Subgroups — #14772782

Order 2816.

622 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	M₁₆×Dic₂₂	C₂	1	2	yes	maximal
2	C₂×#10622292	C₂	1	1	yes	maximal, characteristic
3	#13069391	C₂	1	2	yes	maximal
4	#13069388	C₂	1	4	yes	maximal
5	#13068755	C₂	1	2	yes	maximal
6	#13068754	C₂	1	1	yes	maximal, characteristic
7	#13068521	C₂	1	2	yes	maximal
8	C₄×C₁₁×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#108	C₂²	1	2	yes	—
11	C₄×C₁₁×M₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66886	C₂²	1	4	yes	—
14	#10622731	C₂²	1	4	yes	—
15	M₁₆×Dic₁₁	C₂²	1	4	yes	—
16	C₂×#66879	C₂²	1	2	yes	—
17	C₂×Dic₁₁⋊C₈	C₂²	1	2	yes	—
18	#10622537	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
20	#10622314	—	2	4	no	—
21	#10622292	—	2	2	no	—
22	#10622292	C₄	1	4	yes	—
23	#10622292	C₄	1	4	yes	—
24	C₈×Dic₂₂	—	2	2	no	—
25	C₁₁×#341	C₂²	1	1	yes	characteristic
26	C₁₁×#341	C₂²	1	1	yes	characteristic
27	#10622196	C₂²	1	4	yes	—
28	#10622190	C₂²	1	2	yes	—
29	#10622190	C₂²	1	2	yes	—
30	C₂²×Dic₂₂	Q₈	1	1	yes	characteristic
31	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
32	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₁	Q₈	1	1	yes	characteristic
37	C₂×Dic₁₁⋊C₄	Q₈	1	2	yes	—
38	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
39	C₂×#4318	C₂³	1	4	yes	—
40	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
41	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₂	D₄	1	2	yes	—
44	C₄×Dic₂₂	D₄	1	2	yes	—
45	C₁₁×#108	—	2	4	no	—
46	C₁₁×#97	—	2	2	no	—
47	C₄×C₈×C₁₁	—	2	2	no	—
48	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
50	#66886	—	2	8	no	—
51	#66879	—	2	4	no	—
52	#66879	C₂×C₄	1	8	yes	—
53	Dic₁₁⋊C₈	—	2	4	no	—
54	Dic₁₁⋊C₈	—	2	4	no	—
55	Dic₁₁⋊C₈	C₂×C₄	1	8	yes	—
56	C₈×Dic₁₁	—	2	4	no	—
57	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
59	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
61	#22747	—	11	1	no	maximal, Sylow
62	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₁	C₂×Q₈	1	2	yes	—
66	C₂²×Dic₁₁	C₂×Q₈	1	2	yes	—
67	C₂×Dic₂₂	—	4	2	no	—
68	C₂×Dic₂₂	C₄⋊C₄	1	2	yes	—
69	C₂×Dic₂₂	C₄⋊C₄	1	2	yes	—
70	C₁₁×M₁₆	—	2	2	no	—
71	C₁₁×M₁₆	—	2	2	no	—
72	C₁₁×M₁₆	—	2	2	no	—
73	C₁₁×M₁₆	#41	1	4	yes	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
85	C₄⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
86	Dic₁₁⋊C₄	C₄⋊C₄	1	4	yes	—
87	Dic₁₁⋊C₄	C₄⋊C₄	1	4	yes	—
88	C₄×Dic₁₁	C₂²×C₄	1	4	yes	—
89	C₄×Dic₁₁	C₂×D₄	1	4	yes	—
90	#4318	—	2	4	no	—
91	#4318	—	2	4	no	—
92	C₂×C₁₁⋊C₈	—	2	4	no	—
93	C₂×C₁₁⋊C₈	—	2	4	no	—
94	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
95	Q₈×M₁₆	—	11	2	no	—
96	C₂×#446	—	11	1	no	—
97	#1950	—	11	4	no	—
98	#1950	—	11	2	no	—
99	#1838	—	11	2	no	—
100	#1837	—	11	1	no	—
101	#1778	—	11	2	no	—
102	C₄×#108	D₁₁	1	1	yes	characteristic
103	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₁	—	2	2	no	—
105	C₂×C₄×C₁₁	—	4	1	no	—
106	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
114	C₂×Dic₁₁	—	4	2	no	—
115	C₂×Dic₁₁	—	4	2	no	—
116	C₂×Dic₁₁	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₁₁	C₂×C₄⋊C₄	1	4	yes	—
118	Dic₂₂	—	2	4	no	—
119	Dic₂₂	—	4	2	no	—
120	C₈×C₁₁	—	2	2	no	—
121	C₈×C₁₁	—	2	2	no	—
122	C₈×C₁₁	—	2	4	no	—
123	C₈×C₁₁	—	4	2	no	—
124	C₁₁⋊C₈	—	2	4	no	—
125	C₁₁⋊C₈	—	4	4	no	—
126	C₂×C₄×Q₈	—	11	1	no	—
127	#446	—	11	4	no	—
128	#446	—	11	4	no	—
129	#446	—	22	2	no	—
130	#446	—	22	4	no	—
131	C₈×Q₈	—	22	2	no	—
132	C₂×#108	—	11	4	no	—
133	C₂×#108	C₂×D₁₁	1	2	yes	—
134	#423	—	11	4	no	—
135	#423	—	11	2	no	—
136	C₂×C₄⋊C₈	—	11	2	no	—
137	C₂×C₄⋊C₈	—	11	1	no	—
138	C₄×M₁₆	—	11	4	no	—
139	C₄×M₁₆	C₂×D₁₁	1	2	yes	—
140	C₂×#97	—	11	2	no	—
141	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
