Subgroups — #14772709

Order 2816.

822 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	#13069379	C₂	1	3	yes	maximal
4	#13069377	C₂	1	3	yes	maximal
5	#13068743	C₂	1	3	yes	maximal
6	C₄×#66872	C₂	1	3	yes	maximal
7	Q₈×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₁	—	2	1	no	—
9	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₁	—	2	1	no	—
11	C₁₁×#389	—	2	3	no	—
12	C₁₁×#384	C₂²	1	3	yes	—
13	C₄×C₁₁×C₄⋊C₄	C₂²	1	3	yes	—
14	Q₈×C₁₁⋊C₈	C₄	1	8	yes	complemented
15	#10622598	—	2	3	no	—
16	C₂×#66872	C₂²	1	3	yes	—
17	C₂×#66872	C₂²	1	3	yes	—
18	#10622548	C₂²	1	3	yes	—
19	C₄×C₄⋊Dic₁₁	C₂²	1	3	yes	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	3	yes	—
21	C₂×C₄×C₁₁⋊C₈	C₂²	1	3	yes	—
22	#10622167	C₂²	1	6	yes	—
23	#10622167	C₂²	1	3	yes	—
24	#10622167	C₂²	1	1	yes	characteristic
25	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₁	—	4	4	no	—
27	C₄×Q₈×C₁₁	C₂×C₄	1	4	yes	—
28	C₄×Q₈×C₁₁	Q₈	1	4	yes	—
29	C₂×C₁₁×C₄⋊C₄	—	2	3	no	—
30	C₂×C₁₁×C₄⋊C₄	—	2	3	no	—
31	C₂×C₁₁×C₄⋊C₄	D₄	1	3	yes	—
32	C₂×C₁₁×C₄⋊C₄	C₂³	1	3	yes	—
33	C₂×C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	—	2	3	no	—
35	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
36	Q₈×Dic₁₁	—	4	4	no	—
37	C₂×C₄⋊Dic₁₁	—	2	3	no	—
38	C₂×C₄⋊Dic₁₁	—	2	3	no	—
39	C₂×C₄⋊Dic₁₁	C₂³	1	3	yes	—
40	C₂×C₄⋊Dic₁₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₁₁	—	2	3	no	—
42	C₂²×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₁₁⋊C₈	C₂³	1	3	yes	—
44	C₁₁×#95	—	2	3	no	—
45	C₁₁⋊#95	—	2	3	no	—
46	#66872	—	2	6	no	—
47	#66872	—	2	6	no	—
48	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	12	yes	—
49	C₄×C₁₁⋊C₈	C₂×C₄	1	12	yes	—
50	#22738	—	11	1	no	maximal, Sylow
51	C₂×Q₈×C₁₁	C₄⋊C₄	1	4	yes	—
52	C₂×Q₈×C₁₁	SD₁₆	1	4	yes	—
53	C₂×Q₈×C₁₁	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₁₁	—	2	1	no	—
55	C₂²×C₄×C₁₁	—	2	3	no	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₁	—	2	1	no	—
60	C₂²×Dic₁₁	—	2	3	no	—
61	C₁₁×C₄⋊C₄	—	2	1	no	—
62	C₁₁×C₄⋊C₄	—	2	3	no	—
63	C₁₁×C₄⋊C₄	—	2	3	no	—
64	C₁₁×C₄⋊C₄	—	4	6	no	—
65	C₁₁×C₄⋊C₄	—	8	3	no	—
66	C₁₁×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
67	C₁₁×C₄⋊C₄	#41	1	2	yes	—
68	C₄²×C₁₁	—	4	6	no	—
69	C₄²×C₁₁	C₂²×C₄	1	6	yes	—
70	C₄²×C₁₁	C₂×Q₈	1	6	yes	—
71	C₄⋊Dic₁₁	—	2	3	no	—
72	C₄⋊Dic₁₁	—	2	1	no	—
73	C₄⋊Dic₁₁	—	2	3	no	—
74	C₄⋊Dic₁₁	—	4	6	no	—
75	C₄⋊Dic₁₁	—	8	3	no	—
76	C₄⋊Dic₁₁	#41	1	2	yes	—
77	C₄×Dic₁₁	—	4	6	no	—
78	C₂×C₁₁⋊C₈	—	2	3	no	—
79	C₂×C₁₁⋊C₈	—	2	3	no	—
80	C₂×C₁₁⋊C₈	—	2	1	no	—
81	C₂×C₁₁⋊C₈	—	2	1	no	—
82	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₁₁⋊C₈	C₂²×C₄	1	6	yes	—
84	C₂×C₈×Q₈	—	11	1	no	—
85	Q₈×C₄⋊C₄	—	11	1	no	—
86	Q₈×C₄⋊C₄	D₁₁	1	1	yes	characteristic
87	#1948	—	11	3	no	—
88	#1948	—	11	3	no	—
89	#1834	—	11	3	no	—
90	C₄×#106	—	11	3	no	—
91	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₁₁	#106	1	16	yes	complemented
93	C₂×C₄×C₁₁	—	2	3	no	—
