Subgroups — #27295737

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273474	C₂	1	2	yes	maximal
2	#20272789	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810647	—	2	2	no	—
5	#16810376	—	2	2	no	—
6	#16810376	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	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	2	yes	—
12	C₂×#11719917	—	4	2	no	—
13	C₄⋊#88322	—	2	4	no	—
14	C₄⋊#88322	—	4	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	4	no	—
19	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₄²×C₅₃	D₈	1	1	yes	characteristic
36	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
37	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
38	#11719917	—	8	4	no	—
39	C₂×#88322	—	4	4	no	—
40	C₁₆×C₅₃	—	2	2	no	—
41	C₁₆×C₅₃	M₁₆	1	4	yes	complemented
42	C₂³×C₅₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
50	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	—	2	1	no	—
55	C₈×C₅₃	#108	1	2	yes	—
56	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
57	#88322	—	8	4	no	—
58	#12023	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	—	2	1	no	—
60	C₂²×C₅₃	—	2	1	no	—
61	C₂²×C₅₃	#367	1	2	yes	—
62	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	#367	1	2	yes	—
66	C₄×C₅₃	#368	1	2	yes	—
67	C₄×C₅₃	#423	1	1	yes	characteristic
68	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
69	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
70	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
71	#1567	—	53	2	no	—
72	#1372	—	53	4	no	—
73	C₂×C₅₃	—	2	2	no	—
74	C₂×C₅₃	#1567	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
76	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
77	C₂²×C₁₆	Dic₅₃	1	2	yes	—
78	#423	—	106	2	no	—
79	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
80	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
81	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
83	#335	—	53	4	no	—
84	#335	—	106	2	no	—
85	C₅₃	#12023	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	212	2	no	—
87	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
88	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
89	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
92	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
93	C₄⋊C₈	—	106	4	no	—
94	C₄⋊C₈	—	212	2	no	—
95	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
96	C₄×C₈	D₂₁₂	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
100	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
101	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
102	M₁₆	—	424	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	212	4	no	—
106	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
108	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
109	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
110	C₄²	D₄₂₄	1	1	yes	characteristic
111	C₄²	Dic₂₁₂	1	1	yes	characteristic
112	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
113	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	#11719917	1	4	yes	complemented
116	C₂³	#13133151	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	#13133151	1	1	yes	characteristic
121	C₂×C₄	#13133151	1	1	yes	characteristic
122	C₂×C₄	#13133151	1	1	yes	characteristic
123	C₂×C₄	#13133162	1	4	yes	—
124	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
126	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	424	4	no	—
130	C₈	#13133153	1	2	yes	—
131	C₈	C₂×#11719917	1	2	yes	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#16810451	1	2	yes	—
135	C₂²	C₂×#13133151	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	#16810451	1	2	yes	—
139	C₄	#16810453	1	2	yes	—
140	C₄	#16810647	1	1	yes	characteristic
141	C₄	C₂×#13133153	1	1	yes	characteristic
142	C₄	C₂×#13133162	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	#20273474	1	1	yes	characteristic
145	C₂	C₂×#16810451	1	1	yes	characteristic
146	C₂	C₂×#16810453	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	#107
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#12023
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20272789	C₂	semidirect
41. C₁₆×C₅₃	M₁₆	semidirect
85. C₅₃	#12023	semidirect
115. C₁₆	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20273474	2
#20272789	2
C₂×C₄×C₁₆×C₅₃	2
#12023	53