Subgroups — #27275282

Order 13568.

438 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	#20274568	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810373	C₂	1	2	yes	maximal
4	#20272549	C₂	1	2	yes	maximal
5	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133128	—	2	1	no	—
8	C₂×C₄×#88322	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₄	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	#16810378	—	2	1	no	—
13	#16810373	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×#97×C₅₃	—	2	1	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	2	yes	—
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	C₂×C₄	1	4	yes	—
24	#13133128	—	4	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	—	2	4	no	—
27	C₄×#88322	—	2	4	no	—
28	C₄×#88322	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
37	#97×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	—	2	4	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	SD₁₆	1	2	yes	—
51	C₄²×C₅₃	SD₁₆	1	2	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	C₄²	1	2	yes	—
61	C₂×#88322	C₄²	1	2	yes	—
62	C₂×#88322	M₁₆	1	4	yes	—
63	C₂×#88322	M₁₆	1	4	yes	—
64	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	#106	1	4	yes	—
71	C₂×C₄×C₅₃	#108	1	4	yes	—
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#135	1	2	yes	—
81	C₈×C₅₃	—	4	4	no	—
82	#88322	—	2	2	no	—
83	#88322	—	2	4	no	—
84	#88322	—	4	4	no	—
85	#88322	#97	1	4	yes	complemented
86	#8273	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₅₃	#423	1	2	yes	—
89	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
90	C₂²×C₅₃	#427	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×#108	1	2	yes	—
92	C₄×C₅₃	—	2	4	no	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₅₃	#432	1	2	yes	—
98	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
99	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
100	#1767	—	53	1	no	—
101	#1767	D₅₃	1	1	yes	characteristic
102	C₄²×C₈	—	53	1	no	—
103	C₂×#334	—	53	2	no	—
104	#1277	—	53	2	no	—
105	C₂×C₅₃	#1566	1	2	yes	—
106	C₂×C₅₃	#1568	1	2	yes	—
107	C₂×C₅₃	#1767	1	1	yes	characteristic
108	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
109	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
112	C₂×#97	—	2	1	no	—
113	C₂×#97	—	106	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	2	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₄³	C₂×D₅₃	1	1	yes	characteristic
119	#336	—	2	1	no	—
120	#336	—	106	1	no	—
121	#334	—	53	8	no	—
122	C₅₃	#8273	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	53	1	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	53	2	no	—
127	C₂²×C₈	—	106	2	no	—
128	C₂×C₄²	Dic₁₀₆	1	2	yes	—
129	C₂×C₄²	C₄×D₅₃	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	53	4	no	—
135	C₄⋊C₈	—	106	2	no	—
136	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
137	#97	—	4	2	no	—
138	#97	—	212	2	no	—
139	C₄×C₈	—	53	4	no	—
140	C₄×C₈	—	106	4	no	—
141	C₄×C₈	—	106	4	no	—
142	C₄×C₈	—	106	2	no	—
143	C₂²×C₄	#11719919	1	2	yes	—
144	C₂²×C₄	#11719919	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	106	4	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#11719923	1	2	yes	—
166	C₄²	#11719924	1	2	yes	—
167	C₄²	C₂×C₄×D₅₃	1	2	yes	—
168	C₄²	#11719945	1	2	yes	—
169	C₂³	C₂×#11719919	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#13133148	1	4	yes	—
176	C₂×C₄	#13133152	1	4	yes	—
177	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
178	C₂×C₄	#13133174	1	1	yes	characteristic
179	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
180	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
181	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
182	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
183	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
185	C₂×C₄	#13133237	1	2	yes	—
186	C₈	—	4	4	no	—
187	C₈	—	53	4	no	—
188	C₈	—	106	2	no	—
189	C₈	—	106	4	no	—
190	C₈	—	212	4	no	—
191	C₂²	C₄×#11719919	1	1	yes	characteristic
192	C₂²	#16810646	1	2	yes	—
193	C₂²	C₂×#13133148	1	1	yes	characteristic
194	C₂²	#16810659	1	1	yes	characteristic
195	C₂²	C₂×#13133152	1	2	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	#97×D₅₃	1	1	yes	characteristic
200	C₄	#16810575	1	1	yes	characteristic
201	C₄	#16810678	1	2	yes	—
202	C₄	C₄×#11719923	1	2	yes	—
203	C₄	C₄×#11719924	1	2	yes	—
204	C₂	#20273470	1	2	yes	—
205	C₂	#20273476	1	2	yes	—
206	C₂	#20274569	1	1	yes	characteristic
207	C₂	C₄×#13133148	1	1	yes	characteristic
208	C₂	C₄×#13133152	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1767	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1767	D₅₃
hypocenter	C₅₃	#8273
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
13. #16810373	C₄	semidirect
85. #88322	#97	semidirect
122. C₅₃	#8273	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₄²×#88322	2
C₂×#16810373	2
#20272549	2
C₅₃×#1767	2
#8273	53