Subgroups — #14669697

Order 2816.

478 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	C₂×#10622302	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	2	yes	maximal
4	#13066694	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#4313	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₁	C₂²	1	1	yes	characteristic
8	C₂×#66925	C₄	1	1	yes	characteristic
9	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×#66879	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
14	C₂×#66867	C₄	1	1	yes	characteristic
15	#10622302	—	2	4	no	—
16	(C₂×D₁₁)⋊C₁₆	—	2	4	no	—
17	Dic₁₁⋊C₁₆	—	2	4	no	—
18	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×(C₂×D₁₁)⋊C₄	C₈	1	2	yes	—
23	C₂×Dic₁₁⋊C₄	C₈	1	2	yes	—
24	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×#4313	—	2	2	no	—
28	C₂×#4313	—	2	2	no	—
29	C₂×#4313	—	2	2	no	—
30	C₂×C₈×D₁₁	—	2	2	no	—
31	C₂×C₈×D₁₁	—	2	2	no	—
32	C₂×C₈×D₁₁	D₄	1	2	yes	—
33	C₂×C₈×D₁₁	Q₈	1	2	yes	—
34	#66925	—	2	2	no	—
35	#66925	—	2	2	no	—
36	C₂×C₁₁×C₁₆	—	2	2	no	—
37	C₂×C₁₁×C₁₆	—	2	2	no	—
38	C₂×C₁₁×C₁₆	—	2	2	no	—
39	C₄×C₈×C₁₁	—	2	2	no	—
40	#66879	—	2	2	no	—
41	C₈×Dic₁₁	D₄	1	4	yes	—
42	C₂×C₁₁⋊C₁₆	—	2	2	no	—
43	C₂×C₁₁⋊C₁₆	—	2	2	no	—
44	C₂×C₁₁⋊C₁₆	—	2	2	no	—
45	#66867	—	2	2	no	—
46	#12784	—	11	1	no	maximal, Sylow
47	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
51	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×D₁₁	—	2	1	no	—
53	C₂×C₄×D₁₁	—	2	1	no	—
54	C₂×C₄×D₁₁	—	2	1	no	—
55	C₂×C₄×D₁₁	—	2	1	no	—
56	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₁₁	M₁₆	1	2	yes	—
58	C₂×C₈×C₁₁	—	2	1	no	—
59	C₂×C₈×C₁₁	—	2	1	no	—
60	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
61	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₁₁	#41	1	2	yes	—
65	C₄²×C₁₁	—	2	1	no	—
66	C₄²×C₁₁	—	2	1	no	—
67	(C₂×D₁₁)⋊C₄	—	2	4	no	—
68	Dic₁₁⋊C₄	—	2	4	no	—
69	C₄×Dic₁₁	C₂²⋊C₄	1	2	yes	—
70	C₄×Dic₁₁	M₁₆	1	2	yes	—
71	C₂×C₁₁⋊C₈	—	2	1	no	—
72	C₂×C₁₁⋊C₈	—	2	1	no	—
73	C₂×C₁₁⋊C₈	—	2	1	no	—
74	C₂×C₁₁⋊C₈	—	2	1	no	—
75	C₂×C₁₁⋊C₈	C₂²⋊C₄	1	2	yes	—
76	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
77	#4313	—	2	8	no	—
78	C₈×D₁₁	—	2	4	no	—
79	C₈×D₁₁	—	2	4	no	—
80	C₁₁×C₁₆	—	4	2	no	—
81	C₁₁⋊C₁₆	—	4	2	no	—
82	C₂×#405	—	11	1	no	—
83	C₂×C₄⋊C₁₆	—	11	2	no	—
84	C₂×C₂²⋊C₁₆	—	11	2	no	—
85	#1374	—	11	1	no	—
86	#1374	D₁₁	1	1	yes	characteristic, complemented
87	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
88	C₂²×D₁₁	—	2	1	no	—
89	C₂²×D₁₁	—	2	1	no	—
90	C₂²×D₁₁	—	2	2	no	—
91	C₂²×D₁₁	—	2	2	no	—
92	C₂²×D₁₁	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	2	no	—
