Subgroups — #14665074

Order 2816.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 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	#13070372	C₂	1	1	yes	maximal, characteristic
3	#13066675	C₂	1	2	yes	maximal, complemented
4	#13066672	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic
6	#10623001	—	2	1	no	—
7	C₂×C₁₁×C₄⋊C₈	—	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	C₂×#66876	—	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×#335	—	2	1	no	—
18	C₁₁×#335	C₂²	1	2	yes	—
19	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
20	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₁	—	4	1	no	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
25	#66979	—	2	2	no	—
26	#66977	—	4	1	no	—
27	#66977	—	4	1	no	—
28	#66923	—	2	2	no	—
29	#66922	—	2	1	no	—
30	#66922	—	2	1	no	—
31	C₄×Dic₂₂	—	4	1	no	—
32	C₄×Dic₂₂	—	4	1	no	—
33	C₁₁×C₄⋊C₈	—	2	2	no	—
34	C₁₁×C₄⋊C₈	—	4	1	no	—
35	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
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	#66876	—	4	2	no	—
46	C₂×C₁₁⋊C₁₆	—	2	2	no	—
47	#11942	—	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	—	4	2	no	—
55	C₂×C₈×C₁₁	—	8	2	no	—
56	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
59	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
60	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₁	D₈	1	1	yes	characteristic
64	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
65	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₁	—	4	2	no	—
67	C₄⋊Dic₁₁	—	2	2	no	—
68	C₄⋊Dic₁₁	—	2	2	no	—
69	C₄⋊Dic₁₁	—	4	1	no	—
70	C₄⋊Dic₁₁	—	4	1	no	—
71	Dic₁₁⋊C₄	—	4	2	no	—
72	Dic₁₁⋊C₄	—	4	2	no	—
73	C₄×Dic₁₁	—	4	2	no	—
74	C₁₁⋊C₁₆	—	4	2	no	—
75	#3160	—	11	1	no	—
76	#2150	—	11	1	no	—
77	C₂×#335	D₁₁	1	1	yes	characteristic
78	#1365	—	11	2	no	—
79	#1364	—	11	2	no	—
80	C₂³×C₁₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₁	—	2	1	no	—
82	C₂×C₄×C₁₁	—	2	1	no	—
83	C₂×C₄×C₁₁	—	4	1	no	—
84	C₂×C₄×C₁₁	#101	1	2	yes	—
85	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₁	#103	1	2	yes	—
89	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×Dic₁₁	—	4	2	no	—
93	C₂×Dic₁₁	—	4	2	no	—
94	Dic₂₂	—	8	2	no	—
95	Dic₂₂	—	8	1	no	—
96	Dic₂₂	—	8	1	no	—
97	C₈×C₁₁	—	2	1	no	—
98	C₈×C₁₁	—	2	1	no	—
99	C₈×C₁₁	—	8	2	no	—
100	C₈×C₁₁	#103	1	2	yes	—
101	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
102	C₈×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
103	#533	—	22	1	no	—
104	C₂×M₃₂	—	22	1	no	—
105	#500	—	11	1	no	—
106	#500	—	11	1	no	—
107	#499	—	11	2	no	—
108	#499	—	22	1	no	—
109	#498	—	22	1	no	—
110	#427	—	11	2	no	—
111	#427	—	22	1	no	—
112	C₂×C₄⋊C₈	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
114	C₄⋊C₁₆	—	11	2	no	—
115	C₄⋊C₁₆	—	22	1	no	—
116	#335	—	2	1	no	—
117	#335	C₂×D₁₁	1	2	yes	—
118	C₂²×C₁₁	—	2	1	no	—
119	C₂²×C₁₁	—	2	1	no	—
120	C₂²×C₁₁	#332	1	2	yes	—
121	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
122	C₄×C₁₁	—	2	1	no	—
123	C₄×C₁₁	—	2	1	no	—
124	C₄×C₁₁	—	2	1	no	—
125	C₄×C₁₁	#332	1	2	yes	—
126	C₄×C₁₁	C₂×#101	1	1	yes	characteristic
127	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
128	C₄×C₁₁	#509	1	1	yes	characteristic
129	C₄×C₁₁	#510	1	1	yes	characteristic
