Subgroups — #14665207

Order 2816.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070415	C₂	1	1	yes	maximal, characteristic
2	#13067375	C₂	1	1	yes	maximal, characteristic
3	#13066674	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×C₄⋊Q₈	—	2	1	no	—
6	C₁₁×#500	—	2	1	no	—
7	C₁₁×#500	—	2	1	no	—
8	C₁₁×#500	C₄	1	2	yes	—
9	C₁₁×#500	C₂²	1	2	yes	—
10	C₂×C₁₁×#107	—	2	1	no	—
11	C₂×C₁₁×#107	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622537	—	2	1	no	—
14	C₂²⋊(C₁₁⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
16	C₄×C₁₁⋊C₁₆	—	2	1	no	—
17	#10622096	—	2	1	no	—
18	#10622096	C₂²	1	2	yes	—
19	C₂²×Q₈×C₁₁	—	4	1	no	—
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
22	C₁₁×C₄⋊Q₈	—	2	2	no	—
23	C₁₁×C₄⋊Q₈	—	2	2	no	—
24	C₁₁×C₄⋊Q₈	—	4	1	no	—
25	C₁₁×C₄⋊Q₈	—	4	1	no	—
26	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
28	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
29	C₂×#4318	—	4	1	no	—
30	C₁₁×#107	—	2	2	no	—
31	C₁₁×#107	—	2	2	no	—
32	C₁₁×#107	—	2	2	no	—
33	C₁₁×#107	C₂×C₄	1	4	yes	—
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	C₂×C₁₁⋊C₁₆	—	2	2	no	—
39	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
40	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
41	#11972	—	11	1	no	maximal, Sylow
42	C₂×Q₈×C₁₁	—	4	2	no	—
43	C₂×Q₈×C₁₁	—	4	2	no	—
44	C₂×Q₈×C₁₁	—	8	4	no	—
45	C₂²×C₄×C₁₁	—	4	2	no	—
46	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₁	—	2	2	no	—
50	C₂×C₈×C₁₁	—	2	2	no	—
51	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₁₁×C₄⋊C₄	—	2	4	no	—
56	C₁₁×C₄⋊C₄	—	4	2	no	—
57	C₁₁×C₄⋊C₄	—	4	2	no	—
58	C₁₁×C₄⋊C₄	—	4	2	no	—
59	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₁	D₈	1	1	yes	characteristic
61	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
62	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
63	#4318	—	8	2	no	—
64	C₂×C₁₁⋊C₈	—	4	2	no	—
65	C₁₁⋊C₁₆	—	4	2	no	—
66	C₂×#500	D₁₁	1	1	yes	characteristic
67	#2162	—	11	1	no	—
68	#1567	—	11	1	no	—
69	#1365	—	11	4	no	—
70	C₂³×C₁₁	#102	1	1	yes	characteristic
71	Q₈×C₁₁	—	8	4	no	—
72	Q₈×C₁₁	—	8	4	no	—
73	C₂×C₄×C₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	1	no	—
75	C₂×C₄×C₁₁	—	2	1	no	—
76	C₂×C₄×C₁₁	—	4	4	no	—
77	C₂×C₄×C₁₁	—	8	4	no	—
78	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₁	SD₃₂	1	2	yes	—
82	C₂×C₄×C₁₁	Q₃₂	1	2	yes	—
83	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
86	C₈×C₁₁	—	2	1	no	—
87	C₈×C₁₁	—	2	1	no	—
88	C₈×C₁₁	#101	1	2	yes	—
89	C₈×C₁₁	#137	1	2	yes	—
90	C₁₁⋊C₈	—	8	2	no	—
91	C₂×C₄⋊Q₈	—	2	1	no	—
92	#500	—	2	1	no	—
93	#500	—	2	1	no	—
94	#500	Dic₁₁	1	2	yes	—
95	#500	C₂×D₁₁	1	2	yes	—
96	C₂×#107	—	2	1	no	—
97	C₂×#107	Dic₁₁	1	2	yes	—
98	#423	—	22	1	no	—
99	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
100	C₄⋊C₁₆	—	11	2	no	—
101	C₂²⋊C₁₆	—	22	1	no	—
102	C₄×C₁₆	—	22	1	no	—
103	#335	—	11	2	no	—
104	#335	—	22	1	no	—
105	C₂²×C₁₁	—	2	1	no	—
106	C₂²×C₁₁	—	2	1	no	—
107	C₂²×C₁₁	#358	1	2	yes	—
108	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
