Subgroups — #14671818

Order 2816.

318 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622143	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622142	C₂	1	1	yes	maximal, characteristic
3	#13069349	C₂	1	2	yes	maximal
4	#13066708	C₂	1	2	yes	maximal
5	C₂×C₁₁×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×#66882	C₂²	1	2	yes	—
10	C₂×#66882	C₂²	1	2	yes	—
11	#10622541	—	2	2	no	—
12	C₁₁×#346	—	2	2	no	—
13	#10622189	—	2	2	no	—
14	#10622143	—	2	2	no	—
15	#10622142	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₁	—	2	2	no	—
20	C₂×C₄⋊Dic₁₁	—	2	2	no	—
21	C₂×Dic₁₁⋊C₄	—	4	2	no	—
22	C₂×C₁₁×C₁₆	—	2	1	no	—
23	C₂×C₁₁×C₁₆	—	2	1	no	—
24	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
26	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
27	C₂×C₁₁×C₁₆	Q₈	1	2	yes	—
28	C₄×C₈×C₁₁	—	2	1	no	—
29	C₄×C₈×C₁₁	—	2	1	no	—
30	C₁₁⋊#95	—	4	2	no	—
31	#66882	—	2	2	no	—
32	#66882	—	2	2	no	—
33	#66882	—	2	2	no	—
34	#66882	C₂×C₄	1	4	yes	—
35	#13168	—	11	1	no	maximal, Sylow
36	C₂²×C₄×C₁₁	D₈	1	2	yes	—
37	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₁	—	4	2	no	—
39	C₂²×Dic₁₁	—	4	2	no	—
40	C₂×C₈×C₁₁	—	2	1	no	—
41	C₂×C₈×C₁₁	—	2	1	no	—
42	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
49	C₄²×C₁₁	—	2	1	no	—
50	C₄²×C₁₁	—	2	1	no	—
51	C₄⋊Dic₁₁	—	2	4	no	—
52	C₄⋊Dic₁₁	—	4	2	no	—
53	C₄⋊Dic₁₁	—	4	2	no	—
54	C₄⋊Dic₁₁	—	4	2	no	—
55	Dic₁₁⋊C₄	—	8	4	no	—
56	C₁₁×C₁₆	—	2	2	no	—
57	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	complemented
58	C₂×#367	—	11	1	no	—
59	C₂×#366	—	11	1	no	—
60	C₂×#346	D₁₁	1	1	yes	characteristic
61	#1942	—	11	2	no	—
62	#1380	—	11	2	no	—
63	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₁	—	2	2	no	—
65	C₂×C₄×C₁₁	—	2	2	no	—
66	C₂×C₄×C₁₁	—	2	2	no	—
67	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
73	C₂×Dic₁₁	—	4	4	no	—
74	C₂×Dic₁₁	—	8	2	no	—
75	C₂×Dic₁₁	—	8	2	no	—
76	C₂×Dic₁₁	—	8	2	no	—
77	C₂×Dic₁₁	—	8	2	no	—
78	C₂×Dic₁₁	—	8	2	no	—
79	C₈×C₁₁	—	2	2	no	—
80	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₁	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
83	C₈×C₁₁	#125	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
86	C₂×#107	—	11	2	no	—
87	C₂×#107	—	11	2	no	—
88	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
89	#389	—	22	2	no	—
90	#367	—	22	2	no	—
91	#366	—	11	4	no	—
92	#346	—	2	2	no	—
93	#340	—	22	2	no	—
94	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
95	C₂²×C₁₁	#466	1	1	yes	characteristic
96	C₂²×C₁₁	#500	1	1	yes	characteristic
97	C₂²×C₁₁	#509	1	2	yes	—
98	C₂²×C₁₁	#510	1	2	yes	—
99	C₄×C₁₁	—	4	2	no	—
100	C₄×C₁₁	#389	1	1	yes	characteristic
101	C₄×C₁₁	C₄×Q₁₆	1	1	yes	characteristic
102	C₄×C₁₁	#467	1	1	yes	characteristic
103	C₄×C₁₁	#499	1	1	yes	characteristic
104	Dic₁₁	—	8	4	no	—
105	Dic₁₁	—	16	2	no	—
106	C₂²×C₈	D₄₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	22	2	no	—
109	C₂×C₄⋊C₄	—	22	2	no	—