94	C₂×C₄×C₁₁	—	2	3	no	—
95	C₂×C₄×C₁₁	—	2	2	no	—
96	C₂×C₄×C₁₁	—	4	1	no	—
97	C₂×C₄×C₁₁	—	4	1	no	—
98	C₂×C₄×C₁₁	—	4	3	no	—
99	C₂×C₄×C₁₁	—	4	3	no	—
100	C₂×C₄×C₁₁	—	4	3	no	—
101	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
103	C₂×C₄×C₁₁	C₂×SD₁₆	1	6	yes	—
104	C₂×C₄×C₁₁	C₂×SD₁₆	1	6	yes	—
105	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₁₁	—	2	2	no	—
112	C₂×Dic₁₁	—	4	1	no	—
113	C₂×Dic₁₁	—	4	1	no	—
114	C₂×Dic₁₁	—	4	3	no	—
115	C₂×Dic₁₁	—	4	3	no	—
116	C₂×Dic₁₁	—	4	3	no	—
117	C₁₁⋊C₈	—	2	6	no	—
118	C₁₁⋊C₈	—	2	2	no	—
119	C₂×C₄×Q₈	—	2	1	no	—
120	C₂×C₄×Q₈	—	22	1	no	—
121	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
122	C₈×Q₈	—	11	8	no	—
123	C₂×#106	—	11	3	no	—
124	C₂×#106	—	11	3	no	—
125	C₂×C₄⋊C₈	—	11	3	no	—
126	C₂×C₄×C₈	—	11	3	no	—
127	#389	—	2	3	no	—
128	#389	—	22	3	no	—
129	#384	—	11	3	no	—
130	#384	C₂×D₁₁	1	3	yes	—
131	C₄×C₄⋊C₄	—	11	3	no	—
132	C₄×C₄⋊C₄	C₂×D₁₁	1	3	yes	—
133	#340	—	11	3	no	—
134	#340	—	11	6	no	—
135	#340	—	11	1	no	—
136	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₁	#548	1	1	yes	characteristic
138	C₂²×C₁₁	#548	1	1	yes	characteristic
139	C₂²×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
140	C₂²×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
141	C₂²×C₁₁	#576	1	1	yes	characteristic
142	C₂²×C₁₁	#578	1	1	yes	characteristic
143	C₄×C₁₁	—	2	6	no	—
144	C₄×C₁₁	—	4	2	no	—
145	C₄×C₁₁	—	8	3	no	—
146	C₄×C₁₁	C₂×#106	1	12	yes	—
147	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
148	C₄×C₁₁	#519	1	1	yes	characteristic
149	C₄×C₁₁	#557	1	1	yes	characteristic
150	C₄×C₁₁	#557	1	1	yes	characteristic
151	Dic₁₁	—	4	2	no	—
152	Dic₁₁	—	8	3	no	—
153	C₂²×Q₈	C₂²⋊D₁₁	1	1	yes	characteristic
154	C₂²×C₈	—	11	1	no	—
155	C₂²×C₈	—	11	3	no	—
156	C₄×Q₈	—	4	4	no	—
157	C₄×Q₈	—	44	4	no	—
158	C₄×Q₈	Dic₂₂	1	4	yes	—
159	C₄×Q₈	C₄×D₁₁	1	4	yes	—
160	C₂×C₄⋊C₄	—	2	3	no	—
161	C₂×C₄⋊C₄	—	2	3	no	—
162	C₂×C₄⋊C₄	—	11	1	no	—
163	C₂×C₄⋊C₄	—	11	3	no	—
164	C₂×C₄⋊C₄	—	22	3	no	—
165	C₂×C₄⋊C₄	—	22	3	no	—
166	C₂×C₄⋊C₄	C₂²⋊D₁₁	1	3	yes	—
167	C₂×C₄⋊C₄	C₂²×D₁₁	1	3	yes	—
168	C₂×C₄⋊C₄	C₂²×D₁₁	1	1	yes	characteristic
169	C₂×C₄²	—	2	3	no	—
170	C₂×C₄²	—	22	3	no	—
171	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
172	#106	—	22	6	no	—
173	#106	—	22	6	no	—
174	C₄⋊C₈	—	11	12	no	—
175	C₄×C₈	—	11	12	no	—
176	#95	—	2	3	no	—
177	#95	—	22	3	no	—
178	C₂×C₁₁	#2351	1	1	yes	characteristic
179	C₂×C₁₁	C₂²×#106	1	1	yes	characteristic
180	C₂×C₁₁	#2913	1	1	yes	characteristic
181	C₂×C₁₁	#3359	1	1	yes	characteristic
182	C₂×C₁₁	#3359	1	1	yes	characteristic
183	C₂×C₁₁	#3362	1	1	yes	characteristic
184	C₂×C₁₁	#3362	1	1	yes	characteristic
185	C₂×Q₈	Dic₁₁⋊C₄	1	4	yes	—
186	C₂×Q₈	#4324	1	4	yes	—
187	C₂×Q₈	#4325	1	4	yes	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	3	no	—
190	C₂²×C₄	—	22	1	no	—
191	C₂²×C₄	—	22	3	no	—
192	C₂²×C₄	C₂×C₂²⋊D₁₁	1	3	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₁₁	1	3	yes	—