95	C₂×C₄×C₁₁	—	2	2	no	—
96	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
102	C₂×Dic₁₁	—	2	1	no	—
103	C₂×Dic₁₁	—	2	1	no	—
104	C₂×Dic₁₁	—	2	2	no	—
105	C₂×Dic₁₁	—	2	2	no	—
106	C₂×Dic₁₁	C₂²⋊C₈	1	4	yes	—
107	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
108	C₄×D₁₁	—	2	2	no	—
109	C₄×D₁₁	—	2	2	no	—
110	C₄×D₁₁	—	2	2	no	—
111	C₄×D₁₁	—	2	2	no	—
112	C₈×C₁₁	—	2	2	no	—
113	C₈×C₁₁	#121	1	2	yes	—
114	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
115	C₁₁⋊C₈	—	2	2	no	—
116	C₁₁⋊C₈	—	2	2	no	—
117	C₂²×M₁₆	—	11	1	no	—
118	C₂³×C₈	—	11	1	no	—
119	C₂×#117	—	11	1	no	—
120	C₂²×C₁₆	—	11	2	no	—
121	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
122	#405	—	22	4	no	—
123	C₂×#97	—	11	1	no	—
124	C₂×#97	—	11	1	no	—
125	C₂×C₄×C₈	—	11	1	no	—
126	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	22	4	no	—
128	C₂²⋊C₁₆	—	22	4	no	—
129	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₁₁	#431	1	1	yes	characteristic
132	C₂²×C₁₁	#504	1	2	yes	—
133	C₂²×C₁₁	#504	1	2	yes	—
134	C₂×D₁₁	—	2	2	no	—
135	C₂×D₁₁	—	2	4	no	—
136	C₂×D₁₁	—	4	2	no	—
137	C₂×D₁₁	—	4	1	no	—
138	C₂×D₁₁	—	4	1	no	—
139	C₄×C₁₁	—	4	2	no	—
140	C₄×C₁₁	#380	1	1	yes	characteristic
141	C₄×C₁₁	#407	1	1	yes	characteristic
142	C₄×C₁₁	#423	1	1	yes	characteristic
143	C₄×C₁₁	#432	1	1	yes	characteristic
144	Dic₁₁	—	2	2	no	—
145	Dic₁₁	—	2	4	no	—
146	C₂³×C₄	—	11	1	no	—
147	C₂×M₁₆	—	22	2	no	—
148	C₂×M₁₆	—	22	2	no	—
149	C₂×M₁₆	—	22	2	no	—
150	C₂²×C₈	—	11	1	no	—
151	C₂²×C₈	—	11	1	no	—
152	C₂²×C₈	—	11	2	no	—
153	C₂²×C₈	—	11	2	no	—
154	C₂²×C₈	—	22	2	no	—
155	C₂²×C₈	—	22	2	no	—
156	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
158	#117	—	22	2	no	—
159	#117	—	22	2	no	—
160	C₂×C₄⋊C₄	—	11	2	no	—
161	C₂×C₂²⋊C₄	—	11	2	no	—
162	C₂×C₄²	—	11	1	no	—
163	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	22	2	no	—
168	C₂×C₁₆	—	22	2	no	—
169	C₂×C₁₆	—	22	2	no	—
170	#97	—	22	2	no	—
171	#97	—	22	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	11	4	no	—
174	C₂×C₁₁	#1797	1	1	yes	characteristic
175	C₂×C₁₁	#2113	1	2	yes	—
176	C₂×C₁₁	#2151	1	2	yes	—
177	C₂×C₁₁	#2164	1	1	yes	characteristic
178	C₂×C₁₁	#2164	1	1	yes	characteristic
179	D₁₁	—	4	2	no	—
180	C₂⁴	—	11	1	no	—
181	C₂²×C₄	—	11	2	no	—
182	C₂²×C₄	—	11	1	no	—
183	C₂²×C₄	—	11	2	no	—
184	C₂²×C₄	—	11	2	no	—
185	C₂²×C₄	—	22	1	no	—
186	C₂²×C₄	—	22	1	no	—
187	C₂²×C₄	—	22	1	no	—
188	C₂²×C₄	—	22	1	no	—
189	C₂²×C₄	C₈×D₁₁	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
191	M₁₆	—	22	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	11	2	no	—
