Subgroups — #14653899

Order 2816.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068768	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13066499	C₂	1	4	yes	maximal
4	C₁₁×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₁₁×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₁₁×C₄⋊Q₈	C₄	1	2	yes	—
8	C₁₁×#384	—	2	1	no	—
9	C₁₁×#384	—	2	1	no	—
10	C₁₁×#384	C₄	1	2	yes	—
11	C₁₁×#384	C₂²	1	2	yes	—
12	C₄³×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
14	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
15	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
18	#10622162	—	2	1	no	—
19	C₂²×Q₈×C₁₁	—	2	2	no	—
20	C₁₁×C₄⋊Q₈	—	4	4	no	—
21	C₁₁×C₄⋊Q₈	—	4	2	no	—
22	C₁₁×C₄⋊Q₈	—	4	2	no	—
23	C₁₁×C₄⋊Q₈	—	4	2	no	—
24	C₁₁×C₄⋊Q₈	—	4	2	no	—
25	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²×C₁₁⋊C₈	—	2	2	no	—
39	C₂²×C₁₁⋊C₈	—	2	2	no	—
40	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
42	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
43	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
44	C₄⋊(C₁₁⋊C₈)	Q₈	1	4	yes	complemented
45	C₄×C₁₁⋊C₈	—	4	2	no	—
46	#9898	—	11	1	no	maximal, Sylow
47	C₂×Q₈×C₁₁	—	4	4	no	—
48	C₂×Q₈×C₁₁	—	4	4	no	—
49	C₂×Q₈×C₁₁	—	4	4	no	—
50	C₂²×C₄×C₁₁	—	2	4	no	—
51	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₁₁×C₄⋊C₄	—	4	4	no	—
59	C₁₁×C₄⋊C₄	—	4	4	no	—
60	C₁₁×C₄⋊C₄	—	4	4	no	—
61	C₁₁×C₄⋊C₄	—	4	4	no	—
62	C₁₁×C₄⋊C₄	—	4	2	no	—
63	C₁₁×C₄⋊C₄	—	4	2	no	—
64	C₁₁×C₄⋊C₄	—	4	2	no	—
65	C₁₁×C₄⋊C₄	—	4	2	no	—
66	C₄²×C₁₁	—	2	2	no	—
67	C₄²×C₁₁	—	2	2	no	—
68	C₄²×C₁₁	—	2	2	no	—
69	C₄²×C₁₁	D₈	1	2	yes	—
70	C₄²×C₁₁	D₈	1	2	yes	—
71	C₄²×C₁₁	SD₁₆	1	2	yes	—
72	C₄²×C₁₁	SD₁₆	1	2	yes	—
73	C₄²×C₁₁	SD₁₆	1	2	yes	—
74	C₄²×C₁₁	Q₁₆	1	2	yes	—
75	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₁	#41	1	2	yes	—
77	C₂×C₁₁⋊C₈	—	2	4	no	—
78	C₂×C₁₁⋊C₈	—	4	2	no	—
79	C₂×C₁₁⋊C₈	—	4	2	no	—
80	C₂×C₁₁⋊C₈	—	4	2	no	—
81	C₂×C₁₁⋊C₈	—	4	2	no	—
82	C₂×C₁₁⋊C₈	—	4	2	no	—
83	#2868	D₁₁	1	1	yes	characteristic
84	#1842	—	11	1	no	—
85	C₄×C₄⋊C₈	—	11	1	no	—
86	#1296	—	11	4	no	—
87	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	Q₈×C₁₁	—	8	8	no	—
89	C₂×C₄×C₁₁	—	2	1	no	—
90	C₂×C₄×C₁₁	—	2	1	no	—
91	C₂×C₄×C₁₁	—	2	1	no	—
92	C₂×C₄×C₁₁	—	2	1	no	—
93	C₂×C₄×C₁₁	—	2	1	no	—
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	4	4	no	—
96	C₂×C₄×C₁₁	—	4	4	no	—
97	C₂×C₄×C₁₁	—	4	4	no	—
98	C₂×C₄×C₁₁	#100	1	2	yes	—
