—	2	4	no	—
71	C₂×C₈×C₁₁	—	2	4	no	—
72	C₂×C₈×C₁₁	—	2	2	no	—
73	C₂×C₈×C₁₁	—	4	1	no	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
81	C₁₁×C₂²⋊C₄	C₂×C₈	1	4	yes	—
82	C₁₁×C₂²⋊C₄	C₂×C₈	1	4	yes	—
83	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₁₁⋊C₁₆	—	2	2	no	—
88	C₁₁⋊C₁₆	—	2	4	no	—
89	C₁₁⋊C₁₆	—	2	2	no	—
90	#2931	D₁₁	1	1	yes	characteristic
91	#2169	—	11	2	no	—
92	#2168	—	11	4	no	—
93	D₄×C₁₆	—	11	2	no	—
94	#2163	—	11	1	no	—
95	#2162	—	11	1	no	—
96	#2113	—	11	4	no	—
97	C₂³×C₁₁	C₂²×C₈	1	4	yes	—
98	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
99	Q₈×C₁₁	—	4	2	no	—
100	D₄×C₁₁	—	2	4	no	—
101	D₄×C₁₁	—	4	2	no	—
102	D₄×C₁₁	—	4	2	no	—
103	D₄×C₁₁	—	4	4	no	—
104	C₂×C₄×C₁₁	—	2	1	no	—
105	C₂×C₄×C₁₁	—	2	4	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	4	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	4	1	no	—
110	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₁₁	—	2	4	no	—
118	C₈×C₁₁	—	2	1	no	—
119	C₈×C₁₁	—	2	1	no	—
120	C₈×C₁₁	—	2	1	no	—
121	C₈×C₁₁	H(5,𝔽₂)	1	2	yes	—
122	C₂×#131	C₂×D₁₁	1	2	yes	—
123	#532	Dic₁₁	1	1	yes	characteristic
124	C₂×M₃₂	—	11	4	no	—
125	C₂²×C₁₆	—	11	4	no	—
126	#436	Dic₁₁	1	2	yes	—
127	#436	Dic₁₁	1	2	yes	—
128	C₈×D₄	C₂×D₁₁	1	2	yes	—
129	C₈×D₄	C₂×D₁₁	1	2	yes	—
130	#423	Dic₁₁	1	1	yes	characteristic
131	#408	Dic₁₁	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
133	C₄⋊C₁₆	—	11	2	no	—
134	C₄⋊C₁₆	—	11	2	no	—
135	C₂²⋊C₁₆	—	11	2	no	—
136	C₂²⋊C₁₆	—	11	2	no	—
137	C₂²⋊C₁₆	—	11	4	no	—
138	C₂²⋊C₁₆	—	11	4	no	—
139	#346	—	11	2	no	—
140	C₄×C₁₆	—	11	1	no	—
141	C₄×C₁₆	—	11	1	no	—
142	C₂²×C₁₁	—	2	1	no	—
143	C₂²×C₁₁	—	2	4	no	—
144	C₂²×C₁₁	—	2	1	no	—
145	C₂²×C₁₁	—	2	4	no	—
146	C₂²×C₁₁	—	2	1	no	—
147	C₂²×C₁₁	—	2	4	no	—
148	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
149	C₄×C₁₁	—	2	1	no	—
150	C₄×C₁₁	—	2	1	no	—
151	C₄×C₁₁	—	2	1	no	—
152	C₄×C₁₁	—	2	1	no	—
153	C₄×C₁₁	—	2	4	no	—
154	C₄×C₁₁	#504	1	2	yes	—
155	C₄×C₁₁	#518	1	1	yes	characteristic
156	C₄×C₁₁	#568	1	1	yes	characteristic
157	C₂×#41	C₂×Dic₁₁	1	2	yes	—
158	#131	—	4	4	no	—
159	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
160	C₂×M₁₆	C₂×Dic₁₁	1	4	yes	—
161	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₁₁	1	4	yes	—
164	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
165	#127	C₁₁⋊C₈	1	1	yes	characteristic
166	#124	C₁₁⋊C₈	1	2	yes	—
167	#121	C₁₁⋊C₈	1	2	yes	—
168	#121	C₁₁⋊C₈	1	2	yes	—
169	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
170	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
171	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
172	M₃₂	—	44	4	no	—
173	C₂×C₁₆	—	11	2	no	—
174	C₂×C₁₆	—	11	2	no	—
175	C₂×C₁₆	—	11	4	no	—
176	C₂×C₁₆	—	44	4	no	—
177	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
178	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
179	C₂²⋊C₈	C₂×Dic₁₁	1	4	yes	—
180	C₂²⋊C₈	C₂×Dic₁₁	1	4	yes	—
181	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
182	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
185	C₂×C₁₁	—	4	1	no	—
186	C₂×C₁₁	—	4	4	no	—
187	C₂×C₁₁	#2972	1	1	yes	characteristic
188	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
189	C₂×C₁₁	#3407	1	1	yes	characteristic
190	#41	—	4	2	no	—
191	#41	—	4	2	no	—
192	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
193	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
194	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
195	C₂×D₄	C₂×C₁₁⋊C₈	1	4	yes	—
196	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
197	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₁₁	1	4	yes	—
200	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
201	M₁₆	—	4	4	no	—
202	M₁₆	—	4	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
211	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
212	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
213	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
214	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
215	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
216	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
217	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
218	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
219	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
220	C₁₆	—	22	2	no	—
221	C₁₆	—	22	4	no	—
222	C₁₆	—	22	2	no	—
223	C₁₁	#33495	1	1	yes	characteristic, complemented, Sylow
224	C₂³	C₂²×C₁₁⋊C₈	1	4	yes	—
225	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
242	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
243	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	#67037	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	#10622753	1	2	yes	—
262	C₄	#10623329	1	1	yes	characteristic
263	C₄	#10623350	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	#13077640	1	1	yes	characteristic
267	C₂	C₂×#10622753	1	1	yes	characteristic
268	C₂	#13081719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2931	C₂
prime radical	C₁₁×#532	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2931	D₁₁
hypocenter	C₁₁	#33495
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #13070448	C₂	semidirect
2. #13070442	C₂	semidirect
3. D₄×C₁₁⋊C₁₆	C₂	semidirect
4. #13070419	C₂	semidirect
5. #13070415	C₂	semidirect
6. #13070230	C₂	semidirect
223. C₁₁	#33495	semidirect

Maximal subgroups

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

subgroup	index
#13070448	2
#13070442	2
D₄×C₁₁⋊C₁₆	2
#13070419	2
#13070415	2
#13070230	2
C₁₁×#2931	2
#33495	11