Subgroups — #14645014

Order 2816.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068485	C₂	1	2	yes	maximal
2	C₂×#10622095	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#13066457	C₂	1	2	yes	maximal
5	C₄×C₁₁×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#106	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
10	C₂×#66867	—	2	2	no	—
11	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₁₁×#340	C₄	1	2	yes	—
14	#10622162	—	2	2	no	—
15	#10622095	C₄	1	4	yes	complemented
16	C₈×C₁₁⋊C₈	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₁	—	2	2	no	—
21	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
22	C₂²×C₁₁⋊C₈	—	2	1	no	—
23	C₂²×C₁₁⋊C₈	—	2	2	no	—
24	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
25	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
26	C₁₁×#106	—	2	4	no	—
27	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
29	C₁₁×#95	—	2	2	no	—
30	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
31	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
32	#66867	—	4	4	no	—
33	C₄×C₁₁⋊C₈	—	2	1	no	—
34	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₁₁⋊C₈	Q₈	1	2	yes	—
37	#8283	—	11	1	no	maximal, Sylow
38	C₂²×C₄×C₁₁	—	2	4	no	—
39	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
42	C₂×C₈×C₁₁	—	2	2	no	—
43	C₂×C₈×C₁₁	C₄²	1	2	yes	—
44	C₂×C₈×C₁₁	C₄²	1	2	yes	—
45	C₂×C₈×C₁₁	M₁₆	1	4	yes	—
46	C₁₁×C₄⋊C₄	—	2	4	no	—
47	C₁₁×C₄⋊C₄	—	2	4	no	—
48	C₄²×C₁₁	—	4	2	no	—
49	C₄²×C₁₁	—	4	2	no	—
50	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₁₁	#41	1	1	yes	characteristic
54	C₂×C₁₁⋊C₈	—	2	4	no	—
55	C₂×C₁₁⋊C₈	—	2	2	no	—
56	C₂×C₁₁⋊C₈	—	2	4	no	—
57	C₂×C₁₁⋊C₈	—	2	2	no	—
58	C₂×C₁₁⋊C₈	—	4	2	no	—
59	C₂×C₁₁⋊C₈	—	4	2	no	—
60	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₁₁⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₁₁⋊C₈	SD₁₆	1	4	yes	—
63	C₄×#106	D₁₁	1	1	yes	characteristic
64	#1770	—	11	2	no	—
65	C₂×#334	—	11	1	no	—
66	C₂×C₈²	—	11	1	no	—
67	#1279	—	11	2	no	—
68	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	4	4	no	—
71	C₂×C₄×C₁₁	—	4	4	no	—
72	C₂×C₄×C₁₁	—	4	4	no	—
73	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
76	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₁₁	#131	1	2	yes	—
81	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₁	#135	1	2	yes	—
83	C₈×C₁₁	—	2	2	no	—
84	C₈×C₁₁	#97	1	4	yes	complemented
85	C₁₁⋊C₈	—	2	4	no	—
86	C₁₁⋊C₈	—	2	2	no	—
87	C₁₁⋊C₈	—	4	4	no	—
88	C₁₁⋊C₈	#106	1	4	yes	complemented
89	C₂×#106	Dic₁₁	1	2	yes	—
90	C₂×C₄⋊C₈	—	11	2	no	—
91	C₂×#97	—	22	2	no	—
92	C₂×C₄×C₈	—	11	1	no	—
93	C₂×C₄×C₈	—	11	1	no	—
94	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
95	C₄×C₄⋊C₄	C₂×D₁₁	1	2	yes	—
96	#340	Dic₁₁	1	2	yes	—
97	#336	—	22	2	no	—
98	#334	—	11	4	no	—
99	C₈²	—	11	4	no	—
100	C₂²×C₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₁	#436	1	2	yes	—
102	C₂²×C₁₁	C₄×SD₁₆	1	2	yes	—
103	C₂²×C₁₁	#443	1	2	yes	—
104	C₄×C₁₁	—	2	1	no	—
105	C₄×C₁₁	—	2	1	no	—
106	C₄×C₁₁	—	8	4	no	—
107	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
108	C₄×C₁₁	#405	1	1	yes	characteristic
109	C₄×C₁₁	C₂×#106	1	1	yes	characteristic
110	C₄×C₁₁	#427	1	1	yes	characteristic
