Subgroups — #14664997

Order 2816.

414 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	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
3	#13066659	C₂	1	4	yes	maximal, complemented
4	C₁₁×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#135	—	2	1	no	—
6	C₂×C₁₁×#135	C₄	1	2	yes	—
7	C₁₁×#532	—	2	1	no	—
8	C₁₁×#495	—	2	1	no	—
9	C₁₁×#495	C₂²	1	2	yes	—
10	C₁₁×#494	C₄	1	1	yes	characteristic
11	C₁₁×#493	C₄	1	1	yes	characteristic
12	C₁₁×#492	—	2	1	no	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
15	#10622538	—	2	1	no	—
16	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
17	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
18	#10622096	—	2	1	no	—
19	#10622095	C₂²	1	2	yes	—
20	C₂×C₁₁×#41	—	2	2	no	—
21	C₁₁×#135	—	4	4	no	—
22	C₁₁×#135	—	4	2	no	—
23	C₁₁×#135	—	4	2	no	—
24	C₂×C₁₁×Q₁₆	—	2	1	no	—
25	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₁×SD₁₆	—	2	2	no	—
27	C₂×C₁₁×SD₁₆	—	2	2	no	—
28	C₂×C₁₁×D₈	—	2	1	no	—
29	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
32	C₁₁×C₄⋊Q₈	—	2	1	no	—
33	C₁₁×#127	—	2	1	no	—
34	C₁₁×#124	—	2	2	no	—
35	C₁₁×#121	—	4	1	no	—
36	C₁₁×#121	—	4	1	no	—
37	C₄×D₄×C₁₁	—	4	1	no	—
38	C₄×D₄×C₁₁	—	4	1	no	—
39	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₂²⋊(C₁₁⋊C₈)	—	4	1	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	C₂×C₁₁⋊C₁₆	—	4	2	no	—
47	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
48	C₄×C₁₁⋊C₈	—	4	1	no	—
49	#11924	—	11	1	no	maximal, Sylow
50	C₁₁×#41	—	4	4	no	—
51	C₁₁×#41	—	4	4	no	—
52	C₂×Q₈×C₁₁	—	2	2	no	—
53	C₂×D₄×C₁₁	—	2	2	no	—
54	C₂×D₄×C₁₁	—	4	2	no	—
55	C₂×D₄×C₁₁	—	4	1	no	—
56	C₂×D₄×C₁₁	—	4	1	no	—
57	C₂²×C₄×C₁₁	—	4	2	no	—
58	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
61	C₁₁×Q₁₆	—	4	2	no	—
62	C₁₁×Q₁₆	—	4	2	no	—
63	C₁₁×SD₁₆	—	4	4	no	—
64	C₁₁×SD₁₆	—	4	2	no	—
65	C₁₁×SD₁₆	—	4	2	no	—
66	C₁₁×D₈	—	4	2	no	—
67	C₁₁×D₈	—	4	2	no	—
68	C₂×C₈×C₁₁	—	2	2	no	—
69	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	D₈	1	2	yes	—
72	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
75	C₁₁×C₄⋊C₄	—	4	1	no	—
76	C₁₁×C₄⋊C₄	—	4	1	no	—
77	C₁₁×C₂²⋊C₄	—	4	2	no	—
78	C₁₁×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₁₁	D₈	1	1	yes	characteristic
81	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
82	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₁⋊C₈	—	4	2	no	—
84	C₁₁⋊C₁₆	—	4	2	no	—
85	#3147	D₁₁	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	—	11	1	no	—
87	#1564	—	11	1	no	—
88	#1360	—	11	4	no	—
89	C₂³×C₁₁	—	4	2	no	—
90	C₂³×C₁₁	#102	1	1	yes	characteristic
91	Q₈×C₁₁	—	4	2	no	—
92	Q₈×C₁₁	—	4	2	no	—
93	D₄×C₁₁	—	4	2	no	—
94	D₄×C₁₁	—	4	2	no	—
95	D₄×C₁₁	—	8	2	no	—
96	D₄×C₁₁	—	8	4	no	—
97	D₄×C₁₁	—	8	2	no	—
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	4	1	no	—
100	C₂×C₄×C₁₁	—	4	2	no	—
101	C₂×C₄×C₁₁	—	8	4	no	—
102	C₂×C₄×C₁₁	#100	1	2	yes	—
103	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₁	#102	1	2	yes	—
105	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
106	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₈×C₁₁	—	2	1	no	—
111	C₈×C₁₁	—	2	1	no	—
112	C₈×C₁₁	#102	1	2	yes	—
113	C₈×C₁₁	C₂×D₈	1	1	yes	characteristic
114	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₁₁⋊C₈	—	8	2	no	—
116	C₂×#135	—	2	1	no	—
117	C₂×#135	Dic₁₁	1	2	yes	—
118	#532	—	2	1	no	—
119	C₂²×C₁₆	—	22	1	no	—
120	#495	—	2	1	no	—
121	#495	C₂×D₁₁	1	2	yes	—
122	#494	Dic₁₁	1	1	yes	characteristic
123	#493	Dic₁₁	1	1	yes	characteristic
124	#492	—	2	1	no	—
125	#431	—	22	1	no	—
126	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	11	2	no	—
128	C₄⋊C₁₆	—	22	1	no	—
129	#335	—	22	1	no	—