99	C₂×C₄×C₁₁	#101	1	2	yes	—
100	C₂×C₄×C₁₁	#102	1	2	yes	—
101	C₂×C₄×C₁₁	#102	1	2	yes	—
102	C₂×C₄×C₁₁	#102	1	2	yes	—
103	C₂×C₄×C₁₁	#102	1	2	yes	—
104	C₂×C₄×C₁₁	#103	1	2	yes	—
105	C₂×C₄×C₁₁	#103	1	2	yes	—
106	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#137	1	2	yes	—
117	C₁₁⋊C₈	—	4	4	no	—
118	C₁₁⋊C₈	—	8	2	no	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×C₄⋊Q₈	—	2	1	no	—
121	C₂×C₄⋊Q₈	Dic₁₁	1	2	yes	—
122	C₂×C₄⋊C₈	—	11	2	no	—
123	C₂×C₄⋊C₈	—	11	2	no	—
124	C₂×C₄⋊C₈	—	22	1	no	—
125	C₂×C₄⋊C₈	—	22	1	no	—
126	C₂×C₄×C₈	—	22	1	no	—
127	#384	—	2	1	no	—
128	#384	—	2	1	no	—
129	#384	Dic₁₁	1	2	yes	—
130	#384	C₂×D₁₁	1	2	yes	—
131	C₄³	C₂×D₁₁	1	1	yes	characteristic
132	#336	—	22	1	no	—
133	C₂²×C₁₁	#391	1	1	yes	characteristic
134	C₂²×C₁₁	C₂×#100	1	1	yes	characteristic
135	C₂²×C₁₁	C₂×#101	1	1	yes	characteristic
136	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
137	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
138	C₂²×C₁₁	C₂×#103	1	1	yes	characteristic
139	C₂²×C₁₁	#419	1	1	yes	characteristic
140	C₄×C₁₁	—	2	2	no	—
141	C₄×C₁₁	—	2	2	no	—
142	C₄×C₁₁	—	2	2	no	—
143	C₄×C₁₁	—	8	4	no	—
144	C₄×C₁₁	#331	1	4	yes	—
145	C₄×C₁₁	#332	1	4	yes	—
146	C₄×C₁₁	#474	1	2	yes	—
147	C₄×C₁₁	#475	1	2	yes	—
148	C₄×C₁₁	#486	1	1	yes	characteristic
149	C₄×C₁₁	#487	1	1	yes	characteristic
150	C₄×C₁₁	#498	1	1	yes	characteristic
151	C₄×C₁₁	#500	1	1	yes	characteristic
152	C₂²×Q₈	—	2	2	no	—
153	C₂²×C₈	—	22	2	no	—
154	C₂²×C₈	—	22	2	no	—
155	C₄⋊Q₈	—	4	4	no	—
156	C₄⋊Q₈	—	4	2	no	—
157	C₄⋊Q₈	—	4	2	no	—
158	C₄⋊Q₈	—	4	2	no	—
159	C₄⋊Q₈	—	4	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₂×Dic₁₁	1	4	yes	—
166	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
173	C₄⋊C₈	—	11	4	no	—
174	C₄⋊C₈	—	22	2	no	—
175	C₄⋊C₈	—	22	4	no	—
176	C₄⋊C₈	—	44	2	no	—
177	C₄⋊C₈	—	44	2	no	—
178	C₄×C₈	—	44	2	no	—
179	C₂×C₁₁	C₂×#331	1	1	yes	characteristic
180	C₂×C₁₁	C₂×#332	1	1	yes	characteristic
181	C₂×C₁₁	#1546	1	1	yes	characteristic
182	C₂×C₁₁	#1548	1	1	yes	characteristic
183	C₂×C₁₁	#1988	1	1	yes	characteristic
184	C₂×C₁₁	#1990	1	1	yes	characteristic
185	C₂×C₁₁	#1997	1	1	yes	characteristic
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
197	C₂×C₈	—	22	4	no	—
198	C₂×C₈	—	44	2	no	—
199	C₂×C₈	—	44	2	no	—
200	C₂×C₈	—	44	2	no	—
201	C₂×C₈	—	44	2	no	—
202	C₂×C₈	—	44	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	#4323	1	2	yes	—
