Subgroups — #14669695

Order 2816.

478 conjugacy classes of subgroups.

The subgroup lattice has 314 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×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	2	yes	maximal
4	#13066693	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×C₄⋊C₁₆	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	2	no	—
16	#10622302	—	2	2	no	—
17	C₁₁×C₄⋊C₁₆	—	2	2	no	—
18	(C₂×D₁₁)⋊C₁₆	—	2	4	no	—
19	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
20	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×(C₂×D₁₁)⋊C₄	C₈	1	1	yes	characteristic
25	C₂×(C₂×D₁₁)⋊C₄	C₈	1	1	yes	characteristic
26	C₂×Dic₁₁⋊C₄	C₈	1	1	yes	characteristic
27	C₂×Dic₁₁⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×#4313	—	2	2	no	—
32	C₂×#4313	—	2	2	no	—
33	C₂×#4313	—	2	2	no	—
34	C₂×C₈×D₁₁	—	2	1	no	—
35	C₂×C₈×D₁₁	—	2	1	no	—
36	C₂×C₈×D₁₁	—	2	1	no	—
37	C₂×C₈×D₁₁	—	2	1	no	—
38	C₂×C₈×D₁₁	D₄	1	2	yes	—
39	C₂×C₈×D₁₁	Q₈	1	2	yes	—
40	#66925	—	2	2	no	—
41	#66925	—	2	2	no	—
42	C₂×C₁₁×C₁₆	—	2	2	no	—
43	C₂×C₁₁×C₁₆	—	2	2	no	—
44	C₂×C₁₁×C₁₆	—	2	2	no	—
45	C₄×C₈×C₁₁	D₄	1	2	yes	—
46	C₄×C₈×C₁₁	D₄	1	2	yes	—
47	#66879	—	2	2	no	—
48	C₈×Dic₁₁	—	2	1	no	—
49	C₈×Dic₁₁	—	2	1	no	—
50	C₂×C₁₁⋊C₁₆	—	2	2	no	—
51	C₂×C₁₁⋊C₁₆	—	2	2	no	—
52	C₂×C₁₁⋊C₁₆	—	2	2	no	—
53	#66867	—	2	2	no	—
54	#12784	—	11	1	no	maximal, Sylow
55	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₁₁	—	2	1	no	—
63	C₂×C₄×D₁₁	—	2	1	no	—
64	C₂×C₄×D₁₁	—	2	1	no	—
65	C₂×C₄×D₁₁	—	2	1	no	—
66	C₂×C₄×D₁₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₁₁	M₁₆	1	2	yes	—
68	C₂×C₈×C₁₁	—	2	1	no	—
69	C₂×C₈×C₁₁	—	2	1	no	—
70	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
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×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
77	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₁₁	M₁₆	1	2	yes	—
79	(C₂×D₁₁)⋊C₄	—	2	2	no	—
80	(C₂×D₁₁)⋊C₄	—	2	2	no	—
81	Dic₁₁⋊C₄	—	2	2	no	—
82	Dic₁₁⋊C₄	—	2	2	no	—
83	C₄×Dic₁₁	—	2	1	no	—
84	C₄×Dic₁₁	—	2	1	no	—
85	C₂×C₁₁⋊C₈	—	2	1	no	—
86	C₂×C₁₁⋊C₈	—	2	1	no	—
87	C₂×C₁₁⋊C₈	—	2	1	no	—
88	C₂×C₁₁⋊C₈	—	2	1	no	—
89	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
90	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
91	#4313	—	2	8	no	—
92	C₈×D₁₁	—	2	2	no	—
93	C₈×D₁₁	—	2	2	no	—
94	C₈×D₁₁	—	2	2	no	—
95	C₈×D₁₁	—	2	2	no	—
96	C₁₁×C₁₆	—	4	2	no	—
97	C₁₁⋊C₁₆	—	4	2	no	—
98	C₂×#405	—	11	1	no	—
99	C₂×C₄⋊C₁₆	—	11	1	no	—
100	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic, complemented
101	C₂×C₂²⋊C₁₆	—	11	2	no	—
102	#1374	—	11	2	no	—
103	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
104	C₂²×D₁₁	—	2	1	no	—
105	C₂²×D₁₁	—	2	1	no	—
106	C₂²×D₁₁	—	2	2	no	—
107	C₂²×D₁₁	—	2	2	no	—
108	C₂²×D₁₁	C₄⋊C₈	1	2	yes	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
113	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
119	C₂×Dic₁₁	—	2	1	no	—
120	C₂×Dic₁₁	—	2	1	no	—
121	C₂×Dic₁₁	—	2	1	no	—
122	C₂×Dic₁₁	—	2	1	no	—
