Subgroups — #14907014

Order 2816.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622753	C₂	1	2	yes	maximal, complemented
2	#13070458	C₂	1	4	yes	maximal, complemented
3	#13070453	C₂	1	4	yes	maximal, complemented
4	#13070382	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070244	C₂	1	2	yes	maximal, complemented
6	#13070213	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#2955	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#575	C₄	1	1	yes	characteristic
9	C₂×C₁₁×#131	C₂²	1	2	yes	—
10	C₁₁×#444	C₄	1	4	yes	—
11	C₁₁×#443	C₂²	1	4	yes	—
12	C₁₁×#430	C₄	1	1	yes	characteristic
13	C₁₁×#421	C₄	1	2	yes	—
14	C₁₁×#405	C₂²	1	1	yes	characteristic
15	#10622753	—	2	2	no	—
16	#10622753	—	2	2	no	—
17	#10622753	C₂²	1	4	yes	complemented
18	#10622753	C₂²	1	4	yes	complemented
19	C₂×#66876	—	2	2	no	—
20	C₂×#66876	C₂²	1	2	yes	—
21	C₂×#66876	C₂²	1	1	yes	characteristic
22	C₂²×C₁₁⋊C₁₆	—	2	2	no	—
23	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
24	#10622199	C₂²	1	8	yes	complemented
25	#10622101	C₂²	1	4	yes	complemented
26	#10622099	C₂²	1	2	yes	complemented
27	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	complemented
28	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
29	C₁₁×#137	C₈	1	4	yes	—
30	C₁₁×AGL(1,ℤ/8)	C₈	1	4	yes	—
31	C₁₁×#135	C₂×C₄	1	4	yes	—
32	C₁₁×#131	—	2	2	no	—
33	C₁₁×#131	—	2	2	no	—
34	C₁₁×#131	C₂×C₄	1	4	yes	—
35	C₁₁×#131	C₂³	1	4	yes	—
36	C₂×C₁₁×M₁₆	—	2	2	no	—
37	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₁×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	—	2	2	no	—
40	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
41	C₁₁×#117	C₂×C₄	1	1	yes	characteristic
42	C₁₁×#108	C₂×C₄	1	4	yes	—
43	C₁₁×#104	C₂×C₄	1	8	yes	—
44	C₁₁×#97	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₁	C₂³	1	2	yes	—
46	#66876	—	2	2	no	—
47	#66876	—	2	2	no	—
48	#66876	—	2	4	no	—
49	#66876	—	2	4	no	—
50	#66876	C₂³	1	4	yes	—
51	C₂×C₁₁⋊C₁₆	—	2	2	no	—
52	C₂×C₁₁⋊C₁₆	—	2	4	no	—
53	C₂×C₁₁⋊C₁₆	—	2	4	no	—
54	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
55	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
56	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
59	#33441	—	11	1	no	maximal, Sylow
60	C₁₁×#41	—	2	2	no	—
61	C₁₁×#41	—	2	2	no	—
62	C₁₁×#41	C₂×C₈	1	4	yes	—
63	C₁₁×#41	C₂²×C₄	1	4	yes	—
64	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
65	C₂×D₄×C₁₁	—	2	2	no	—
66	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₁₁	—	2	2	no	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₁₁×Q₁₆	C₂×C₈	1	4	yes	—
70	C₁₁×SD₁₆	C₂×C₈	1	8	yes	—
71	C₁₁×D₈	C₂×C₈	1	4	yes	—
72	C₁₁×M₁₆	—	2	4	no	—
73	C₁₁×M₁₆	—	2	4	no	—
74	C₁₁×M₁₆	—	2	2	no	—
75	C₁₁×M₁₆	C₂×C₈	1	4	yes	—
76	C₁₁×M₁₆	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₁	—	2	4	no	—
78	C₂×C₈×C₁₁	—	2	4	no	—
79	C₂×C₈×C₁₁	—	2	2	no	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₁	#41	1	2	yes	—
84	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
85	C₁₁×C₄⋊C₄	—	2	1	no	—
86	C₁₁×C₂²⋊C₄	—	2	1	no	—
87	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
88	C₁₁⋊C₁₆	—	2	4	no	—
89	C₁₁⋊C₁₆	—	2	2	no	—
90	C₁₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
91	C₁₁⋊C₁₆	C₂×D₄	1	2	yes	—
92	C₁₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
93	C₂×#504	—	11	2	no	—
94	#2955	D₁₁	1	1	yes	characteristic
95	#2171	—	11	4	no	—
96	#2170	—	11	4	no	—
97	#2153	—	11	1	no	—
98	#2117	—	11	2	no	—
99	#2108	—	11	1	no	—
100	C₂³×C₁₁	—	2	2	no	—
101	C₂³×C₁₁	#131	1	1	yes	characteristic
102	Q₈×C₁₁	—	2	2	no	—
103	Q₈×C₁₁	C₂²×C₈	1	4	yes	—
104	D₄×C₁₁	—	2	4	no	—
105	D₄×C₁₁	—	2	4	no	—
106	D₄×C₁₁	—	2	2	no	—
107	D₄×C₁₁	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	2	4	no	—
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₁₁	—	2	4	no	—
117	C₈×C₁₁	C₄×D₄	1	2	yes	—
118	C₈×C₁₁	C₂²×C₈	1	4	yes	—
119	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
121	#575	Dic₁₁	1	1	yes	characteristic
122	C₂×#131	C₂×D₁₁	1	2	yes	—
123	#504	—	11	4	no	—
124	#504	—	11	4	no	—
125	#504	—	22	2	no	—
