Subgroups — #27283123

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810372	C₂	1	4	yes	maximal
4	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810372	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₄	1	4	yes	—
12	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#384	—	2	1	no	—
14	C₅₃×#384	—	2	1	no	—
15	C₅₃×#384	—	2	2	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	2	4	no	—
21	C₄⋊#88322	—	4	4	no	—
22	C₄⋊#88322	D₄	1	4	yes	—
23	C₄×#88322	—	4	2	no	—
24	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₄⋊Q₈×C₅₃	—	2	4	no	—
26	C₄⋊Q₈×C₅₃	—	2	4	no	—
27	C₄⋊Q₈×C₅₃	—	4	8	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₅₃	—	2	8	no	—
38	C₂×Q₈×C₅₃	—	4	8	no	—
39	C₂²×C₄×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	4	8	no	—
46	C₄⋊C₄×C₅₃	—	4	8	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄⋊C₄×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	—	2	4	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	SD₁₆	1	2	yes	—
53	C₄²×C₅₃	SD₁₆	1	4	yes	—
54	C₄²×C₅₃	Q₁₆	1	4	yes	—
55	C₄²×C₅₃	Q₁₆	1	2	yes	—
56	C₄²×C₅₃	C₂×D₄	1	2	yes	—
57	C₄²×C₅₃	#41	1	2	yes	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	—	4	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂×#88322	—	4	4	no	—
63	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	Q₈×C₅₃	—	8	8	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	4	8	no	—
70	C₂×C₄×C₅₃	#100	1	4	yes	—
71	C₂×C₄×C₅₃	#103	1	4	yes	—
72	C₂×C₄×C₅₃	#103	1	4	yes	—
73	C₂×C₄×C₅₃	#103	1	4	yes	—
74	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₅₃	#121	1	2	yes	—
76	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₅₃	#137	1	2	yes	—
83	#88322	—	4	4	no	—
84	#88322	—	8	2	no	—
85	#9691	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#390	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#100	1	2	yes	—
88	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×#103	1	2	yes	—
90	C₂²×C₅₃	#419	1	1	yes	characteristic
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	8	4	no	—
94	C₄×C₅₃	#333	1	8	yes	complemented
95	C₄×C₅₃	#461	1	2	yes	—
96	C₄×C₅₃	#462	1	2	yes	—
97	C₄×C₅₃	#487	1	2	yes	—
98	C₄×C₅₃	#492	1	1	yes	characteristic
99	C₄×C₅₃	#494	1	1	yes	characteristic
100	#2868	D₅₃	1	1	yes	characteristic
101	#1842	—	53	1	no	—
102	C₄×C₄⋊C₈	—	53	1	no	—
103	C₂×#333	—	53	4	no	—
104	C₂×C₅₃	C₂×#333	1	2	yes	—
105	C₂×C₅₃	#1548	1	2	yes	—
106	C₂×C₅₃	#1956	1	1	yes	characteristic
107	C₂×C₅₃	#1965	1	1	yes	characteristic
108	C₂×C₅₃	#1981	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	Dic₅₃	1	4	yes	—
110	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
111	C₂×C₄⋊C₈	—	53	2	no	—
112	C₂×C₄⋊C₈	—	53	2	no	—
113	C₂×C₄⋊C₈	—	106	2	no	—
114	C₂×C₄×C₈	—	106	1	no	—
115	#384	—	2	1	no	—
116	#384	—	2	1	no	—
117	#384	—	2	2	no	—
118	C₄³	C₂×D₅₃	1	1	yes	characteristic
119	#336	—	106	1	no	—
120	#333	—	106	8	no	—
121	C₅₃	#9691	1	1	yes	characteristic, complemented, Sylow
122	C₂²×Q₈	C₂×Dic₅₃	1	4	yes	—
123	C₂²×C₈	—	106	2	no	—
124	C₂²×C₈	—	106	2	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	4	8	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
136	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	53	4	no	—
138	C₄⋊C₈	—	106	4	no	—
139	C₄⋊C₈	—	106	2	no	—
140	C₄⋊C₈	—	212	4	no	—
141	C₄×C₈	—	212	2	no	—
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	—	4	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
146	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂×C₈	—	106	4	no	—
151	C₂×C₈	—	212	2	no	—
152	C₂×C₈	—	212	4	no	—
153	C₂×C₈	—	212	2	no	—
154	C₂×C₈	—	212	2	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	#11719924	1	2	yes	—
163	C₄²	#11719924	1	4	yes	—
164	C₄²	#11719925	1	4	yes	—
165	C₄²	#11719925	1	2	yes	—
166	C₄²	D₄×D₅₃	1	2	yes	—
167	C₄²	#11719947	1	2	yes	—
168	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	#13133134	1	4	yes	—
176	C₂×C₄	#13133141	1	4	yes	—
177	C₂×C₄	#13133141	1	4	yes	—
178	C₂×C₄	#13133141	1	4	yes	—
179	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
180	C₂×C₄	#13133188	1	2	yes	—
181	C₂×C₄	#13133204	1	1	yes	characteristic
182	C₂×C₄	#13133213	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719924	1	2	yes	—
185	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
186	C₂×C₄	C₂×#11719925	1	2	yes	—
187	C₂×C₄	#13133248	1	2	yes	—
188	C₈	—	424	2	no	—
189	C₈	—	212	4	no	—
190	C₂²	#16810526	1	1	yes	characteristic
191	C₂²	C₂×#13133134	1	2	yes	—
192	C₂²	C₂×#13133141	1	2	yes	—
193	C₂²	C₂×#13133141	1	1	yes	characteristic
194	C₂²	#16810632	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	8	4	no	—
198	C₄	#16810372	1	8	yes	complemented
199	C₄	#16810825	1	2	yes	—
200	C₄	#16810830	1	2	yes	—
201	C₄	#16810973	1	2	yes	—
202	C₄	#16810997	1	1	yes	characteristic
203	C₄	#16811002	1	1	yes	characteristic
204	C₂	C₂×#16810372	1	2	yes	—
205	C₂	#20273396	1	2	yes	—
206	C₂	#20275518	1	1	yes	characteristic
207	C₂	#20275556	1	1	yes	characteristic
208	C₂	#20275639	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#9691
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
94. C₄×C₅₃	#333	semidirect
121. C₅₃	#9691	semidirect
198. C₄	#16810372	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄×C₄⋊#88322	2
C₂×#16810372	2
C₅₃×#2868	2
#9691	53