123	C₂×Dic₁₁	—	2	2	no	—
124	C₂×Dic₁₁	—	2	2	no	—
125	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
126	C₄×D₁₁	—	2	2	no	—
127	C₄×D₁₁	—	2	2	no	—
128	C₄×D₁₁	—	2	2	no	—
129	C₄×D₁₁	—	2	2	no	—
130	C₈×C₁₁	—	2	2	no	—
131	C₈×C₁₁	#121	1	1	yes	characteristic
132	C₈×C₁₁	#121	1	1	yes	characteristic
133	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
135	C₁₁⋊C₈	—	2	2	no	—
136	C₁₁⋊C₈	—	2	2	no	—
137	C₂²×M₁₆	—	11	1	no	—
138	C₂³×C₈	—	11	1	no	—
139	C₂×#117	—	11	1	no	—
140	C₂²×C₁₆	—	11	2	no	—
141	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
142	#405	—	22	2	no	—
143	#405	—	22	2	no	—
144	C₂×#97	—	11	1	no	—
145	C₂×#97	—	11	1	no	—
146	C₂×C₄×C₈	—	11	1	no	—
147	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	2	no	—
149	C₄⋊C₁₆	—	22	2	no	—
150	C₂²⋊C₁₆	—	22	4	no	—
151	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₂²×C₁₁	#431	1	1	yes	characteristic
154	C₂²×C₁₁	#504	1	2	yes	—
155	C₂²×C₁₁	#504	1	2	yes	—
156	C₂×D₁₁	—	2	2	no	—
157	C₂×D₁₁	—	2	4	no	—
158	C₂×D₁₁	—	4	1	no	—
159	C₂×D₁₁	—	4	1	no	—
160	C₂×D₁₁	—	4	2	no	—
161	C₄×C₁₁	—	2	2	no	—
162	C₄×C₁₁	—	2	2	no	—
163	C₄×C₁₁	#380	1	1	yes	characteristic
164	C₄×C₁₁	#407	1	1	yes	characteristic
165	C₄×C₁₁	#423	1	1	yes	characteristic
166	C₄×C₁₁	#432	1	1	yes	characteristic
167	Dic₁₁	—	2	2	no	—
168	Dic₁₁	—	4	1	no	—
169	Dic₁₁	—	4	1	no	—
170	C₂³×C₄	—	11	1	no	—
171	C₂×M₁₆	—	22	2	no	—
172	C₂×M₁₆	—	22	2	no	—
173	C₂×M₁₆	—	22	2	no	—
174	C₂²×C₈	—	11	1	no	—
175	C₂²×C₈	—	11	1	no	—
176	C₂²×C₈	—	11	2	no	—
177	C₂²×C₈	—	11	2	no	—
178	C₂²×C₈	—	22	1	no	—
179	C₂²×C₈	—	22	1	no	—
180	C₂²×C₈	—	22	1	no	—
181	C₂²×C₈	—	22	1	no	—
182	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
184	#117	—	22	2	no	—
185	#117	—	22	2	no	—
186	C₂×C₄⋊C₄	—	11	1	no	—
187	C₂×C₄⋊C₄	—	11	1	no	—
188	C₂×C₂²⋊C₄	—	11	1	no	—
189	C₂×C₂²⋊C₄	—	11	1	no	—
190	C₂×C₄²	—	11	1	no	—
191	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
192	C₂×C₁₆	—	2	2	no	—
193	C₂×C₁₆	—	2	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	22	2	no	—
196	C₂×C₁₆	—	22	2	no	—
197	C₂×C₁₆	—	22	2	no	—
198	#97	—	22	2	no	—
199	#97	—	22	2	no	—
200	C₄×C₈	—	22	1	no	—
201	C₄×C₈	—	22	1	no	—
202	C₄×C₈	D₄₄	1	2	yes	—
203	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
204	C₂×C₁₁	#1797	1	1	yes	characteristic
205	C₂×C₁₁	#2113	1	1	yes	characteristic
206	C₂×C₁₁	#2113	1	1	yes	characteristic
207	C₂×C₁₁	#2151	1	2	yes	—
208	C₂×C₁₁	#2164	1	2	yes	—
209	D₁₁	—	4	2	no	—
210	C₂⁴	—	11	1	no	—
211	C₂²×C₄	—	11	1	no	—
212	C₂²×C₄	—	11	2	no	—
213	C₂²×C₄	—	11	2	no	—
214	C₂²×C₄	—	11	1	no	—
215	C₂²×C₄	—	11	1	no	—
216	C₂²×C₄	—	22	1	no	—
217	C₂²×C₄	—	22	1	no	—
218	C₂²×C₄	—	22	1	no	—
219	C₂²×C₄	—	22	1	no	—
220	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
221	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
223	M₁₆	—	22	8	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	11	2	no	—
227	C₂×C₈	—	11	2	no	—
228	C₂×C₈	—	22	1	no	—
229	C₂×C₈	—	22	1	no	—
230	C₂×C₈	—	22	1	no	—
