Subgroups — #14665058

Order 2816.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079130	C₂	1	1	yes	maximal, characteristic
2	#13067364	C₂	1	1	yes	maximal, characteristic
3	#13066673	C₂	1	2	yes	maximal, complemented
4	#13066670	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#10623001	—	2	1	no	—
7	C₂²×C₁₁×C₁₆	—	2	1	no	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	#10622719	—	2	1	no	—
10	#10622719	C₄	1	2	yes	—
11	#10622538	—	2	1	no	—
12	#10622296	—	2	1	no	—
13	#10622296	C₄	1	2	yes	—
14	#10622295	C₂²	1	1	yes	characteristic
15	#10622295	C₂²	1	1	yes	characteristic
16	#10622293	—	2	1	no	—
17	C₁₁×C₄⋊C₁₆	—	2	1	no	—
18	C₁₁×C₄⋊C₁₆	C₂²	1	2	yes	—
19	#10622096	C₂²	1	2	yes	—
20	#10622095	—	2	1	no	—
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
24	#66979	—	2	2	no	—
25	#66977	—	4	1	no	—
26	#66977	—	4	1	no	—
27	#66923	—	2	2	no	—
28	#66922	—	2	1	no	—
29	#66922	—	2	1	no	—
30	C₄×Dic₂₂	—	4	1	no	—
31	C₄×Dic₂₂	—	4	1	no	—
32	C₂×C₁₁×C₁₆	—	2	2	no	—
33	C₂×C₁₁×C₁₆	—	4	2	no	—
34	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²⋊(C₁₁⋊C₈)	—	4	1	no	—
39	#66882	—	2	1	no	—
40	#66882	—	2	1	no	—
41	#66882	C₂×C₄	1	2	yes	—
42	#66882	C₂×C₄	1	2	yes	—
43	#66881	—	2	2	no	—
44	#66881	—	2	2	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
46	C₄×C₁₁⋊C₈	—	4	1	no	—
47	#11939	—	11	1	no	maximal, Sylow
48	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
51	C₂×Dic₂₂	—	4	1	no	—
52	C₂×Dic₂₂	—	4	1	no	—
53	C₂×C₈×C₁₁	—	2	2	no	—
54	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₁	D₈	1	1	yes	characteristic
62	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
63	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
64	C₂²⋊Dic₁₁	—	4	2	no	—
65	C₄⋊Dic₁₁	—	2	2	no	—
66	C₄⋊Dic₁₁	—	2	2	no	—
67	C₄⋊Dic₁₁	—	4	1	no	—
68	C₄⋊Dic₁₁	—	4	1	no	—
69	Dic₁₁⋊C₄	—	4	2	no	—
70	Dic₁₁⋊C₄	—	4	2	no	—
71	C₄×Dic₁₁	—	4	2	no	—
72	C₂×C₁₁⋊C₈	—	4	2	no	—
73	C₁₁×C₁₆	—	4	2	no	—
74	#3160	—	11	1	no	—
75	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic
76	#1564	—	11	1	no	—
77	#1365	—	11	2	no	—
78	#1364	—	11	2	no	—
79	C₂³×C₁₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₁₁	—	2	2	no	—
81	C₂×C₄×C₁₁	—	4	1	no	—
82	C₂×C₄×C₁₁	#101	1	2	yes	—
83	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
84	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₁	#103	1	2	yes	—
87	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₁₁	—	4	2	no	—
91	C₂×Dic₁₁	—	4	2	no	—
92	Dic₂₂	—	8	2	no	—
93	Dic₂₂	—	8	2	no	—
94	C₈×C₁₁	—	2	1	no	—
95	C₈×C₁₁	—	2	1	no	—
96	C₈×C₁₁	#103	1	2	yes	—
97	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₁₁⋊C₈	—	8	2	no	—
100	#533	—	22	1	no	—
101	C₂²×C₁₆	—	2	1	no	—
102	#500	—	11	1	no	—
103	#500	—	11	1	no	—
104	#499	—	11	2	no	—
105	#499	—	22	1	no	—
106	#498	—	22	1	no	—
107	#431	—	22	1	no	—
108	#427	—	11	2	no	—
109	#427	—	22	1	no	—
110	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
111	C₄⋊C₁₆	—	2	1	no	—
112	C₄⋊C₁₆	C₂×D₁₁	1	2	yes	—
113	#335	—	11	2	no	—
114	#334	—	22	1	no	—
115	C₂²×C₁₁	—	2	1	no	—
116	C₂²×C₁₁	—	2	1	no	—
117	C₂²×C₁₁	#332	1	2	yes	—
118	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
119	C₄×C₁₁	—	2	1	no	—
120	C₄×C₁₁	—	2	1	no	—
121	C₄×C₁₁	—	2	1	no	—
122	C₄×C₁₁	#332	1	2	yes	—
123	C₄×C₁₁	C₂×#101	1	1	yes	characteristic
124	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
