Subgroups — #27273477

Order 13568.

734 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	C₂×#16810676	C₂	1	2	yes	maximal
2	#20272542	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	4	no	—
5	#16810676	—	2	4	no	—
6	C₂×C₄⋊#88322	C₄	1	4	yes	—
7	C₂×C₂²⋊#88322	C₄	1	4	yes	—
8	C₂×C₄×#88322	C₂²	1	4	yes	—
9	C₂³×C₈×C₅₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	—	2	4	no	—
12	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂²×#88322	C₂×C₄	1	8	yes	—
15	C₄⋊#88322	—	2	8	no	—
16	C₂²⋊#88322	—	2	8	no	—
17	C₄×#88322	—	2	4	no	—
18	C₄×#88322	—	2	4	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	—	2	4	no	—
22	C₂²×C₈×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	4	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	4	no	—
32	C₄×C₈×C₅₃	—	2	4	no	—
33	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	8	no	—
35	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	—	2	16	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
49	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
50	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×D₄	1	2	yes	—
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×#88322	—	2	8	no	—
57	C₂×#88322	—	2	16	no	—
58	C₂³×C₅₃	—	2	4	no	—
59	C₂³×C₅₃	#95	1	2	yes	—
60	C₂³×C₅₃	C₄×C₈	1	2	yes	—
61	C₂³×C₅₃	#97	1	2	yes	—
62	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	8	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	#95	1	4	yes	—
67	C₂×C₄×C₅₃	#95	1	2	yes	—
68	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₅₃	#97	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
71	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	#131	1	8	yes	—
79	C₈×C₅₃	—	2	8	no	—
80	#88322	—	4	8	no	—
81	#7849	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	4	no	—
83	C₂²×C₅₃	—	2	4	no	—
84	C₂²×C₅₃	—	2	4	no	—
85	C₂²×C₅₃	#336	1	4	yes	—
86	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
89	C₂²×C₅₃	#405	1	4	yes	—
90	C₄×C₅₃	—	2	4	no	—
91	C₄×C₅₃	#336	1	4	yes	—
92	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
93	C₄×C₅₃	#408	1	4	yes	—
94	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
95	C₄×C₅₃	#424	1	4	yes	—
96	C₂×#431	—	53	2	no	—
97	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
98	#1273	—	53	4	no	—
99	C₂×C₅₃	—	2	4	no	—
100	C₂×C₅₃	#1450	1	2	yes	—
101	C₂×C₅₃	#1452	1	2	yes	—
102	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
103	C₂×C₅₃	#1729	1	2	yes	—
104	C₂³×C₈	Dic₅₃	1	2	yes	—
105	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
106	#431	—	106	4	no	—
107	#431	—	106	4	no	—
108	C₂×C₄⋊C₈	—	53	4	no	—
109	C₂×C₂²⋊C₈	—	53	4	no	—
110	C₂×C₄×C₈	—	2	4	no	—
111	C₂×C₄×C₈	—	53	4	no	—
112	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
114	C₅₃	#7849	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₄	C₄×D₅₃	1	2	yes	—
116	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	53	8	no	—
120	C₂²×C₈	#88322	1	8	yes	complemented
121	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
122	C₂×C₄²	Dic₁₀₆	1	2	yes	—
123	C₂×C₄²	C₄×D₅₃	1	2	yes	—
124	C₂×C₄²	D₂₁₂	1	2	yes	—
125	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
127	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊C₈	—	106	8	no	—
129	C₂²⋊C₈	—	106	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	106	4	no	—
133	C₄×C₈	—	106	4	no	—
134	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	C₂×#88322	1	4	yes	—
137	C₂²×C₄	#11719917	1	4	yes	—
138	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
139	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₄	#11719919	1	4	yes	—
141	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
142	C₂²×C₄	#11719921	1	4	yes	—
143	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
146	C₂×C₈	—	2	16	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	106	8	no	—
150	C₂×C₈	—	106	16	no	—
151	C₂×C₈	C₂×#88322	1	8	yes	—
152	C₄²	#11719919	1	4	yes	—
153	C₄²	C₄⋊Dic₅₃	1	2	yes	—
154	C₄²	#11719921	1	4	yes	—
155	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
156	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
157	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
158	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
159	C₂³	—	2	4	no	—
160	C₂³	#13133124	1	2	yes	—
161	C₂³	C₄×#88322	1	2	yes	—
162	C₂³	#13133128	1	2	yes	—
163	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	#13133124	1	4	yes	—
168	C₂×C₄	#13133124	1	2	yes	—
169	C₂×C₄	C₄×#88322	1	2	yes	—
170	C₂×C₄	#13133128	1	2	yes	—
171	C₂×C₄	C₂²⋊#88322	1	8	yes	—
172	C₂×C₄	C₄⋊#88322	1	8	yes	—
173	C₂×C₄	C₂×#11719921	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
175	C₂×C₄	C₂×#11719919	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×#88322	1	2	yes	—
178	C₂×C₄	C₂×#11719917	1	2	yes	—
179	C₂×C₄	#13133223	1	8	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	212	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	#16810378	1	4	yes	—
186	C₂²	C₂×#13133124	1	1	yes	characteristic
187	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
188	C₂²	C₂×#13133128	1	1	yes	characteristic
189	C₂²	#16810578	1	4	yes	—
190	C₄	—	2	4	no	—
191	C₄	#16810378	1	4	yes	—
192	C₄	C₂×C₂²⋊#88322	1	2	yes	—
193	C₄	#16810589	1	4	yes	—
194	C₄	C₂×C₄⋊#88322	1	2	yes	—
195	C₄	#16810649	1	4	yes	—
196	C₂	—	2	4	no	—
197	C₂	#20272998	1	2	yes	—
198	C₂	#20273007	1	2	yes	—
199	C₂	C₂×#16810378	1	1	yes	characteristic
200	C₂	#20274411	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#7849
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20272542	C₂	semidirect
23. C₂²×C₈×C₅₃	C₈	semidirect
114. C₅₃	#7849	semidirect
120. C₂²×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20272542	2
C₂²×C₄×C₈×C₅₃	2
#7849	53