110	C₂×C₄⋊C₄	—	44	2	no	—
111	C₂×C₄²	D₄₄	1	1	yes	characteristic
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	Dic₂₂	1	2	yes	—
115	C₂×C₁₆	Dic₂₂	1	2	yes	—
116	C₂×C₁₆	C₄×D₁₁	1	2	yes	—
117	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
118	#107	—	11	4	no	—
119	#107	—	22	2	no	—
120	#107	—	22	2	no	—
121	#107	—	22	2	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	#95	—	44	2	no	—
125	C₂×C₁₁	#1942	1	1	yes	characteristic
126	C₂×C₁₁	#2176	1	1	yes	characteristic
127	C₂×C₁₁	#2177	1	1	yes	characteristic
128	C₂×C₁₁	#2220	1	2	yes	—
129	C₂×C₁₁	#2255	1	2	yes	—
130	C₂²×C₄	—	44	2	no	—
131	C₂²×C₄	—	44	2	no	—
132	C₂²×C₄	D₈₈	1	2	yes	—
133	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	Dic₄₄	1	2	yes	—
137	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
140	C₂×C₈	#4339	1	1	yes	characteristic
141	C₂×C₈	#4339	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
143	C₄⋊C₄	—	22	4	no	—
144	C₄⋊C₄	—	44	2	no	—
145	C₄⋊C₄	—	44	2	no	—
146	C₄⋊C₄	—	44	2	no	—
147	C₄⋊C₄	—	88	4	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	2	1	no	—
150	C₁₆	—	2	2	no	—
151	C₁₆	Dic₁₁⋊C₄	1	4	yes	complemented
152	C₁₁	#13168	1	1	yes	characteristic, complemented, Sylow
153	C₂³	C₂×D₈₈	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	44	4	no	—
158	C₂×C₄	—	88	2	no	—
159	C₂×C₄	—	88	2	no	—
160	C₂×C₄	—	88	2	no	—
161	C₂×C₄	—	88	2	no	—
162	C₂×C₄	—	88	2	no	—
163	C₂×C₄	#66922	1	1	yes	characteristic
164	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
165	C₂×C₄	#66957	1	1	yes	characteristic
166	C₂×C₄	#66957	1	1	yes	characteristic
167	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
168	C₂×C₄	#66983	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₄×Dic₂₂	1	1	yes	characteristic
171	C₈	#66923	1	1	yes	characteristic
172	C₈	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
173	C₈	#66977	1	1	yes	characteristic
174	C₂²	#10622295	1	1	yes	characteristic
175	C₂²	C₄×D₈₈	1	1	yes	characteristic
176	C₂²	#10622521	1	2	yes	—
177	C₂²	#10622522	1	2	yes	—
178	C₂²	#10622728	1	1	yes	characteristic
179	C₄	—	4	2	no	—
180	C₄	—	88	4	no	—
181	C₄	—	176	2	no	—
182	C₄	#10622296	1	1	yes	characteristic
183	C₄	C₄×Dic₄₄	1	1	yes	characteristic
184	C₄	#10622541	1	1	yes	characteristic
185	C₄	#10622729	1	1	yes	characteristic
186	C₂	#13069349	1	1	yes	characteristic
187	C₂	#13070488	1	1	yes	characteristic
188	C₂	#13070493	1	1	yes	characteristic
189	C₂	#13070729	1	2	yes	—
190	C₂	#13070902	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#346	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#346	D₁₁
hypocenter	C₁₁	#13168
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
15. #10622142	C₄	semidirect
57. C₁₁×C₁₆	C₄⋊C₄	semidirect
151. C₁₆	Dic₁₁⋊C₄	semidirect
152. C₁₁	#13168	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622143	2
C₂×#10622142	2
#13069349	2
#13066708	2
C₂×C₁₁×#346	2
#13168	11