125	C₄×C₁₁	#508	1	2	yes	—
126	Dic₁₁	—	8	2	no	—
127	Dic₁₁	—	8	2	no	—
128	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
130	C₄⋊Q₈	—	22	1	no	—
131	C₄⋊Q₈	—	22	1	no	—
132	#125	—	22	2	no	—
133	C₂²⋊Q₈	—	44	1	no	—
134	C₂²⋊Q₈	—	44	1	no	—
135	C₄×Q₈	—	44	1	no	—
136	C₄×Q₈	—	44	1	no	—
137	#117	—	22	2	no	—
138	C₂×C₄²	D₄₄	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	4	2	no	—
141	#107	—	11	2	no	—
142	#107	—	11	2	no	—
143	#107	—	22	1	no	—
144	#107	—	22	1	no	—
145	#106	—	22	2	no	—
146	#106	—	22	2	no	—
147	C₄⋊C₈	—	22	2	no	—
148	C₂²⋊C₈	—	44	1	no	—
149	C₄×C₈	—	44	1	no	—
150	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
151	C₄×C₈	D₄₄	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
153	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
154	C₂×C₁₁	—	2	2	no	—
155	C₂×C₁₁	C₂×#332	1	1	yes	characteristic
156	C₂×C₁₁	#2138	1	1	yes	characteristic
157	C₂×C₁₁	#2148	1	1	yes	characteristic
158	C₂×Q₈	—	44	1	no	—
159	C₂×Q₈	—	44	1	no	—
160	C₂²×C₄	#4314	1	1	yes	characteristic
161	C₂²×C₄	D₈₈	1	1	yes	characteristic
162	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	44	2	no	—
165	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
166	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
167	C₂×C₈	#4324	1	2	yes	—
168	C₂×C₈	#4326	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
171	C₄⋊C₄	—	22	2	no	—
172	C₄⋊C₄	—	22	2	no	—
173	C₄⋊C₄	—	44	2	no	—
174	C₄⋊C₄	—	44	2	no	—
175	C₄⋊C₄	—	44	1	no	—
176	C₄⋊C₄	—	44	1	no	—
177	C₂²⋊C₄	—	44	2	no	—
178	C₄²	—	44	2	no	—
179	C₄²	#4314	1	1	yes	characteristic
180	C₄²	D₈₈	1	1	yes	characteristic
181	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
182	C₄²	C₂×D₄₄	1	1	yes	characteristic
183	C₁₆	—	4	2	no	—
184	C₁₁	#11939	1	1	yes	characteristic, complemented, Sylow
185	C₂³	#66885	1	1	yes	characteristic
186	Q₈	—	88	2	no	—
187	Q₈	—	88	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	44	2	no	—
191	C₂×C₄	—	44	2	no	—
192	C₂×C₄	#66874	1	2	yes	—
193	C₂×C₄	#66885	1	1	yes	characteristic
194	C₂×C₄	#66885	1	1	yes	characteristic
195	C₂×C₄	#66885	1	1	yes	characteristic
196	C₂×C₄	#66888	1	2	yes	—
197	C₂×C₄	C₂×#4314	1	1	yes	characteristic
198	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
199	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	88	2	no	—
203	C₈	#66874	1	2	yes	—
204	C₈	C₂×#4324	1	1	yes	characteristic
205	C₈	C₂×#4326	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#10622124	1	2	yes	—
209	C₂²	C₂×#66885	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	88	2	no	—
214	C₄	—	88	2	no	—
215	C₄	#10622124	1	2	yes	—
216	C₄	#10622517	1	2	yes	—
217	C₄	C₂×#66874	1	1	yes	characteristic
218	C₄	C₂×#66888	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₂×#10622124	1	1	yes	characteristic
221	C₂	#13070324	1	1	yes	characteristic
222	C₂	#13070366	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#66885
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
socle	C₂²×C₁₁	C₂×#102
prime radical	C₂³×C₁₁	#102
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	#11939
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #13066673	C₂	semidirect
4. #13066670	C₂	semidirect
184. C₁₁	#11939	semidirect

Maximal subgroups

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

subgroup	index
#13079130	2
#13067364	2
#13066673	2
#13066670	2
C₂×C₁₁×C₄⋊C₁₆	2
#11939	11