Subgroups — #13079953

Order 1408.

354 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#529	C₂	1	1	yes	maximal, characteristic
2	#10623001	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622676	C₂	1	2	yes	maximal, complemented
4	#10622667	C₂	1	2	yes	maximal, complemented
5	#10622664	C₂	1	2	yes	maximal, complemented
6	#10622654	C₂	1	2	yes	maximal
7	#10622644	C₂	1	1	yes	maximal, characteristic, complemented
8	#10622633	C₂	1	1	yes	maximal, characteristic, complemented
9	#10622632	C₂	1	1	yes	maximal, characteristic, complemented
10	#10622620	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₁×#126	—	2	1	no	—
13	C₁₁×#125	C₂²	1	1	yes	characteristic
14	C₁₁×#124	C₂²	1	1	yes	characteristic
15	C₁₁×#123	—	2	1	no	—
16	C₁₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₁×#121	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₁	—	2	1	no	—
20	C₄×D₄×C₁₁	C₂²	1	1	yes	characteristic
21	C₁₁×#117	—	2	1	no	—
22	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
23	C₂×#4326	C₂²	1	2	yes	—
24	C₂×#4324	C₂²	1	2	yes	—
25	#66979	—	2	1	no	—
26	#66977	—	2	1	no	—
27	#66977	C₂²	1	2	yes	—
28	C₂²×C₁₁⋊C₈	C₂²	1	2	yes	—
29	#66923	—	2	1	no	—
30	#66922	C₂²	1	1	yes	characteristic
31	#66922	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₂	—	2	1	no	—
33	C₄×Dic₂₂	C₂²	1	2	yes	—
34	#66899	C₂²	1	2	yes	—
35	#66896	C₂²	1	2	yes	—
36	#66874	C₂²	1	2	yes	—
37	#66874	C₂²	1	2	yes	—
38	#66872	C₂²	1	2	yes	—
39	#66871	C₂²	1	2	yes	—
40	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₁	—	2	1	no	—
45	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	—	2	1	no	—
47	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₂	—	2	1	no	—
51	C₂×Dic₂₂	C₂³	1	2	yes	—
52	C₁₁×C₄⋊C₄	—	2	1	no	—
53	C₁₁×C₄⋊C₄	—	2	1	no	—
54	C₁₁×C₄⋊C₄	—	2	1	no	—
55	C₁₁×C₄⋊C₄	—	2	1	no	—
56	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
57	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
58	C₁₁×C₂²⋊C₄	—	2	1	no	—
59	C₁₁×C₂²⋊C₄	—	2	1	no	—
60	C₁₁×C₂²⋊C₄	—	2	1	no	—
61	C₁₁×C₂²⋊C₄	—	2	1	no	—
62	C₁₁×C₂²⋊C₄	—	2	1	no	—
63	C₄²×C₁₁	—	2	1	no	—
64	C₄²×C₁₁	D₄	1	1	yes	characteristic
65	C₄²×C₁₁	D₄	1	1	yes	characteristic
66	C₄²×C₁₁	C₂³	1	1	yes	characteristic
67	C₄²×C₁₁	C₂³	1	1	yes	characteristic
68	C₂²⋊Dic₁₁	—	2	2	no	—
69	#4326	—	4	2	no	—
70	#4324	—	4	2	no	—
71	C₄⋊Dic₁₁	—	2	1	no	—
72	C₄⋊Dic₁₁	—	2	1	no	—
73	C₄⋊Dic₁₁	—	2	1	no	—
74	C₄⋊Dic₁₁	C₂³	1	2	yes	—
75	Dic₁₁⋊C₄	—	2	2	no	—
76	Dic₁₁⋊C₄	—	2	2	no	—
77	C₄×Dic₁₁	—	2	2	no	—
78	C₂×C₁₁⋊C₈	—	4	2	no	—
79	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
80	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
81	#3248	—	11	1	no	maximal, Sylow
82	C₂³×C₁₁	—	2	1	no	—
83	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₁	—	2	1	no	—
85	Q₈×C₁₁	—	2	1	no	—
86	D₄×C₁₁	—	2	1	no	—
87	D₄×C₁₁	—	2	1	no	—
88	D₄×C₁₁	—	4	2	no	—
89	C₂×C₄×C₁₁	—	2	2	no	—
90	C₂×C₄×C₁₁	—	2	2	no	—
91	C₂×C₄×C₁₁	—	2	1	no	—
92	C₂×C₄×C₁₁	—	2	1	no	—
93	C₂×C₄×C₁₁	—	2	1	no	—
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	4	2	no	—
97	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
103	C₂×Dic₁₁	—	2	2	no	—
104	C₂×Dic₁₁	—	2	2	no	—
105	Dic₂₂	—	2	2	no	—
106	Dic₂₂	—	2	2	no	—
107	Dic₂₂	—	4	2	no	—
108	C₁₁⋊C₈	—	2	2	no	—
109	C₁₁⋊C₈	—	2	2	no	—
110	#533	—	11	1	no	—
111	#529	D₁₁	1	1	yes	characteristic
112	#479	—	11	2	no	—
113	#475	—	11	1	no	—
114	#474	—	11	1	no	—
115	#467	—	11	2	no	—
116	#465	—	11	2	no	—
117	#464	—	11	2	no	—
118	#462	—	11	1	no	—
119	#461	—	11	1	no	—
120	C₂×C₄⋊C₈	—	11	1	no	—
121	C₂²×C₁₁	—	2	1	no	—
122	C₂²×C₁₁	—	2	1	no	—
123	C₂²×C₁₁	—	4	1	no	—
124	C₂²×C₁₁	—	4	1	no	—
125	C₂²×C₁₁	—	4	1	no	—
126	C₂²×C₁₁	#137	1	2	yes	—
127	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₁₁	—	2	1	no	—
129	C₄×C₁₁	—	2	1	no	—
