Subgroups — #14670242

Order 2816.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#66876	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622149	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#13066697	C₂	1	2	yes	maximal, complemented
5	C₁₁×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊Dic₁₁	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₂²	1	2	yes	—
12	C₂×#66876	—	2	1	no	—
13	C₂×#66876	—	2	4	no	—
14	C₂×#66876	C₂²	1	2	yes	—
15	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
16	C₁₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₁₁×C₂²⋊C₁₆	—	2	1	no	—
18	#10622188	C₄	1	2	yes	—
19	#10622149	—	2	4	no	—
20	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	—	2	2	no	—
22	C₂²×C₈×C₁₁	—	2	2	no	—
23	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₁₁	—	2	1	no	—
30	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
31	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
32	C₁₁×M₃₂	—	2	2	no	—
33	C₁₁×M₃₂	—	2	2	no	—
34	C₂×C₁₁×C₁₆	—	2	1	no	—
35	C₂×C₁₁×C₁₆	—	2	1	no	—
36	C₁₁⋊#95	—	2	1	no	—
37	C₂²⋊(C₁₁⋊C₈)	C₈	1	2	yes	—
38	C₂²⋊(C₁₁⋊C₈)	C₈	1	2	yes	—
39	C₈×Dic₁₁	—	2	2	no	—
40	C₈×Dic₁₁	—	2	2	no	—
41	#66876	—	2	4	no	—
42	#66876	—	2	2	no	—
43	#66876	D₄	1	4	yes	—
44	C₂×C₁₁⋊C₁₆	—	2	4	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	#12914	—	11	1	no	maximal, Sylow
47	C₂⁴×C₁₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	—	2	2	no	—
49	C₂²×C₄×C₁₁	—	2	2	no	—
50	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	—	2	1	no	—
58	C₂×C₈×C₁₁	—	2	1	no	—
59	C₂×C₈×C₁₁	—	2	2	no	—
60	C₂×C₈×C₁₁	—	2	2	no	—
61	C₂×C₈×C₁₁	—	2	1	no	—
62	C₂×C₈×C₁₁	—	2	1	no	—
63	C₂×C₈×C₁₁	—	2	2	no	—
64	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₁₁	#41	1	2	yes	—
70	C₂²⋊Dic₁₁	—	2	4	no	—
71	C₄×Dic₁₁	—	2	2	no	—
72	C₄×Dic₁₁	—	2	2	no	—
73	C₂×C₁₁⋊C₈	—	2	2	no	—
74	C₂×C₁₁⋊C₈	—	2	2	no	—
75	C₂×C₁₁⋊C₈	C₂×C₈	1	4	yes	—
76	C₁₁×C₁₆	—	4	1	no	—
77	C₁₁×C₁₆	—	4	1	no	—
78	C₁₁⋊C₁₆	—	2	4	no	—
79	C₂²×M₃₂	—	11	1	no	—
80	C₂×#364	—	11	2	no	—
81	#2112	D₁₁	1	1	yes	characteristic
82	C₈×C₂²⋊C₄	—	11	1	no	—
83	#1376	—	11	2	no	—
84	C₂³×C₁₁	—	2	2	no	—
85	C₂³×C₁₁	—	2	2	no	—
86	C₂³×C₁₁	—	4	1	no	—
87	C₂³×C₁₁	C₄⋊C₈	1	1	yes	characteristic
88	C₂³×C₁₁	C₄⋊C₈	1	1	yes	characteristic
89	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₁	—	2	2	no	—
91	C₂×C₄×C₁₁	—	2	1	no	—
92	C₂×C₄×C₁₁	—	2	1	no	—
93	C₂×C₄×C₁₁	—	2	1	no	—
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	2	2	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₁	#131	1	2	yes	—
103	C₂×Dic₁₁	—	2	4	no	—
104	C₂×Dic₁₁	—	4	2	no	—
105	C₂×Dic₁₁	—	4	2	no	—
106	C₈×C₁₁	—	2	1	no	—
107	C₈×C₁₁	—	2	1	no	—
108	C₈×C₁₁	#120	1	1	yes	characteristic
109	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
110	C₈×C₁₁	#123	1	1	yes	characteristic
111	C₁₁⋊C₈	—	2	4	no	—
112	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
113	C₂×M₃₂	—	11	2	no	—
114	C₂×M₃₂	—	22	4	no	—
115	C₂×M₃₂	—	22	1	no	—
116	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
117	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
118	C₂²×C₁₆	—	22	1	no	—
119	C₂×C₂²⋊C₈	—	11	1	no	—
120	C₂×C₄×C₈	—	11	2	no	—
121	C₄×C₂²⋊C₄	—	11	1	no	—
122	#364	—	22	4	no	—
123	C₂²⋊C₁₆	—	2	1	no	—
124	C₂²⋊C₁₆	—	2	1	no	—
125	#336	—	11	2	no	—
126	C₂²×C₁₁	—	2	1	no	—
127	C₂²×C₁₁	—	2	1	no	—
128	C₂²×C₁₁	—	2	1	no	—
129	C₂²×C₁₁	—	2	1	no	—
130	C₂²×C₁₁	—	2	1	no	—
131	C₂²×C₁₁	—	2	1	no	—
