Subgroups — #14668652

Order 2816.

606 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622719	C₂	1	1	yes	maximal, characteristic
3	#13066707	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₁×#427	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₁₁×#117	—	2	1	no	—
7	C₂×#66979	—	2	1	no	—
8	C₁₁×#427	—	2	2	no	—
9	C₁₁×#427	C₄	1	4	yes	complemented
10	C₂×C₁₁×#107	C₂²	1	2	yes	—
11	C₂×C₁₁×#106	—	2	1	no	—
12	#10622719	—	2	2	no	—
13	#10622719	C₄	1	4	yes	complemented
14	C₂×#66882	C₂²	1	2	yes	—
15	C₂×#66881	—	2	1	no	—
16	C₂²×C₁₁⋊C₁₆	—	2	2	no	—
17	C₂²×C₁₁⋊C₁₆	—	2	2	no	—
18	C₂²×C₁₁⋊C₁₆	C₄	1	4	yes	complemented
19	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₁	D₄	1	2	yes	—
25	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	D₄	1	2	yes	—
27	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
29	C₁₁×#117	—	2	4	no	—
30	C₁₁×#117	—	4	2	no	—
31	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₁×C₂²⋊C₄	—	4	1	no	—
33	C₂×C₄²×C₁₁	—	4	1	no	—
34	C₂×C₂²⋊Dic₁₁	—	4	1	no	—
35	#66979	—	2	4	no	—
36	#66979	—	4	2	no	—
37	C₂×C₄⋊Dic₁₁	—	2	2	no	—
38	C₂×C₄×Dic₁₁	—	4	1	no	—
39	C₁₁×#107	—	2	2	no	—
40	C₁₁×#107	C₂×C₄	1	4	yes	—
41	C₁₁×#106	—	2	4	no	—
42	#66882	—	2	2	no	—
43	#66882	C₂×C₄	1	4	yes	—
44	#66881	—	2	4	no	—
45	C₂×C₁₁⋊C₁₆	—	2	8	no	—
46	C₂×C₁₁⋊C₁₆	—	2	2	no	—
47	C₂×C₁₁⋊C₁₆	—	2	2	no	—
48	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
49	#12556	—	11	1	no	maximal, Sylow
50	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	—	4	2	no	—
52	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
58	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₁	—	4	2	no	—
61	C₂×C₈×C₁₁	C₄²	1	4	yes	—
62	C₂×C₈×C₁₁	C₂²⋊C₄	1	4	yes	—
63	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
74	C₁₁×C₄⋊C₄	—	2	4	no	—
75	C₁₁×C₄⋊C₄	—	4	2	no	—
76	C₁₁×C₂²⋊C₄	—	4	4	no	—
77	C₄²×C₁₁	—	4	4	no	—
78	C₂²⋊Dic₁₁	—	4	4	no	—
79	C₄⋊Dic₁₁	—	2	4	no	—
80	C₄⋊Dic₁₁	—	4	2	no	—
81	C₄×Dic₁₁	—	4	4	no	—
82	C₁₁⋊C₁₆	—	2	4	no	—
83	C₂³×C₁₆	—	11	1	no	—
84	C₂×#427	—	11	1	no	—
85	C₂×#427	D₁₁	1	1	yes	characteristic
86	#1380	—	11	4	no	—
87	C₂³×C₁₁	—	2	4	no	—
88	C₂³×C₁₁	#102	1	2	yes	—
89	C₂³×C₁₁	#102	1	2	yes	—
90	C₂³×C₁₁	#107	1	2	yes	—
91	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₁	—	4	4	no	—
93	C₂×C₄×C₁₁	—	8	4	no	—
94	C₂×C₄×C₁₁	#95	1	4	yes	—
95	C₂×C₄×C₁₁	#102	1	2	yes	—
96	C₂×C₄×C₁₁	#102	1	2	yes	—
97	C₂×C₄×C₁₁	#103	1	4	yes	—
98	C₂×C₄×C₁₁	#103	1	4	yes	—
99	C₂×C₄×C₁₁	#106	1	4	yes	—
100	C₂×C₄×C₁₁	#107	1	2	yes	—
101	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×Dic₁₁	—	4	4	no	—
108	C₂×Dic₁₁	—	8	4	no	—
109	C₈×C₁₁	#95	1	4	yes	—
110	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
111	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
115	C₂×#117	—	2	1	no	—
116	C₂×#117	—	22	1	no	—
117	C₂²×C₁₆	—	11	2	no	—
118	C₂²×C₁₆	—	11	4	no	—
119	C₂²×C₁₆	—	22	2	no	—
120	C₂²×C₁₆	—	22	2	no	—
121	#427	—	2	2	no	—
122	#427	—	11	4	no	—
123	#427	—	22	2	no	—
124	#427	Dic₁₁	1	4	yes	complemented
125	C₂×#107	—	11	2	no	—
126	C₂×#107	C₂×D₁₁	1	2	yes	—
127	C₂×#106	—	2	1	no	—
128	C₂×#106	—	22	1	no	—
129	C₂²×C₁₁	—	2	4	no	—
130	C₂²×C₁₁	—	2	4	no	—
131	C₂²×C₁₁	—	2	4	no	—
132	C₂²×C₁₁	#340	1	4	yes	—
133	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
134	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
136	C₂²×C₁₁	#508	1	4	yes	—
137	C₄×C₁₁	—	8	4	no	—
138	C₄×C₁₁	#340	1	4	yes	—
139	C₄×C₁₁	C₂×#95	1	1	yes	characteristic
140	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
141	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
142	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
143	Dic₁₁	—	8	4	no	—
144	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
145	C₂²×C₈	Dic₂₂	1	2	yes	—
146	C₂²×C₈	C₄×D₁₁	1	2	yes	—
147	C₂²×C₈	C₄×D₁₁	1	2	yes	—
148	C₂²×C₈	D₄₄	1	2	yes	—
149	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
150	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₁₁	1	2	yes	—
