Subgroups — #14668808

Order 2816.

514 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×#10622099	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#13066694	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×#117	C₄	1	1	yes	characteristic
9	C₁₁×#405	—	2	2	no	—
10	C₁₁×#405	—	2	2	no	—
11	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
12	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
17	C₂²⋊(C₁₁⋊C₁₆)	C₄	1	8	yes	complemented
18	#10622099	—	2	2	no	—
19	C₄×C₁₁⋊C₁₆	—	2	2	no	—
20	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₁×M₁₆	—	2	2	no	—
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	C₈	1	4	yes	—
24	C₂²×C₈×C₁₁	—	2	1	no	—
25	C₂²×C₈×C₁₁	—	2	1	no	—
26	C₂²×C₈×C₁₁	—	2	2	no	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₁₁×#117	—	2	2	no	—
33	C₁₁×#117	—	2	2	no	—
34	C₂×C₁₁×C₄⋊C₄	C₈	1	2	yes	—
35	C₂×C₁₁×C₂²⋊C₄	C₈	1	2	yes	—
36	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₁₁×#97	—	2	2	no	—
39	C₁₁×#97	—	2	2	no	—
40	C₄×C₈×C₁₁	—	2	1	no	—
41	C₄×C₈×C₁₁	—	2	1	no	—
42	C₄×C₈×C₁₁	D₄	1	2	yes	—
43	C₄×C₈×C₁₁	D₄	1	2	yes	—
44	C₂×C₁₁⋊C₁₆	—	2	2	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	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	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
50	#12600	—	11	1	no	maximal, Sylow
51	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	—	2	1	no	—
53	C₂²×C₄×C₁₁	—	2	1	no	—
54	C₂²×C₄×C₁₁	—	2	1	no	—
55	C₂²×C₄×C₁₁	—	2	1	no	—
56	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₁₁×M₁₆	—	2	8	no	—
63	C₂×C₈×C₁₁	—	2	4	no	—
64	C₂×C₈×C₁₁	—	2	4	no	—
65	C₂×C₈×C₁₁	—	2	1	no	—
66	C₂×C₈×C₁₁	—	2	1	no	—
67	C₂×C₈×C₁₁	—	2	1	no	—
68	C₂×C₈×C₁₁	—	2	1	no	—
69	C₂×C₈×C₁₁	C₄²	1	2	yes	—
70	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	#41	1	2	yes	—
78	C₁₁×C₄⋊C₄	—	2	4	no	—
79	C₁₁×C₂²⋊C₄	—	2	4	no	—
80	C₄²×C₁₁	—	2	1	no	—
81	C₄²×C₁₁	—	2	1	no	—
82	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₁₁	M₁₆	1	2	yes	—
84	C₁₁⋊C₁₆	—	2	4	no	—
85	C₁₁⋊C₁₆	—	2	4	no	—
86	C₂×#405	D₁₁	1	1	yes	characteristic
87	C₂×C₂²⋊C₁₆	—	11	2	no	—
88	C₂×#346	—	11	1	no	—
89	C₂×C₄×C₁₆	—	11	1	no	—
90	#1374	—	11	2	no	—
91	C₂³×C₁₁	—	2	1	no	—
92	C₂³×C₁₁	—	2	1	no	—
93	C₂³×C₁₁	—	2	4	no	—
94	C₂³×C₁₁	C₄×C₈	1	2	yes	—
95	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₁	—	2	1	no	—
97	C₂×C₄×C₁₁	—	2	1	no	—
98	C₂×C₄×C₁₁	—	2	4	no	—
99	C₂×C₄×C₁₁	—	2	2	no	—
100	C₂×C₄×C₁₁	—	2	2	no	—
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	—	2	2	no	—
103	C₂×C₄×C₁₁	—	2	2	no	—
104	C₂×C₄×C₁₁	—	2	4	no	—
105	C₂×C₄×C₁₁	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₁₁	C₂²⋊C₈	1	4	yes	—
107	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
113	C₈×C₁₁	—	2	2	no	—
114	C₈×C₁₁	—	2	2	no	—
115	C₈×C₁₁	C₂²⋊C₈	1	4	yes	—
116	C₈×C₁₁	C₄×D₄	1	2	yes	—
117	C₈×C₁₁	C₄×D₄	1	2	yes	—
118	C₂²×M₁₆	Dic₁₁	1	1	yes	characteristic
119	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
120	C₂×#117	Dic₁₁	1	1	yes	characteristic
121	C₂²×C₁₆	—	11	2	no	—
122	C₂²×C₁₆	—	11	2	no	—
123	#405	—	2	2	no	—
124	#405	—	2	2	no	—
125	C₂×#97	Dic₁₁	1	1	yes	characteristic
126	C₂×#97	Dic₁₁	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
129	C₂²⋊C₁₆	—	11	8	no	—
130	#346	—	22	2	no	—
131	C₄×C₁₆	—	22	2	no	—
132	C₂²×C₁₁	—	2	4	no	—
133	C₂²×C₁₁	—	2	2	no	—
134	C₂²×C₁₁	—	4	2	no	—
135	C₂²×C₁₁	—	4	2	no	—
136	C₂²×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₁	#431	1	1	yes	characteristic
