Subgroups — #14722796

Order 2816.

710 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	#13077249	C₂	1	2	yes	maximal
2	#13067383	C₂	1	8	yes	maximal, complemented
3	#13067364	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×M₁₆	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
7	C₄×C₁₁×M₁₆	C₄	1	4	yes	—
8	C₄×C₁₁×M₁₆	C₂²	1	4	yes	—
9	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
10	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
12	#10622820	—	2	2	no	—
13	#10622539	—	2	2	no	—
14	#10622539	C₂²	1	4	yes	—
15	#10622538	—	2	2	no	—
16	#10622538	C₂²	1	4	yes	—
17	#10622537	C₂²	1	4	yes	—
18	C₄×#4318	—	2	2	no	—
19	#10622096	C₂²	1	8	yes	complemented
20	#10622095	C₂²	1	8	yes	complemented
21	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
24	C₂×C₁₁×M₁₆	—	2	8	no	—
25	C₂×C₁₁×M₁₆	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₁	D₄	1	2	yes	—
31	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
34	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
36	C₂×#4318	—	2	4	no	—
37	C₁₁×#97	C₂×C₄	1	4	yes	—
38	C₁₁×#97	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₁₁	C₂³	1	4	yes	—
41	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
42	C₂²⋊(C₁₁⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
44	C₄⋊(C₁₁⋊C₈)	C₂³	1	8	yes	—
45	#66867	—	2	4	no	—
46	C₄×C₁₁⋊C₈	—	2	4	no	—
47	#17795	—	11	1	no	maximal, Sylow
48	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	—	2	4	no	—
50	C₂²×C₄×C₁₁	—	4	2	no	—
51	C₂²×C₄×C₁₁	—	4	2	no	—
52	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
64	C₁₁×M₁₆	—	2	16	no	—
65	C₂×C₈×C₁₁	—	2	4	no	—
66	C₂×C₈×C₁₁	M₁₆	1	8	yes	complemented
67	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
69	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₁	C₂×D₄	1	2	yes	—
75	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
77	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
79	#4318	—	4	8	no	—
80	C₂×C₁₁⋊C₈	—	2	8	no	—
81	#2920	—	11	2	no	—
82	C₂×C₄×M₁₆	D₁₁	1	1	yes	characteristic
83	#1569	—	11	8	no	—
84	#1564	—	11	4	no	—
85	C₂³×C₁₁	—	2	2	no	—
86	C₂³×C₁₁	—	4	1	no	—
87	C₂³×C₁₁	—	4	1	no	—
88	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁	—	2	2	no	—
91	C₂×C₄×C₁₁	—	2	4	no	—
92	C₂×C₄×C₁₁	—	2	4	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	4	no	—
95	C₂×C₄×C₁₁	—	2	4	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	—	4	1	no	—
98	C₂×C₄×C₁₁	—	4	2	no	—
99	C₂×C₄×C₁₁	—	4	1	no	—
100	C₂×C₄×C₁₁	—	4	2	no	—
101	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₁₁	—	2	4	no	—
112	C₈×C₁₁	C₂×M₁₆	1	8	yes	—
113	C₁₁⋊C₈	—	4	8	no	—
114	C₂²×M₁₆	Dic₁₁	1	2	yes	—
115	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
116	#432	—	11	4	no	—
117	#432	—	22	2	no	—
118	#431	—	11	4	no	—
119	#431	—	22	2	no	—
120	#423	—	11	4	no	—
121	#407	—	22	2	no	—
122	C₄×M₁₆	—	22	2	no	—
123	C₄×M₁₆	Dic₁₁	1	4	yes	—
124	C₄×M₁₆	C₂×D₁₁	1	4	yes	—
125	C₂×#97	Dic₁₁	1	1	yes	characteristic
126	C₂×#97	Dic₁₁	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
128	#335	—	11	8	no	—
129	#334	—	11	8	no	—
130	C₂²×C₁₁	—	2	1	no	—
131	C₂²×C₁₁	—	2	1	no	—
132	C₂²×C₁₁	—	2	2	no	—
133	C₂²×C₁₁	—	2	2	no	—
134	C₂²×C₁₁	—	2	2	no	—
135	C₂²×C₁₁	—	4	4	no	—
136	C₂²×C₁₁	#423	1	2	yes	—
137	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₁₁	—	2	2	no	—
139	C₄×C₁₁	—	2	2	no	—
140	C₄×C₁₁	—	2	1	no	—
141	C₄×C₁₁	—	2	2	no	—
142	C₄×C₁₁	—	2	1	no	—
143	C₄×C₁₁	#423	1	2	yes	—
