Subgroups — #28302068

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290262	C₂	1	2	yes	maximal
2	#20289918	C₂	1	2	yes	maximal
3	#20289543	C₂	1	2	yes	maximal
4	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811639	C₂²	1	4	yes	—
6	#16811486	C₄	1	4	yes	complemented
7	#16811484	C₄	1	4	yes	complemented
8	#16811464	—	2	2	no	—
9	C₈×#88324	—	2	2	no	—
10	C₂×#13133128	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
17	C₄⋊#88324	C₂×C₄	1	8	yes	—
18	C₂²⋊#88324	—	2	4	no	—
19	C₄×#88324	—	2	4	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	—	2	2	no	—
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
25	#13133128	—	2	2	no	—
26	#13133127	—	2	2	no	—
27	#13133127	C₈	1	4	yes	complemented
28	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₈×Dic₅₃	D₄	1	2	yes	—
31	C₈×Dic₅₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#97×C₅₃	C₈	1	4	yes	complemented
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	2	1	no	—
39	C₂²×Dic₅₃	#41	1	2	yes	—
40	C₂×#88324	—	2	8	no	—
41	C₂×C₈×C₅₃	C₄²	1	2	yes	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	C₂×C₈	1	2	yes	—
49	C₄×Dic₅₃	—	2	1	no	—
50	C₄×Dic₅₃	—	4	4	no	—
51	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
52	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
53	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×#88322	—	2	1	no	—
56	C₂×#88322	—	2	1	no	—
57	C₂×#88322	C₄²	1	1	yes	characteristic
58	C₂×#88322	C₄²	1	1	yes	characteristic
59	C₂×#88322	C₄⋊C₄	1	2	yes	—
60	C₂×#88322	C₂×C₈	1	2	yes	—
61	C₂×#88322	M₁₆	1	2	yes	—
62	C₂³×C₅₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	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	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	4	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	4	2	no	—
74	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
75	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
76	#88324	—	4	8	no	—
77	C₈×C₅₃	C₄×C₈	1	4	yes	—
78	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
79	#88322	—	2	1	no	—
80	#88322	—	2	1	no	—
81	#88322	C₄⋊C₈	1	4	yes	—
82	#8268	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₅₃	#431	1	1	yes	characteristic
86	C₂²×C₅₃	#431	1	1	yes	characteristic
87	C₄×C₅₃	—	4	2	no	—
88	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
90	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	Dic₅₃	—	2	2	no	—
93	Dic₅₃	—	4	2	no	—
94	Dic₅₃	#444	1	4	yes	—
95	C₄×#97	—	53	1	no	—
96	#1564	—	53	2	no	—
97	#1450	—	53	2	no	—
98	#1276	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	#1773	1	2	yes	—
103	#431	—	53	4	no	—
104	C₂×#97	—	53	1	no	—
105	C₂×#97	—	53	1	no	—
106	C₂×#97	—	53	1	no	—
107	C₂×#97	#6157	1	1	yes	characteristic
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₄³	—	53	1	no	—
111	#335	—	53	4	no	—
112	#334	—	53	4	no	—
113	#322	—	106	2	no	—
114	C₈²	—	106	2	no	—
115	C₅₃	#8268	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	53	1	no	—
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
119	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	—	106	2	no	—
123	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
124	C₄⋊C₈	—	53	8	no	—
125	C₂²⋊C₈	—	106	4	no	—
126	#97	—	53	4	no	—
127	#97	—	106	2	no	—
128	#97	—	106	2	no	—
129	#97	#88324	1	4	yes	complemented
130	C₄×C₈	—	53	2	no	—
131	C₄×C₈	—	53	2	no	—
132	C₄×C₈	—	53	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	106	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	53	2	no	—
137	C₂²×C₄	—	106	1	no	—
138	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	106	1	no	—
145	C₂×C₈	—	106	1	no	—
146	C₂×C₈	—	106	8	no	—
147	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
148	C₂×C₈	#11719936	1	2	yes	—
149	C₂×C₈	C₄×#6157	1	2	yes	—
150	C₂×C₈	C₄⋊#6157	1	2	yes	—
151	C₂×C₈	C₂×#88324	1	2	yes	—
152	C₂×C₈	C₂×#88324	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	53	2	no	—
155	C₄²	—	53	1	no	—
156	C₄²	—	53	1	no	—
157	C₄²	—	53	2	no	—
158	C₄²	—	106	1	no	—
159	C₄²	—	212	4	no	—
160	C₄²	C₂×#88324	1	2	yes	—
161	C₂³	#13133296	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	53	2	no	—
165	C₂×C₄	—	53	2	no	—
166	C₂×C₄	—	106	4	no	—
167	C₂×C₄	—	106	2	no	—
168	C₂×C₄	—	212	2	no	—
169	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
171	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
172	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
173	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
175	C₈	—	53	4	no	—
176	C₈	—	106	1	no	—
177	C₈	—	106	1	no	—
178	C₈	—	212	8	no	—
179	C₈	C₄×#88324	1	4	yes	—
180	C₈	C₄⋊#88324	1	4	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
183	C₂²	#16811639	1	1	yes	characteristic
184	C₂²	#16811642	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	—	53	4	no	—
187	C₄	—	106	2	no	—
188	C₄	—	212	2	no	—
189	C₄	C₂×C₄×#88324	1	1	yes	characteristic
190	C₄	C₄×#11719936	1	1	yes	characteristic
191	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
192	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₄×C₄⋊#88324	1	1	yes	characteristic
196	C₂	#20290521	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#97×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#8268
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
6. #16811486	C₄	semidirect
7. #16811484	C₄	semidirect
27. #13133127	C₈	semidirect
35. #97×C₅₃	C₈	semidirect
115. C₅₃	#8268	semidirect
129. #97	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290262	2
#20289918	2
#20289543	2
#97×Dic₅₃	2
#8268	53