Subgroups — #13077002

Order 1408.

400 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#389	C₂	1	1	yes	maximal, characteristic
2	#10622598	C₂	1	1	yes	maximal, characteristic
3	#10622594	C₂	1	2	yes	maximal
4	#10622591	C₂	1	2	yes	maximal
5	#10622590	C₂	1	1	yes	maximal, characteristic
6	#10622590	C₂	1	1	yes	maximal, characteristic
7	#10622586	C₂	1	1	yes	maximal, characteristic
8	C₄²×Dic₁₁	C₂	1	1	yes	maximal, characteristic
9	#10622541	C₂	1	1	yes	maximal, characteristic
10	#10622266	C₂	1	2	yes	maximal
11	#10622265	C₂	1	2	yes	maximal
12	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₁	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₁⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₁₁⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
27	C₁₁×#95	C₂²	1	2	yes	—
28	C₁₁⋊#95	C₂²	1	2	yes	—
29	C₁₁⋊#95	C₂²	1	2	yes	—
30	C₁₁⋊#95	C₂²	1	2	yes	—
31	C₁₁⋊#95	C₂²	1	2	yes	—
32	C₁₁⋊#95	C₂²	1	2	yes	—
33	C₁₁⋊#95	C₂²	1	2	yes	—
34	C₁₁⋊#95	C₂²	1	2	yes	—
35	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₁	C₂³	1	2	yes	—
41	C₂²×Dic₁₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₁	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₁	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₁	C₂³	1	1	yes	characteristic
46	C₁₁×C₄⋊C₄	—	2	2	no	—
47	C₁₁×C₄⋊C₄	—	2	4	no	—
48	C₁₁×C₄⋊C₄	—	2	2	no	—
49	C₄²×C₁₁	—	2	2	no	—
50	C₄⋊Dic₁₁	—	2	4	no	—
51	C₄⋊Dic₁₁	—	2	2	no	—
52	C₄⋊Dic₁₁	—	2	2	no	—
53	Dic₁₁⋊C₄	—	2	4	no	—
54	Dic₁₁⋊C₄	—	2	4	no	—
55	C₄×Dic₁₁	—	2	2	no	—
56	C₄×Dic₁₁	—	2	2	no	—
57	C₄×Dic₁₁	—	2	2	no	—
58	C₄×Dic₁₁	—	2	2	no	—
59	C₄×Dic₁₁	—	2	2	no	—
60	C₄×Dic₁₁	Q₈	1	4	yes	—
61	#2864	—	11	1	no	maximal, Sylow
62	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₁₁	—	2	1	no	—
64	C₂×C₄×C₁₁	—	2	1	no	—
65	C₂×C₄×C₁₁	—	2	1	no	—
66	C₂×C₄×C₁₁	—	2	1	no	—
67	C₂×C₄×C₁₁	—	2	2	no	—
68	C₂×C₄×C₁₁	—	2	2	no	—
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	2	2	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	—	2	2	no	—
73	C₂×C₄×C₁₁	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₁₁	#41	1	2	yes	—
75	C₂×C₄×C₁₁	#41	1	2	yes	—
76	C₂×C₄×C₁₁	#41	1	2	yes	—
77	C₂×C₄×C₁₁	#41	1	2	yes	—
78	C₂×Dic₁₁	—	2	1	no	—
79	C₂×Dic₁₁	—	2	1	no	—
80	C₂×Dic₁₁	—	2	1	no	—
81	C₂×Dic₁₁	—	2	1	no	—
82	C₂×Dic₁₁	—	2	1	no	—
83	C₂×Dic₁₁	—	2	1	no	—
84	C₂×Dic₁₁	—	2	1	no	—
85	C₂×Dic₁₁	—	2	1	no	—
86	C₂×Dic₁₁	—	2	2	no	—
87	C₂×Dic₁₁	—	2	2	no	—
88	C₂×Dic₁₁	—	2	2	no	—
89	C₂×Dic₁₁	—	2	2	no	—
90	C₂×Dic₁₁	—	2	2	no	—
91	C₂×Dic₁₁	—	2	2	no	—
92	C₂×Dic₁₁	C₂×Q₈	1	2	yes	—
93	C₂×Dic₁₁	C₂×Q₈	1	2	yes	—
94	C₂×Dic₁₁	#41	1	2	yes	—
95	C₂×Dic₁₁	#41	1	2	yes	—
96	#389	—	11	1	no	—
97	#389	—	11	1	no	—
98	#389	—	11	1	no	—
99	#389	D₁₁	1	1	yes	characteristic
100	#387	—	11	2	no	—
101	#387	—	11	2	no	—
102	#387	—	11	2	no	—
103	#387	—	11	2	no	—
104	#382	—	11	1	no	—
105	#382	—	11	1	no	—
106	C₄³	—	11	1	no	—
107	C₂²×C₁₁	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
110	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₁	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₁	—	2	2	no	—
115	C₄×C₁₁	—	2	2	no	—
116	C₄×C₁₁	—	4	2	no	—
117	C₄×C₁₁	—	4	2	no	—
118	C₄×C₁₁	#125	1	4	yes	—
119	Dic₁₁	—	2	2	no	—
120	Dic₁₁	—	2	2	no	—
121	Dic₁₁	—	2	2	no	—
122	Dic₁₁	—	2	2	no	—
123	Dic₁₁	—	4	2	no	—
