Subgroups — #27304618

Order 13568.

202 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13133123	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13133123	C₂	1	2	yes	maximal
3	C₄×#13133123	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133123	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#13133123	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	—	2	2	no	—
18	C₂×C₁₆×C₅₃	—	2	2	no	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	—	2	1	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
38	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
39	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₁₆×C₅₃	—	2	1	no	—
42	C₁₆×C₅₃	—	2	1	no	—
43	C₁₆×C₅₃	M₁₆	1	2	yes	—
44	C₁₆×C₅₃	#41	1	2	yes	—
45	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	C₃₂	1	4	yes	complemented
50	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	M₃₂	1	2	yes	—
59	C₈×C₅₃	#117	1	1	yes	characteristic
60	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	#13639	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	C₂×C₃₂	1	2	yes	—
65	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
69	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
70	C₄×C₅₃	M₆₄	1	2	yes	—
71	C₄×C₅₃	#431	1	1	yes	characteristic
72	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
73	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
74	C₄⋊C₃₂	—	53	2	no	—
75	C₂²⋊C₃₂	—	53	2	no	—
76	C₄×C₃₂	—	53	2	no	—
77	C₂×C₅₃	—	2	2	no	—
78	C₂×C₅₃	#2162	1	1	yes	characteristic
79	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
80	C₂×C₅₃	C₂×M₆₄	1	1	yes	characteristic
81	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
83	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
84	C₂×C₃₂	—	53	4	no	—
85	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
86	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
87	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
88	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
89	C₅₃	#13639	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	#88322	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
92	C₂×C₄²	#88322	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
96	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
99	C₄×C₈	#88322	1	1	yes	characteristic
100	C₄×C₈	#88322	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
103	C₃₂	—	106	4	no	—
104	C₂²×C₄	#11719908	1	1	yes	characteristic
105	C₂²×C₄	#11719908	1	1	yes	characteristic
106	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	C₂×#88322	1	1	yes	characteristic
111	C₂×C₈	C₂×#88322	1	1	yes	characteristic
112	C₂×C₈	C₂×#88322	1	1	yes	characteristic
113	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₄²	#11719908	1	1	yes	characteristic
115	C₄²	#11719908	1	1	yes	characteristic
116	C₄²	C₂×#88322	1	1	yes	characteristic
117	C₄²	C₂×#88322	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	2	1	no	—
120	C₁₆	#11719917	1	2	yes	—
121	C₁₆	#11719945	1	2	yes	—
122	C₂³	C₂×#11719908	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	#13133123	1	4	yes	complemented
127	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
128	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
129	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
130	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
131	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
132	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	#13133157	1	2	yes	—
136	C₈	#13133176	1	1	yes	characteristic
137	C₈	C₂×#11719917	1	1	yes	characteristic
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×#13133123	1	2	yes	—
141	C₂²	C₂²×#11719908	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	C₂×#13133123	1	2	yes	—
145	C₄	C₂×#13133123	1	2	yes	—
146	C₄	#16810457	1	2	yes	—
147	C₄	#16810676	1	1	yes	characteristic
148	C₄	C₂×#13133157	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#20276514	1	1	yes	characteristic
151	C₂	C₂²×#13133123	1	1	yes	characteristic
152	C₂	C₂×#16810457	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₃₂
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#13639
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13133123	C₂	semidirect
3. C₄×#13133123	C₂	semidirect
49. C₂×C₄×C₅₃	C₃₂	semidirect
89. C₅₃	#13639	semidirect
126. C₂×C₄	#13133123	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13133123	2
C₂²⋊#13133123	2
C₄×#13133123	2
C₂×C₄×C₁₆×C₅₃	2
#13639	53