109	C₄×C₁₁	—	2	1	no	—
110	C₄×C₁₁	—	2	1	no	—
111	C₄×C₁₁	—	8	4	no	—
112	C₄×C₁₁	#358	1	2	yes	—
113	C₄×C₁₁	#361	1	2	yes	—
114	C₄×C₁₁	C₂×#101	1	1	yes	characteristic
115	C₄×C₁₁	#419	1	1	yes	characteristic
116	C₄×C₁₁	C₂×SD₃₂	1	1	yes	characteristic
117	C₄×C₁₁	C₂×Q₃₂	1	1	yes	characteristic
118	C₂²×Q₈	—	4	1	no	—
119	C₂×M₁₆	—	44	1	no	—
120	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	2	no	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	4	1	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
128	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
129	C₂×C₁₆	—	22	2	no	—
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	C₂×Dic₁₁	1	4	yes	—
134	C₄⋊C₈	—	22	2	no	—
135	C₄⋊C₈	—	44	1	no	—
136	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
139	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
140	C₂×C₁₁	—	2	2	no	—
141	C₂×C₁₁	#1546	1	1	yes	characteristic
142	C₂×C₁₁	C₂×#358	1	1	yes	characteristic
143	C₂×C₁₁	C₂×#361	1	1	yes	characteristic
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	8	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	#4323	1	1	yes	characteristic
149	C₂²×C₄	#4324	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
151	M₁₆	—	88	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	44	2	no	—
155	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄²	#4323	1	1	yes	characteristic
164	C₄²	#4324	1	1	yes	characteristic
165	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
167	C₁₆	—	44	2	no	—
168	C₁₁	#11972	1	1	yes	characteristic, complemented, Sylow
169	C₂³	#66896	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#66892	1	2	yes	—
178	C₂×C₄	#66893	1	2	yes	—
179	C₂×C₄	#66896	1	1	yes	characteristic
180	C₂×C₄	#66896	1	1	yes	characteristic
181	C₂×C₄	#66896	1	1	yes	characteristic
182	C₂×C₄	C₂×#4323	1	1	yes	characteristic
183	C₂×C₄	C₂×#4324	1	1	yes	characteristic
184	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	88	2	no	—
188	C₈	#66900	1	2	yes	—
189	C₈	#66995	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#10622202	1	2	yes	—
193	C₂²	C₂×#66896	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	#10622201	1	2	yes	—
198	C₄	#10622202	1	2	yes	—
199	C₄	C₂×#66892	1	1	yes	characteristic
200	C₄	C₂×#66893	1	1	yes	characteristic
201	C₄	#10622838	1	1	yes	characteristic
202	C₄	C₂×#66900	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#13067288	1	1	yes	characteristic
205	C₂	C₂×#10622202	1	1	yes	characteristic
206	C₂	C₂×#10622201	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#66896
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#500	C₂
socle	C₂²×C₁₁	C₂×#102
prime radical	C₂³×C₁₁	#102
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₂×#500	D₁₁
hypocenter	C₁₁	#11972
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #13066674	C₂	semidirect
168. C₁₁	#11972	semidirect

Maximal subgroups

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

subgroup	index
#13070415	2
#13067375	2
#13066674	2
C₂×C₁₁×#500	2
#11972	11