130	C₄×C₁₁	—	2	1	no	—
131	C₄×C₁₁	—	2	1	no	—
132	C₄×C₁₁	—	4	1	no	—
133	C₄×C₁₁	—	4	1	no	—
134	C₄×C₁₁	—	4	1	no	—
135	C₄×C₁₁	#135	1	2	yes	—
136	C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
138	Dic₁₁	—	4	2	no	—
139	Dic₁₁	—	4	2	no	—
140	C₂×Q₁₆	—	11	2	no	—
141	C₂×SD₁₆	—	11	2	no	—
142	C₂²×C₈	—	11	2	no	—
143	C₄⋊Q₈	—	11	1	no	—
144	C₄⋊Q₈	—	11	1	no	—
145	#126	—	2	1	no	—
146	#125	—	22	1	no	—
147	#125	C₂×D₁₁	1	1	yes	characteristic
148	#124	C₂×D₁₁	1	1	yes	characteristic
149	#123	—	2	1	no	—
150	C₂²⋊Q₈	—	11	2	no	—
151	C₂²⋊Q₈	—	22	1	no	—
152	C₂²⋊Q₈	C₂×D₁₁	1	1	yes	characteristic
153	#121	C₂×D₁₁	1	1	yes	characteristic
154	C₄×Q₈	—	11	2	no	—
155	C₄×Q₈	—	22	1	no	—
156	C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	C₂×D₁₁	1	1	yes	characteristic
159	#117	—	2	1	no	—
160	#117	—	22	1	no	—
161	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
162	#107	—	11	2	no	—
163	#106	—	11	2	no	—
164	C₄⋊C₈	—	11	1	no	—
165	C₄⋊C₈	—	11	1	no	—
166	C₄⋊C₈	—	11	2	no	—
167	#103	—	11	2	no	—
168	#103	—	11	2	no	—
169	#103	—	11	2	no	—
170	#102	—	11	2	no	—
171	C₂×C₁₁	—	2	2	no	—
172	C₂×C₁₁	—	8	1	no	—
173	C₂×C₁₁	#537	1	1	yes	characteristic
174	C₂×C₁₁	C₂×#135	1	1	yes	characteristic
175	C₂×C₁₁	C₂×#137	1	1	yes	characteristic
176	C₂×Q₈	—	11	2	no	—
177	C₂×Q₈	—	22	1	no	—
178	C₂×Q₈	C₂²×D₁₁	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂²×D₁₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
185	Q₁₆	—	44	2	no	—
186	SD₁₆	—	44	2	no	—
187	C₂×C₈	—	11	2	no	—
188	C₂×C₈	—	11	2	no	—
189	C₂×C₈	—	44	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	11	2	no	—
195	C₄⋊C₄	—	22	1	no	—
196	C₄⋊C₄	—	22	1	no	—
197	C₄⋊C₄	—	22	2	no	—
198	C₄⋊C₄	—	22	2	no	—
199	C₄⋊C₄	—	22	1	no	—
200	C₄⋊C₄	C₂²×D₁₁	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₁₁	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	22	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	22	2	no	—
210	C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
211	C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
212	C₄²	C₂²×D₁₁	1	1	yes	characteristic
213	C₄²	C₂²×D₁₁	1	1	yes	characteristic
214	C₁₁	#3248	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	1	no	—
216	C₂³	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
217	Q₈	—	2	1	no	—
218	Q₈	—	2	1	no	—
219	Q₈	—	22	2	no	—
220	Q₈	—	22	2	no	—
221	Q₈	—	44	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	1	no	—
224	D₄	—	4	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	22	2	no	—
234	C₂×C₄	—	22	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
241	C₈	—	22	2	no	—
242	C₈	—	22	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	#67004	1	2	yes	—
249	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	44	2	no	—
258	C₄	—	44	2	no	—
259	C₄	#67003	1	2	yes	—
260	C₄	#67042	1	1	yes	characteristic
261	C₄	#67043	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂×#67003	1	1	yes	characteristic
265	C₂	C₂×#67004	1	1	yes	characteristic
266	C₂	#10623364	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₁
commutator, cosocle	C₂×C₄×C₁₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₁
Fitting	C₁₁×#529	C₂
socle	C₂²×C₁₁	C₂²×D₄
prime radical	C₁₁×#121	C₂²
hypercenter	#529	D₁₁
hypocenter	C₁₁	#3248
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
2. #10623001	C₂	semidirect
3. #10622676	C₂	semidirect
4. #10622667	C₂	semidirect
5. #10622664	C₂	semidirect
7. #10622644	C₂	semidirect
8. #10622633	C₂	semidirect
9. #10622632	C₂	semidirect
10. #10622620	C₂	semidirect
11. C₂×C₄⋊(C₁₁⋊C₈)	C₂	semidirect
214. C₁₁	#3248	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×#529	2
#10623001	2
#10622676	2
#10622667	2
#10622664	2
#10622654	2
#10622644	2
#10622633	2
#10622632	2
#10622620	2
C₂×C₄⋊(C₁₁⋊C₈)	2
#3248	11