144	C₄×C₁₁	#430	1	2	yes	—
145	C₄×C₁₁	C₂²×M₁₆	1	2	yes	—
146	C₄×C₁₁	#575	1	2	yes	—
147	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
148	C₂³×C₄	D₄₄	1	1	yes	characteristic
149	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
150	C₂×M₁₆	—	2	8	no	—
151	C₂×M₁₆	—	22	4	no	—
152	C₂×M₁₆	C₂×Dic₁₁	1	8	yes	—
153	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
154	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
155	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
156	C₂×C₄²	Dic₂₂	1	2	yes	—
157	C₂×C₄²	D₄₄	1	1	yes	characteristic
158	C₂×C₄²	D₄₄	1	2	yes	—
159	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
161	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
162	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
163	C₄⋊C₈	—	11	8	no	—
164	C₄⋊C₈	—	22	4	no	—
165	C₂²⋊C₈	—	22	4	no	—
166	C₂²⋊C₈	—	22	4	no	—
167	#97	—	22	4	no	—
168	#97	C₂×Dic₁₁	1	4	yes	—
169	#97	C₂×Dic₁₁	1	4	yes	—
170	C₄×C₈	—	22	4	no	—
171	C₄×C₈	C₂×Dic₁₁	1	4	yes	—
172	C₄×C₈	C₂²×D₁₁	1	4	yes	—
173	C₂×C₁₁	—	2	2	no	—
174	C₂×C₁₁	—	4	2	no	—
175	C₂×C₁₁	C₂×#423	1	1	yes	characteristic
176	C₂×C₁₁	#2911	1	1	yes	characteristic
177	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
178	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
184	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
189	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
190	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄₄	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
194	M₁₆	—	2	16	no	—
195	M₁₆	—	44	8	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	22	8	no	—
198	C₂×C₈	#4318	1	8	yes	complemented
199	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
200	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
201	C₄²	C₄⋊Dic₁₁	1	2	yes	—
202	C₄²	C₄⋊Dic₁₁	1	2	yes	—
203	C₄²	C₄⋊Dic₁₁	1	2	yes	—
204	C₄²	C₂×Dic₂₂	1	2	yes	—
205	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
206	C₄²	C₂×D₄₄	1	2	yes	—
207	C₄²	C₂×D₄₄	1	1	yes	characteristic
208	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
209	C₄²	C₂²×Dic₁₁	1	2	yes	—
210	C₄²	C₂³×D₁₁	1	1	yes	characteristic
211	C₁₁	#17795	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊Dic₁₁	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×#4318	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
233	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
235	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	44	8	no	—
240	C₈	C₂×#4318	1	8	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#10622537	1	2	yes	—
248	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	#10622537	1	2	yes	—
255	C₄	#10622738	1	2	yes	—
256	C₄	#10623279	1	2	yes	—
257	C₄	C₂²×#4318	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#10622537	1	1	yes	characteristic
261	C₂	#13077206	1	1	yes	characteristic
262	C₂	C₂×#10622738	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₄×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₁
hypocenter	C₁₁	#17795
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13067383	C₂	semidirect
3. #13067364	C₂	semidirect
19. #10622096	C₂²	semidirect
20. #10622095	C₂²	semidirect
66. C₂×C₈×C₁₁	M₁₆	semidirect
198. C₂×C₈	#4318	semidirect
211. C₁₁	#17795	semidirect

Maximal subgroups

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

subgroup	index
#13077249	2
#13067383	2
#13067364	2
C₂×C₄×C₁₁×M₁₆	2
#17795	11