Subgroups — #14646852

Order 2816.

438 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068744	C₂	1	1	yes	maximal, characteristic
2	#13068621	C₂	1	2	yes	maximal
3	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
4	#13066500	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	—	2	2	no	—
8	C₈×Q₈×C₁₁	C₄	1	4	yes	complemented
9	C₂×C₁₁×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	—	2	1	no	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66899	—	2	2	no	—
14	#10622548	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
16	#10622189	—	2	1	no	—
17	#10622189	—	2	1	no	—
18	#10622167	—	2	2	no	—
19	#10622096	—	2	2	no	—
20	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	—	2	1	no	—
22	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₁	—	2	1	no	—
25	C₄×Q₈×C₁₁	—	2	1	no	—
26	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
27	C₄×Q₈×C₁₁	Q₈	1	2	yes	—
28	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	—	2	1	no	—
31	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₁	—	2	2	no	—
33	C₂×C₄⋊Dic₁₁	—	2	2	no	—
34	C₂×C₄⋊Dic₁₁	—	2	2	no	—
35	C₂²×C₁₁⋊C₈	—	2	2	no	—
36	C₁₁×C₄⋊C₈	—	2	4	no	—
37	C₁₁×C₄⋊C₈	—	2	1	no	—
38	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₁	—	2	4	no	—
40	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₁	D₄	1	2	yes	—
42	#66899	—	4	4	no	—
43	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
45	#8649	—	11	1	no	maximal, Sylow
46	C₂×Q₈×C₁₁	—	2	1	no	—
47	C₂×Q₈×C₁₁	—	2	1	no	—
48	C₂×Q₈×C₁₁	—	2	1	no	—
49	C₂×Q₈×C₁₁	C₄⋊C₄	1	2	yes	—
50	C₂×Q₈×C₁₁	D₈	1	2	yes	—
51	C₂×Q₈×C₁₁	Q₁₆	1	2	yes	—
52	C₂²×C₄×C₁₁	—	2	1	no	—
53	C₂²×C₄×C₁₁	—	2	1	no	—
54	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
57	C₂²×Dic₁₁	—	4	2	no	—
58	C₂×C₈×C₁₁	—	2	2	no	—
59	C₂×C₈×C₁₁	—	2	2	no	—
60	C₂×C₈×C₁₁	—	2	2	no	—
61	C₂×C₈×C₁₁	—	4	2	no	—
62	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
64	C₁₁×C₄⋊C₄	—	2	2	no	—
65	C₁₁×C₄⋊C₄	—	2	1	no	—
66	C₁₁×C₄⋊C₄	—	2	2	no	—
67	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
68	C₄²×C₁₁	—	2	2	no	—
69	C₄²×C₁₁	—	2	2	no	—
70	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₁₁	#41	1	1	yes	characteristic
74	C₄⋊Dic₁₁	—	4	2	no	—
75	C₄⋊Dic₁₁	—	4	2	no	—
76	C₄⋊Dic₁₁	—	4	2	no	—
77	C₄⋊Dic₁₁	—	4	2	no	—
78	C₄⋊Dic₁₁	—	8	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×C₈×Q₈	D₁₁	1	1	yes	characteristic
83	#1834	—	11	1	no	—
84	#1803	—	11	2	no	—
85	C₂×#335	—	11	1	no	—
86	#1296	—	11	2	no	—
87	C₂³×C₁₁	#121	1	1	yes	characteristic
88	Q₈×C₁₁	—	2	2	no	—
89	Q₈×C₁₁	—	2	2	no	—
90	Q₈×C₁₁	—	2	2	no	—
91	Q₈×C₁₁	#107	1	4	yes	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	2	no	—
95	C₂×C₄×C₁₁	—	2	2	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	—	4	2	no	—
98	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
107	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
108	C₂×C₄×C₁₁	#135	1	2	yes	—
109	C₂×C₄×C₁₁	#137	1	2	yes	—
110	C₂×Dic₁₁	—	8	2	no	—
111	C₂×Dic₁₁	—	8	2	no	—
112	C₂×Dic₁₁	—	8	2	no	—
113	C₈×C₁₁	—	2	2	no	—
114	C₈×C₁₁	—	2	2	no	—
115	C₈×C₁₁	—	4	2	no	—
116	C₁₁⋊C₈	—	8	2	no	—
117	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
118	C₈×Q₈	—	2	2	no	—
119	C₈×Q₈	Dic₁₁	1	4	yes	complemented
120	C₂×C₄⋊C₈	—	2	1	no	—
121	C₂×C₄⋊C₈	—	11	2	no	—
122	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
123	C₂×#103	—	22	2	no	—
124	C₂×C₄×C₈	—	2	1	no	—
125	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
126	#384	—	11	2	no	—
127	#340	—	22	1	no	—
128	#340	—	22	1	no	—
129	#340	—	22	2	no	—
130	#335	—	22	2	no	—
131	C₂²×C₁₁	#380	1	1	yes	characteristic
132	C₂²×C₁₁	#464	1	2	yes	—
133	C₂²×C₁₁	#466	1	1	yes	characteristic
134	C₂²×C₁₁	#467	1	1	yes	characteristic
135	C₂²×C₁₁	#471	1	2	yes	—
136	C₄×C₁₁	—	2	4	no	—
137	C₄×C₁₁	—	2	1	no	—
138	C₄×C₁₁	—	2	1	no	—
