Subgroups — #28307764

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#88324⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
6	C₄×D₅₃⋊C₈	C₄	1	1	yes	characteristic
7	C₄×D₅₃⋊C₈	C₄	1	1	yes	characteristic
8	#16811509	C₂²	1	2	yes	—
9	C₄×#11719910	C₄	1	2	yes	—
10	C₈×#88324	C₄	1	2	yes	—
11	C₈×#88324	C₂²	1	2	yes	—
12	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
13	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	C₄×#11719908	C₄	1	1	yes	characteristic
15	C₁₆×Dic₅₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
17	C₂×D₅₃⋊C₈	C₈	1	2	yes	—
18	C₂×D₅₃⋊C₈	C₈	1	2	yes	—
19	D₅₃⋊C₁₆	—	2	4	no	—
20	C₂×#11719910	C₂×C₄	1	4	yes	—
21	C₄×#88324	C₈	1	2	yes	—
22	C₄×#88324	C₈	1	2	yes	—
23	C₄×#88324	C₂×C₄	1	2	yes	—
24	C₄×#88324	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719908	C₂×C₄	1	2	yes	—
29	C₁₆×D₅₃	—	2	4	no	—
30	C₄×#88322	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×#88324	C₂×C₈	1	4	yes	—
39	C₂×#88324	C₂×C₈	1	4	yes	—
40	D₅₃⋊C₈	—	2	4	no	—
41	D₅₃⋊C₈	C₁₆	1	8	yes	complemented
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
46	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
47	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×#88322	C₄²	1	1	yes	characteristic
49	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
50	C₈×D₅₃	M₁₆	1	4	yes	—
51	C₈×D₅₃	M₁₆	1	4	yes	—
52	#11719910	—	2	4	no	—
53	C₁₆×C₅₃	—	2	2	no	—
54	#11719908	—	2	2	no	—
55	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
60	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₄×D₅₃	#97	1	2	yes	—
63	C₄×D₅₃	#97	1	2	yes	—
64	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
65	C₄×D₅₃	M₃₂	1	4	yes	—
66	#88324	—	2	4	no	—
67	#88324	C₂×C₁₆	1	8	yes	complemented
68	C₈×C₅₃	#97	1	2	yes	—
69	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
70	#88322	#97	1	2	yes	—
71	#88322	C₂×M₁₆	1	2	yes	—
72	C₂×#1312	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₂×D₅₃	#322	1	2	yes	—
75	C₂×D₅₃	C₄×C₁₆	1	2	yes	—
76	C₂×D₅₃	#346	1	2	yes	—
77	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
78	C₄×C₅₃	#346	1	2	yes	—
79	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
80	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
81	Dic₅₃	#322	1	2	yes	—
82	Dic₅₃	C₂²×C₁₆	1	2	yes	—
83	Dic₅₃	C₂×M₃₂	1	2	yes	—
84	C₂×C₄×C₁₆	—	53	1	no	—
85	C₂×C₄×C₁₆	—	53	1	no	—
86	C₂×C₈²	—	53	1	no	—
87	#1312	—	53	4	no	—
88	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
89	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
90	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
91	D₅₃	#1312	1	4	yes	complemented
92	C₂²×C₁₆	—	53	2	no	—
93	C₂²×C₁₆	—	53	2	no	—
94	C₂×C₄×C₈	—	53	1	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	C₂×C₄×C₈	—	53	1	no	—
97	C₄×C₁₆	—	53	2	no	—
98	C₄×C₁₆	—	53	2	no	—
99	C₄×C₁₆	—	53	2	no	—
100	C₄×C₁₆	—	53	1	no	—
101	C₄×C₁₆	#6157	1	1	yes	characteristic
102	C₈²	—	53	2	no	—
103	C₈²	—	53	2	no	—
104	C₅₃	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	53	2	no	—
106	C₂²×C₈	—	53	1	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	2	no	—
109	C₂×C₄²	—	53	1	no	—
110	C₂×C₁₆	—	53	4	no	—
111	C₂×C₁₆	—	53	2	no	—
112	C₂×C₁₆	—	106	4	no	—
113	C₂×C₁₆	—	106	4	no	—
114	C₂×C₁₆	C₂×#6157	1	2	yes	—
115	C₄×C₈	—	53	2	no	—
116	C₄×C₈	—	53	2	no	—
117	C₄×C₈	—	53	2	no	—
118	C₄×C₈	—	53	2	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	—	53	1	no	—
122	C₄×C₈	C₂×#6157	1	1	yes	characteristic
123	C₂²×C₄	—	53	1	no	—
124	C₂²×C₄	—	53	1	no	—
125	C₂²×C₄	—	53	1	no	—
126	C₂×C₈	—	53	4	no	—
127	C₂×C₈	—	53	8	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	—	53	4	no	—
131	C₂×C₈	—	53	4	no	—
132	C₂×C₈	—	53	4	no	—
133	C₂×C₈	—	106	4	no	—
134	C₂×C₈	C₄×#6157	1	1	yes	characteristic
135	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
136	C₄²	—	53	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	53	1	no	—
139	C₄²	C₄×#6157	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	106	4	no	—
142	C₁₆	—	106	2	no	—
143	C₂³	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	2	no	—
148	C₂×C₄	—	53	2	no	—
149	C₂×C₄	—	53	4	no	—
150	C₂×C₄	—	53	4	no	—
151	C₂×C₄	C₈×#6157	1	1	yes	characteristic
152	C₂×C₄	C₈×#6157	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
154	C₈	—	53	8	no	—
155	C₈	—	53	2	no	—
156	C₈	—	53	2	no	—
157	C₈	—	106	4	no	—
158	C₈	#13133271	1	2	yes	—
159	C₈	D₅₃⋊M₁₆	1	2	yes	—
160	C₂²	—	53	2	no	—
161	C₂²	—	53	2	no	—
162	C₂²	—	53	2	no	—
163	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
164	C₄	—	53	2	no	—
165	C₄	—	53	2	no	—
166	C₄	—	53	2	no	—
167	C₄	C₁₆×#6157	1	2	yes	—
168	C₄	#16811510	1	2	yes	—
169	C₄	C₂×#13133271	1	1	yes	characteristic
170	C₄	#16811582	1	1	yes	characteristic
171	C₂	—	53	4	no	—
172	C₂	#20289911	1	1	yes	characteristic
173	C₂	C₂×C₁₆×#6157	1	1	yes	characteristic
174	C₂	C₂×#16811510	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	C₂×#1312
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #88324⋊C₁₆	C₂	semidirect
41. D₅₃⋊C₈	C₁₆	semidirect
67. #88324	C₂×C₁₆	semidirect
91. D₅₃	#1312	semidirect
104. C₅₃	C₂×#1312	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₅₃⋊C₁₆	2
C₈×D₅₃⋊C₈	2
#88324⋊C₁₆	2
C₄×C₁₆×D₅₃	2
C₂×#1312	53