Subgroups — #27299414

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	2	yes	maximal
2	C₄×#13133157	C₂	1	2	yes	maximal
3	#20272797	C₂	1	2	yes	maximal
4	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	#16810444	C₄	1	8	yes	complemented
7	#16810399	—	2	2	no	—
8	C₄×#11719908	—	2	2	no	—
9	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
11	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#376	C₄	1	1	yes	characteristic
16	#13133157	—	2	4	no	—
17	#13133157	C₂×C₄	1	8	yes	—
18	C₂×#11719908	—	2	4	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	#97×C₅₃	—	2	2	no	—
27	#97×C₅₃	C₈	1	4	yes	—
28	#97×C₅₃	C₈	1	4	yes	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×C₈×C₅₃	Q₈	1	2	yes	—
33	#95×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	—	2	1	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	C₄²	1	2	yes	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	2	yes	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄²×C₅₃	C₄²	1	2	yes	—
54	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅₃	C₂×C₈	1	2	yes	—
56	C₄²×C₅₃	C₂×C₈	1	2	yes	—
57	#11719908	—	2	8	no	—
58	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	4	2	no	—
62	C₂×C₄×C₅₃	—	4	2	no	—
63	C₂×C₄×C₅₃	—	4	2	no	—
64	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	C₄×C₈	1	4	yes	—
73	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
74	C₈×C₅₃	C₄×D₄	1	2	yes	—
75	C₈×C₅₃	C₄×Q₈	1	2	yes	—
76	#12656	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₃	#431	1	1	yes	characteristic
81	C₄×C₅₃	—	2	4	no	—
82	C₄×C₅₃	—	4	2	no	—
83	C₄×C₅₃	—	4	2	no	—
84	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₅₃	#431	1	1	yes	characteristic
88	C₂×#364	—	53	2	no	—
89	C₄×M₃₂	—	53	2	no	—
90	#1726	D₅₃	1	1	yes	characteristic
91	#1376	—	53	2	no	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₅₃	#2153	1	2	yes	—
96	C₂×M₃₂	—	53	4	no	—
97	C₂×#97	Dic₅₃	1	1	yes	characteristic
98	C₂×#97	Dic₅₃	1	1	yes	characteristic
99	C₂×#97	Dic₅₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	#376	Dic₅₃	1	1	yes	characteristic
104	#364	—	53	8	no	—
105	#346	—	106	2	no	—
106	C₄×C₁₆	—	106	2	no	—
107	C₅₃	#12656	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
115	M₃₂	—	53	8	no	—
116	M₃₂	—	106	4	no	—
117	C₂×C₁₆	—	106	4	no	—
118	#97	—	2	2	no	—
119	#97	#88322	1	4	yes	—
120	#97	#88322	1	4	yes	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	Dic₁₀₆	1	2	yes	—
123	C₄×C₈	C₄×D₅₃	1	4	yes	—
124	C₄×C₈	D₂₁₂	1	2	yes	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	#11719917	1	2	yes	—
129	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	C₂×#88322	1	2	yes	—
133	C₂×C₈	C₂×#88322	1	2	yes	—
134	C₂×C₈	C₂×#88322	1	2	yes	—
135	C₂×C₈	C₂×#88322	1	2	yes	—
136	C₂×C₈	#11719917	1	2	yes	—
137	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
138	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
139	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
141	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
142	C₂×C₈	#11719945	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂×#88322	1	2	yes	—
146	C₄²	C₂×#88322	1	2	yes	—
147	C₄²	C₄×Dic₅₃	1	2	yes	—
148	C₄²	C₄⋊Dic₅₃	1	2	yes	—
149	C₁₆	—	106	8	no	—
150	C₂³	C₂×#11719917	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₄⋊#88322	1	4	yes	—
157	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	#13133176	1	1	yes	characteristic
160	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
161	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
162	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	C₄×#88322	1	4	yes	—
165	C₈	C₄⋊#88322	1	4	yes	—
166	C₈	C₄×D₂₁₂	1	2	yes	—
167	C₈	C₄×Dic₁₀₆	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	C₄×#11719917	1	1	yes	characteristic
170	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
171	C₂²	#16810676	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
176	C₄	C₂×C₄×#88322	1	1	yes	characteristic
177	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
178	C₄	#16810676	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₄⋊#88322	1	1	yes	characteristic
182	C₂	#20276481	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12656
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #16810444	C₄	semidirect
107. C₅₃	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
C₄×#13133157	2
#20272797	2
C₅₃×#1726	2
#12656	53