142	#341	—	11	4	no	—
143	#341	—	11	2	no	—
144	#341	—	11	2	no	—
145	#341	C₂×D₁₁	1	1	yes	characteristic
146	#341	C₂×D₁₁	1	1	yes	characteristic
147	C₂²×C₁₁	—	2	1	no	—
148	C₂²×C₁₁	—	2	1	no	—
149	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₁	#557	1	2	yes	—
151	C₄×C₁₁	—	2	1	no	—
152	C₄×C₁₁	—	2	1	no	—
153	C₄×C₁₁	—	2	1	no	—
154	C₄×C₁₁	#430	1	2	yes	—
155	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
156	C₄×C₁₁	#519	1	1	yes	characteristic
157	C₄×C₁₁	#548	1	2	yes	—
158	Dic₁₁	—	2	4	no	—
159	Dic₁₁	—	2	4	no	—
160	C₂²×Q₈	—	11	1	no	—
161	C₂×M₁₆	—	11	4	no	—
162	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
163	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
164	C₂²×C₈	—	11	2	no	—
165	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
167	C₄×Q₈	—	11	2	no	—
168	C₄×Q₈	—	11	2	no	—
169	C₄×Q₈	—	11	2	no	—
170	C₄×Q₈	—	11	2	no	—
171	C₂×C₄⋊C₄	—	11	2	no	—
172	C₂×C₄⋊C₄	—	11	1	no	—
173	C₂×C₄²	—	11	2	no	—
174	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
175	#108	—	2	4	no	—
176	#108	—	22	8	no	—
177	C₄⋊C₈	—	11	8	no	—
178	C₄⋊C₈	—	11	2	no	—
179	C₄⋊C₈	—	11	2	no	—
180	C₄⋊C₈	—	22	4	no	—
181	C₄⋊C₈	—	22	4	no	—
182	C₄⋊C₈	—	22	2	no	—
183	C₄⋊C₈	—	22	2	no	—
184	#97	—	2	2	no	—
185	#97	—	11	8	no	—
186	#97	—	22	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	22	4	no	—
189	C₄×C₈	C₄×D₁₁	1	2	yes	—
190	C₄×C₈	C₄×D₁₁	1	2	yes	—
191	C₂×C₁₁	—	2	2	no	—
192	C₂×C₁₁	#2351	1	1	yes	characteristic
193	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
194	C₂×C₁₁	#2916	1	1	yes	characteristic
195	C₂×Q₈	—	11	2	no	—
196	C₂×Q₈	—	11	2	no	—
197	C₂×Q₈	—	44	2	no	—
198	C₂²×C₄	—	11	2	no	—
199	C₂²×C₄	—	11	2	no	—
200	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
201	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	22	4	no	—
207	M₁₆	—	22	4	no	—
208	M₁₆	#4339	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	11	4	no	—
214	C₂×C₈	—	22	4	no	—
215	C₂×C₈	—	22	4	no	—
216	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
218	C₄⋊C₄	—	11	4	no	—
219	C₄⋊C₄	—	11	4	no	—
220	C₄⋊C₄	—	11	2	no	—
221	C₄⋊C₄	—	11	2	no	—
222	C₄²	—	11	4	no	—
223	C₄²	—	11	4	no	—
224	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
226	C₄²	D₄×D₁₁	1	1	yes	characteristic
227	C₄²	D₄×D₁₁	1	1	yes	characteristic
228	C₁₁	#22747	1	1	yes	characteristic, complemented, Sylow
229	C₂³	C₂×Q₈×D₁₁	1	1	yes	characteristic
230	Q₈	—	22	4	no	—
231	Q₈	—	44	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	11	4	no	—
235	C₂×C₄	—	11	4	no	—
236	C₂×C₄	—	44	2	no	—
237	C₂×C₄	—	44	2	no	—
238	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
243	C₂×C₄	C₂×#4339	1	2	yes	—
244	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
245	C₂×C₄	#67043	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	—	4	2	no	—
250	C₈	—	22	4	no	—
251	C₈	—	44	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
255	C₂²	#10623098	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	22	4	no	—
260	C₄	—	22	4	no	—
261	C₄	#10622506	1	2	yes	—
262	C₄	C₄×#4339	1	1	yes	characteristic
263	C₄	#10623047	1	1	yes	characteristic
264	C₄	#10623083	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#13071536	1	1	yes	characteristic
267	C₂	C₂×#10622506	1	1	yes	characteristic
268	C₂	#13077224	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₄×C₁₁×#108	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂²×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄×#108	D₁₁
hypocenter	C₁₁	#22747
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
228. C₁₁	#22747	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₂	2
C₂×#10622292	2
#13069391	2
#13069388	2
#13068755	2
#13068754	2
#13068521	2
C₄×C₁₁×#108	2
#22747	11