Subgroups — #14909127

Order 2816.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622308	C₂	1	1	yes	maximal, characteristic
2	#13070859	C₂	1	4	yes	maximal, complemented
3	#13070841	C₂	1	4	yes	maximal, complemented
4	#13070331	C₂	1	4	yes	maximal, complemented
5	C₄×#66876	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#533	—	2	1	no	—
8	C₂²×D₈₈	—	2	1	no	—
9	C₂²×D₈₈	C₂²	1	2	yes	—
10	C₂×#66927	—	2	1	no	—
11	C₁₁×#500	C₂²	1	2	yes	complemented
12	C₁₁×#499	—	2	1	no	—
13	C₁₁×#499	C₂²	1	2	yes	complemented
14	C₁₁×#498	—	2	1	no	—
15	C₁₁×#427	—	2	1	no	—
16	C₁₁×#427	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	C₂×#66876	C₂²	1	2	yes	—
19	#10622308	—	2	1	no	—
20	#10622308	—	2	1	no	—
21	#10622308	C₂²	1	2	yes	—
22	#10622308	C₂²	1	2	yes	—
23	#10622129	C₂²	1	16	yes	complemented
24	#10622099	C₂²	1	2	yes	complemented
25	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	complemented
26	C₂²×D₄₄	—	2	2	no	—
27	C₂²×D₄₄	—	4	2	no	—
28	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₁₁×C₄⋊Q₈	—	2	2	no	—
31	C₁₁×#125	—	2	2	no	—
32	C₁₁×C₂²⋊Q₈	—	4	2	no	—
33	C₄×Q₈×C₁₁	—	4	2	no	—
34	C₁₁×#117	—	2	2	no	—
35	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
36	C₂×D₈₈	—	2	8	no	—
37	C₂×D₈₈	—	2	2	no	—
38	C₂×D₈₈	—	2	2	no	—
39	C₂×D₈₈	—	2	2	no	—
40	C₂×D₈₈	—	4	4	no	—
41	C₂×D₈₈	C₂³	1	4	yes	—
42	#66927	—	2	2	no	—
43	#66927	—	2	2	no	—
44	#66927	—	4	2	no	—
45	C₁₁×#107	—	2	2	no	—
46	C₁₁×#107	C₂³	1	4	yes	—
47	C₁₁×#106	—	2	2	no	—
48	C₁₁×#106	—	2	2	no	—
49	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
53	#66876	—	2	4	no	—
54	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
55	#33704	—	11	1	no	maximal, Sylow
56	C₂³×D₁₁	—	4	2	no	—
57	C₂×Q₈×C₁₁	—	4	2	no	—
58	C₂²×C₄×C₁₁	D₈	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×D₄₄	—	2	4	no	—
61	C₂×D₄₄	—	4	4	no	—
62	C₂×D₄₄	—	4	4	no	—
63	C₂×D₄₄	—	4	4	no	—
64	C₂×D₄₄	—	4	2	no	—
65	C₂×D₄₄	—	4	4	no	—
66	C₂×D₄₄	—	8	8	no	—
67	C₂×C₈×C₁₁	SD₁₆	1	4	yes	—
68	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	#41	1	4	yes	—
72	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
73	C₁₁×C₄⋊C₄	—	2	4	no	—
74	C₁₁×C₄⋊C₄	—	4	4	no	—
75	C₁₁×C₄⋊C₄	—	4	2	no	—
76	C₁₁×C₂²⋊C₄	—	4	2	no	—
77	C₄²×C₁₁	—	4	2	no	—
78	C₄²×C₁₁	D₈	1	2	yes	—
79	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
81	D₈₈	—	2	8	no	—
82	D₈₈	—	4	4	no	—
83	D₈₈	—	4	4	no	—
84	D₈₈	—	4	4	no	—
85	C₁₁⋊C₁₆	—	2	4	no	—
86	#3160	D₁₁	1	1	yes	characteristic, complemented
87	C₂×#493	—	11	1	no	—
88	#2244	—	11	4	no	—
89	#2240	—	11	4	no	—
90	#2140	—	11	4	no	—
91	C₄×M₃₂	—	11	1	no	—
92	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
93	C₂²×D₁₁	—	4	4	no	—
94	C₂²×D₁₁	—	8	8	no	—
95	C₂²×D₁₁	—	8	4	no	—
96	Q₈×C₁₁	—	8	4	no	—
97	C₂×C₄×C₁₁	—	2	4	no	—
98	C₂×C₄×C₁₁	—	4	4	no	—
99	C₂×C₄×C₁₁	#124	1	2	yes	—
100	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
105	D₄₄	—	4	4	no	—
106	D₄₄	—	4	4	no	—
107	D₄₄	—	8	8	no	—
108	D₄₄	—	8	4	no	—
109	D₄₄	—	8	8	no	—
110	C₈×C₁₁	#124	1	2	yes	—
111	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
112	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
113	C₈×C₁₁	C₂×#41	1	2	yes	—
114	C₂²×D₈	—	11	2	no	—
115	C₂²×D₈	—	22	1	no	—
116	#533	—	2	1	no	—
117	C₂×#127	—	22	1	no	—
118	C₂×M₃₂	—	11	2	no	—
119	#500	C₂×D₁₁	1	2	yes	complemented
120	#499	—	2	1	no	—
121	#499	C₂×D₁₁	1	2	yes	complemented
122	#498	—	2	1	no	—
123	#493	—	11	2	no	—
124	#493	—	11	2	no	—
125	#493	—	22	1	no	—
126	#493	—	22	1	no	—
127	#427	—	2	1	no	—
128	#427	C₂×D₁₁	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
130	#357	—	11	16	no	—
131	#346	—	11	2	no	—
132	C₄×C₁₆	—	11	2	no	—
133	C₂²×C₁₁	—	2	2	no	—
134	C₂²×C₁₁	#486	1	2	yes	—
