Subgroups — #27302103

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	#20276474	C₂	1	2	yes	maximal
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
4	#20272804	C₂	1	2	yes	maximal
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	#16810444	—	2	2	no	—
10	C₄⋊#11719908	—	2	4	no	—
11	C₄×#11719908	—	2	2	no	—
12	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
13	C₅₃×#427	C₄	1	1	yes	characteristic
14	C₅₃×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#339	C₄	1	2	yes	—
17	#13133157	—	2	2	no	—
18	#13133157	—	2	2	no	—
19	C₂×#11719908	—	2	1	no	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	—	2	1	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	D₄	1	2	yes	—
25	C₂×#11719908	Q₈	1	2	yes	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	#117×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#108×C₅₃	—	2	2	no	—
32	#107×C₅₃	C₈	1	2	yes	—
33	#106×C₅₃	C₈	1	2	yes	—
34	#97×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₅₃	—	2	2	no	—
41	M₁₆×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
58	#11719908	—	2	2	no	—
59	#11719908	—	2	2	no	—
60	#11719908	—	4	2	no	—
61	C₂³×C₅₃	#131	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
71	C₈×C₅₃	—	4	2	no	—
72	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
73	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₈×C₅₃	#121	1	1	yes	characteristic
75	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₅₃	#125	1	1	yes	characteristic
77	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	#13161	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	#424	1	1	yes	characteristic
81	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₅₃	#446	1	1	yes	characteristic
83	C₄×C₅₃	—	2	4	no	—
84	C₄×C₅₃	—	4	4	no	—
85	C₄×C₅₃	#389	1	1	yes	characteristic
86	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₅₃	#436	1	1	yes	characteristic
88	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
89	C₂×#364	—	53	1	no	—
90	#2151	—	53	2	no	—
91	C₂×C₄×C₁₆	—	53	1	no	—
92	#1772	D₅₃	1	1	yes	characteristic
93	#1379	—	53	2	no	—
94	C₂×C₅₃	—	2	1	no	—
95	C₂×C₅₃	—	2	1	no	—
96	C₂×C₅₃	#1939	1	1	yes	characteristic
97	C₂×C₅₃	#2170	1	2	yes	—
98	C₂×M₃₂	—	53	2	no	—
99	C₂²×C₁₆	—	53	1	no	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂×#108	Dic₅₃	1	1	yes	characteristic
102	#427	Dic₅₃	1	1	yes	characteristic
103	#405	C₂×D₅₃	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	#364	—	106	2	no	—
106	C₄⋊C₁₆	—	106	4	no	—
107	C₄×C₁₆	—	106	2	no	—
108	#339	Dic₅₃	1	2	yes	—
109	C₅₃	#13161	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
111	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	#117	C₂×Dic₅₃	1	2	yes	—
114	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
115	M₃₂	—	106	2	no	—
116	M₃₂	—	106	2	no	—
117	C₂×C₁₆	—	53	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	106	1	no	—
120	C₂×C₁₆	—	106	1	no	—
121	C₂×C₁₆	—	106	1	no	—
122	C₂×C₁₆	—	106	1	no	—
123	C₂×C₁₆	—	106	2	no	—
124	#108	—	2	2	no	—
125	#107	#88322	1	2	yes	—
126	#106	#88322	1	2	yes	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	Dic₁₀₆	1	2	yes	—
130	C₄×C₈	D₂₁₂	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂×#88322	1	1	yes	characteristic
137	C₂×C₈	C₂×#88322	1	1	yes	characteristic
138	C₂×C₈	#11719917	1	2	yes	—
139	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
142	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
143	C₂×C₈	#11719947	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
145	C₂×C₈	#11719949	1	1	yes	characteristic
146	C₄⋊C₄	C₂×#88322	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊Dic₅₃	1	2	yes	—
151	C₁₆	—	106	2	no	—
152	C₁₆	—	106	2	no	—
153	C₁₆	—	212	2	no	—
154	C₂³	#13133224	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
160	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
162	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
163	C₂×C₄	#13133224	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊#88322	1	2	yes	—
166	C₈	C₄⋊#88322	1	2	yes	—
167	C₈	#13133192	1	1	yes	characteristic
168	C₈	#13133197	1	1	yes	characteristic
169	C₈	#13133208	1	1	yes	characteristic
170	C₈	#13133216	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#16810650	1	1	yes	characteristic
173	C₂²	D₄×#88322	1	1	yes	characteristic
174	C₂²	#16810754	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	4	4	no	—
177	C₄	#16810522	1	1	yes	characteristic
178	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
179	C₄	#16810701	1	1	yes	characteristic
180	C₄	Q₈×#88322	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#20275434	1	1	yes	characteristic
184	C₂	#20276552	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#13161
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
109. C₅₃	#13161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20276474	2
C₂×C₄×#11719908	2
#20272804	2
C₅₃×#1772	2
#13161	53