Subgroups — #14646883

Order 2816.

418 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622292	C₂	1	1	yes	maximal, characteristic
2	#13068744	C₂	1	1	yes	maximal, characteristic
3	#13068622	C₂	1	2	yes	maximal
4	#13066500	C₂	1	2	yes	maximal
5	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×#66880	—	2	2	no	—
10	C₂×#66879	—	2	1	no	—
11	C₂×Dic₁₁⋊C₈	—	2	1	no	—
12	#10622548	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
14	#10622292	—	2	2	no	—
15	#10622292	C₄	1	4	yes	complemented
16	C₁₁×#335	—	2	2	no	—
17	#10622189	—	2	1	no	—
18	#10622189	—	2	1	no	—
19	#10622189	—	2	2	no	—
20	C₂²×Dic₂₂	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	—	2	2	no	—
22	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₁	—	2	2	no	—
26	C₂×C₄⋊Dic₁₁	—	2	2	no	—
27	C₂×C₄⋊Dic₁₁	—	2	2	no	—
28	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
29	C₂×Dic₁₁⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₁₁	—	2	1	no	—
31	C₂²×C₁₁⋊C₈	—	2	1	no	—
32	C₄×Dic₂₂	—	2	1	no	—
33	C₄×Dic₂₂	—	2	1	no	—
34	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
35	C₄×Dic₂₂	Q₈	1	2	yes	—
36	C₁₁×C₄⋊C₈	—	2	2	no	—
37	C₁₁×C₄⋊C₈	—	2	2	no	—
38	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₁	D₄	1	2	yes	—
40	#66880	—	4	4	no	—
41	#66879	—	2	4	no	—
42	Dic₁₁⋊C₈	—	2	4	no	—
43	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
44	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
45	#8655	—	11	1	no	maximal, Sylow
46	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
49	C₂²×Dic₁₁	—	2	1	no	—
50	C₂²×Dic₁₁	—	2	1	no	—
51	C₂²×Dic₁₁	—	4	2	no	—
52	C₂×Dic₂₂	—	2	1	no	—
53	C₂×Dic₂₂	—	2	2	no	—
54	C₂×Dic₂₂	—	2	2	no	—
55	C₂×Dic₂₂	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	—	2	2	no	—
58	C₂×C₈×C₁₁	—	4	2	no	—
59	C₂×C₈×C₁₁	—	4	2	no	—
60	C₂×C₈×C₁₁	—	4	2	no	—
61	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₁₁	#41	1	1	yes	characteristic
67	C₄⋊Dic₁₁	—	2	1	no	—
68	C₄⋊Dic₁₁	—	4	2	no	—
69	C₄⋊Dic₁₁	—	4	2	no	—
70	C₄⋊Dic₁₁	—	4	2	no	—
71	C₄⋊Dic₁₁	—	4	2	no	—
72	C₄⋊Dic₁₁	—	8	2	no	—
73	C₄⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
74	Dic₁₁⋊C₄	—	2	2	no	—
75	Dic₁₁⋊C₄	—	2	2	no	—
76	C₄×Dic₁₁	—	2	2	no	—
77	C₄×Dic₁₁	—	2	2	no	—
78	C₂×C₁₁⋊C₈	—	2	2	no	—
79	C₂×C₁₁⋊C₈	—	4	2	no	—
80	C₂×#446	—	11	1	no	—
81	#1834	—	11	1	no	—
82	#1803	—	11	2	no	—
83	C₂×#335	D₁₁	1	1	yes	characteristic
84	#1296	—	11	2	no	—
85	C₂³×C₁₁	#121	1	1	yes	characteristic
86	C₂×C₄×C₁₁	—	2	1	no	—
87	C₂×C₄×C₁₁	—	2	1	no	—
88	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
91	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₁₁	#135	1	2	yes	—
95	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₁₁	#137	1	2	yes	—
97	C₂×C₄×C₁₁	#137	1	2	yes	—
98	C₂×Dic₁₁	—	2	2	no	—
99	C₂×Dic₁₁	—	2	2	no	—
100	C₂×Dic₁₁	—	4	2	no	—
101	C₂×Dic₁₁	—	4	1	no	—
102	C₂×Dic₁₁	—	4	1	no	—
103	C₂×Dic₁₁	—	8	2	no	—
104	C₂×Dic₁₁	—	8	2	no	—
105	C₂×Dic₁₁	—	8	2	no	—
106	Dic₂₂	—	2	4	no	—
107	Dic₂₂	—	4	1	no	—
108	Dic₂₂	—	4	1	no	—
109	C₈×C₁₁	—	2	2	no	—
110	C₈×C₁₁	—	2	2	no	—
111	C₈×C₁₁	—	8	2	no	—
112	C₁₁⋊C₈	—	4	2	no	—
113	C₂×C₄×Q₈	—	11	1	no	—
114	#446	—	11	4	no	—
115	#446	—	22	2	no	—
116	C₂×C₄⋊C₈	—	11	1	no	—
117	C₂×C₄⋊C₈	—	22	1	no	—
118	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
119	C₂×#103	—	22	2	no	—
120	C₂×#97	—	22	1	no	—
121	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
122	#384	—	11	2	no	—
123	#340	—	22	2	no	—
124	#340	—	22	1	no	—
125	#340	—	22	1	no	—
126	#335	—	2	2	no	—
127	C₂²×C₁₁	#380	1	1	yes	characteristic
128	C₂²×C₁₁	#464	1	2	yes	—
129	C₂²×C₁₁	#468	1	2	yes	—
130	C₂²×C₁₁	#472	1	1	yes	characteristic
131	C₂²×C₁₁	#473	1	1	yes	characteristic
132	C₄×C₁₁	—	2	1	no	—
133	C₄×C₁₁	—	2	1	no	—
134	C₄×C₁₁	#416	1	1	yes	characteristic
135	C₄×C₁₁	#419	1	1	yes	characteristic
136	C₄×C₁₁	#428	1	2	yes	—
137	C₄×C₁₁	#469	1	1	yes	characteristic