139	C₂²×C₁₁	#504	1	4	yes	—
140	C₄×C₁₁	—	2	4	no	—
141	C₄×C₁₁	—	2	2	no	—
142	C₄×C₁₁	—	4	2	no	—
143	C₄×C₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₁₁	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₄×C₁₁	#431	1	1	yes	characteristic
147	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₁₁⋊C₈	1	4	yes	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₄×D₁₁	1	4	yes	—
155	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
156	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
159	#117	—	2	2	no	—
160	#117	—	2	2	no	—
161	C₂×C₄⋊C₄	C₁₁⋊C₈	1	2	yes	—
162	C₂×C₂²⋊C₄	C₁₁⋊C₈	1	2	yes	—
163	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
165	C₂×C₁₆	—	11	4	no	—
166	C₂×C₁₆	—	11	4	no	—
167	C₂×C₁₆	—	22	2	no	—
168	C₂×C₁₆	—	22	2	no	—
169	C₂×C₁₆	—	22	2	no	—
170	C₂×C₁₆	—	22	2	no	—
171	#97	—	2	2	no	—
172	#97	—	2	2	no	—
173	C₄×C₈	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
176	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
177	C₂×C₁₁	—	4	2	no	—
178	C₂×C₁₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
179	C₂×C₁₁	#2108	1	2	yes	—
180	C₂×C₁₁	#2113	1	2	yes	—
181	C₂×C₁₁	#2164	1	2	yes	—
182	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
188	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
190	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
191	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
193	M₁₆	—	2	8	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
201	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
202	C₂×C₈	#4318	1	2	yes	—
203	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
206	C₂×C₈	#4339	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	#4318	1	2	yes	—
214	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
215	C₁₆	—	22	4	no	—
216	C₁₆	—	22	4	no	—
217	C₁₁	#12600	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	2	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₄×C₁₁⋊C₈	1	2	yes	—
222	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×C₁₁⋊C₈	1	2	yes	—
233	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	4	yes	—
234	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#4318	1	1	yes	characteristic
236	C₂×C₄	C₂×#4318	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
238	C₂×C₄	#66979	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂²⋊(C₁₁⋊C₈)	1	4	yes	—
243	C₈	C₄×C₂²⋊D₁₁	1	2	yes	—
244	C₈	C₄×C₂²⋊D₁₁	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂×C₄×C₁₁⋊C₈	1	1	yes	characteristic
250	C₂²	#10622538	1	1	yes	characteristic
251	C₂²	#10622753	1	4	yes	—
252	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	C₄×#4318	1	1	yes	characteristic
257	C₄	#10622538	1	1	yes	characteristic
258	C₄	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
259	C₄	C₄×C₂²⋊Dic₁₁	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
262	C₂	#13070213	1	2	yes	—
263	C₂	#13070230	1	2	yes	—
264	C₂	#13070423	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂×C₄×C₈
Fitting	C₂×C₁₁×#405	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#405	D₁₁
hypocenter	C₁₁	#12600
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₂×#10622099	C₂	semidirect
3. C₂×C₄×C₁₁⋊C₁₆	C₂	semidirect
4. #13066694	C₂	semidirect
17. C₂²⋊(C₁₁⋊C₁₆)	C₄	semidirect
217. C₁₁	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₁⋊C₁₆)	2
C₂×#10622099	2
C₂×C₄×C₁₁⋊C₁₆	2
#13066694	2
C₂×C₁₁×#405	2
#12600	11