Subgroups — #14669702

Order 2816.

486 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	#13070372	C₂	1	2	yes	maximal, complemented
2	C₂×#10622191	C₂	1	2	yes	maximal
3	#13066700	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₁	C₄	1	1	yes	characteristic
8	C₄×C₁₁×M₁₆	—	2	4	no	—
9	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
10	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66876	C₂²	1	4	yes	—
14	#10622191	—	2	4	no	—
15	C₄⋊(C₁₁⋊C₁₆)	—	2	4	no	—
16	C₂³×C₄×C₁₁	C₈	1	2	yes	—
17	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₁×M₁₆	—	2	2	no	—
19	C₂×C₁₁×M₁₆	—	2	2	no	—
20	C₂×C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₁₁×M₁₆	—	2	2	no	—
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
24	C₂×C₁₁×M₁₆	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	—	2	2	no	—
30	C₂×C₄²×C₁₁	—	2	1	no	—
31	C₂×C₄²×C₁₁	—	2	1	no	—
32	C₂×C₄²×C₁₁	C₈	1	2	yes	—
33	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₁₁×#97	—	2	2	no	—
36	C₁₁×#97	—	2	2	no	—
37	C₄×C₈×C₁₁	—	2	2	no	—
38	C₄×C₈×C₁₁	D₄	1	4	yes	—
39	#66876	—	2	4	no	—
40	#66876	—	2	4	no	—
41	C₂×C₁₁⋊C₁₆	—	2	4	no	—
42	#12785	—	11	1	no	maximal, Sylow
43	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	—	2	1	no	—
45	C₂²×C₄×C₁₁	—	2	1	no	—
46	C₂²×C₄×C₁₁	—	2	1	no	—
47	C₂²×C₄×C₁₁	—	2	1	no	—
48	C₂²×C₄×C₁₁	—	2	2	no	—
49	C₂²×C₄×C₁₁	—	2	2	no	—
50	C₂²×C₄×C₁₁	—	4	4	no	—
51	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
57	C₁₁×M₁₆	—	2	8	no	—
58	C₁₁×M₁₆	—	2	4	no	—
59	C₁₁×M₁₆	—	2	4	no	—
60	C₂×C₈×C₁₁	—	2	1	no	—
61	C₂×C₈×C₁₁	—	2	1	no	—
62	C₂×C₈×C₁₁	—	2	1	no	—
63	C₂×C₈×C₁₁	—	2	1	no	—
64	C₂×C₈×C₁₁	—	2	1	no	—
65	C₂×C₈×C₁₁	—	2	1	no	—
66	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
69	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₁₁	#41	1	2	yes	—
73	C₄²×C₁₁	—	2	2	no	—
74	C₄²×C₁₁	—	2	2	no	—
75	C₄²×C₁₁	—	2	1	no	—
76	C₄²×C₁₁	—	2	1	no	—
77	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
79	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
80	C₁₁⋊C₁₆	—	4	4	no	—
81	C₂×C₄×M₁₆	D₁₁	1	1	yes	characteristic
82	#2150	—	11	2	no	—
83	C₂×#349	—	11	2	no	—
84	#1377	—	11	2	no	—
85	C₂³×C₁₁	—	2	1	no	—
86	C₂³×C₁₁	—	2	1	no	—
87	C₂³×C₁₁	—	4	2	no	—
88	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
89	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₁	—	2	1	no	—
91	C₂×C₄×C₁₁	—	2	1	no	—
92	C₂×C₄×C₁₁	—	2	4	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	2	no	—
95	C₂×C₄×C₁₁	—	2	2	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	—	2	2	no	—
98	C₂×C₄×C₁₁	—	4	4	no	—
99	C₂×C₄×C₁₁	—	4	4	no	—
100	C₂×C₄×C₁₁	—	4	2	no	—
101	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
104	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
110	C₈×C₁₁	—	2	2	no	—
111	C₈×C₁₁	—	2	2	no	—
112	C₈×C₁₁	—	2	2	no	—
113	C₈×C₁₁	#121	1	2	yes	—
114	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
115	C₂²×M₁₆	Dic₁₁	1	1	yes	characteristic
116	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
117	C₂²×C₄²	Dic₁₁	1	1	yes	characteristic
118	C₂×M₃₂	—	11	4	no	—
119	C₄×M₁₆	—	2	4	no	—
120	C₂×#97	Dic₁₁	1	1	yes	characteristic
121	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
122	C₂×C₄×C₈	Dic₁₁	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
124	C₄⋊C₁₆	—	22	4	no	—