111	C₄×C₁₁	C₄×SD₁₆	1	2	yes	—
112	C₄×C₁₁	#446	1	2	yes	—
113	C₂²×C₈	—	11	1	no	—
114	C₂²×C₈	—	11	1	no	—
115	C₂²×C₈	—	22	1	no	—
116	C₂²×C₈	—	22	2	no	—
117	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×Dic₁₁	1	4	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
122	#106	—	2	4	no	—
123	C₄⋊C₈	—	11	4	no	—
124	C₄⋊C₈	—	22	2	no	—
125	#97	—	44	4	no	—
126	C₄×C₈	—	11	2	no	—
127	C₄×C₈	—	11	2	no	—
128	C₄×C₈	—	11	2	no	—
129	C₄×C₈	—	22	1	no	—
130	C₄×C₈	C₄×D₁₁	1	2	yes	—
131	C₄×C₈	C₄×D₁₁	1	2	yes	—
132	#95	—	2	2	no	—
133	C₂×C₁₁	#1590	1	4	yes	—
134	C₂×C₁₁	#1770	1	1	yes	characteristic
135	C₂×C₁₁	#1772	1	1	yes	characteristic
136	C₂×C₁₁	C₄×#106	1	1	yes	characteristic
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
139	C₂²×C₄	#4341	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	11	2	no	—
143	C₂×C₈	—	11	4	no	—
144	C₂×C₈	—	11	2	no	—
145	C₂×C₈	—	22	4	no	—
146	C₂×C₈	—	22	2	no	—
147	C₂×C₈	—	22	2	no	—
148	C₂×C₈	—	22	4	no	—
149	C₂×C₈	—	44	2	no	—
150	C₂×C₈	—	44	2	no	—
151	C₂×C₈	#4318	1	4	yes	—
152	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
153	C₂×C₈	C₄×Dic₁₁	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄²	—	4	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
159	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
160	C₄²	Q₈×D₁₁	1	1	yes	characteristic
161	C₄²	#4343	1	1	yes	characteristic
162	C₁₁	#8283	1	1	yes	characteristic, complemented, Sylow
163	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#66940	1	1	yes	characteristic
169	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
170	C₂×C₄	#66945	1	1	yes	characteristic
171	C₂×C₄	#66946	1	1	yes	characteristic
172	C₂×C₄	SD₁₆×D₁₁	1	2	yes	—
173	C₂×C₄	#66969	1	2	yes	—
174	C₂×C₄	C₂×#4318	1	2	yes	—
175	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
176	C₂×C₄	Q₈×Dic₁₁	1	1	yes	characteristic
177	C₂×C₄	#67001	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	11	4	no	—
180	C₈	—	22	4	no	—
181	C₈	—	22	2	no	—
182	C₈	—	44	4	no	—
183	C₈	#66867	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₁₁	1	1	yes	characteristic
185	C₂²	#10622625	1	2	yes	—
186	C₂²	SD₁₆×Dic₁₁	1	2	yes	—
187	C₂²	#10622809	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	8	4	no	—
191	C₄	#10622465	1	2	yes	—
192	C₄	D₁₁×#106	1	1	yes	characteristic
193	C₄	#10622473	1	1	yes	characteristic
194	C₄	C₂×#66867	1	1	yes	characteristic
195	C₄	#10622610	1	1	yes	characteristic
196	C₄	#10622637	1	2	yes	—
197	C₂	#13067507	1	4	yes	—
198	C₂	#13068485	1	1	yes	characteristic
199	C₂	#13068497	1	1	yes	characteristic
200	C₂	#106×Dic₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₄×C₁₁×#106	C₂
socle, prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₄×#106	D₁₁
hypocenter	C₁₁	#8283
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
15. #10622095	C₄	semidirect
16. C₈×C₁₁⋊C₈	C₄	semidirect
84. C₈×C₁₁	#97	semidirect
88. C₁₁⋊C₈	#106	semidirect
162. C₁₁	#8283	semidirect
183. C₈	#66867	semidirect

Maximal subgroups

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

subgroup	index
#13068485	2
C₂×#10622095	2
C₂×C₈×C₁₁⋊C₈	2
#13066457	2
C₄×C₁₁×#106	2
#8283	11