Subgroups — #14907347

Order 2816.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070525	C₂	1	2	yes	maximal, complemented
2	#13070442	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#13070419	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070419	C₂	1	2	yes	maximal, complemented
6	C₄×#66876	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10622099	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₁×#41	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₁×#41	C₄	1	1	yes	characteristic
10	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₁₁	C₄	1	2	yes	—
13	C₈×D₄×C₁₁	C₄	1	1	yes	characteristic
14	C₈×D₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₁₁	C₂²	1	2	yes	—
16	C₁₁×#431	C₄	1	1	yes	characteristic
17	C₁₁×#431	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
20	C₂×#66876	C₂²	1	2	yes	—
21	C₂²×C₁₁⋊C₁₆	C₂²	1	4	yes	—
22	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
23	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
24	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
25	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
26	#10622099	C₂²	1	2	yes	—
27	#10622099	C₂²	1	1	yes	characteristic
28	#10622099	C₂²	1	4	yes	—
29	#10622099	C₂²	1	1	yes	characteristic
30	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
31	C₂×C₁₁×#41	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₁	C₈	1	1	yes	characteristic
37	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₁₁	C₈	1	2	yes	—
39	C₄×D₄×C₁₁	C₈	1	1	yes	characteristic
40	C₄×D₄×C₁₁	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₁₁×#117	C₈	1	1	yes	characteristic
43	C₁₁×#117	C₈	1	2	yes	—
44	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₁₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
51	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
52	C₁₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₁	C₂³	1	2	yes	—
60	#66876	—	2	4	no	—
61	C₂×C₁₁⋊C₁₆	—	2	8	no	—
62	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
64	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
65	#33484	—	11	1	no	maximal, Sylow
66	C₁₁×#41	—	2	4	no	—
67	C₂×Q₈×C₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₁₁	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₁	—	2	4	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	2	4	no	—
79	C₂×C₈×C₁₁	—	2	2	no	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
86	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
88	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
89	C₁₁×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
90	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
91	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
92	C₁₁×C₂²⋊C₄	C₂×C₈	1	2	yes	—
93	C₄²×C₁₁	C₂×C₈	1	2	yes	—
94	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₁₁	C₂×C₈	1	2	yes	—
96	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₁⋊C₁₆	—	2	4	no	—
101	C₁₁⋊C₁₆	—	2	2	no	—
102	C₁₁⋊C₁₆	#41	1	4	yes	complemented
103	C₈×#41	D₁₁	1	1	yes	characteristic
104	#2183	—	11	2	no	—
105	#2168	—	11	4	no	—
106	D₄×C₁₆	—	11	2	no	—
107	#2163	—	11	1	no	—
108	#2163	—	11	2	no	—
109	C₄×M₃₂	—	11	1	no	—
110	C₂×#346	—	11	2	no	—
111	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
112	C₂³×C₁₁	C₂²×C₈	1	2	yes	—
113	Q₈×C₁₁	—	2	2	no	—
114	D₄×C₁₁	—	2	4	no	—
115	D₄×C₁₁	M₃₂	1	4	yes	complemented
116	C₂×C₄×C₁₁	—	2	4	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	—	2	1	no	—
119	C₂×C₄×C₁₁	—	2	2	no	—
120	C₂×C₄×C₁₁	—	2	1	no	—
121	C₂×C₄×C₁₁	—	2	2	no	—
122	C₂×C₄×C₁₁	—	4	2	no	—
123	C₂×C₄×C₁₁	—	4	1	no	—
124	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₁₁	—	2	2	no	—
134	C₈×C₁₁	—	2	1	no	—
135	C₈×C₁₁	—	2	1	no	—
136	C₈×C₁₁	—	2	2	no	—
137	C₈×C₁₁	#131	1	2	yes	—
138	C₈×C₁₁	C₂×#41	1	2	yes	—
139	C₄×#41	Dic₁₁	1	1	yes	characteristic
140	C₂×M₃₂	—	11	2	no	—
141	C₂²×C₁₆	—	11	4	no	—
142	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
143	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
144	C₈×D₄	Dic₁₁	1	2	yes	—
145	C₈×D₄	Dic₁₁	1	1	yes	characteristic
146	C₈×D₄	C₂×D₁₁	1	1	yes	characteristic
147	C₈×D₄	C₂×D₁₁	1	2	yes	—
148	#431	Dic₁₁	1	1	yes	characteristic
149	#431	Dic₁₁	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
151	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
152	C₄⋊C₁₆	—	11	2	no	—
153	C₄⋊C₁₆	—	11	4	no	—
154	C₂²⋊C₁₆	—	11	4	no	—
155	C₂²⋊C₁₆	—	11	2	no	—
156	#346	—	11	1	no	—
157	#346	—	11	2	no	—
158	#346	—	11	4	no	—
159	#346	—	11	1	no	—
160	C₄×C₁₆	—	11	2	no	—
161	C₂²×C₁₁	—	2	2	no	—
162	C₂²×C₁₁	—	2	2	no	—
163	C₂²×C₁₁	—	2	1	no	—
164	C₂²×C₁₁	—	2	1	no	—
165	C₂²×C₁₁	—	2	1	no	—
166	C₂²×C₁₁	C₂×M₃₂	1	4	yes	—
167	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₁₁	—	2	1	no	—
169	C₄×C₁₁	—	2	2	no	—