132	C₂²×C₁₁	—	2	1	no	—
133	C₂²×C₁₁	—	4	4	no	—
134	C₂²×C₁₁	#364	1	2	yes	complemented
135	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₁	#435	1	2	yes	—
137	C₄×C₁₁	—	2	1	no	—
138	C₄×C₁₁	—	2	1	no	—
139	C₄×C₁₁	#385	1	1	yes	characteristic
140	C₄×C₁₁	#424	1	1	yes	characteristic
141	C₄×C₁₁	C₈×D₄	1	2	yes	—
142	Dic₁₁	—	4	4	no	—
143	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	11	2	no	—
147	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
148	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
149	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	—	22	1	no	—
154	C₂×C₄²	—	11	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	11	4	no	—
158	M₃₂	—	22	4	no	—
159	M₃₂	—	22	2	no	—
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	—	22	4	no	—
163	C₂×C₁₆	—	22	2	no	—
164	C₂²⋊C₈	—	11	2	no	—
165	C₂²⋊C₈	—	11	2	no	—
166	C₄×C₈	—	22	2	no	—
167	C₄×C₈	—	22	2	no	—
168	#95	—	22	1	no	—
169	C₂×C₁₁	—	2	2	no	—
170	C₂×C₁₁	—	2	2	no	—
171	C₂×C₁₁	—	4	1	no	—
172	C₂×C₁₁	#1850	1	1	yes	characteristic
173	C₂×C₁₁	C₂×#364	1	1	yes	characteristic
174	C₂×C₁₁	#2153	1	1	yes	characteristic
175	C₂⁴	#4313	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	22	2	no	—
179	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
180	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
181	C₂²×C₄	#4313	1	1	yes	characteristic
182	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	11	4	no	—
193	C₂×C₈	—	22	2	no	—
194	C₂×C₈	—	22	2	no	—
195	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
196	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
198	C₂×C₈	D₄×D₁₁	1	2	yes	—
199	C₂×C₈	#4341	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
201	C₂²⋊C₄	—	22	4	no	—
202	C₄²	—	22	2	no	—
203	C₄²	—	22	2	no	—
204	C₁₆	—	4	1	no	—
205	C₁₆	—	4	1	no	—
206	C₁₆	—	22	4	no	—
207	C₁₁	#12914	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	Dic₁₁⋊C₈	1	1	yes	characteristic
212	C₂³	Dic₁₁⋊C₈	1	1	yes	characteristic
213	C₂³	C₂×#4313	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	22	4	no	—
222	C₂×C₄	—	44	2	no	—
223	C₂×C₄	—	44	2	no	—
224	C₂×C₄	Dic₁₁⋊C₈	1	1	yes	characteristic
225	C₂×C₄	Dic₁₁⋊C₈	1	1	yes	characteristic
226	C₂×C₄	(C₂²⋊D₁₁)⋊C₄	1	2	yes	—
227	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
228	C₂×C₄	#66960	1	2	yes	—
229	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	22	4	no	—
233	C₈	#66931	1	2	yes	—
234	C₈	#66988	1	1	yes	characteristic
235	C₈	#66990	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#10622149	1	2	yes	complemented
245	C₂²	#10622343	1	2	yes	—
246	C₂²	C₂×Dic₁₁⋊C₈	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	44	4	no	—
250	C₄	(C₂²⋊D₁₁)⋊C₈	1	2	yes	—
251	C₄	#10622559	1	1	yes	characteristic
252	C₄	#10622732	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#13068793	1	1	yes	characteristic
257	C₂	C₂×#10622149	1	1	yes	characteristic
258	C₂	#13070381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2112	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₁	M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2112	D₁₁
hypocenter	C₁₁	#12914
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. C₂×#10622149	C₂	semidirect
4. #13066697	C₂	semidirect
134. C₂²×C₁₁	#364	semidirect
207. C₁₁	#12914	semidirect
244. C₂²	#10622149	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#66876	2
C₂×#10622149	2
C₈×C₂²⋊Dic₁₁	2
#13066697	2
C₁₁×#2112	2
#12914	11