Subgroups — #27275250

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	2	yes	maximal
2	#20272998	C₂	1	2	yes	maximal
3	#20272548	C₂	1	2	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	4	yes	—
6	#16810376	C₄	1	4	yes	complemented
7	#16810374	C₄	1	4	yes	complemented
8	#16810344	—	2	2	no	—
9	C₈×#88322	—	2	2	no	—
10	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊#88322	C₂×C₄	1	8	yes	—
16	C₂²⋊#88322	—	2	4	no	—
17	C₄×#88322	—	2	4	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	—	2	2	no	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₅₃	—	2	4	no	—
25	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
26	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	#41	1	2	yes	—
34	C₂×C₈×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	C₄²	1	2	yes	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
43	C₄²×C₅₃	—	2	8	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	C₂×C₈	1	4	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×#88322	—	2	8	no	—
50	C₂³×C₅₃	#117	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	8	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	C₄×C₈	1	2	yes	—
62	C₈×C₅₃	C₄×C₈	1	2	yes	—
63	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
64	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
65	#88322	—	4	8	no	—
66	#8266	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₅₃	#431	1	2	yes	—
70	C₄×C₅₃	—	2	8	no	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
74	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₅₃	#443	1	4	yes	—
76	C₄²×C₈	D₅₃	1	1	yes	characteristic
77	#1564	—	53	2	no	—
78	#1450	—	53	2	no	—
79	#1276	—	53	2	no	—
80	C₂×C₅₃	—	2	2	no	—
81	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₅₃	#1772	1	2	yes	—
83	#431	—	53	4	no	—
84	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
85	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₄³	C₂×D₅₃	1	1	yes	characteristic
89	#335	—	53	4	no	—
90	#334	—	53	4	no	—
91	#322	—	106	2	no	—
92	C₈²	—	106	2	no	—
93	C₅₃	#8266	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
99	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
100	C₄⋊C₈	—	53	8	no	—
101	C₂²⋊C₈	—	106	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	106	4	no	—
104	C₄×C₈	#88322	1	8	yes	complemented
105	C₄×C₈	Dic₁₀₆	1	2	yes	—
106	C₄×C₈	C₄×D₅₃	1	2	yes	—
107	C₄×C₈	C₄×D₅₃	1	2	yes	—
108	C₄×C₈	D₂₁₂	1	1	yes	characteristic
109	C₄×C₈	D₂₁₂	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	#11719945	1	2	yes	—
112	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	106	8	no	—
115	C₂×C₈	C₂×#88322	1	4	yes	—
116	C₂×C₈	C₂×#88322	1	4	yes	—
117	C₂×C₈	#11719917	1	4	yes	—
118	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
119	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
120	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
122	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
123	C₄²	—	2	8	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×#88322	1	4	yes	—
126	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
127	C₄²	C₂×C₄×D₅₃	1	2	yes	—
128	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
129	C₂³	#13133176	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
135	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
136	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
137	C₂×C₄	C₂²×#88322	1	2	yes	—
138	C₂×C₄	C₂×#11719917	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	212	8	no	—
141	C₈	C₄×#88322	1	2	yes	—
142	C₈	C₄×#88322	1	2	yes	—
143	C₈	C₄⋊#88322	1	4	yes	—
144	C₈	C₄⋊#88322	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂²	#16810676	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×#88322	1	1	yes	characteristic
151	C₄	C₄×#11719917	1	1	yes	characteristic
152	C₄	C₂×C₄⋊#88322	1	2	yes	—
153	C₄	#16810741	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₄⋊#88322	1	1	yes	characteristic
156	C₂	#20274594	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8266
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #16810376	C₄	semidirect
7. #16810374	C₄	semidirect
25. C₄×C₈×C₅₃	C₈	semidirect
93. C₅₃	#8266	semidirect
104. 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
#20273463	2
#20272998	2
#20272548	2
C₄²×C₈×C₅₃	2
#8266	53