215	C₄²	#4323	1	2	yes	—
216	C₄²	#4324	1	2	yes	—
217	C₄²	#4324	1	2	yes	—
218	C₄²	#4325	1	2	yes	—
219	C₄²	#4326	1	2	yes	—
220	C₄²	Q₈×D₁₁	1	2	yes	—
221	C₄²	#4343	1	2	yes	—
222	C₁₁	#9898	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
224	Q₈	—	8	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#66896	1	2	yes	—
235	C₂×C₄	#66896	1	2	yes	—
236	C₂×C₄	#66896	1	2	yes	—
237	C₂×C₄	#66896	1	2	yes	—
238	C₂×C₄	#66897	1	2	yes	—
239	C₂×C₄	#66899	1	2	yes	—
240	C₂×C₄	#66899	1	2	yes	—
241	C₂×C₄	#66900	1	2	yes	—
242	C₂×C₄	C₂×#4323	1	1	yes	characteristic
243	C₂×C₄	C₂×#4323	1	1	yes	characteristic
244	C₂×C₄	C₂×#4324	1	1	yes	characteristic
245	C₂×C₄	C₂×#4324	1	1	yes	characteristic
246	C₂×C₄	C₂×#4325	1	1	yes	characteristic
247	C₂×C₄	#66995	1	2	yes	—
248	C₂×C₄	C₂×#4326	1	1	yes	characteristic
249	C₂×C₄	#66997	1	1	yes	characteristic
250	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
251	C₂×C₄	#66999	1	2	yes	—
252	C₂×C₄	#67000	1	1	yes	characteristic
253	C₈	—	44	4	no	—
254	C₈	—	88	2	no	—
255	C₂²	C₂×#66896	1	1	yes	characteristic
256	C₂²	C₂×#66896	1	1	yes	characteristic
257	C₂²	C₂×#66897	1	1	yes	characteristic
258	C₂²	C₂×#66899	1	1	yes	characteristic
259	C₂²	#10622838	1	1	yes	characteristic
260	C₂²	C₂×#66900	1	1	yes	characteristic
261	C₂²	#10622843	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	4	no	—
266	C₄	#10622182	1	4	yes	—
267	C₄	#10622185	1	4	yes	—
268	C₄	#10622691	1	1	yes	characteristic
269	C₄	#10622692	1	1	yes	characteristic
270	C₄	#10622700	1	2	yes	—
271	C₄	#10622701	1	1	yes	characteristic
272	C₄	#10622706	1	1	yes	characteristic
273	C₄	#10622707	1	2	yes	—
274	C₂	C₂×#10622182	1	1	yes	characteristic
275	C₂	C₂×#10622185	1	1	yes	characteristic
276	C₂	#13067288	1	1	yes	characteristic
277	C₂	#13067297	1	1	yes	characteristic
278	C₂	#13069569	1	1	yes	characteristic
279	C₂	#13069578	1	1	yes	characteristic
280	C₂	#13069606	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₁
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#2868	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#2868	D₁₁
hypocenter	C₁₁	#9898
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
44. C₄⋊(C₁₁⋊C₈)	Q₈	semidirect
222. C₁₁	#9898	semidirect

Maximal subgroups

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

subgroup	index
#13068768	2
C₄×C₄⋊(C₁₁⋊C₈)	2
#13066499	2
C₁₁×#2868	2
#9898	11