Subgroups — #14764663

Order 2816.

734 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13069553	C₂	1	2	yes	maximal
2	#13069335	C₂	1	2	yes	maximal
3	#13069175	C₂	1	4	yes	maximal
4	#13068759	C₂	1	2	yes	maximal
5	#13068733	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#13068362	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₁×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₄×C₁₁×C₄⋊C₄	C₄	1	2	yes	—
15	C₄×C₁₁×C₄⋊C₄	C₄	1	1	yes	characteristic
16	C₄×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄³×C₁₁	C₂²	1	1	yes	characteristic
19	C₄³×C₁₁	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
21	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
22	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
24	#10622162	C₂²	1	4	yes	—
25	#10622162	C₂²	1	2	yes	—
26	#10622162	C₂²	1	1	yes	characteristic
27	#10622162	C₂²	1	1	yes	characteristic
28	#10622162	C₂²	1	4	yes	—
29	#10622162	C₂²	1	4	yes	—
30	#10622162	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₁	—	2	4	no	—
33	C₄×Q₈×C₁₁	—	2	4	no	—
34	C₄×Q₈×C₁₁	C₈	1	8	yes	complemented
35	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
54	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
55	C₂²×C₁₁⋊C₈	C₂³	1	4	yes	—
56	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
57	C₁₁×#95	C₂×C₄	1	2	yes	—
58	C₁₁×#95	C₂×C₄	1	2	yes	—
59	C₁₁×#95	C₂×C₄	1	2	yes	—
60	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
61	C₄⋊(C₁₁⋊C₈)	—	4	4	no	—
62	C₄×C₁₁⋊C₈	—	2	4	no	—
63	C₄×C₁₁⋊C₈	—	4	2	no	—
64	#22018	—	11	1	no	maximal, Sylow
65	C₂×Q₈×C₁₁	—	2	2	no	—
66	C₂×Q₈×C₁₁	—	2	2	no	—
67	C₂×Q₈×C₁₁	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₁₁×C₄⋊C₄	—	2	4	no	—
80	C₁₁×C₄⋊C₄	—	2	2	no	—
81	C₁₁×C₄⋊C₄	—	2	4	no	—
82	C₁₁×C₄⋊C₄	—	2	2	no	—
83	C₁₁×C₄⋊C₄	C₂×C₈	1	8	yes	—
84	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
85	C₄²×C₁₁	—	2	4	no	—
86	C₄²×C₁₁	—	2	4	no	—
87	C₄²×C₁₁	—	2	1	no	—
88	C₄²×C₁₁	—	2	2	no	—
89	C₄²×C₁₁	—	2	2	no	—
90	C₄²×C₁₁	—	2	1	no	—
91	C₄²×C₁₁	—	2	2	no	—
92	C₄²×C₁₁	—	4	1	no	—
93	C₄²×C₁₁	—	4	1	no	—
94	C₄²×C₁₁	—	4	2	no	—
95	C₄²×C₁₁	—	4	2	no	—
96	C₄²×C₁₁	C₂×C₈	1	8	yes	—
97	C₄²×C₁₁	C₂×C₈	1	4	yes	—
98	C₄²×C₁₁	M₁₆	1	4	yes	—
99	C₄²×C₁₁	#41	1	4	yes	—
100	C₂×C₁₁⋊C₈	—	2	2	no	—
101	C₂×C₁₁⋊C₈	—	2	2	no	—
102	C₂×C₁₁⋊C₈	—	2	4	no	—
103	C₂×C₁₁⋊C₈	—	2	4	no	—
104	C₂×C₁₁⋊C₈	—	2	2	no	—
105	C₂×C₁₁⋊C₈	—	2	2	no	—
106	C₂×C₁₁⋊C₈	—	2	2	no	—
107	C₂×C₁₁⋊C₈	—	2	4	no	—
108	C₂×C₁₁⋊C₈	—	2	2	no	—
109	C₄²×Q₈	D₁₁	1	1	yes	characteristic
110	#1984	—	11	2	no	—
111	#1939	—	11	2	no	—
112	#1916	—	11	4	no	—
113	#1839	—	11	2	no	—
114	#1831	—	11	2	no	—
115	C₄×C₄⋊C₈	—	11	1	no	—
116	#1744	—	11	1	no	—
117	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
118	Q₈×C₁₁	—	2	8	no	—
119	C₂×C₄×C₁₁	—	2	2	no	—
120	C₂×C₄×C₁₁	—	2	1	no	—
121	C₂×C₄×C₁₁	—	2	1	no	—
122	C₂×C₄×C₁₁	—	2	1	no	—
123	C₂×C₄×C₁₁	—	2	2	no	—
124	C₂×C₄×C₁₁	—	2	1	no	—
125	C₂×C₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	1	no	—
129	C₂×C₄×C₁₁	—	2	1	no	—
130	C₂×C₄×C₁₁	—	2	1	no	—
131	C₂×C₄×C₁₁	—	2	2	no	—
132	C₂×C₄×C₁₁	—	2	1	no	—