231	C₂×C₈	—	22	1	no	—
232	C₂×C₈	—	22	2	no	—
233	C₂×C₈	—	22	2	no	—
234	C₂×C₈	—	22	2	no	—
235	C₂×C₈	—	22	2	no	—
236	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
237	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
238	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
239	C₂×C₈	#4341	1	1	yes	characteristic
240	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
241	C₂×C₈	#4343	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
243	C₄⋊C₄	—	22	2	no	—
244	C₄⋊C₄	—	22	2	no	—
245	C₂²⋊C₄	—	22	2	no	—
246	C₂²⋊C₄	—	22	2	no	—
247	C₄²	—	22	1	no	—
248	C₄²	—	22	1	no	—
249	C₄²	#4313	1	2	yes	—
250	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
251	C₁₆	—	4	2	no	—
252	C₁₆	—	44	2	no	—
253	C₁₁	#12784	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	11	2	no	—
255	C₂³	—	22	1	no	—
256	C₂³	—	22	2	no	—
257	C₂³	—	22	2	no	—
258	C₂³	—	22	1	no	—
259	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	11	2	no	—
263	C₂×C₄	—	22	1	no	—
264	C₂×C₄	—	22	1	no	—
265	C₂×C₄	—	22	2	no	—
266	C₂×C₄	—	22	2	no	—
267	C₂×C₄	—	22	2	no	—
268	C₂×C₄	—	22	2	no	—
269	C₂×C₄	—	22	1	no	—
270	C₂×C₄	—	22	1	no	—
271	C₂×C₄	—	22	2	no	—
272	C₂×C₄	—	22	2	no	—
273	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
274	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
276	C₂×C₄	#66945	1	1	yes	characteristic
277	C₂×C₄	C₂×#4313	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
279	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
280	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	22	2	no	—
283	C₈	—	22	2	no	—
284	C₈	#66948	1	1	yes	characteristic
285	C₈	#66950	1	1	yes	characteristic
286	C₈	#66991	1	1	yes	characteristic
287	C₈	#66999	1	1	yes	characteristic
288	C₂²	—	22	4	no	—
289	C₂²	—	22	2	no	—
290	C₂²	—	44	2	no	—
291	C₂²	—	44	1	no	—
292	C₂²	—	44	1	no	—
293	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
294	C₂²	#10622446	1	1	yes	characteristic
295	C₂²	#10622514	1	2	yes	—
296	C₂²	#10622520	1	2	yes	—
297	C₂²	C₂×(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	22	2	no	—
301	C₄	—	44	1	no	—
302	C₄	—	44	1	no	—
303	C₄	#10622448	1	1	yes	characteristic
304	C₄	#10622450	1	1	yes	characteristic
305	C₄	#10622600	1	1	yes	characteristic
306	C₄	#10622739	1	1	yes	characteristic
307	C₂	—	44	2	no	—
308	C₂	#13068593	1	1	yes	characteristic
309	C₂	#13070229	1	1	yes	characteristic
310	C₂	#13070231	1	1	yes	characteristic
311	C₂	#13070373	1	2	yes	—
312	C₂	#13070422	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator, socle	C₂³×C₁₁	C₂²×C₈
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
prime radical	C₂³×D₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	#12784
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
4. #13066693	C₂	semidirect
5. C₂×C₁₁×C₄⋊C₁₆	C₂	semidirect
100. C₂×C₄⋊C₁₆	D₁₁	semidirect
253. 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×C₄⋊(C₁₁⋊C₁₆)	2
C₂×(C₂×D₁₁)⋊C₁₆	2
#13066693	2
C₂×C₁₁×C₄⋊C₁₆	2
#12784	11