Subgroups — #27295663

Order 13568.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	2	yes	maximal
2	#20272785	C₂	1	4	yes	maximal, complemented
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	—	2	2	no	—
5	C₄⋊#11719908	—	2	2	no	—
6	C₄⋊#11719908	C₂²	1	4	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	1	yes	characteristic
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₈²×C₅₃	C₂²	1	2	yes	—
12	#13133157	—	4	4	no	—
13	C₂×#11719908	—	2	4	no	—
14	C₂²×C₈×C₅₃	—	2	1	no	—
15	C₂²×C₈×C₅₃	C₈	1	2	yes	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₅₃	—	2	1	no	—
20	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	2	4	no	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₅₃	D₈	1	2	yes	—
40	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
46	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	#11719908	—	4	4	no	—
48	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#107	1	2	yes	—
55	C₂×C₄×C₅₃	#108	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₅₃	—	2	4	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	2	2	no	—
63	C₈×C₅₃	#107	1	2	yes	—
64	C₈×C₅₃	M₃₂	1	4	yes	complemented
65	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
67	#12004	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	#335	1	2	yes	—
71	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#335	1	2	yes	—
75	C₄×C₅₃	#364	1	2	yes	—
76	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
78	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
79	#2150	—	53	2	no	—
80	C₂×C₈²	D₅₃	1	1	yes	characteristic
81	#1370	—	53	4	no	—
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
84	C₂×C₅₃	#2150	1	1	yes	characteristic
85	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
86	C₂×M₃₂	—	106	2	no	—
87	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
88	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₄⋊C₁₆	—	53	4	no	—
91	C₄⋊C₁₆	—	106	2	no	—
92	C₈²	Dic₅₃	1	2	yes	—
93	C₈²	C₂×D₅₃	1	2	yes	—
94	C₅₃	#12004	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	#88322	1	2	yes	—
97	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
98	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
99	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
100	M₃₂	—	212	4	no	—
101	C₂×C₁₆	—	106	4	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	#88322	1	1	yes	characteristic
104	C₄×C₈	#88322	1	1	yes	characteristic
105	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
106	C₄×C₈	D₂₁₂	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
108	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
109	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
110	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
112	C₂²×C₄	#11719917	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	D₄₂₄	1	2	yes	—
119	C₂×C₈	Dic₂₁₂	1	2	yes	—
120	C₂×C₈	C₂×#88322	1	2	yes	—
121	C₂×C₈	C₂×#88322	1	2	yes	—
122	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
123	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
124	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
125	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
126	C₄²	C₂×#88322	1	1	yes	characteristic
127	C₄²	#11719917	1	1	yes	characteristic
128	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
129	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₁₆	—	212	4	no	—
131	C₂³	C₄⋊#88322	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
135	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
136	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
137	C₂×C₄	#13133151	1	2	yes	—
138	C₂×C₄	#13133153	1	2	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
141	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	#13133151	1	2	yes	—
147	C₈	#13133157	1	4	yes	complemented
148	C₈	C₂×D₄₂₄	1	1	yes	characteristic
149	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#16810376	1	2	yes	—
153	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#16810376	1	2	yes	—
157	C₄	#16810444	1	2	yes	—
158	C₄	C₂×#13133151	1	1	yes	characteristic
159	C₄	C₂×#13133153	1	1	yes	characteristic
160	C₄	C₂×#13133157	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₂×#16810376	1	1	yes	characteristic
163	C₂	#20276471	1	1	yes	characteristic
164	C₂	C₂×#16810444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#12004
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #20272785	C₂	semidirect
64. C₈×C₅₃	M₃₂	semidirect
94. C₅₃	#12004	semidirect
147. C₈	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
#20272785	2
C₂×C₈²×C₅₃	2
#12004	53