Subgroups — #27275396

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#88322	C₂	1	2	yes	maximal
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	#20272553	C₂	1	2	yes	maximal
5	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810376	C₄	1	2	yes	—
10	#16810374	C₄	1	2	yes	—
11	#16810344	—	2	2	no	—
12	#16810344	C₄	1	2	yes	—
13	#16810344	C₄	1	2	yes	—
14	C₈×#88322	—	2	2	no	—
15	C₂×#108×C₅₃	C₄	1	2	yes	—
16	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#341	C₄	1	2	yes	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	Q₈	1	2	yes	—
21	C₄⋊#88322	C₂×C₄	1	4	yes	—
22	C₂²⋊#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	—	2	1	no	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₄×#88322	D₄	1	2	yes	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#108×C₅₃	C₈	1	8	yes	complemented
33	#97×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	M₁₆×C₅₃	—	2	4	no	—
43	M₁₆×C₅₃	C₂×C₈	1	8	yes	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	#41	1	1	yes	characteristic
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	C₄⋊C₄	1	4	yes	—
60	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₅₃	#131	1	2	yes	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	8	no	—
74	C₈×C₅₃	C₄×C₈	1	2	yes	—
75	C₈×C₅₃	C₄×C₈	1	2	yes	—
76	#88322	—	2	4	no	—
77	#88322	—	4	2	no	—
78	#88322	—	4	4	no	—
79	#8294	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₅₃	#405	1	1	yes	characteristic
88	C₄×C₅₃	#430	1	2	yes	—
89	C₄×C₅₃	C₈×D₄	1	2	yes	—
90	C₄×C₅₃	#444	1	2	yes	—
91	C₄×#108	D₅₃	1	1	yes	characteristic
92	#1564	—	53	1	no	—
93	C₈×M₁₆	—	53	2	no	—
94	C₂×#322	—	53	1	no	—
95	#1278	—	53	2	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₅₃	#1773	1	1	yes	characteristic
99	C₂×C₅₃	#1778	1	1	yes	characteristic
100	#431	—	53	2	no	—
101	C₂×#108	Dic₅₃	1	2	yes	—
102	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
103	C₂×C₄×C₈	—	53	1	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	#341	Dic₅₃	1	2	yes	—
107	#335	—	53	2	no	—
108	#334	—	53	2	no	—
109	#322	—	53	2	no	—
110	#322	—	53	2	no	—
111	#322	—	106	2	no	—
112	C₈²	—	106	2	no	—
113	C₅₃	#8294	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	106	1	no	—
117	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	#108	#88322	1	8	yes	complemented
121	C₄⋊C₈	—	53	4	no	—
122	C₂²⋊C₈	—	106	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	53	2	no	—
126	C₄×C₈	—	53	2	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	106	1	no	—
130	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
131	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
133	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
134	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
135	C₂²×C₄	#11719949	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×#88322	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	—	212	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	C₂×#88322	1	4	yes	—
147	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
149	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
151	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
153	C₄²	D₄×D₅₃	1	1	yes	characteristic
154	C₄²	#11719947	1	1	yes	characteristic
155	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄	#13133175	1	1	yes	characteristic
161	C₂×C₄	#13133179	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	#13133182	1	1	yes	characteristic
164	C₂×C₄	C₂²×#88322	1	2	yes	—
165	C₂×C₄	#13133224	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	8	no	—
169	C₈	—	106	4	no	—
170	C₈	—	212	2	no	—
171	C₈	—	212	4	no	—
172	C₈	C₄×#88322	1	2	yes	—
173	C₈	C₄×#88322	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
177	C₂²	Q₈×#88322	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×#88322	1	1	yes	characteristic
181	C₄	#16810579	1	1	yes	characteristic
182	C₄	#16810670	1	2	yes	—
183	C₄	D₄×#88322	1	2	yes	—
184	C₄	#16810744	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×#88322	1	1	yes	characteristic
187	C₂	#20274602	1	1	yes	characteristic
188	C₂	#20274623	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#8294
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
32. #108×C₅₃	C₈	semidirect
113. C₅₃	#8294	semidirect
120. #108	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
M₁₆×#88322	2
C₂×#16810344	2
#20272553	2
C₄×#108×C₅₃	2
#8294	53