126	#504	—	22	2	no	—
127	C₂×M₃₂	—	11	1	no	—
128	C₂×M₃₂	—	11	2	no	—
129	C₂×M₃₂	—	22	2	no	—
130	C₂²×C₁₆	—	11	1	no	—
131	C₂²×C₁₆	—	22	2	no	—
132	#444	Dic₁₁	1	4	yes	—
133	#443	C₂×D₁₁	1	4	yes	—
134	#430	Dic₁₁	1	1	yes	characteristic
135	#421	Dic₁₁	1	2	yes	—
136	#405	C₂×D₁₁	1	1	yes	characteristic
137	#364	—	11	4	no	—
138	#350	—	11	8	no	—
139	#346	—	11	2	no	—
140	C₄×C₁₆	—	11	2	no	—
141	C₂²×C₁₁	—	2	4	no	—
142	C₂²×C₁₁	—	4	1	no	—
143	C₂²×C₁₁	—	4	4	no	—
144	C₂²×C₁₁	C₈×D₄	1	2	yes	—
145	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
146	C₄×C₁₁	—	2	4	no	—
147	C₄×C₁₁	—	4	2	no	—
148	C₄×C₁₁	C₈×D₄	1	2	yes	—
149	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
151	C₂×#41	C₂×Dic₁₁	1	2	yes	—
152	#137	C₁₁⋊C₈	1	4	yes	—
153	AGL(1,ℤ/8)	C₁₁⋊C₈	1	4	yes	—
154	#135	C₂×Dic₁₁	1	4	yes	—
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	#131	C₂×Dic₁₁	1	4	yes	—
158	#131	C₂²×D₁₁	1	4	yes	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
161	C₂×M₁₆	C₂×Dic₁₁	1	1	yes	characteristic
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
164	#117	C₂×Dic₁₁	1	1	yes	characteristic
165	M₃₂	—	11	4	no	—
166	M₃₂	—	22	4	no	—
167	M₃₂	—	22	4	no	—
168	M₃₂	—	22	2	no	—
169	M₃₂	—	22	2	no	—
170	C₂×C₁₆	—	11	2	no	—
171	C₂×C₁₆	—	11	2	no	—
172	C₂×C₁₆	—	11	1	no	—
173	C₂×C₁₆	—	11	1	no	—
174	C₂×C₁₆	—	11	2	no	—
175	C₂×C₁₆	—	22	4	no	—
176	C₂×C₁₆	—	22	4	no	—
177	C₂×C₁₆	—	22	2	no	—
178	#108	C₂×Dic₁₁	1	4	yes	—
179	#104	C₂×Dic₁₁	1	8	yes	—
180	#97	C₂×Dic₁₁	1	2	yes	—
181	C₄×C₈	C₂²×D₁₁	1	2	yes	—
182	C₂×C₁₁	—	2	1	no	—
183	C₂×C₁₁	—	2	2	no	—
184	C₂×C₁₁	—	4	4	no	—
185	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	C₂×C₁₁⋊C₈	1	4	yes	—
189	#41	C₂²×Dic₁₁	1	4	yes	—
190	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
195	Q₁₆	C₂×C₁₁⋊C₈	1	4	yes	—
196	SD₁₆	C₂×C₁₁⋊C₈	1	8	yes	—
197	D₈	C₂×C₁₁⋊C₈	1	4	yes	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	C₂×C₁₁⋊C₈	1	4	yes	—
202	M₁₆	C₂²×Dic₁₁	1	4	yes	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	D₄×D₁₁	1	2	yes	—
207	C₂×C₈	#4341	1	2	yes	—
208	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
209	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
210	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
211	C₄⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₄²	C₂²×Dic₁₁	1	2	yes	—
214	C₁₆	—	11	1	no	—
215	C₁₆	—	11	2	no	—
216	C₁₆	—	11	1	no	—
217	C₁₆	—	22	4	no	—
218	C₁₆	—	22	2	no	—
219	C₁₁	#33441	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	2	2	no	—
221	C₂³	#67001	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×C₁₁⋊C₈	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×C₁₁⋊C₈	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
233	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
234	C₂×C₄	#67001	1	1	yes	characteristic
235	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	C₂²×C₁₁⋊C₈	1	4	yes	—
238	C₈	D₄×Dic₁₁	1	2	yes	—
239	C₈	C₂×D₄×D₁₁	1	1	yes	characteristic
240	C₈	C₂×#4341	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	D₄×C₁₁⋊C₈	1	2	yes	—
245	C₂²	C₂×#67001	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	D₄×C₁₁⋊C₈	1	2	yes	—
249	C₄	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
250	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×D₄×C₁₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2955	C₂
socle	C₂×C₁₁	C₂×C₈×D₄
prime radical	C₁₁×#575	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2955	D₁₁
hypocenter	C₁₁	#33441
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. C₂×#10622753	C₂	semidirect
2. #13070458	C₂	semidirect
3. #13070453	C₂	semidirect
4. #13070382	C₂	semidirect
5. #13070244	C₂	semidirect
6. #13070213	C₂	semidirect
17. #10622753	C₂²	semidirect
18. #10622753	C₂²	semidirect
24. #10622199	C₂²	semidirect
25. #10622101	C₂²	semidirect
26. #10622099	C₂²	semidirect
27. C₄×C₁₁⋊C₁₆	C₂²	semidirect
219. C₁₁	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622753	2
#13070458	2
#13070453	2
#13070382	2
#13070244	2
#13070213	2
C₁₁×#2955	2
#33441	11