130	#334	—	11	2	no	—
131	C₂²×C₁₁	—	2	1	no	—
132	C₂²×C₁₁	—	2	1	no	—
133	C₂²×C₁₁	—	8	2	no	—
134	C₂²×C₁₁	—	8	2	no	—
135	C₂²×C₁₁	—	8	2	no	—
136	C₂²×C₁₁	#331	1	2	yes	—
137	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
138	C₄×C₁₁	—	2	1	no	—
139	C₄×C₁₁	—	2	1	no	—
140	C₄×C₁₁	—	2	1	no	—
141	C₄×C₁₁	—	8	2	no	—
142	C₄×C₁₁	#331	1	2	yes	—
143	C₄×C₁₁	#359	1	2	yes	—
144	C₄×C₁₁	C₂×#100	1	1	yes	characteristic
145	C₄×C₁₁	C₂×#102	1	1	yes	characteristic
146	C₂×#41	—	2	2	no	—
147	#135	—	4	4	no	—
148	#135	—	4	2	no	—
149	#135	—	4	2	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×SD₁₆	—	2	2	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
156	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
158	C₄⋊Q₈	—	2	1	no	—
159	#127	—	2	1	no	—
160	#124	—	2	2	no	—
161	#121	—	4	1	no	—
162	#121	—	4	1	no	—
163	C₄×D₄	—	4	1	no	—
164	C₄×D₄	—	4	1	no	—
165	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
166	C₂×C₁₆	—	22	2	no	—
167	C₂×C₁₆	—	44	2	no	—
168	C₄⋊C₈	—	22	2	no	—
169	C₂²⋊C₈	—	44	1	no	—
170	C₄×C₈	—	44	1	no	—
171	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
172	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
174	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
175	C₂×C₁₁	—	2	2	no	—
176	C₂×C₁₁	—	16	2	no	—
177	C₂×C₁₁	C₂×#331	1	1	yes	characteristic
178	C₂×C₁₁	C₂×#359	1	1	yes	characteristic
179	C₂×C₁₁	#2145	1	1	yes	characteristic
180	#41	—	4	4	no	—
181	#41	—	4	4	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	#4323	1	1	yes	characteristic
189	C₂²×C₄	#4324	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
191	Q₁₆	—	4	2	no	—
192	Q₁₆	—	4	2	no	—
193	SD₁₆	—	4	4	no	—
194	SD₁₆	—	4	2	no	—
195	SD₁₆	—	4	2	no	—
196	D₈	—	4	2	no	—
197	D₈	—	4	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	44	2	no	—
200	C₂×C₈	#4323	1	2	yes	—
201	C₂×C₈	#4324	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
204	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
205	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₄²	#4323	1	1	yes	characteristic
211	C₄²	#4324	1	1	yes	characteristic
212	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
214	C₁₆	—	44	2	no	—
215	C₁₁	#11924	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	4	2	no	—
217	C₂³	#66896	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	#66896	1	1	yes	characteristic
230	C₂×C₄	#66896	1	2	yes	—
231	C₂×C₄	#66896	1	1	yes	characteristic
232	C₂×C₄	#66896	1	1	yes	characteristic
233	C₂×C₄	#66897	1	2	yes	—
234	C₂×C₄	C₂×#4323	1	1	yes	characteristic
235	C₂×C₄	C₂×#4324	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	88	2	no	—
240	C₈	#66896	1	2	yes	—
241	C₈	C₂×#4323	1	1	yes	characteristic
242	C₈	C₂×#4324	1	1	yes	characteristic
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	#10622182	1	2	yes	—
249	C₂²	C₂×#66896	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	#10622182	1	2	yes	—
255	C₄	#10622205	1	2	yes	—
256	C₄	C₂×#66896	1	1	yes	characteristic
257	C₄	C₂×#66897	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×#10622182	1	1	yes	characteristic
261	C₂	C₂×#10622205	1	1	yes	characteristic
262	C₂	#13070352	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#66896
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₁
Fitting, prime radical	C₁₁×#3147	C₂
socle	C₂²×C₁₁	C₂×#102
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	#3147	D₁₁
hypocenter	C₁₁	#11924
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
2. #13067364	C₂	semidirect
3. #13066659	C₂	semidirect
215. C₁₁	#11924	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
#13067364	2
#13066659	2
C₁₁×#3147	2
#11924	11