Subgroups — #14907159

Order 2816.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077492	C₂	1	2	yes	maximal, complemented
2	#13070481	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₁₁	C₂	1	2	yes	maximal, complemented
4	D₁₆×Dic₁₁	C₂	1	2	yes	maximal, complemented
5	C₂×#10622142	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070328	C₂	1	2	yes	maximal, complemented
7	#13070212	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×#508	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×#135	C₂²	1	2	yes	—
10	#10623358	—	2	2	no	—
11	C₁₁×#508	C₄	1	8	yes	complemented
12	C₂×C₁₁×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₁×D₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
16	#10622845	—	2	2	no	—
17	Q₁₆×Dic₁₁	C₂²	1	4	yes	—
18	#10622777	—	2	4	no	—
19	D₈×Dic₁₁	C₂²	1	4	yes	—
20	C₂×#66882	C₂²	1	2	yes	—
21	#10622714	C₂²	1	1	yes	characteristic
22	C₂×#66876	C₂²	1	1	yes	characteristic
23	#10622202	C₂²	1	4	yes	—
24	#10622200	C₂²	1	4	yes	—
25	#10622142	C₂²	1	2	yes	—
26	#10622142	C₂²	1	2	yes	—
27	#10622140	C₂²	1	2	yes	—
28	C₁₆×Dic₁₁	C₂²	1	2	yes	—
29	C₂×C₁₁×#41	—	2	2	no	—
30	C₁₁×#135	—	2	4	no	—
31	C₁₁×#135	C₂×C₄	1	8	yes	complemented
32	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
33	C₂×C₁₁×SD₁₆	—	2	2	no	—
34	C₂×C₁₁×D₈	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
36	Q₈×Dic₁₁	—	2	4	no	—
37	D₄×Dic₁₁	—	2	4	no	—
38	D₄×Dic₁₁	—	4	4	no	—
39	#66979	—	4	2	no	—
40	#66979	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₁	—	2	2	no	—
42	C₂×C₄⋊Dic₁₁	—	4	2	no	—
43	C₂×#4318	D₄	1	1	yes	characteristic
44	C₁₁×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₁₁×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₁₁×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	complemented
48	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
50	#66899	—	2	4	no	—
51	#66896	—	2	4	no	—
52	#66882	—	2	2	no	—
53	#66882	C₂³	1	4	yes	—
54	#66879	D₄	1	2	yes	—
55	C₈×Dic₁₁	C₂³	1	2	yes	—
56	#66876	—	2	2	no	—
57	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
58	#33459	—	11	1	no	maximal, Sylow
59	C₁₁×#41	—	2	8	no	—
60	C₁₁×#41	—	4	4	no	—
61	C₂×Q₈×C₁₁	—	2	2	no	—
62	C₂×D₄×C₁₁	—	2	2	no	—
63	C₂×D₄×C₁₁	—	4	2	no	—
64	C₂²×C₄×C₁₁	—	4	2	no	—
65	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₁₁	—	4	4	no	—
67	C₂²×Dic₁₁	—	4	2	no	—
68	C₁₁×Q₁₆	—	2	2	no	—
69	C₁₁×Q₁₆	C₂²×C₄	1	4	yes	—
70	C₁₁×SD₁₆	—	2	8	no	—
71	C₁₁×D₈	—	2	2	no	—
72	C₁₁×D₈	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₁	#41	1	2	yes	—
76	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₁₁	—	4	4	no	—
78	C₂²⋊Dic₁₁	D₈	1	2	yes	—
79	C₄⋊Dic₁₁	—	2	4	no	—
80	C₄⋊Dic₁₁	—	4	4	no	—
81	C₄⋊Dic₁₁	—	4	2	no	—
82	C₄⋊Dic₁₁	D₈	1	1	yes	characteristic
83	C₄⋊Dic₁₁	D₈	1	1	yes	characteristic
84	C₄×Dic₁₁	—	4	4	no	—
85	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
86	#4318	—	2	2	no	—
87	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