138	C₄×C₁₁	#470	1	1	yes	characteristic
139	C₄×C₁₁	#479	1	2	yes	—
140	Dic₁₁	—	4	2	no	—
141	Dic₁₁	—	4	2	no	—
142	Dic₁₁	—	16	2	no	—
143	C₂²×Q₈	—	11	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	22	1	no	—
146	C₂²×C₈	D₄₄	1	1	yes	characteristic
147	C₂²×C₈	D₄₄	1	1	yes	characteristic
148	C₄×Q₈	—	11	2	no	—
149	C₄×Q₈	—	11	2	no	—
150	C₄×Q₈	—	22	1	no	—
151	C₄×Q₈	—	22	1	no	—
152	C₂×C₄⋊C₄	—	11	1	no	—
153	C₂×C₄⋊C₄	—	22	1	no	—
154	C₂×C₄⋊C₄	—	22	2	no	—
155	C₂×C₄⋊C₄	—	22	2	no	—
156	C₂×C₄⋊C₄	—	22	2	no	—
157	C₂×C₄²	—	22	1	no	—
158	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	11	2	no	—
162	C₄⋊C₈	—	22	4	no	—
163	C₄⋊C₈	—	22	1	no	—
164	#103	—	44	4	no	—
165	#97	—	22	4	no	—
166	C₄×C₈	C₄×D₁₁	1	2	yes	—
167	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
168	C₂×C₁₁	#1698	1	2	yes	—
169	C₂×C₁₁	#1700	1	2	yes	—
170	C₂×C₁₁	#1805	1	1	yes	characteristic
171	C₂×C₁₁	#1809	1	1	yes	characteristic
172	C₂×C₁₁	#1816	1	1	yes	characteristic
173	C₂×Q₈	—	11	2	no	—
174	C₂×Q₈	—	22	1	no	—
175	C₂×Q₈	—	22	2	no	—
176	C₂×Q₈	—	22	2	no	—
177	C₂²×C₄	—	22	1	no	—
178	C₂²×C₄	—	22	1	no	—
179	C₂²×C₄	—	44	2	no	—
180	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
181	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
182	C₂²×C₄	#4343	1	1	yes	characteristic
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	22	2	no	—
189	C₂×C₈	—	44	2	no	—
190	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
191	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
192	C₄⋊C₄	—	11	2	no	—
193	C₄⋊C₄	—	22	2	no	—
194	C₄⋊C₄	—	22	1	no	—
195	C₄⋊C₄	—	22	2	no	—
196	C₄⋊C₄	—	44	2	no	—
197	C₄⋊C₄	—	44	2	no	—
198	C₄⋊C₄	—	44	2	no	—
199	C₄⋊C₄	—	44	2	no	—
200	C₄⋊C₄	—	88	2	no	—
201	C₄²	—	22	2	no	—
202	C₄²	—	22	2	no	—
203	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
204	C₄²	#4341	1	1	yes	characteristic
205	C₄²	Q₈×D₁₁	1	1	yes	characteristic
206	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
207	C₁₁	#8655	1	1	yes	characteristic, complemented, Sylow
208	C₂³	#66948	1	1	yes	characteristic
209	Q₈	—	22	4	no	—
210	Q₈	—	44	1	no	—
211	Q₈	—	44	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	22	2	no	—
215	C₂×C₄	—	22	2	no	—
216	C₂×C₄	—	44	2	no	—
217	C₂×C₄	—	44	1	no	—
218	C₂×C₄	—	44	1	no	—
219	C₂×C₄	—	88	2	no	—
220	C₂×C₄	—	88	2	no	—
221	C₂×C₄	—	88	2	no	—
222	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
223	C₂×C₄	#66945	1	1	yes	characteristic
224	C₂×C₄	#66950	1	1	yes	characteristic
225	C₂×C₄	#66957	1	2	yes	—
226	C₂×C₄	#66962	1	2	yes	—
227	C₂×C₄	#66964	1	2	yes	—
228	C₂×C₄	#66971	1	2	yes	—
229	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	#66991	1	1	yes	characteristic
231	C₂×C₄	#66999	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	8	2	no	—
235	C₈	—	44	2	no	—
236	C₂²	#10622458	1	2	yes	—
237	C₂²	#10622499	1	2	yes	—
238	C₂²	#10622509	1	1	yes	characteristic
239	C₂²	#10622510	1	1	yes	characteristic
240	C₂²	#10622600	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	44	2	no	—
244	C₄	—	44	2	no	—
245	C₄	—	176	2	no	—
246	C₄	#10622442	1	2	yes	—
247	C₄	#10622494	1	2	yes	—
248	C₄	#10622726	1	1	yes	characteristic
249	C₄	#10622733	1	1	yes	characteristic
250	C₄	#10622772	1	1	yes	characteristic
251	C₄	#10622802	1	1	yes	characteristic
252	C₂	#13068157	1	2	yes	—
253	C₂	#13068171	1	2	yes	—
254	C₂	#13068632	1	1	yes	characteristic
255	C₂	#13068650	1	1	yes	characteristic
256	C₂	#13068677	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#66948
commutator	C₄²×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle, prime radical	C₂³×C₁₁	#121
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#8655
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
15. #10622292	C₄	semidirect
207. C₁₁	#8655	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622292	2
#13068744	2
#13068622	2
#13066500	2
C₂×C₁₁×#335	2
#8655	11