Subgroups — #27275387

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
2	#20272996	C₂	1	2	yes	maximal
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	#20272551	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₄	1	2	yes	—
7	C₄×#11719917	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810388	C₄	1	2	yes	—
11	#16810345	—	2	2	no	—
12	C₈×#88322	—	2	2	no	—
13	C₈×#88322	C₄	1	2	yes	—
14	C₈×#88322	C₄	1	2	yes	—
15	C₅₃×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	C₄	1	2	yes	—
18	C₅₃×#334	C₄	1	2	yes	—
19	C₂×#11719917	C₂×C₄	1	4	yes	—
20	C₂²×#88322	—	2	1	no	—
21	C₂²×#88322	Q₈	1	2	yes	—
22	#13133153	C₈	1	8	yes	complemented
23	#13133128	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₄×#88322	C₂×C₄	1	2	yes	—
28	C₄×#88322	D₄	1	1	yes	characteristic
29	C₄×#88322	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
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	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
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	C₂×C₈	1	4	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	#11719917	—	2	4	no	—
55	#11719917	C₂×C₈	1	8	yes	—
56	C₂×#88322	—	2	4	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	C₄⋊C₄	1	2	yes	—
61	C₂×#88322	C₄⋊C₄	1	2	yes	—
62	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
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	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₃	#131	1	2	yes	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	—	4	4	no	—
76	C₈×C₅₃	C₄×C₈	1	2	yes	—
77	C₈×C₅₃	C₄×C₈	1	2	yes	—
78	#88322	—	2	4	no	—
79	#88322	—	2	2	no	—
80	#88322	—	2	8	no	—
81	#88322	#108	1	4	yes	complemented
82	C₈×#108	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₅₃	#405	1	1	yes	characteristic
91	C₄×C₅₃	C₂×#108	1	2	yes	—
92	C₄×C₅₃	C₈×D₄	1	2	yes	—
93	C₄×C₅₃	#443	1	2	yes	—
94	C₄×#108	—	53	1	no	—
95	#1564	D₅₃	1	1	yes	characteristic
96	C₈×M₁₆	—	53	2	no	—
97	C₂×C₈²	—	53	1	no	—
98	#1278	—	53	2	no	—
99	C₂×C₅₃	—	2	2	no	—
100	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅₃	#1772	1	1	yes	characteristic
102	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
103	#431	C₂×D₅₃	1	2	yes	—
104	C₂×#108	—	53	2	no	—
105	C₄×M₁₆	—	53	2	no	—
106	C₂×C₄×C₈	—	53	1	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	#341	—	53	2	no	—
110	#335	Dic₅₃	1	2	yes	—
111	#334	Dic₅₃	1	2	yes	—
112	#322	—	106	2	no	—
113	C₈²	—	53	2	no	—
114	C₈²	—	53	2	no	—
115	C₈²	—	106	2	no	—
116	C₅₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	53	4	no	—
118	C₂²×C₈	—	53	2	no	—
119	C₂²×C₈	—	106	1	no	—
120	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	#108	—	53	8	no	—
124	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	106	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	53	1	no	—
131	C₄×C₈	—	53	1	no	—
132	C₄×C₈	—	106	2	no	—
133	C₄×C₈	—	106	1	no	—
134	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
139	C₂²×C₄	#11719947	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
141	M₁₆	—	53	8	no	—
142	M₁₆	—	106	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	106	4	no	—
148	C₂×C₈	—	106	4	no	—
149	C₂×C₈	—	106	2	no	—
150	C₂×C₈	—	106	4	no	—
151	C₂×C₈	C₈×D₅₃	1	4	yes	—
152	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₄²	D₄×D₅₃	1	1	yes	characteristic
158	C₄²	#11719949	1	1	yes	characteristic
159	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
160	C₂³	#13133182	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133175	1	1	yes	characteristic
166	C₂×C₄	#13133179	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
170	C₂×C₄	#13133225	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	53	4	no	—
175	C₈	—	106	4	no	—
176	C₈	—	106	2	no	—
177	C₈	—	106	8	no	—
178	C₈	C₈×Dic₅₃	1	2	yes	—
179	C₈	C₈×Dic₅₃	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
183	C₂²	#16810747	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
187	C₄	#16810581	1	1	yes	characteristic
188	C₄	#108×D₅₃	1	2	yes	—
189	C₄	#16810688	1	2	yes	—
190	C₄	#16810740	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#20274580	1	1	yes	characteristic
193	C₂	#20274596	1	1	yes	characteristic
194	C₂	#108×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	C₈×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
22. #13133153	C₈	semidirect
81. #88322	#108	semidirect
116. C₅₃	C₈×#108	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133153	2
#20272996	2
C₂×C₈×#88322	2
#20272551	2
C₅₃×#1564	2
C₈×#108	53