88	C₁₁×C₁₆	C₂²×C₄	1	2	yes	—
89	C₁₁×C₁₆	C₂²×C₄	1	2	yes	—
90	C₁₁⋊C₁₆	—	2	2	no	—
91	C₂×#508	D₁₁	1	1	yes	characteristic
92	#2951	—	11	2	no	—
93	#2175	—	11	4	no	—
94	C₄×Q₃₂	—	11	2	no	—
95	C₄×D₁₆	—	11	2	no	—
96	C₂×#366	—	11	1	no	—
97	#2139	—	11	2	no	—
98	#2108	—	11	1	no	—
99	C₂³×C₁₁	—	4	2	no	—
100	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
101	Q₈×C₁₁	—	2	4	no	—
102	Q₈×C₁₁	—	4	2	no	—
103	D₄×C₁₁	—	2	4	no	—
104	D₄×C₁₁	—	4	8	no	—
105	D₄×C₁₁	—	4	2	no	—
106	C₂×C₄×C₁₁	—	4	2	no	—
107	C₂×C₄×C₁₁	—	4	8	no	—
108	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₁₁	#135	1	2	yes	—
111	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₁₁	—	4	4	no	—
113	C₂×Dic₁₁	—	16	4	no	—
114	C₂×Dic₁₁	—	16	2	no	—
115	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
116	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
117	C₈×C₁₁	C₄×D₄	1	2	yes	—
118	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
120	C₁₁⋊C₈	—	2	2	no	—
121	C₂×#135	C₂×D₁₁	1	2	yes	—
122	#520	—	22	2	no	—
123	#508	Dic₁₁	1	8	yes	complemented
124	C₂×Q₃₂	C₂×D₁₁	1	1	yes	characteristic
125	C₂×SD₃₂	C₂×D₁₁	1	2	yes	—
126	C₂×D₁₆	C₂×D₁₁	1	1	yes	characteristic
127	C₂×M₃₂	—	11	1	no	—
128	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
129	#440	—	22	4	no	—
130	C₄×Q₁₆	—	11	4	no	—
131	C₄×D₈	—	11	4	no	—
132	C₂×#107	—	11	2	no	—
133	#417	—	22	2	no	—
134	#405	—	11	1	no	—
135	#366	—	11	2	no	—
136	#366	—	11	2	no	—
137	#358	—	11	4	no	—
138	#357	—	11	4	no	—
139	#346	—	11	2	no	—
140	C₄×C₁₆	—	11	2	no	—
141	C₂²×C₁₁	—	2	1	no	—
142	C₂²×C₁₁	—	2	1	no	—
143	C₂²×C₁₁	—	4	4	no	—
144	C₂²×C₁₁	—	8	2	no	—
145	C₂²×C₁₁	—	8	2	no	—
146	C₂²×C₁₁	C₄×D₈	1	2	yes	—
147	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₁₁	—	4	4	no	—
149	C₄×C₁₁	C₄×D₈	1	2	yes	—
150	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
152	Dic₁₁	—	2	2	no	—
153	Dic₁₁	—	2	2	no	—
154	Dic₁₁	—	8	4	no	—
155	C₂×#41	—	2	2	no	—
156	#135	—	2	4	no	—
157	#135	C₂×Dic₁₁	1	8	yes	complemented
158	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂²×D₁₁	1	2	yes	—
161	C₂×M₁₆	—	11	1	no	—
162	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
163	C₄×Q₈	—	22	4	no	—
164	C₄×D₄	—	22	4	no	—
165	C₄×D₄	—	44	4	no	—
166	#117	—	11	1	no	—
167	#117	—	44	2	no	—
168	C₂×C₄⋊C₄	—	22	2	no	—
169	C₂×C₄⋊C₄	—	44	2	no	—
170	Q₃₂	C₂×Dic₁₁	1	4	yes	complemented
171	SD₃₂	C₂×Dic₁₁	1	8	yes	complemented
172	D₁₆	C₂×Dic₁₁	1	4	yes	complemented
173	M₃₂	—	22	2	no	—
174	C₂×C₁₆	—	11	2	no	—
175	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	complemented
176	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
178	#107	—	11	4	no	—
179	#107	—	22	2	no	—
180	#103	—	22	4	no	—
181	#102	—	22	4	no	—
182	#97	—	11	2	no	—
183	C₄×C₈	—	11	2	no	—
184	C₂×C₁₁	—	2	2	no	—
185	C₂×C₁₁	—	8	4	no	—
186	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
187	C₂×C₁₁	#3415	1	1	yes	characteristic
188	C₂×C₁₁	#3417	1	1	yes	characteristic
189	#41	—	2	8	no	—