194	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
195	C₂×C₈	—	11	2	no	—
196	C₂×C₈	—	11	6	no	—
197	C₂×C₈	—	22	3	no	—
198	C₂×C₈	—	22	1	no	—
199	C₂×C₈	—	22	3	no	—
200	C₂×C₈	—	22	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	3	no	—
203	C₄⋊C₄	—	2	3	no	—
204	C₄⋊C₄	—	4	6	no	—
205	C₄⋊C₄	—	8	3	no	—
206	C₄⋊C₄	—	11	2	no	—
207	C₄⋊C₄	—	22	1	no	—
208	C₄⋊C₄	—	22	3	no	—
209	C₄⋊C₄	—	22	3	no	—
210	C₄⋊C₄	—	44	6	no	—
211	C₄⋊C₄	—	88	3	no	—
212	C₄⋊C₄	Dic₁₁⋊C₄	1	12	yes	—
213	C₄⋊C₄	#4343	1	2	yes	—
214	C₄²	—	4	6	no	—
215	C₄²	—	44	6	no	—
216	C₄²	C₂×Dic₂₂	1	6	yes	—
217	C₄²	C₂×C₄×D₁₁	1	6	yes	—
218	C₁₁	#22738	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
220	Q₈	#66872	1	16	yes	complemented
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	—	2	3	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	3	no	—
227	C₂×C₄	—	4	3	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	22	2	no	—
230	C₂×C₄	—	44	1	no	—
231	C₂×C₄	—	44	1	no	—
232	C₂×C₄	—	44	3	no	—
233	C₂×C₄	—	44	3	no	—
234	C₂×C₄	—	44	3	no	—
235	C₂×C₄	C₂×Dic₁₁⋊C₄	1	6	yes	—
236	C₂×C₄	C₂×Dic₁₁⋊C₄	1	6	yes	—
237	C₂×C₄	C₂×#4324	1	6	yes	—
238	C₂×C₄	C₂×#4325	1	6	yes	—
239	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
241	C₂×C₄	C₂×#4343	1	1	yes	characteristic
242	C₂×C₄	#67040	1	1	yes	characteristic
243	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
244	C₂×C₄	#67043	1	1	yes	characteristic
245	C₈	—	22	2	no	—
246	C₈	—	22	6	no	—
247	C₂²	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
248	C₂²	C₂²×#4324	1	1	yes	characteristic
249	C₂²	C₂²×#4325	1	1	yes	characteristic
250	C₂²	#10623348	1	1	yes	characteristic
251	C₂²	#10623362	1	1	yes	characteristic
252	C₂²	#10623366	1	1	yes	characteristic
253	C₂²	#10623369	1	1	yes	characteristic
254	C₄	—	2	6	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	3	no	—
257	C₄	—	44	2	no	—
258	C₄	—	88	3	no	—
259	C₄	C₂×#66872	1	12	yes	—
260	C₄	#10623100	1	1	yes	characteristic
261	C₄	#10623101	1	1	yes	characteristic
262	C₄	#10623103	1	1	yes	characteristic
263	C₄	C₄×#4343	1	1	yes	characteristic
264	C₂	#13071534	1	1	yes	characteristic
265	C₂	C₂²×#66872	1	1	yes	characteristic
266	C₂	#13077214	1	1	yes	characteristic
267	C₂	#13081254	1	1	yes	characteristic
268	C₂	#13081258	1	1	yes	characteristic
269	C₂	#13081283	1	1	yes	characteristic
270	C₂	#13081287	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	Q₈×C₁₁×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₁	C₂²×D₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₁₁
hypocenter	C₁₁	#22738
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. Q₈×C₁₁⋊C₈	C₄	semidirect
92. Q₈×C₁₁	#106	semidirect
218. C₁₁	#22738	semidirect
220. Q₈	#66872	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₁⋊C₈	2
Q₈×C₄⋊Dic₁₁	2
#13069379	2
#13069377	2
#13068743	2
C₄×#66872	2
Q₈×C₁₁×C₄⋊C₄	2
#22738	11