Subgroups — #27302152

Order 13568.

274 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×#16810453	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
3	#20275500	C₂	1	2	yes	maximal
4	#20272824	C₂	1	2	yes	maximal
5	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810666	—	2	2	no	—
8	C₂×#13133151	C₂²	1	2	yes	—
9	#16810453	C₄	1	4	yes	complemented
10	#16810449	—	2	1	no	—
11	#16810449	—	2	1	no	—
12	#16810391	—	2	2	no	—
13	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#405	C₂²	1	1	yes	characteristic
16	C₅₃×#346	—	2	1	no	—
17	C₄×C₁₆×C₅₃	—	2	1	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	#13133153	—	2	2	no	—
21	#13133153	C₂×C₄	1	4	yes	—
22	#13133151	—	2	2	no	—
23	#13133151	—	2	2	no	—
24	#13133150	—	4	2	no	—
25	#13133148	—	4	2	no	—
26	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	#117×C₅₃	D₄	1	1	yes	characteristic
29	M₃₂×C₅₃	—	2	1	no	—
30	M₃₂×C₅₃	Q₈	1	2	yes	—
31	C₂×C₁₆×C₅₃	—	2	1	no	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
36	#97×C₅₃	—	2	1	no	—
37	C₄×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	4	2	no	—
40	M₁₆×C₅₃	—	2	1	no	—
41	M₁₆×C₅₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²⋊C₄×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄⋊Dic₅₃	—	4	2	no	—
54	C₄⋊Dic₅₃	—	4	2	no	—
55	#11719917	—	2	4	no	—
56	#11719917	—	4	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₁₆×C₅₃	—	2	2	no	—
59	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂³×C₅₃	#135	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
70	C₂×Dic₅₃	—	8	2	no	—
71	C₂×Dic₅₃	—	16	2	no	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₃	#125	1	1	yes	characteristic
77	#88322	—	4	4	no	—
78	#13171	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
82	C₂²×C₅₃	#466	1	1	yes	characteristic
83	C₂²×C₅₃	#499	1	1	yes	characteristic
84	C₄×C₅₃	—	4	1	no	—
85	C₄×C₅₃	—	4	1	no	—
86	C₄×C₅₃	#389	1	1	yes	characteristic
87	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
88	C₄×C₅₃	#467	1	1	yes	characteristic
89	C₄×C₅₃	#500	1	1	yes	characteristic
90	Dic₅₃	—	16	2	no	—
91	C₂×#368	—	53	1	no	—
92	C₂×#366	—	53	1	no	—
93	#2108	D₅₃	1	1	yes	characteristic
94	#1952	—	53	2	no	—
95	#1387	—	53	2	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	#1942	1	1	yes	characteristic
99	C₂×C₅₃	#2221	1	1	yes	characteristic
100	C₂×C₅₃	#2222	1	1	yes	characteristic
101	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
103	C₂×#108	—	53	2	no	—
104	#428	—	106	2	no	—
105	C₂×#107	—	53	2	no	—
106	#405	C₂×D₅₃	1	1	yes	characteristic
107	#368	—	53	4	no	—
108	#366	—	106	1	no	—
109	#366	—	106	1	no	—
110	#346	—	2	1	no	—
111	C₄×C₁₆	—	2	1	no	—
112	#343	—	106	2	no	—
113	C₅₃	#13171	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	106	2	no	—
115	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	#117	D₂₁₂	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	106	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	Dic₁₀₆	1	2	yes	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
123	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
126	#108	—	53	4	no	—
127	#108	—	106	2	no	—
128	#107	—	106	2	no	—
129	#107	—	106	2	no	—
130	#107	—	212	2	no	—
131	#106	—	212	2	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂²×C₄	—	212	2	no	—
135	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	106	4	no	—
138	M₁₆	—	212	2	no	—
139	M₁₆	Dic₂₁₂	1	2	yes	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₈	#11719945	1	1	yes	characteristic
146	C₂×C₈	#11719945	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	212	2	no	—
149	C₄⋊C₄	—	212	2	no	—
150	C₄⋊C₄	D₄₂₄	1	1	yes	characteristic
151	C₄⋊C₄	D₄₂₄	1	1	yes	characteristic
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	#11719919	1	2	yes	—
156	C₁₆	#11719919	1	2	yes	—
157	C₂³	#13133239	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	848	2	no	—
161	C₂×C₄	—	424	2	no	—
162	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
163	C₂×C₄	#13133191	1	1	yes	characteristic
164	C₂×C₄	#13133217	1	1	yes	characteristic
165	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
167	C₂×C₄	#13133239	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	212	4	no	—
170	C₈	C₂×#11719919	1	1	yes	characteristic
171	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
172	C₈	#13133196	1	1	yes	characteristic
173	C₈	#13133209	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×Dic₂₁₂	1	1	yes	characteristic
177	C₂²	#16810859	1	1	yes	characteristic
178	C₂²	#16811025	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	848	2	no	—
182	C₄	#16810521	1	1	yes	characteristic
183	C₄	C₄×D₄₂₄	1	1	yes	characteristic
184	C₄	#16810864	1	1	yes	characteristic
185	C₄	#16811028	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#20275448	1	1	yes	characteristic
189	C₂	#20276835	1	1	yes	characteristic
190	C₂	#20276840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810859
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	#466
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#13171
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
9. #16810453	C₄	semidirect
113. C₅₃	#13171	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
C₂×#16810449	2
#20275500	2
#20272824	2
C₅₃×#2108	2
#13171	53