133	C₂×C₄×C₁₁	—	2	2	no	—
134	C₂×C₄×C₁₁	—	2	2	no	—
135	C₂×C₄×C₁₁	—	2	2	no	—
136	C₂×C₄×C₁₁	—	2	2	no	—
137	C₂×C₄×C₁₁	—	2	2	no	—
138	C₂×C₄×C₁₁	—	2	2	no	—
139	C₂×C₄×C₁₁	#117	1	2	yes	—
140	C₂×C₄×C₁₁	#117	1	2	yes	—
141	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
142	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
143	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
144	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
145	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
146	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
149	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
150	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
152	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
153	C₁₁⋊C₈	—	4	2	no	—
154	C₁₁⋊C₈	—	4	2	no	—
155	C₁₁⋊C₈	—	4	4	no	—
156	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
157	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	11	2	no	—
160	C₂×C₄⋊C₈	—	11	4	no	—
161	C₂×C₄×C₈	—	11	2	no	—
162	C₂×C₄×C₈	—	11	2	no	—
163	C₄×C₄⋊C₄	Dic₁₁	1	2	yes	—
164	C₄×C₄⋊C₄	Dic₁₁	1	1	yes	characteristic
165	C₄×C₄⋊C₄	Dic₁₁	1	2	yes	—
166	C₄×C₄⋊C₄	Dic₁₁	1	1	yes	characteristic
167	C₄×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
168	C₄×C₄⋊C₄	C₂×D₁₁	1	1	yes	characteristic
169	C₄³	C₂×D₁₁	1	2	yes	—
170	C₄³	C₂×D₁₁	1	1	yes	characteristic
171	#336	—	11	1	no	—
172	#336	—	11	2	no	—
173	#336	—	11	4	no	—
174	#336	—	11	1	no	—
175	#336	—	11	4	no	—
176	#336	—	11	4	no	—
177	#336	—	11	2	no	—
178	C₂²×C₁₁	C₂×#117	1	1	yes	characteristic
179	C₂²×C₁₁	#519	1	1	yes	characteristic
180	C₂²×C₁₁	#520	1	1	yes	characteristic
181	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
182	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
183	C₂²×C₁₁	#568	1	1	yes	characteristic
184	C₂²×C₁₁	#568	1	1	yes	characteristic
185	C₄×C₁₁	—	2	4	no	—
186	C₄×C₁₁	—	2	2	no	—
187	C₄×C₁₁	—	2	2	no	—
188	C₄×C₁₁	—	2	2	no	—
189	C₄×C₁₁	—	2	2	no	—
190	C₄×C₁₁	—	2	4	no	—
191	C₄×C₁₁	—	2	2	no	—
192	C₄×C₁₁	—	4	2	no	—
193	C₄×C₁₁	—	4	2	no	—
194	C₄×C₁₁	#431	1	4	yes	—
195	C₄×C₁₁	#541	1	2	yes	—
196	C₄×C₁₁	#542	1	2	yes	—
197	C₂²×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
198	C₂²×C₈	—	11	2	no	—
199	C₂²×C₈	—	11	2	no	—
200	C₂²×C₈	—	11	4	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₄×Q₈	—	2	4	no	—
203	C₄×Q₈	C₁₁⋊C₈	1	8	yes	complemented
204	C₂×C₄⋊C₄	C₂×Dic₁₁	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×Dic₁₁	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×Dic₁₁	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
211	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
212	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
213	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
216	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
219	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
222	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
223	C₄⋊C₈	—	22	4	no	—
224	C₄⋊C₈	—	44	4	no	—
225	C₄×C₈	—	22	4	no	—
226	C₄×C₈	—	44	2	no	—
227	#95	C₂×Dic₁₁	1	2	yes	—
228	#95	C₂×Dic₁₁	1	2	yes	—
