Subgroups — #14671804

Order 2816.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622142	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	#13069350	C₂	1	2	yes	maximal
4	#13066707	C₂	1	2	yes	maximal
5	C₂×C₁₁×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#107	C₂²	1	2	yes	—
8	C₂×#66882	C₂²	1	2	yes	—
9	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#10622598	—	2	2	no	—
12	C₁₁×#366	C₄	1	4	yes	complemented
13	#10622167	—	2	2	no	—
14	#10622142	—	2	2	no	—
15	C₁₆×Dic₁₁	—	2	2	no	—
16	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
17	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄⋊Dic₁₁	—	2	2	no	—
19	C₂×C₄⋊Dic₁₁	—	4	2	no	—
20	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
21	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
24	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
25	C₁₁×#107	—	2	2	no	—
26	C₁₁×#107	C₂×C₄	1	4	yes	—
27	C₁₁⋊#95	—	4	2	no	—
28	#66882	—	2	2	no	—
29	#66882	—	2	2	no	—
30	C₈×Dic₁₁	—	2	1	no	—
31	C₈×Dic₁₁	—	2	1	no	—
32	C₂×C₁₁⋊C₁₆	—	2	1	no	—
33	C₂×C₁₁⋊C₁₆	—	2	1	no	—
34	C₂×C₁₁⋊C₁₆	Q₈	1	2	yes	—
35	#13165	—	11	1	no	maximal, Sylow
36	C₂²×C₄×C₁₁	—	4	2	no	—
37	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₁	—	4	2	no	—
39	C₂²×Dic₁₁	D₈	1	1	yes	characteristic
40	C₂²×Dic₁₁	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
47	C₁₁×C₄⋊C₄	—	2	4	no	—
48	C₁₁×C₄⋊C₄	—	4	2	no	—
49	C₄⋊Dic₁₁	—	4	2	no	—
50	C₄⋊Dic₁₁	—	4	2	no	—
51	C₄⋊Dic₁₁	—	8	4	no	—
52	C₄×Dic₁₁	—	2	1	no	—
53	C₄×Dic₁₁	—	2	1	no	—
54	C₂×C₁₁⋊C₈	—	2	1	no	—
55	C₂×C₁₁⋊C₈	—	2	1	no	—
56	C₂×C₁₁⋊C₈	Q₁₆	1	2	yes	—
57	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	complemented
58	C₁₁⋊C₁₆	—	2	2	no	—
59	C₂×#366	—	11	1	no	—
60	C₂×#366	D₁₁	1	1	yes	characteristic
61	C₂×C₄×C₁₆	—	11	1	no	—
62	#1942	—	11	2	no	—
63	#1380	—	11	2	no	—
64	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₁	—	4	4	no	—
66	C₂×C₄×C₁₁	—	8	2	no	—
67	C₂×C₄×C₁₁	—	8	2	no	—
68	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
73	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
74	C₂×Dic₁₁	—	2	1	no	—
75	C₂×Dic₁₁	—	2	1	no	—
76	C₂×Dic₁₁	—	2	1	no	—
77	C₂×Dic₁₁	—	2	1	no	—
78	C₂×Dic₁₁	—	8	2	no	—
79	C₂×Dic₁₁	—	8	2	no	—
80	C₂×Dic₁₁	—	8	2	no	—
81	C₂×Dic₁₁	D₁₆	1	2	yes	—
82	C₂×Dic₁₁	Q₃₂	1	2	yes	—
83	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₁₁	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₈×C₁₁	#125	1	1	yes	characteristic
87	C₁₁⋊C₈	—	2	2	no	—
88	C₂²×C₁₆	—	11	1	no	—
89	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
90	C₂×#107	—	11	2	no	—
91	C₂×#107	C₂×D₁₁	1	2	yes	—
92	C₂×C₄×C₈	—	11	1	no	—
93	#389	—	22	2	no	—
94	#366	—	22	2	no	—
95	#366	Dic₁₁	1	4	yes	complemented
96	C₄×C₁₆	—	22	2	no	—
97	#340	—	22	2	no	—
98	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₁₁	#466	1	1	yes	characteristic
100	C₂²×C₁₁	#500	1	1	yes	characteristic
101	C₂²×C₁₁	C₂×D₁₆	1	1	yes	characteristic
102	C₂²×C₁₁	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₁₁	#508	1	1	yes	characteristic
104	C₂²×C₁₁	#508	1	1	yes	characteristic
105	C₄×C₁₁	—	8	4	no	—
106	C₄×C₁₁	#389	1	1	yes	characteristic
107	C₄×C₁₁	C₄×Q₁₆	1	1	yes	characteristic
108	C₄×C₁₁	#467	1	1	yes	characteristic
109	C₄×C₁₁	#499	1	1	yes	characteristic
110	Dic₁₁	—	2	2	no	—
111	Dic₁₁	—	2	2	no	—
112	Dic₁₁	—	16	2	no	—
113	C₂²×C₈	—	11	1	no	—
114	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	22	2	no	—
117	C₂×C₄⋊C₄	—	44	2	no	—
118	C₂×C₄²	—	11	1	no	—
