Subgroups — #27277756

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	2	yes	maximal
2	#20272550	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	C₂²	1	4	yes	—
5	#16810378	—	2	4	no	—
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₄³×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×#88322	—	2	4	no	—
12	C₄⋊#88322	—	2	8	no	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	2	yes	—
18	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₃	—	2	4	no	—
21	C₄×C₈×C₅₃	—	2	2	no	—
22	C₄×C₈×C₅₃	—	2	4	no	—
23	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
35	C₄²×C₅₃	—	2	4	no	—
36	C₄²×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	4	no	—
38	C₄²×C₅₃	C₂×C₈	1	2	yes	—
39	C₄²×C₅₃	M₁₆	1	2	yes	—
40	C₄²×C₅₃	#41	1	4	yes	—
41	C₂×#88322	—	4	4	no	—
42	C₂×#88322	—	4	8	no	—
43	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
50	C₂×C₄×C₅₃	#108	1	4	yes	—
51	C₂×C₄×C₅₃	#117	1	2	yes	—
52	C₂×C₄×C₅₃	#124	1	2	yes	—
53	C₂×C₄×C₅₃	#125	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#135	1	4	yes	—
58	C₈×C₅₃	—	2	4	no	—
59	C₈×C₅₃	—	2	4	no	—
60	#88322	—	8	4	no	—
61	#8785	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	#382	1	1	yes	characteristic
63	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂²×C₅₃	#423	1	1	yes	characteristic
65	C₂²×C₅₃	#427	1	2	yes	—
66	C₂²×C₅₃	C₂×#108	1	2	yes	—
67	C₄×C₅₃	—	2	4	no	—
68	C₄×C₅₃	—	2	4	no	—
69	C₄×C₅₃	—	2	2	no	—
70	C₄×C₅₃	#431	1	2	yes	—
71	C₄×C₅₃	#432	1	2	yes	—
72	C₄×C₅₃	#488	1	4	yes	—
73	#1831	—	53	2	no	—
74	C₄²×C₈	D₅₃	1	1	yes	characteristic
75	#1277	—	53	4	no	—
76	C₂×C₅₃	#1564	1	2	yes	—
77	C₂×C₅₃	#1568	1	2	yes	—
78	C₂×C₅₃	#1831	1	1	yes	characteristic
79	C₂×C₅₃	#1835	1	1	yes	characteristic
80	C₂×C₅₃	#1837	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	53	4	no	—
82	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
83	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
84	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
86	C₄³	C₂×D₅₃	1	1	yes	characteristic
87	#336	—	106	4	no	—
88	C₅₃	#8785	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	106	4	no	—
90	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
91	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
92	C₂×C₄²	Dic₁₀₆	1	2	yes	—
93	C₂×C₄²	D₂₁₂	1	2	yes	—
94	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
96	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
97	C₄⋊C₈	—	106	8	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	2	4	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	#88322	1	4	yes	complemented
102	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
103	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
104	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
106	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	212	4	no	—
112	C₂×C₈	—	212	8	no	—
113	C₂×C₈	C₂×#88322	1	4	yes	—
114	C₂×C₈	#11719917	1	4	yes	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	C₂×#88322	1	2	yes	—
119	C₄²	#11719917	1	2	yes	—
120	C₄²	#11719945	1	4	yes	—
121	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊#88322	1	4	yes	—
128	C₂×C₄	#13133153	1	4	yes	—
129	C₂×C₄	#13133176	1	2	yes	—
130	C₂×C₄	#13133205	1	2	yes	—
131	C₂×C₄	#13133209	1	2	yes	—
132	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
133	C₂×C₄	C₂×#11719917	1	2	yes	—
134	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
135	C₂×C₄	#13133239	1	4	yes	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	—	424	4	no	—
139	C₂²	#16810489	1	1	yes	characteristic
140	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
141	C₂²	#16810647	1	1	yes	characteristic
142	C₂²	#16810661	1	2	yes	—
143	C₂²	C₂×#13133153	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₄	#16810676	1	2	yes	—
148	C₄	#16810680	1	2	yes	—
149	C₄	#16810978	1	4	yes	—
150	C₂	#20273463	1	2	yes	—
151	C₂	#20273478	1	2	yes	—
152	C₂	#20274832	1	1	yes	characteristic
153	C₂	#20274846	1	1	yes	characteristic
154	C₂	#20274853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8785
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. C₄×C₈×C₅₃	C₈	semidirect
88. C₅₃	#8785	semidirect
101. C₄×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
#20272550	2
C₄²×C₈×C₅₃	2
#8785	53