130	Dic₁₁	—	8	2	no	—
131	Dic₁₁	—	8	2	no	—
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
134	C₂²×C₈	D₄₄	1	1	yes	characteristic
135	C₄⋊Q₈	—	22	1	no	—
136	C₄⋊Q₈	—	22	1	no	—
137	#125	—	22	2	no	—
138	C₂²⋊Q₈	—	44	1	no	—
139	C₂²⋊Q₈	—	44	1	no	—
140	C₄×Q₈	—	44	1	no	—
141	C₄×Q₈	—	44	1	no	—
142	#117	—	22	2	no	—
143	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
144	M₃₂	—	44	2	no	—
145	C₂×C₁₆	—	22	2	no	—
146	#107	—	11	2	no	—
147	#107	—	11	2	no	—
148	#107	—	22	1	no	—
149	#107	—	22	1	no	—
150	#106	—	22	2	no	—
151	#106	—	22	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
155	C₄×C₈	D₄₄	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
158	C₂×C₁₁	—	2	2	no	—
159	C₂×C₁₁	C₂×#332	1	1	yes	characteristic
160	C₂×C₁₁	#2139	1	1	yes	characteristic
161	C₂×C₁₁	#2148	1	1	yes	characteristic
162	C₂×Q₈	—	44	1	no	—
163	C₂×Q₈	—	44	1	no	—
164	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	#4323	1	1	yes	characteristic
166	C₂²×C₄	#4325	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	8	2	no	—
170	C₂×C₈	#4314	1	2	yes	—
171	C₂×C₈	Dic₄₄	1	2	yes	—
172	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
176	C₄⋊C₄	—	22	2	no	—
177	C₄⋊C₄	—	22	2	no	—
178	C₄⋊C₄	—	44	2	no	—
179	C₄⋊C₄	—	44	2	no	—
180	C₄⋊C₄	—	44	1	no	—
181	C₄⋊C₄	—	44	1	no	—
182	C₂²⋊C₄	—	44	2	no	—
183	C₄²	—	44	2	no	—
184	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
185	C₄²	#4323	1	1	yes	characteristic
186	C₄²	#4325	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
188	C₁₆	—	44	2	no	—
189	C₁₁	#11942	1	1	yes	characteristic, complemented, Sylow
190	C₂³	#66873	1	1	yes	characteristic
191	Q₈	—	88	2	no	—
192	Q₈	—	88	1	no	—
193	Q₈	—	88	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	44	2	no	—
198	C₂×C₄	—	44	2	no	—
199	C₂×C₄	#66873	1	1	yes	characteristic
200	C₂×C₄	#66873	1	1	yes	characteristic
201	C₂×C₄	#66873	1	1	yes	characteristic
202	C₂×C₄	#66880	1	2	yes	—
203	C₂×C₄	#66888	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×#4323	1	1	yes	characteristic
206	C₂×C₄	C₂×#4325	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	8	2	no	—
210	C₈	#66880	1	2	yes	—
211	C₈	C₂×#4314	1	1	yes	characteristic
212	C₈	C₂×Dic₄₄	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#10622120	1	2	yes	—
216	C₂²	C₂×#66873	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	88	2	no	—
221	C₄	—	88	2	no	—
222	C₄	#10622120	1	2	yes	—
223	C₄	C₂×#66880	1	1	yes	characteristic
224	C₄	C₂×#66888	1	1	yes	characteristic
225	C₄	#10622760	1	1	yes	characteristic
226	C₄	#10622792	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#10622120	1	1	yes	characteristic
229	C₂	#13070326	1	1	yes	characteristic
230	C₂	#13070363	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#66873
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#335	C₂
socle	C₂²×C₁₁	C₂×#102
prime radical	C₂³×C₁₁	#102
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₂×#335	D₁₁
hypocenter	C₁₁	#11942
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #13066675	C₂	semidirect
4. #13066672	C₂	semidirect
189. C₁₁	#11942	semidirect

Maximal subgroups

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

subgroup	index
#13079130	2
#13070372	2
#13066675	2
#13066672	2
C₂×C₁₁×#335	2
#11942	11