Subgroups — #27275227

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133146	C₂	1	2	yes	maximal
2	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	#20272542	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133146	C₄	1	4	yes	—
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×#13133127	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810377	C₄	1	4	yes	—
13	#16810344	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₈²×C₅₃	—	2	2	no	—
17	C₂²×#88322	C₂×C₄	1	4	yes	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	#13133146	—	2	8	no	—
24	#13133127	—	2	4	no	—
25	#13133127	C₈	1	8	yes	complemented
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₄×#88322	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₄²	1	2	yes	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₄×Dic₅₃	—	2	2	no	—
51	C₄×Dic₅₃	—	2	4	no	—
52	C₄×Dic₅₃	—	2	4	no	—
53	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	C₂×C₈	1	4	yes	—
60	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
68	C₂×C₄×C₅₃	#131	1	4	yes	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	1	no	—
74	C₂×Dic₅₃	—	2	2	no	—
75	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
76	C₈×C₅₃	—	2	8	no	—
77	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
78	#88322	—	2	4	no	—
79	#88322	—	4	4	no	—
80	#8262	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
84	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₅₃	#405	1	2	yes	—
86	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
87	C₄×C₅₃	#431	1	2	yes	—
88	C₄×C₅₃	#435	1	4	yes	—
89	C₄×C₅₃	#446	1	4	yes	—
90	Dic₅₃	—	2	4	no	—
91	Dic₅₃	—	4	1	no	—
92	Dic₅₃	—	4	1	no	—
93	C₄×C₄⋊C₈	—	53	2	no	—
94	C₄×#97	—	53	1	no	—
95	C₂×#322	—	53	1	no	—
96	C₂×C₈²	D₅₃	1	1	yes	characteristic
97	#1273	—	53	2	no	—
98	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
99	C₂×C₅₃	#1453	1	1	yes	characteristic
100	C₂×C₅₃	#1455	1	1	yes	characteristic
101	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	#1771	1	2	yes	—
103	C₂×C₄⋊C₈	—	53	4	no	—
104	C₂×#97	—	53	2	no	—
105	C₂×#97	—	53	2	no	—
106	C₂×C₄×C₈	—	53	2	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄³	—	53	1	no	—
111	#336	—	53	4	no	—
112	#322	—	106	2	no	—
113	C₈²	—	2	2	no	—
114	C₅₃	#8262	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	53	4	no	—
117	C₂²×C₈	C₄×D₅₃	1	4	yes	—
118	C₂²×C₈	C₄×D₅₃	1	2	yes	—
119	C₂×C₄²	—	53	1	no	—
120	C₂×C₄²	—	53	2	no	—
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	—	53	2	no	—
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊C₈	—	106	8	no	—
125	#97	—	53	8	no	—
126	#97	—	106	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	106	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	—	106	2	no	—
132	C₄×C₈	Dic₁₀₆	1	2	yes	—
133	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
134	C₂²×C₄	—	53	2	no	—
135	C₂²×C₄	—	53	1	no	—
136	C₂²×C₄	—	53	1	no	—
137	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	106	4	no	—
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	—	106	4	no	—
147	C₂×C₈	C₈×D₅₃	1	4	yes	—
148	C₂×C₈	#11719912	1	8	yes	—
149	C₂×C₈	#11719912	1	4	yes	—
150	C₂×C₈	#11719919	1	4	yes	—
151	C₄²	—	53	4	no	—
152	C₄²	—	106	2	no	—
153	C₄²	—	106	4	no	—
154	C₄²	—	106	4	no	—
155	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
156	C₄²	#11719945	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂³	C₄²×D₅₃	1	1	yes	characteristic
159	C₂×C₄	—	53	4	no	—
160	C₂×C₄	—	106	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	1	no	—
163	C₂×C₄	—	106	2	no	—
164	C₂×C₄	—	106	1	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
167	C₂×C₄	#13133174	1	1	yes	characteristic
168	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
169	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
171	C₂×C₄	C₂×#11719912	1	4	yes	—
172	C₂×C₄	C₂×#11719912	1	2	yes	—
173	C₂×C₄	#13133223	1	4	yes	—
174	C₈	—	2	8	no	—
175	C₈	—	106	4	no	—
176	C₈	—	212	4	no	—
177	C₈	#13133146	1	8	yes	complemented
178	C₂²	C₄×#11719919	1	1	yes	characteristic
179	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
180	C₂²	C₄×#11719912	1	2	yes	—
181	C₂²	C₄×#11719912	1	1	yes	characteristic
182	C₂²	#16810577	1	2	yes	—
183	C₄	—	106	4	no	—
184	C₄	—	212	1	no	—
185	C₄	—	212	1	no	—
186	C₄	C₂×#13133146	1	2	yes	—
187	C₄	#16810674	1	2	yes	—
188	C₄	#16810694	1	4	yes	—
189	C₄	#16810753	1	4	yes	—
190	C₂	C₈×#11719912	1	2	yes	—
191	C₂	#20273009	1	1	yes	characteristic
192	C₂	#20273017	1	1	yes	characteristic
193	C₂	C₄×#13133146	1	1	yes	characteristic
194	C₂	#20274589	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#8262
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
25. #13133127	C₈	semidirect
77. C₈×C₅₃	C₄⋊C₈	semidirect
114. C₅₃	#8262	semidirect
177. C₈	#13133146	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133146	2
C₄×#13133127	2
C₂×#16810344	2
#20272542	2
C₂×C₈²×C₅₃	2
#8262	53