139	C₄×C₁₁	—	4	2	no	—
140	C₄×C₁₁	#416	1	1	yes	characteristic
141	C₄×C₁₁	#419	1	1	yes	characteristic
142	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
143	C₄×C₁₁	#427	1	1	yes	characteristic
144	C₄×C₁₁	#466	1	1	yes	characteristic
145	C₄×C₁₁	#467	1	1	yes	characteristic
146	C₄×C₁₁	#479	1	2	yes	—
147	Dic₁₁	—	16	2	no	—
148	C₂²×Q₈	D₄₄	1	1	yes	characteristic
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	22	2	no	—
151	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	Dic₂₂	1	2	yes	—
156	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	22	2	no	—
159	C₂×C₄⋊C₄	—	22	2	no	—
160	C₂×C₄⋊C₄	—	22	2	no	—
161	C₂×C₄⋊C₄	D₄₄	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	1	no	—
166	C₄⋊C₈	—	22	2	no	—
167	C₄⋊C₈	—	22	2	no	—
168	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
169	#103	—	44	4	no	—
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
172	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
173	C₂×C₁₁	#1682	1	2	yes	—
174	C₂×C₁₁	#1684	1	2	yes	—
175	C₂×C₁₁	#1805	1	1	yes	characteristic
176	C₂×C₁₁	#1808	1	1	yes	characteristic
177	C₂×C₁₁	#1815	1	1	yes	characteristic
178	C₂×Q₈	—	2	1	no	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	D₈₈	1	2	yes	—
182	C₂×Q₈	Dic₄₄	1	2	yes	—
183	C₂×Q₈	C₄⋊Dic₁₁	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	44	2	no	—
187	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
188	C₂²×C₄	#4339	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	44	2	no	—
195	C₂×C₈	—	44	2	no	—
196	C₂×C₈	—	44	2	no	—
197	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	44	2	no	—
203	C₄⋊C₄	—	44	2	no	—
204	C₄⋊C₄	—	44	2	no	—
205	C₄⋊C₄	—	44	2	no	—
206	C₄⋊C₄	—	88	2	no	—
207	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
211	C₄²	#4339	1	1	yes	characteristic
212	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
214	C₁₁	#8649	1	1	yes	characteristic, complemented, Sylow
215	C₂³	#66983	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	#66882	1	4	yes	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	88	2	no	—
227	C₂×C₄	—	88	2	no	—
228	C₂×C₄	—	88	2	no	—
229	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
230	C₂×C₄	#66957	1	1	yes	characteristic
231	C₂×C₄	#66957	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
233	C₂×C₄	#66977	1	1	yes	characteristic
234	C₂×C₄	#66977	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
236	C₂×C₄	#66979	1	1	yes	characteristic
237	C₂×C₄	#66983	1	1	yes	characteristic
238	C₂×C₄	#66995	1	2	yes	—
239	C₂×C₄	#67003	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	88	2	no	—
245	C₂²	#10622641	1	2	yes	—
246	C₂²	#10622728	1	1	yes	characteristic
247	C₂²	#10622729	1	1	yes	characteristic
248	C₂²	#10622754	1	2	yes	—
249	C₂²	#10622825	1	1	yes	characteristic
250	C₄	—	2	4	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	176	2	no	—
255	C₄	#10622635	1	2	yes	—
256	C₄	C₂×#66882	1	1	yes	characteristic
257	C₄	#10622719	1	1	yes	characteristic
258	C₄	#10622728	1	1	yes	characteristic
259	C₄	#10622729	1	1	yes	characteristic
260	C₄	#10622838	1	1	yes	characteristic
261	C₄	#10622846	1	1	yes	characteristic
262	C₂	#13068060	1	2	yes	—
263	C₂	#13068074	1	2	yes	—
264	C₂	#13068631	1	1	yes	characteristic
265	C₂	#13068645	1	1	yes	characteristic
266	C₂	#13068675	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#66983
commutator	C₄²×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	#121
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	#8649
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₁	C₄	semidirect
119. C₈×Q₈	Dic₁₁	semidirect
214. C₁₁	#8649	semidirect

Maximal subgroups

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

subgroup	index
#13068744	2
#13068621	2
C₂×#10622096	2
#13066500	2
C₂×C₈×Q₈×C₁₁	2
#8649	11