125	#349	—	22	4	no	—
126	C₂²×C₁₁	—	2	2	no	—
127	C₂²×C₁₁	—	2	2	no	—
128	C₂²×C₁₁	—	4	2	no	—
129	C₂²×C₁₁	—	4	2	no	—
130	C₂²×C₁₁	—	4	2	no	—
131	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂²×C₁₁	#431	1	1	yes	characteristic
134	C₄×C₁₁	—	2	2	no	—
135	C₄×C₁₁	—	2	2	no	—
136	C₄×C₁₁	—	4	2	no	—
137	C₄×C₁₁	#349	1	4	yes	complemented
138	C₄×C₁₁	#380	1	1	yes	characteristic
139	C₄×C₁₁	#407	1	1	yes	characteristic
140	C₄×C₁₁	#423	1	1	yes	characteristic
141	C₄×C₁₁	#432	1	1	yes	characteristic
142	C₂³×C₄	C₁₁⋊C₈	1	2	yes	—
143	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	Dic₂₂	1	2	yes	—
150	C₂×M₁₆	D₄₄	1	2	yes	—
151	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₁₁⋊C₈	1	2	yes	—
159	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
161	M₃₂	—	22	4	no	—
162	M₃₂	—	22	4	no	—
163	C₂×C₁₆	—	22	4	no	—
164	#97	—	2	2	no	—
165	#97	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂²⋊D₁₁	1	4	yes	—
168	C₂×C₁₁	—	2	1	no	—
169	C₂×C₁₁	—	2	1	no	—
170	C₂×C₁₁	—	4	2	no	—
171	C₂×C₁₁	#1797	1	1	yes	characteristic
172	C₂×C₁₁	C₂×#349	1	2	yes	—
173	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
183	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
184	C₂²×C₄	#4318	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	M₁₆	—	2	4	no	—
189	M₁₆	—	2	4	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	#4318	1	2	yes	—
197	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
199	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
201	C₂×C₈	#4339	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	#4318	1	1	yes	characteristic
208	C₄²	#4318	1	1	yes	characteristic
209	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
210	C₁₆	—	44	4	no	—
211	C₁₁	#12785	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
216	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
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	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
229	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
230	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
231	C₂×C₄	C₂×#4318	1	1	yes	characteristic
232	C₂×C₄	C₂×#4318	1	1	yes	characteristic
233	C₂×C₄	C₂×#4318	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
235	C₂×C₄	#66979	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	#66977	1	2	yes	—
241	C₈	#66983	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
248	C₂²	#10622538	1	1	yes	characteristic
249	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#10622191	1	4	yes	complemented
254	C₄	#10622537	1	1	yes	characteristic
255	C₄	#10622539	1	1	yes	characteristic
256	C₄	#10622820	1	1	yes	characteristic
257	C₄	#10622825	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#13068596	1	1	yes	characteristic
262	C₂	C₂×#10622191	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₁
commutator	C₂³×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₁
hypocenter	C₁₁	#12785
LCS	G ▷ C₂³×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13070372	C₂	semidirect
3. #13066700	C₂	semidirect
137. C₄×C₁₁	#349	semidirect
211. C₁₁	#12785	semidirect
253. C₄	#10622191	semidirect

Maximal subgroups

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

subgroup	index
#13070372	2
C₂×#10622191	2
#13066700	2
C₂×C₄×C₁₁×M₁₆	2
#12785	11