124	Dic₁₁	—	4	2	no	—
125	C₂×C₄⋊C₄	—	11	1	no	—
126	C₂×C₄⋊C₄	—	11	1	no	—
127	C₂×C₄⋊C₄	—	11	2	no	—
128	C₂×C₄⋊C₄	—	11	2	no	—
129	C₂×C₄⋊C₄	—	11	2	no	—
130	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
133	C₂×C₄²	—	11	1	no	—
134	C₂×C₄²	—	11	1	no	—
135	C₂×C₄²	—	11	1	no	—
136	C₂×C₄²	—	11	1	no	—
137	C₂×C₄²	—	11	1	no	—
138	C₂×C₄²	—	11	1	no	—
139	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
140	#95	—	11	2	no	—
141	#95	—	11	2	no	—
142	#95	—	11	2	no	—
143	#95	—	11	2	no	—
144	#95	—	11	2	no	—
145	#95	—	11	2	no	—
146	#95	—	11	2	no	—
147	#95	C₂×D₁₁	1	2	yes	—
148	C₂×C₁₁	C₂×#125	1	1	yes	characteristic
149	C₂×C₁₁	#529	1	1	yes	characteristic
150	C₂×C₁₁	#529	1	1	yes	characteristic
151	C₂×C₁₁	#529	1	1	yes	characteristic
152	C₂×C₁₁	#529	1	1	yes	characteristic
153	C₂×C₁₁	#533	1	1	yes	characteristic
154	C₂×C₁₁	#533	1	1	yes	characteristic
155	C₂²×C₄	—	11	1	no	—
156	C₂²×C₄	—	11	1	no	—
157	C₂²×C₄	—	11	1	no	—
158	C₂²×C₄	—	11	1	no	—
159	C₂²×C₄	—	11	2	no	—
160	C₂²×C₄	—	11	2	no	—
161	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
162	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
163	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	22	4	no	—
170	C₄⋊C₄	—	22	4	no	—
171	C₄⋊C₄	—	22	2	no	—
172	C₄⋊C₄	—	22	2	no	—
173	C₄⋊C₄	—	22	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	11	4	no	—
176	C₄²	—	22	2	no	—
177	C₄²	—	22	2	no	—
178	C₄²	—	22	2	no	—
179	C₄²	—	22	2	no	—
180	C₄²	—	22	2	no	—
181	C₁₁	#2864	1	1	yes	characteristic, complemented, Sylow
182	C₂³	C₂³×D₁₁	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	11	2	no	—
194	C₂×C₄	—	11	2	no	—
195	C₂×C₄	—	11	2	no	—
196	C₂×C₄	—	11	2	no	—
197	C₂×C₄	—	22	1	no	—
198	C₂×C₄	—	22	1	no	—
199	C₂×C₄	—	22	1	no	—
200	C₂×C₄	—	22	1	no	—
201	C₂×C₄	—	22	1	no	—
202	C₂×C₄	—	22	1	no	—
203	C₂×C₄	—	22	1	no	—
204	C₂×C₄	—	22	1	no	—
205	C₂×C₄	—	22	2	no	—
206	C₂×C₄	—	22	2	no	—
207	C₂×C₄	—	22	2	no	—
208	C₂×C₄	—	22	2	no	—
209	C₂×C₄	—	22	2	no	—
210	C₂×C₄	—	22	2	no	—
211	C₂×C₄	#4339	1	2	yes	—
212	C₂×C₄	#4341	1	2	yes	—
213	C₂×C₄	#4341	1	2	yes	—
214	C₂×C₄	Q₈×D₁₁	1	2	yes	—
215	C₂×C₄	#4343	1	2	yes	—
216	C₂²	C₂×#4339	1	1	yes	characteristic
217	C₂²	C₂×#4341	1	1	yes	characteristic
218	C₂²	C₂×#4341	1	1	yes	characteristic
219	C₂²	C₂×Q₈×D₁₁	1	1	yes	characteristic
220	C₂²	C₂×#4343	1	1	yes	characteristic
221	C₂²	#41×D₁₁	1	1	yes	characteristic
222	C₂²	#41×D₁₁	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	22	2	no	—
228	C₄	—	22	2	no	—
229	C₄	—	22	2	no	—
230	C₄	—	22	2	no	—
231	C₄	—	44	2	no	—
232	C₄	—	44	2	no	—
233	C₄	#66942	1	4	yes	—
234	C₂	C₂×#66942	1	1	yes	characteristic
235	C₂	#10623088	1	1	yes	characteristic
236	C₂	#10623111	1	1	yes	characteristic
237	C₂	#10623131	1	1	yes	characteristic
238	C₂	#10623155	1	1	yes	characteristic
239	C₂	#10623219	1	1	yes	characteristic
240	C₂	#10623259	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁
commutator, socle, prime radical, cosocle	C₂³×C₁₁	C₂⁴
Fitting	C₁₁×#389	C₂
hypercenter	#389	D₁₁
hypocenter	C₁₁	#2864
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
181. C₁₁	#2864	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×#389	2
#10622598	2
#10622594	2
#10622591	2
#10622590	2
#10622586	2
C₄²×Dic₁₁	2
#10622541	2
#10622266	2
#10622265	2
#2864	11