Subgroups — #28300039

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290429	C₂	1	2	yes	maximal
2	C₂×#16811464	C₂	1	2	yes	maximal
3	C₂×C₈×#88324	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88324	C₂²	1	4	yes	—
6	#16811489	C₄	1	8	yes	—
7	#16811464	C₄	1	8	yes	complemented
8	C₈×#88324	C₄	1	8	yes	complemented
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂²×#88324	C₂×C₄	1	8	yes	—
17	C₄×#88324	C₂×C₄	1	8	yes	—
18	C₄×#88324	C₂×C₄	1	8	yes	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
23	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
24	C₄×#88322	C₈	1	4	yes	complemented
25	C₈×Dic₅₃	C₈	1	4	yes	complemented
26	C₈×Dic₅₃	C₈	1	4	yes	complemented
27	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
28	C₈×Dic₅₃	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	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
33	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₄²	1	2	yes	—
36	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
37	C₂×#88324	—	2	16	no	—
38	C₂×#88324	C₄²	1	16	yes	—
39	C₂×C₈×C₅₃	C₄²	1	2	yes	—
40	C₂×C₈×C₅₃	C₄²	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₄²×C₅₃	C₂×C₈	1	2	yes	—
46	C₄²×C₅₃	C₂×C₈	1	2	yes	—
47	C₄×Dic₅₃	—	2	8	no	—
48	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
49	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
50	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
51	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
52	C₂×#88322	C₄²	1	2	yes	—
53	C₂×#88322	C₄²	1	2	yes	—
54	C₂×#88322	C₂×C₈	1	2	yes	—
55	C₂×#88322	C₂×C₈	1	2	yes	—
56	C₂×#88322	C₂×C₈	1	2	yes	—
57	C₂×#88322	C₂×C₈	1	2	yes	—
58	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×Dic₅₃	—	2	4	no	—
68	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
69	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
70	C₂×Dic₅₃	#131	1	8	yes	—
71	#88324	—	2	16	no	—
72	C₈×C₅₃	C₄×C₈	1	4	yes	—
73	C₈×C₅₃	C₄×C₈	1	4	yes	—
74	#88322	C₄×C₈	1	4	yes	—
75	#88322	C₄×C₈	1	4	yes	—
76	#7841	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₄³	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
80	C₂²×C₅₃	#405	1	4	yes	—
81	C₄×C₅₃	—	2	4	no	—
82	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
85	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
86	Dic₅₃	—	2	4	no	—
87	Dic₅₃	#405	1	8	yes	—
88	#1744	—	53	2	no	—
89	C₄²×C₈	—	53	1	no	—
90	C₂×#322	—	53	2	no	—
91	C₂×C₈²	—	53	2	no	—
92	C₂×C₅₃	#1450	1	2	yes	—
93	C₂×C₅₃	#1450	1	2	yes	—
94	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
95	C₂×C₅₃	#1726	1	2	yes	—
96	C₂×C₄×C₈	—	53	4	no	—
97	C₂×C₄×C₈	—	53	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	C₂×C₄×C₈	—	53	1	no	—
102	C₂×C₄×C₈	#6157	1	1	yes	characteristic
103	C₄³	—	53	1	no	—
104	#336	—	53	8	no	—
105	#322	—	53	8	no	—
106	C₈²	—	53	8	no	—
107	C₅₃	#7841	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	—	53	1	no	—
110	C₂²×C₈	—	53	8	no	—
111	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
112	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
113	C₂×C₄²	—	53	1	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₄²	—	53	4	no	—
116	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
117	C₄×C₈	—	53	8	no	—
118	C₄×C₈	—	53	8	no	—
119	C₄×C₈	—	53	4	no	—
120	C₄×C₈	—	53	4	no	—
121	C₄×C₈	—	53	4	no	—
122	C₄×C₈	—	53	4	no	—
123	C₄×C₈	—	53	4	no	—
124	C₄×C₈	#88324	1	4	yes	complemented
125	C₂²×C₄	—	53	2	no	—
126	C₂²×C₄	—	53	2	no	—
127	C₂²×C₄	C₄×#6157	1	2	yes	—
128	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
129	C₂×C₈	—	53	2	no	—
130	C₂×C₈	—	53	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	53	2	no	—
135	C₂×C₈	—	53	16	no	—
136	C₂×C₈	—	106	16	no	—
137	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
138	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
139	C₂×C₈	C₄×#6157	1	2	yes	—
140	C₂×C₈	C₄×#6157	1	2	yes	—
141	C₂×C₈	C₂×#88324	1	2	yes	—
142	C₂×C₈	C₂×#88324	1	2	yes	—
143	C₄²	—	53	2	no	—
144	C₄²	—	53	2	no	—
145	C₄²	—	53	2	no	—
146	C₄²	—	53	2	no	—
147	C₄²	—	106	8	no	—
148	C₄²	D₅₃⋊C₈	1	2	yes	—
149	C₄²	C₂×#88324	1	2	yes	—
150	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	53	4	no	—
153	C₂×C₄	—	53	8	no	—
154	C₂×C₄	—	53	4	no	—
155	C₂×C₄	—	106	4	no	—
156	C₂×C₄	C₄×#88324	1	4	yes	—
157	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
159	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
160	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
161	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
163	C₈	—	53	4	no	—
164	C₈	—	53	4	no	—
165	C₈	—	106	16	no	—
166	C₈	C₄×#88324	1	4	yes	—
167	C₈	C₄×#88324	1	4	yes	—
168	C₂²	C₄²×#6157	1	1	yes	characteristic
169	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
170	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
171	C₂²	#16811586	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	53	8	no	—
174	C₄	—	106	4	no	—
175	C₄	C₂×C₄×#88324	1	1	yes	characteristic
176	C₄	C₂×C₄×#88324	1	1	yes	characteristic
177	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
178	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
179	C₂	#20289918	1	2	yes	—
180	C₂	#20289919	1	2	yes	—
181	C₂	C₄²×#88324	1	1	yes	characteristic
182	C₂	#20290371	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7841
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #16811464	C₄	semidirect
8. C₈×#88324	C₄	semidirect
24. C₄×#88322	C₈	semidirect
25. C₈×Dic₅₃	C₈	semidirect
26. C₈×Dic₅₃	C₈	semidirect
32. C₄×C₈×C₅₃	C₈	semidirect
107. C₅₃	#7841	semidirect
124. C₄×C₈	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290429	2
C₂×#16811464	2
C₂×C₈×#88324	2
C₄×C₈×Dic₅₃	2
#7841	53