190	#41	—	4	4	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	44	4	no	—
196	C₂²×C₄	—	44	2	no	—
197	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
198	Q₁₆	—	2	2	no	—
199	Q₁₆	C₂²×Dic₁₁	1	4	yes	—
200	SD₁₆	—	2	8	no	—
201	D₈	—	2	2	no	—
202	D₈	C₂²×Dic₁₁	1	4	yes	—
203	M₁₆	—	22	2	no	—
204	C₂×C₈	—	11	2	no	—
205	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
206	C₂×C₈	#4341	1	2	yes	—
207	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
208	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
209	C₄⋊C₄	—	11	1	no	—
210	C₄⋊C₄	—	11	1	no	—
211	C₄⋊C₄	—	22	4	no	—
212	C₄⋊C₄	—	44	4	no	—
213	C₄⋊C₄	—	44	2	no	—
214	C₂²⋊C₄	—	11	2	no	—
215	C₂²⋊C₄	—	44	4	no	—
216	C₄²	—	11	2	no	—
217	C₄²	—	44	4	no	—
218	C₁₆	—	22	2	no	—
219	C₁₆	C₂²×Dic₁₁	1	2	yes	—
220	C₁₆	C₂²×Dic₁₁	1	2	yes	—
221	C₁₁	#33459	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	4	2	no	—
223	C₂³	D₈×D₁₁	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	11	2	no	—
232	C₂×C₄	—	11	2	no	—
233	C₂×C₄	—	44	4	no	—
234	C₂×C₄	—	176	4	no	—
235	C₂×C₄	—	176	2	no	—
236	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
237	C₂×C₄	#66965	1	2	yes	—
238	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
239	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
240	C₈	—	22	2	no	—
241	C₈	D₄×Dic₁₁	1	2	yes	—
242	C₈	C₂×#4341	1	1	yes	characteristic
243	C₈	C₂³×Dic₁₁	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	D₈×Dic₁₁	1	2	yes	—
250	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
251	C₄	—	4	4	no	—
252	C₄	—	22	2	no	—
253	C₄	—	22	2	no	—
254	C₄	—	88	4	no	—
255	C₄	D₈×Dic₁₁	1	2	yes	—
256	C₄	C₂×#66965	1	1	yes	characteristic
257	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	4	no	—
260	C₂	C₂×D₈×Dic₁₁	1	1	yes	characteristic
261	C₂	#13081754	1	1	yes	characteristic
262	C₂	#13081766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₁
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting, prime radical	C₂×C₁₁×#508	C₂
socle	C₂²×C₁₁	C₂²×D₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×#508	D₁₁
hypocenter	C₁₁	#33459
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #13077492	C₂	semidirect
2. #13070481	C₂	semidirect
3. Q₃₂×Dic₁₁	C₂	semidirect
4. D₁₆×Dic₁₁	C₂	semidirect
5. C₂×#10622142	C₂	semidirect
6. #13070328	C₂	semidirect
7. #13070212	C₂	semidirect
11. C₁₁×#508	C₄	semidirect
31. C₁₁×#135	C₂×C₄	semidirect
44. C₁₁×Q₃₂	C₂×C₄	semidirect
45. C₁₁×SD₃₂	C₂×C₄	semidirect
46. C₁₁×D₁₆	C₂×C₄	semidirect
47. C₂×C₁₁×C₁₆	C₂×C₄	semidirect
123. #508	Dic₁₁	semidirect
157. #135	C₂×Dic₁₁	semidirect
170. Q₃₂	C₂×Dic₁₁	semidirect
171. SD₃₂	C₂×Dic₁₁	semidirect
172. D₁₆	C₂×Dic₁₁	semidirect
175. C₂×C₁₆	C₂×Dic₁₁	semidirect
221. C₁₁	#33459	semidirect

Maximal subgroups

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

subgroup	index
#13077492	2
#13070481	2
Q₃₂×Dic₁₁	2
D₁₆×Dic₁₁	2
C₂×#10622142	2
#13070328	2
#13070212	2
C₂×C₁₁×#508	2
#33459	11