135	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₁₁	—	8	4	no	—
137	C₂×D₁₁	—	8	8	no	—
138	C₂×D₁₁	—	16	8	no	—
139	C₄×C₁₁	—	2	2	no	—
140	C₄×C₁₁	—	8	4	no	—
141	C₄×C₁₁	#486	1	2	yes	—
142	C₄×C₁₁	#509	1	4	yes	—
143	C₄×C₁₁	C₂×#124	1	1	yes	characteristic
144	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂²×D₄	—	22	2	no	—
146	C₂²×D₄	—	44	2	no	—
147	C₂×D₈	—	11	4	no	—
148	C₂×D₈	—	22	8	no	—
149	C₂×D₈	—	22	2	no	—
150	C₂×D₈	—	22	2	no	—
151	C₂×D₈	—	22	2	no	—
152	C₂×D₈	—	44	4	no	—
153	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	2	no	—
156	#127	—	22	2	no	—
157	#127	—	22	2	no	—
158	#127	—	44	2	no	—
159	#125	—	2	2	no	—
160	C₂²⋊Q₈	—	4	2	no	—
161	C₄×Q₈	—	4	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
164	M₃₂	—	22	4	no	—
165	C₂×C₁₆	—	11	4	no	—
166	#107	—	2	2	no	—
167	#107	C₂²×D₁₁	1	4	yes	—
168	#106	—	2	2	no	—
169	#106	—	2	2	no	—
170	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
172	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
174	C₂×C₁₁	—	2	2	no	—
175	C₂×C₁₁	C₂×#486	1	1	yes	characteristic
176	C₂×C₁₁	C₂×#509	1	2	yes	—
177	D₁₁	—	16	4	no	—
178	C₂⁴	—	44	2	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂×D₄	—	22	4	no	—
181	C₂×D₄	—	44	4	no	—
182	C₂×D₄	—	44	4	no	—
183	C₂×D₄	—	44	2	no	—
184	C₂×D₄	—	44	4	no	—
185	C₂×D₄	—	44	4	no	—
186	C₂×D₄	—	88	8	no	—
187	C₂²×C₄	#4323	1	2	yes	—
188	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
189	D₈	—	22	8	no	—
190	D₈	—	44	4	no	—
191	D₈	—	44	4	no	—
192	D₈	—	44	4	no	—
193	C₂×C₈	#4325	1	4	yes	—
194	C₂×C₈	#4343	1	4	yes	—
195	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
198	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	#4323	1	2	yes	—
205	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
206	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
207	C₁₆	—	22	4	no	—
208	C₁₁	#33704	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	44	4	no	—
210	C₂³	—	88	8	no	—
211	C₂³	—	88	4	no	—
212	C₂³	C₂×#4323	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	D₄	—	44	4	no	—
215	D₄	—	44	4	no	—
216	D₄	—	88	8	no	—
217	D₄	—	88	8	no	—
218	D₄	—	88	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₂×#4323	1	2	yes	—
222	C₂×C₄	C₂×#4323	1	2	yes	—
223	C₂×C₄	C₂×#4323	1	1	yes	characteristic
224	C₂×C₄	C₂×#4325	1	2	yes	—
225	C₂×C₄	#67000	1	2	yes	—
226	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
227	C₈	C₂×#4325	1	2	yes	—
228	C₈	C₂×#4325	1	2	yes	—
229	C₈	#67000	1	2	yes	—
230	C₈	C₂×#4343	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	88	8	no	—
233	C₂²	—	88	4	no	—
234	C₂²	—	176	8	no	—
235	C₂²	#10622701	1	2	yes	—
236	C₂²	C₂²×#4323	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	#10622701	1	2	yes	—
240	C₄	#10622806	1	4	yes	—
241	C₄	C₂²×#4325	1	1	yes	characteristic
242	C₄	C₂×#67000	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	176	4	no	—
245	C₂	C₂×#10622701	1	1	yes	characteristic
246	C₂	C₂×#10622806	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#3160	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂×#10622308	C₂
hypercenter	#3160	D₁₁
hypocenter	C₁₁	#33704
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13070859	C₂	semidirect
3. #13070841	C₂	semidirect
4. #13070331	C₂	semidirect
5. C₄×#66876	C₂	semidirect
6. C₁₁×#3160	C₂	semidirect
11. C₁₁×#500	C₂²	semidirect
13. C₁₁×#499	C₂²	semidirect
23. #10622129	C₂²	semidirect
24. #10622099	C₂²	semidirect
25. C₄×C₁₁⋊C₁₆	C₂²	semidirect
86. #3160	D₁₁	semidirect
119. #500	C₂×D₁₁	semidirect
121. #499	C₂×D₁₁	semidirect
208. C₁₁	#33704	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622308	2
#13070859	2
#13070841	2
#13070331	2
C₄×#66876	2
C₁₁×#3160	2
#33704	11