Subgroups — #27302701

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20272807	C₂	1	4	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	—	2	1	no	—
6	#16811025	—	2	2	no	—
7	#16811025	C₄	1	4	yes	complemented
8	C₂×#13133151	—	2	1	no	—
9	C₂×#13133151	C₂²	1	2	yes	—
10	C₂×#13133151	C₂²	1	2	yes	—
11	C₂×#13133149	—	2	1	no	—
12	#16810493	—	2	1	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#346	—	2	2	no	—
16	#13133209	—	2	4	no	—
17	#13133209	—	4	2	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	4	2	no	—
21	C₂×#11719919	—	4	2	no	—
22	#13133151	—	2	2	no	—
23	#13133151	—	2	4	no	—
24	#13133151	—	4	2	no	—
25	#13133151	C₂×C₄	1	4	yes	—
26	#13133149	—	2	4	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	—	2	2	no	—
32	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₂²×C₄×C₅₃	D₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	4	2	no	—
38	C₂²×Dic₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₅₃	SD₁₆	1	2	yes	—
48	C₄⋊Dic₅₃	—	2	4	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄⋊Dic₅₃	—	4	2	no	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	8	4	no	—
53	#11719919	—	4	8	no	—
54	C₁₆×C₅₃	—	2	4	no	—
55	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	#102	1	4	yes	—
59	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₅₃	—	4	4	no	—
66	C₂×Dic₅₃	—	8	2	no	—
67	C₂×Dic₅₃	—	8	2	no	—
68	C₂×Dic₅₃	—	8	2	no	—
69	C₂×Dic₅₃	—	8	4	no	—
70	C₈×C₅₃	#103	1	4	yes	—
71	C₈×C₅₃	C₄×Q₈	1	2	yes	—
72	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	#13284	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
75	C₂²×C₅₃	#486	1	1	yes	characteristic
76	C₂²×C₅₃	#500	1	1	yes	characteristic
77	C₂²×C₅₃	#509	1	2	yes	—
78	C₂²×C₅₃	#510	1	2	yes	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	#391	1	1	yes	characteristic
81	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
82	C₄×C₅₃	#487	1	1	yes	characteristic
83	C₄×C₅₃	#498	1	1	yes	characteristic
84	Dic₅₃	—	8	4	no	—
85	Dic₅₃	—	16	2	no	—
86	C₂×#499	—	53	1	no	—
87	C₂×#346	D₅₃	1	1	yes	characteristic
88	#1845	—	53	1	no	—
89	#1380	—	53	4	no	—
90	C₂×C₅₃	#1990	1	1	yes	characteristic
91	C₂×C₅₃	#2139	1	2	yes	—
92	C₂×C₅₃	#2244	1	1	yes	characteristic
93	C₂×C₅₃	#2245	1	1	yes	characteristic
94	C₂×C₅₃	#2255	1	2	yes	—
95	C₂×#125	—	106	1	no	—
96	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
97	#499	—	53	4	no	—
98	#499	—	106	2	no	—
99	C₂×#107	—	53	2	no	—
100	C₂×#107	—	53	2	no	—
101	C₂×#107	—	106	1	no	—
102	C₂×#106	—	106	1	no	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	#384	—	106	1	no	—
105	#346	—	2	2	no	—
106	C₅₃	#13284	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	#125	—	106	4	no	—
110	#125	—	212	2	no	—
111	C₂×C₄⋊C₄	—	106	2	no	—
112	C₂×C₄⋊C₄	—	106	2	no	—
113	C₂×C₄⋊C₄	—	212	2	no	—
114	C₂×C₄⋊C₄	—	212	2	no	—
115	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
119	#107	—	53	4	no	—
120	#107	—	106	2	no	—
121	#107	—	106	4	no	—
122	#107	—	212	2	no	—
123	#106	—	106	4	no	—
124	C₄×C₈	C₄×D₅₃	1	2	yes	—
125	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
126	C₂²×C₄	—	212	2	no	—
127	C₂²×C₄	—	212	2	no	—
128	C₂²×C₄	D₄₂₄	1	2	yes	—
129	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
130	C₂×C₈	#11719913	1	2	yes	—
131	C₂×C₈	Dic₂₁₂	1	2	yes	—
132	C₂×C₈	#11719921	1	2	yes	—
133	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
134	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
135	C₂×C₈	#11719945	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₄⋊C₄	—	106	4	no	—
138	C₄⋊C₄	—	424	4	no	—
139	C₄⋊C₄	—	212	2	no	—
140	C₄⋊C₄	—	212	2	no	—
141	C₄⋊C₄	—	212	8	no	—
142	C₄⋊C₄	—	212	2	no	—
143	C₄²	#11719913	1	2	yes	—
144	C₄²	#11719921	1	2	yes	—
145	C₁₆	—	2	4	no	—
146	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	424	4	no	—
150	C₂×C₄	—	424	2	no	—
151	C₂×C₄	—	424	2	no	—
152	C₂×C₄	—	424	2	no	—
153	C₂×C₄	—	212	4	no	—
154	C₂×C₄	#13133139	1	4	yes	—
155	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
156	C₂×C₄	#13133205	1	1	yes	characteristic
157	C₂×C₄	#13133217	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
159	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
161	C₈	#13133142	1	4	yes	—
162	C₈	C₄×Dic₁₀₆	1	2	yes	—
163	C₈	#13133196	1	2	yes	—
164	C₂²	C₂×#13133139	1	1	yes	characteristic
165	C₂²	#16810970	1	1	yes	characteristic
166	C₂²	#16811028	1	1	yes	characteristic
167	C₂²	#16811061	1	2	yes	—
168	C₂²	#16811066	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	848	2	no	—
171	C₄	—	424	4	no	—
172	C₄	#16810529	1	1	yes	characteristic
173	C₄	C₂×#13133142	1	1	yes	characteristic
174	C₄	#16810974	1	1	yes	characteristic
175	C₄	#16811021	1	1	yes	characteristic
176	C₂	#20275676	1	1	yes	characteristic
177	C₂	#20276428	1	2	yes	—
178	C₂	#20276957	1	1	yes	characteristic
179	C₂	#20276961	1	1	yes	characteristic
180	C₂	#20277001	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#13284
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
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
7. #16811025	C₄	semidirect
106. C₅₃	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811025	2
#20274875	2
#20272807	2
C₂×C₅₃×#346	2
#13284	53