195	C₂×C₈	—	11	2	no	—
196	C₂×C₈	—	22	1	no	—
197	C₂×C₈	—	22	1	no	—
198	C₂×C₈	—	22	1	no	—
199	C₂×C₈	—	22	1	no	—
200	C₂×C₈	—	22	4	no	—
201	C₂×C₈	—	22	4	no	—
202	C₂×C₈	#4313	1	2	yes	—
203	C₂×C₈	#4339	1	2	yes	—
204	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
205	C₂×C₈	D₄×D₁₁	1	2	yes	—
206	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
207	C₄⋊C₄	—	22	4	no	—
208	C₂²⋊C₄	—	22	4	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	11	2	no	—
212	C₄²	—	11	2	no	—
213	C₁₆	—	4	2	no	—
214	C₁₆	—	44	2	no	—
215	C₁₁	#12784	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	11	2	no	—
217	C₂³	—	22	2	no	—
218	C₂³	—	22	2	no	—
219	C₂³	—	22	1	no	—
220	C₂³	—	22	1	no	—
221	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	11	4	no	—
226	C₂×C₄	—	11	2	no	—
227	C₂×C₄	—	22	2	no	—
228	C₂×C₄	—	22	2	no	—
229	C₂×C₄	—	22	1	no	—
230	C₂×C₄	—	22	1	no	—
231	C₂×C₄	—	22	2	no	—
232	C₂×C₄	—	22	2	no	—
233	C₂×C₄	—	22	2	no	—
234	C₂×C₄	—	22	2	no	—
235	C₂×C₄	#66925	1	1	yes	characteristic
236	C₂×C₄	C₂²⋊C₄×D₁₁	1	1	yes	characteristic
237	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
238	C₂×C₄	C₂×#4313	1	1	yes	characteristic
239	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	22	2	no	—
243	C₈	—	22	2	no	—
244	C₈	#66937	1	2	yes	—
245	C₈	#66949	1	2	yes	—
246	C₂²	—	22	4	no	—
247	C₂²	—	22	2	no	—
248	C₂²	—	44	1	no	—
249	C₂²	—	44	1	no	—
250	C₂²	—	44	2	no	—
251	C₂²	#10622298	1	1	yes	characteristic
252	C₂²	D₁₁×C₂²⋊C₈	1	1	yes	characteristic
253	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
254	C₂²	#10622514	1	2	yes	—
255	C₂²	#10622520	1	2	yes	—
256	C₄	—	4	2	no	—
257	C₄	—	22	2	no	—
258	C₄	—	22	4	no	—
259	C₄	#10622280	1	1	yes	characteristic
260	C₄	#10622318	1	1	yes	characteristic
261	C₄	#10622339	1	1	yes	characteristic
262	C₄	#10622450	1	1	yes	characteristic
263	C₂	—	44	2	no	—
264	C₂	#13068595	1	1	yes	characteristic
265	C₂	#13070228	1	2	yes	—
266	C₂	#13070373	1	2	yes	—
267	C₂	#13070421	1	1	yes	characteristic
268	C₂	#13070424	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator, socle	C₂³×C₁₁	C₂²×C₈
Fitting	C₁₁×#1374	C₂
prime radical	C₂³×D₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1374	D₁₁
hypocenter	C₁₁	#12784
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₁⋊C₁₆	C₂	semidirect
4. #13066694	C₂	semidirect
5. C₁₁×#1374	C₂	semidirect
86. #1374	D₁₁	semidirect
215. C₁₁	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622302	2
C₂×Dic₁₁⋊C₁₆	2
C₂×(C₂×D₁₁)⋊C₁₆	2
#13066694	2
C₁₁×#1374	2
#12784	11