Subgroups — #27293134

Order 13568.

250 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
3	#20272636	C₂	1	2	yes	maximal, complemented
4	#20272636	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₄	1	2	yes	—
7	C₄⋊#11719908	C₄	1	2	yes	—
8	C₂²⋊#11719908	C₄	1	2	yes	—
9	#16810399	C₄	1	2	yes	—
10	#16810399	C₂²	1	2	yes	—
11	#16810399	C₂²	1	1	yes	characteristic
12	#16810399	C₂²	1	1	yes	characteristic
13	C₅₃×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#322	C₄	1	2	yes	—
17	C₅₃×#322	C₂²	1	2	yes	—
18	C₂×#11719908	—	2	4	no	—
19	C₂×#11719908	C₂×C₄	1	2	yes	—
20	C₂×#11719908	C₂×C₄	1	4	yes	—
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₅₃	—	2	1	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	1	no	—
27	C₂²⋊C₈×C₅₃	—	2	1	no	—
28	#97×C₅₃	—	2	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
38	M₁₆×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄²	1	1	yes	characteristic
48	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
49	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	#11719908	—	2	2	no	—
52	#11719908	—	2	4	no	—
53	#11719908	M₁₆	1	4	yes	complemented
54	C₂³×C₅₃	#97	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	4	2	no	—
68	C₈×C₅₃	—	4	2	no	—
69	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
70	C₈×C₅₃	#131	1	2	yes	—
71	#11413	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	#347	1	2	yes	—
75	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	1	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
82	C₄×C₅₃	#405	1	1	yes	characteristic
83	#2163	—	53	1	no	—
84	C₂×#346	—	53	1	no	—
85	#1451	D₅₃	1	1	yes	characteristic
86	#1313	—	53	2	no	—
87	#1313	—	53	2	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	#1451	1	1	yes	characteristic
90	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
91	C₂×C₅₃	#2110	1	1	yes	characteristic
92	C₂²×C₁₆	—	53	2	no	—
93	#431	Dic₅₃	1	1	yes	characteristic
94	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₄⋊C₁₆	—	53	2	no	—
97	C₂²⋊C₁₆	—	53	2	no	—
98	#346	—	53	2	no	—
99	#346	—	53	1	no	—
100	#346	—	53	1	no	—
101	#346	—	53	2	no	—
102	#322	Dic₅₃	1	2	yes	—
103	#322	C₂×D₅₃	1	2	yes	—
104	C₅₃	#11413	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	2	1	no	—
106	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂×C₁₆	—	53	2	no	—
110	C₂×C₁₆	—	53	2	no	—
111	C₂×C₁₆	—	53	4	no	—
112	C₂×C₁₆	—	106	4	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₂²⋊C₈	—	2	1	no	—
115	#97	—	2	1	no	—
116	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
117	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
119	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
120	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	C₂²×C₄	#11719917	1	1	yes	characteristic
123	C₂²×C₄	#11719917	1	1	yes	characteristic
124	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
134	C₄²	#11719917	1	1	yes	characteristic
135	C₄²	#11719917	1	1	yes	characteristic
136	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
137	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
138	C₁₆	—	53	4	no	—
139	C₁₆	—	106	2	no	—
140	C₁₆	—	106	4	no	—
141	C₂³	#13133128	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#13133128	1	1	yes	characteristic
147	C₂×C₄	#13133128	1	1	yes	characteristic
148	C₂×C₄	#13133128	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
151	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	2	no	—
156	C₈	M₁₆×D₅₃	1	2	yes	—
157	C₈	#13133225	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#16810401	1	2	yes	—
161	C₂²	C₂×#13133128	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
168	C₄	#16810581	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#20273002	1	1	yes	characteristic
171	C₂	C₂×#16810401	1	1	yes	characteristic
172	C₂	#20276320	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#11413
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #20272636	C₂	semidirect
4. #20272636	C₂	semidirect
53. #11719908	M₁₆	semidirect
104. C₅₃	#11413	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
C₂×#16810399	2
#20272636	2
C₅₃×#1451	2
#11413	53