Subgroups — #27300438

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	C₄⋊#11719908	—	2	2	no	—
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	4	no	—
21	C₄×#88322	—	2	2	no	—
22	C₈×Dic₅₃	—	2	2	no	—
23	C₈×Dic₅₃	—	2	2	no	—
24	C₈×Dic₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	—	2	4	no	—
30	C₄×C₈×C₅₃	D₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
36	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	#41	1	2	yes	—
42	C₂×C₈×C₅₃	#41	1	2	yes	—
43	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₅₃	M₁₆	1	2	yes	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	—	2	2	no	—
47	C₄×Dic₅₃	—	2	2	no	—
48	C₄×Dic₅₃	—	2	4	no	—
49	C₄×Dic₅₃	C₁₆	1	4	yes	complemented
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	M₁₆	1	4	yes	—
53	C₁₆×C₅₃	—	4	2	no	—
54	#11719908	—	4	2	no	—
55	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#117	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₅₃	—	2	2	no	—
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
70	C₂×Dic₅₃	M₃₂	1	4	yes	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	#124	1	1	yes	characteristic
73	C₈×C₅₃	#124	1	1	yes	characteristic
74	C₈×C₅₃	#125	1	2	yes	—
75	#88322	—	2	4	no	—
76	#12879	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₅₃	#431	1	2	yes	—
79	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
80	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
81	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	—
84	C₄×C₅₃	#382	1	1	yes	characteristic
85	C₄×C₅₃	#423	1	1	yes	characteristic
86	C₄×C₅₃	#432	1	2	yes	—
87	Dic₅₃	—	2	4	no	—
88	Dic₅₃	—	2	4	no	—
89	C₂×C₄⋊C₁₆	—	53	1	no	—
90	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
91	C₄²×C₈	—	53	1	no	—
92	#1374	—	53	4	no	—
93	C₂×C₅₃	#1831	1	1	yes	characteristic
94	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
95	C₂×C₅₃	#2150	1	1	yes	characteristic
96	C₂×C₅₃	#2162	1	2	yes	—
97	C₂×C₅₃	#2163	1	2	yes	—
98	C₂²×C₁₆	—	53	2	no	—
99	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
100	C₂×C₄×C₈	—	53	2	no	—
101	C₂×C₄×C₈	—	53	2	no	—
102	C₂×C₄×C₈	—	53	1	no	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₄³	—	53	1	no	—
105	C₄⋊C₁₆	—	2	2	no	—
106	C₄⋊C₁₆	—	106	2	no	—
107	C₅₃	#12879	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	53	2	no	—
109	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	C₂×C₄²	—	53	1	no	—
112	C₂×C₄²	—	53	2	no	—
113	C₂×C₄²	—	53	2	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	106	4	no	—
119	C₂×C₁₆	—	106	2	no	—
120	C₄×C₈	—	106	4	no	—
121	C₄×C₈	—	106	2	no	—
122	C₄×C₈	—	106	2	no	—
123	C₄×C₈	—	106	2	no	—
124	C₄×C₈	Dic₁₀₆	1	2	yes	—
125	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
126	C₂²×C₄	—	53	2	no	—
127	C₂²×C₄	—	53	2	no	—
128	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	53	4	no	—
133	C₂×C₈	—	106	2	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	#11719919	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
137	C₂×C₈	#11719947	1	2	yes	—
138	C₂×C₈	#11719949	1	2	yes	—
139	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₄²	—	53	4	no	—
141	C₄²	—	106	4	no	—
142	C₄²	—	106	2	no	—
143	C₄²	—	106	2	no	—
144	C₄²	—	106	2	no	—
145	C₄²	#11719912	1	2	yes	—
146	C₄²	#11719919	1	2	yes	—
147	C₁₆	—	4	2	no	—
148	C₁₆	—	212	2	no	—
149	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	53	4	no	—
152	C₂×C₄	—	53	4	no	—
153	C₂×C₄	—	106	2	no	—
154	C₂×C₄	—	106	2	no	—
155	C₂×C₄	—	106	2	no	—
156	C₂×C₄	—	106	2	no	—
157	C₂×C₄	#13133146	1	2	yes	—
158	C₂×C₄	#13133146	1	2	yes	—
159	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
160	C₂×C₄	#13133156	1	2	yes	—
161	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
162	C₂×C₄	#13133175	1	2	yes	—
163	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
164	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	106	4	no	—
167	C₈	#13133204	1	1	yes	characteristic
168	C₈	#13133206	1	1	yes	characteristic
169	C₈	#13133208	1	2	yes	—
170	C₂²	C₂×#13133146	1	1	yes	characteristic
171	C₂²	#16810675	1	2	yes	—
172	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
173	C₂²	M₃₂×D₅₃	1	2	yes	—
174	C₂²	C₂×#13133156	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	106	4	no	—
177	C₄	—	106	4	no	—
178	C₄	#16810441	1	4	yes	—
179	C₄	#16810488	1	1	yes	characteristic
180	C₄	#16810646	1	1	yes	characteristic
181	C₄	#16810679	1	2	yes	—
182	C₂	#20274831	1	1	yes	characteristic
183	C₂	C₂×#16810441	1	1	yes	characteristic
184	C₂	#20276470	1	1	yes	characteristic
185	C₂	#20276513	1	2	yes	—
186	C₂	#20276517	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12879
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
49. C₄×Dic₅₃	C₁₆	semidirect
107. C₅₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₄×C₈×Dic₅₃	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#12879	53