229	#95	C₂×Dic₁₁	1	2	yes	—
230	C₂×C₁₁	#2320	1	1	yes	characteristic
231	C₂×C₁₁	C₂×#431	1	1	yes	characteristic
232	C₂×C₁₁	#2923	1	1	yes	characteristic
233	C₂×C₁₁	#2982	1	1	yes	characteristic
234	C₂×C₁₁	#2983	1	1	yes	characteristic
235	C₂×C₁₁	#2988	1	1	yes	characteristic
236	C₂×C₁₁	#2989	1	1	yes	characteristic
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	C₂×C₁₁⋊C₈	1	4	yes	—
240	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
250	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
251	C₂×C₈	—	22	2	no	—
252	C₂×C₈	—	22	2	no	—
253	C₂×C₈	—	22	4	no	—
254	C₂×C₈	—	22	2	no	—
255	C₂×C₈	—	22	4	no	—
256	C₂×C₈	—	22	2	no	—
257	C₂×C₈	—	22	4	no	—
258	C₂×C₈	—	22	2	no	—
259	C₂×C₈	—	22	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	C₂×C₁₁⋊C₈	1	8	yes	—
265	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	C₂×C₁₁⋊C₈	1	8	yes	—
278	C₄²	C₂×C₁₁⋊C₈	1	4	yes	—
279	C₄²	#4318	1	4	yes	—
280	C₄²	#4339	1	4	yes	—
281	C₁₁	#22018	1	1	yes	characteristic, complemented, Sylow
282	C₂³	C₂³×Dic₁₁	1	1	yes	characteristic
283	Q₈	—	2	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
305	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
306	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
307	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
308	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
309	C₂×C₄	C₂×#4318	1	2	yes	—
310	C₂×C₄	C₂×#4318	1	2	yes	—
311	C₂×C₄	C₂×#4318	1	2	yes	—
312	C₂×C₄	#66979	1	2	yes	—
313	C₂×C₄	#66979	1	2	yes	—
314	C₂×C₄	C₂×#4339	1	2	yes	—
315	C₂×C₄	#67040	1	2	yes	—
316	C₂×C₄	#67042	1	1	yes	characteristic
317	C₂×C₄	#67043	1	1	yes	characteristic
318	C₈	—	44	2	no	—
319	C₈	—	44	4	no	—
320	C₈	—	44	2	no	—
321	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
322	C₂²	C₂²×#4318	1	1	yes	characteristic
323	C₂²	C₂×#66979	1	1	yes	characteristic
324	C₂²	#10623343	1	1	yes	characteristic
325	C₂²	#10623350	1	1	yes	characteristic
326	C₂²	#10623350	1	1	yes	characteristic
327	C₂²	#10623358	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	4	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	#10622538	1	4	yes	—
338	C₄	#10623113	1	2	yes	—
339	C₄	#10623114	1	2	yes	—
340	C₂	#13071267	1	1	yes	characteristic
341	C₂	C₂×#10622538	1	1	yes	characteristic
342	C₂	#13077267	1	1	yes	characteristic
343	C₂	#13077697	1	1	yes	characteristic
344	C₂	#13077701	1	1	yes	characteristic
345	C₂	#13077727	1	1	yes	characteristic
346	C₂	#13077733	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₁
commutator	C₂²×C₁₁	C₂³×C₈
Fitting	C₄²×Q₈×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	C₂³×C₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄²×Q₈	D₁₁
hypocenter	C₁₁	#22018
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
34. C₄×Q₈×C₁₁	C₈	semidirect
203. C₄×Q₈	C₁₁⋊C₈	semidirect
281. C₁₁	#22018	semidirect

Maximal subgroups

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

subgroup	index
#13069553	2
#13069335	2
#13069175	2
#13068759	2
#13068733	2
C₄×C₄⋊(C₁₁⋊C₈)	2
#13068362	2
C₄²×Q₈×C₁₁	2
#22018	11