152	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
153	#117	—	2	4	no	—
154	#117	—	4	2	no	—
155	#117	—	22	4	no	—
156	#117	—	44	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	22	2	no	—
159	C₂×C₂²⋊C₄	—	4	1	no	—
160	C₂×C₂²⋊C₄	—	44	1	no	—
161	C₂×C₄²	—	4	1	no	—
162	C₂×C₄²	—	44	1	no	—
163	C₂×C₁₆	—	11	4	no	—
164	C₂×C₁₆	—	22	8	no	—
165	C₂×C₁₆	—	22	2	no	—
166	C₂×C₁₆	—	22	2	no	—
167	#107	—	2	2	no	—
168	#107	—	11	4	no	—
169	#107	—	22	2	no	—
170	#107	C₂×Dic₁₁	1	4	yes	—
171	#106	—	2	4	no	—
172	#106	—	22	4	no	—
173	C₂×C₁₁	—	2	4	no	—
174	C₂×C₁₁	C₂×#340	1	1	yes	characteristic
175	C₂×C₁₁	#2138	1	2	yes	—
176	C₂×C₁₁	#2138	1	2	yes	—
177	C₂×C₁₁	#2158	1	2	yes	—
178	C₂⁴	#4323	1	1	yes	characteristic
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	44	2	no	—
181	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
182	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
183	C₂²×C₄	#4323	1	1	yes	characteristic
184	C₂²×C₄	#4324	1	2	yes	—
185	C₂²×C₄	#4325	1	2	yes	—
186	C₂²×C₄	#4326	1	2	yes	—
187	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
188	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
189	C₂×C₈	C₄×Dic₁₁	1	4	yes	—
190	C₂×C₈	Dic₁₁⋊C₄	1	2	yes	—
191	C₂×C₈	Dic₁₁⋊C₄	1	4	yes	—
192	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
193	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
194	C₂×C₈	(C₂×D₁₁)⋊C₄	1	4	yes	—
195	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
196	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
198	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
200	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	22	4	no	—
205	C₄⋊C₄	—	44	2	no	—
206	C₂²⋊C₄	—	4	4	no	—
207	C₂²⋊C₄	—	44	4	no	—
208	C₄²	—	4	4	no	—
209	C₄²	—	44	4	no	—
210	C₁₆	—	22	4	no	—
211	C₁₁	#12556	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	4	no	—
213	C₂³	#66871	1	2	yes	—
214	C₂³	#66873	1	2	yes	—
215	C₂³	#66896	1	2	yes	—
216	C₂³	C₂×#4323	1	1	yes	characteristic
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	—	44	4	no	—
220	C₂×C₄	—	88	4	no	—
221	C₂×C₄	#66871	1	2	yes	—
222	C₂×C₄	#66872	1	4	yes	—
223	C₂×C₄	#66873	1	2	yes	—
224	C₂×C₄	#66874	1	4	yes	—
225	C₂×C₄	C₁₁⋊#95	1	4	yes	—
226	C₂×C₄	#66896	1	2	yes	—
227	C₂×C₄	#66899	1	4	yes	—
228	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₂×#4324	1	1	yes	characteristic
231	C₂×C₄	C₂×#4325	1	1	yes	characteristic
232	C₂×C₄	C₂×#4326	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
234	C₈	C₁₁⋊#95	1	4	yes	—
235	C₈	C₂×C₄×Dic₁₁	1	1	yes	characteristic
236	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
237	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
238	C₈	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	#10622167	1	4	yes	—
243	C₂²	C₂×#66871	1	1	yes	characteristic
244	C₂²	C₂×#66873	1	1	yes	characteristic
245	C₂²	#10622807	1	4	yes	—
246	C₂²	C₂×#66896	1	1	yes	characteristic
247	C₄	—	8	4	no	—
248	C₄	—	88	4	no	—
249	C₄	#10622167	1	4	yes	—
250	C₄	C₂×#66872	1	1	yes	characteristic
251	C₄	C₂×#66874	1	1	yes	characteristic
252	C₄	C₂×C₁₁⋊#95	1	1	yes	characteristic
253	C₄	C₂×#66899	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	C₂×#10622167	1	1	yes	characteristic
256	C₂	#13070321	1	2	yes	—
257	C₂	#13070323	1	2	yes	—
258	C₂	#13070398	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#4323
commutator	C₈×C₁₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#427	C₂
socle	C₂³×C₁₁	C₂×D₈
prime radical	C₂⁴×C₁₁	D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#427	D₁₁
hypocenter	C₁₁	#12556
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #13066707	C₂	semidirect
9. C₁₁×#427	C₄	semidirect
13. #10622719	C₄	semidirect
18. C₂²×C₁₁⋊C₁₆	C₄	semidirect
124. #427	Dic₁₁	semidirect
211. C₁₁	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₁⋊C₁₆	2
C₂×#10622719	2
#13066707	2
C₂×C₁₁×#427	2
#12556	11