119	C₂×C₁₆	—	11	2	no	—
120	C₂×C₁₆	—	22	1	no	—
121	C₂×C₁₆	—	22	1	no	—
122	C₂×C₁₆	Dic₂₂	1	2	yes	—
123	C₂×C₁₆	D₄₄	1	2	yes	—
124	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
125	#107	—	2	2	no	—
126	#107	—	22	2	no	—
127	#107	—	22	2	no	—
128	#107	C₂×Dic₁₁	1	4	yes	—
129	C₄×C₈	—	22	1	no	—
130	C₄×C₈	—	22	1	no	—
131	#95	—	44	2	no	—
132	C₂×C₁₁	#1942	1	1	yes	characteristic
133	C₂×C₁₁	C₄×D₁₆	1	1	yes	characteristic
134	C₂×C₁₁	C₄×Q₃₂	1	1	yes	characteristic
135	C₂×C₁₁	#2215	1	1	yes	characteristic
136	C₂×C₁₁	#2216	1	1	yes	characteristic
137	C₂×C₁₁	#2252	1	1	yes	characteristic
138	C₂×C₁₁	#2253	1	1	yes	characteristic
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	11	1	no	—
141	C₂²×C₄	—	11	1	no	—
142	C₂²×C₄	—	44	2	no	—
143	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
144	C₂×C₈	—	11	2	no	—
145	C₂×C₈	—	22	1	no	—
146	C₂×C₈	—	22	1	no	—
147	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
148	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
149	C₂×C₈	#4341	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
151	C₂×C₈	#4343	1	1	yes	characteristic
152	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	44	2	no	—
156	C₄⋊C₄	—	44	2	no	—
157	C₄⋊C₄	—	88	4	no	—
158	C₄²	—	22	1	no	—
159	C₄²	—	22	1	no	—
160	C₁₆	—	22	2	no	—
161	C₁₆	C₄⋊Dic₁₁	1	4	yes	complemented
162	C₁₁	#13165	1	1	yes	characteristic, complemented, Sylow
163	C₂³	D₈×D₁₁	1	1	yes	characteristic
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	11	2	no	—
168	C₂×C₄	—	11	2	no	—
169	C₂×C₄	—	22	1	no	—
170	C₂×C₄	—	22	1	no	—
171	C₂×C₄	—	22	1	no	—
172	C₂×C₄	—	22	1	no	—
173	C₂×C₄	—	88	2	no	—
174	C₂×C₄	—	88	2	no	—
175	C₂×C₄	—	88	2	no	—
176	C₂×C₄	#66941	1	1	yes	characteristic
177	C₂×C₄	#66948	1	1	yes	characteristic
178	C₂×C₄	#66965	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₁₁	1	1	yes	characteristic
180	C₂×C₄	#66972	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
182	C₈	—	22	2	no	—
183	C₈	#66943	1	1	yes	characteristic
184	C₈	#66950	1	1	yes	characteristic
185	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
186	C₈	Q₈×Dic₁₁	1	1	yes	characteristic
187	C₂²	#10622487	1	1	yes	characteristic
188	C₂²	#10622496	1	1	yes	characteristic
189	C₂²	D₁₁×D₁₆	1	1	yes	characteristic
190	C₂²	#10622525	1	1	yes	characteristic
191	C₂²	D₁₁×Q₃₂	1	1	yes	characteristic
192	C₂²	#10622532	1	1	yes	characteristic
193	C₂²	D₈×Dic₁₁	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	22	2	no	—
196	C₄	—	22	2	no	—
197	C₄	—	176	2	no	—
198	C₄	#10622491	1	1	yes	characteristic
199	C₄	#10622500	1	1	yes	characteristic
200	C₄	#10622598	1	1	yes	characteristic
201	C₄	Q₁₆×Dic₁₁	1	1	yes	characteristic
202	C₂	#13069350	1	1	yes	characteristic
203	C₂	D₁₆×Dic₁₁	1	1	yes	characteristic
204	C₂	Q₃₂×Dic₁₁	1	1	yes	characteristic
205	C₂	#13070703	1	1	yes	characteristic
206	C₂	#13070708	1	1	yes	characteristic
207	C₂	#13070890	1	1	yes	characteristic
208	C₂	#13070893	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₁
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#366	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#366	D₁₁
hypocenter	C₁₁	#13165
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
12. C₁₁×#366	C₄	semidirect
57. C₁₁×C₁₆	C₄⋊C₄	semidirect
95. #366	Dic₁₁	semidirect
161. C₁₆	C₄⋊Dic₁₁	semidirect
162. C₁₁	#13165	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622142	2
C₂×C₁₆×Dic₁₁	2
#13069350	2
#13066707	2
C₂×C₁₁×#366	2
#13165	11