170	C₄×C₁₁	—	2	1	no	—
171	C₄×C₁₁	—	2	2	no	—
172	C₄×C₁₁	—	2	1	no	—
173	C₄×C₁₁	—	2	2	no	—
174	C₄×C₁₁	—	2	1	no	—
175	C₄×C₁₁	C₂×M₃₂	1	2	yes	—
176	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
177	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
178	C₂×#41	C₁₁⋊C₈	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
180	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
182	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
183	C₄×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
184	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
185	C₄×D₄	C₁₁⋊C₈	1	2	yes	—
186	C₄×D₄	C₁₁⋊C₈	1	1	yes	characteristic
187	C₄×D₄	C₂×Dic₁₁	1	2	yes	—
188	C₄×D₄	C₂×Dic₁₁	1	1	yes	characteristic
189	#117	C₁₁⋊C₈	1	1	yes	characteristic
190	#117	C₁₁⋊C₈	1	2	yes	—
191	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
192	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
193	M₃₂	—	22	4	no	—
194	C₂×C₁₆	—	11	2	no	—
195	C₂×C₁₆	—	11	2	no	—
196	C₂×C₁₆	—	11	4	no	—
197	C₂×C₁₆	—	22	8	no	—
198	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
199	C₄⋊C₈	C₂×Dic₁₁	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₁₁	1	1	yes	characteristic
201	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
202	C₂²⋊C₈	C₂×Dic₁₁	1	2	yes	—
203	C₂²⋊C₈	C₂×Dic₁₁	1	2	yes	—
204	C₂²⋊C₈	C₂×Dic₁₁	1	2	yes	—
205	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
206	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
207	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
208	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₁₁	1	2	yes	—
212	C₂×C₁₁	—	2	4	no	—
213	C₂×C₁₁	—	4	1	no	—
214	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
215	C₂×C₁₁	C₂²×M₃₂	1	1	yes	characteristic
216	C₂×C₁₁	#3407	1	1	yes	characteristic
217	#41	—	2	4	no	—
218	C₂×Q₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
219	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
220	C₂×D₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
222	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
232	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
233	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
234	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
237	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
238	C₄⋊C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
239	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
240	C₄⋊C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
241	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
242	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
243	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
244	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
245	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
246	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
247	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
248	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
249	C₄²	C₂²×Dic₁₁	1	2	yes	—
250	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
251	C₁₆	—	11	4	no	—
252	C₁₆	—	22	4	no	—
253	C₁₆	—	22	2	no	—
254	C₁₁	#33484	1	1	yes	characteristic, complemented, Sylow
255	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
256	C₂³	C₂²×C₁₁⋊C₈	1	2	yes	—
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	#66876	1	4	yes	complemented
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
271	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
273	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
275	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	1	no	—
281	C₈	#67001	1	2	yes	—
282	C₈	#41×D₁₁	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×#66876	1	4	yes	—
289	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	C₂×#66876	1	2	yes	—
298	C₄	C₂×#67001	1	1	yes	characteristic
299	C₄	#41×Dic₁₁	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	4	1	no	—
302	C₂	#41×C₁₁⋊C₈	1	1	yes	characteristic
303	C₂	C₂²×#66876	1	1	yes	characteristic
304	C₂	#13081719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₈×C₁₁×#41	C₂
prime radical	C₂×C₁₁×#41	C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₈×#41	D₁₁
hypocenter	C₁₁	#33484
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #13070525	C₂	semidirect
2. #13070442	C₂	semidirect
3. D₄×C₁₁⋊C₁₆	C₂	semidirect
4. #13070419	C₂	semidirect
5. #13070419	C₂	semidirect
6. C₄×#66876	C₂	semidirect
7. C₂×#10622099	C₂	semidirect
102. C₁₁⋊C₁₆	#41	semidirect
115. D₄×C₁₁	M₃₂	semidirect
254. C₁₁	#33484	semidirect
259. D₄	#66876	semidirect

Maximal subgroups

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

subgroup	index
#13070525	2
#13070442	2
D₄×C₁₁⋊C₁₆	2
#13070419	2
C₄×#66876	2
C₂×#10622099	2
C₈×C₁₁×#41	2
#33484	11