Subgroups — #28310188

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811512	C₂	1	2	yes	maximal
2	C₂×#16811511	C₂	1	2	yes	maximal
3	C₂×#16811509	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	4	yes	—
6	#16811512	C₄	1	8	yes	complemented
7	#16811511	C₄	1	8	yes	complemented
8	#16811509	C₄	1	8	yes	complemented
9	C₂×C₄×#88322	C₂²	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	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719910	C₂×C₄	1	8	yes	—
17	C₂×#11719910	C₂×C₄	1	8	yes	—
18	C₂×#11719910	C₂×C₄	1	8	yes	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#88322	C₂×C₄	1	4	yes	—
25	C₈×Dic₅₃	C₈	1	4	yes	—
26	C₈×Dic₅₃	C₈	1	4	yes	—
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	—
33	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
36	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₄²	1	2	yes	—
38	C₂×C₈×C₅₃	C₄²	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₄²×C₅₃	C₄²	1	2	yes	—
44	C₄²×C₅₃	C₂×C₈	1	2	yes	—
45	C₄×Dic₅₃	—	2	8	no	—
46	C₄×Dic₅₃	C₄²	1	2	yes	—
47	C₄×Dic₅₃	C₄²	1	2	yes	—
48	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
49	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
50	C₂×#88322	C₄²	1	2	yes	—
51	C₂×#88322	C₂×C₈	1	2	yes	—
52	C₂×#88322	C₂×C₈	1	2	yes	—
53	C₂×#88322	C₂²×C₄	1	2	yes	—
54	C₂×#88322	C₂²×C₄	1	2	yes	—
55	C₂×#88322	C₂²×C₄	1	2	yes	—
56	#11719910	C₄²	1	16	yes	complemented
57	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
60	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
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×Dic₅₃	—	2	4	no	—
67	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
68	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
69	C₂×Dic₅₃	M₃₂	1	8	yes	—
70	C₈×C₅₃	C₄×C₈	1	4	yes	—
71	C₈×C₅₃	C₄×C₈	1	4	yes	—
72	#88322	C₄×C₈	1	4	yes	—
73	#88322	C₂×C₄²	1	4	yes	—
74	C₄×#346	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	C₄³	1	1	yes	characteristic
81	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
84	Dic₅₃	—	2	4	no	—
85	Dic₅₃	#346	1	8	yes	complemented
86	C₂×#346	—	53	2	no	—
87	C₂×#346	—	53	2	no	—
88	C₂×C₄×C₁₆	—	53	2	no	—
89	C₄²×C₈	—	53	1	no	—
90	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
91	C₂×C₅₃	C₂×#346	1	2	yes	—
92	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
93	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
94	C₂²×C₁₆	—	53	4	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	C₂×C₄×C₈	—	53	1	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	#6157	1	1	yes	characteristic
101	C₄³	—	53	1	no	—
102	#346	—	53	8	no	—
103	#346	—	53	8	no	—
104	C₄×C₁₆	—	53	8	no	—
105	C₅₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	53	1	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
109	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
110	C₂×C₄²	—	53	1	no	—
111	C₂×C₄²	—	53	1	no	—
112	C₂×C₄²	—	53	4	no	—
113	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
114	C₂×C₁₆	—	53	8	no	—
115	C₂×C₁₆	—	53	8	no	—
116	C₂×C₁₆	—	53	8	no	—
117	C₄×C₈	—	53	4	no	—
118	C₄×C₈	—	53	4	no	—
119	C₄×C₈	—	53	4	no	—
120	C₄×C₈	—	53	4	no	—
121	C₄×C₈	—	53	4	no	—
122	C₄×C₈	#88324	1	4	yes	—
123	C₂²×C₄	—	53	2	no	—
124	C₂²×C₄	—	53	2	no	—
125	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
126	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
127	C₂×C₈	—	53	2	no	—
128	C₂×C₈	—	53	2	no	—
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	D₅₃⋊C₈	1	2	yes	—
134	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
135	C₂×C₈	C₄×#6157	1	2	yes	—
136	C₂×C₈	C₄×#6157	1	2	yes	—
137	C₂×C₈	C₂×#88324	1	2	yes	—
138	C₂×C₈	C₂×#88324	1	2	yes	—
139	C₄²	—	53	2	no	—
140	C₄²	—	53	2	no	—
141	C₄²	—	53	2	no	—
142	C₄²	—	53	2	no	—
143	C₄²	—	106	8	no	—
144	C₄²	C₄×#6157	1	2	yes	—
145	C₄²	C₂×#88324	1	2	yes	—
146	C₁₆	—	53	16	no	—
147	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	53	4	no	—
150	C₂×C₄	—	53	8	no	—
151	C₂×C₄	—	53	4	no	—
152	C₂×C₄	—	106	4	no	—
153	C₂×C₄	C₄×#88324	1	4	yes	—
154	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
157	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
158	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
159	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
160	C₈	—	53	4	no	—
161	C₈	—	53	4	no	—
162	C₈	C₄×#88324	1	4	yes	—
163	C₈	C₄×#88324	1	4	yes	—
164	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
165	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
166	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
167	C₂²	D₅₃⋊M₃₂	1	4	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	53	8	no	—
170	C₄	—	106	4	no	—
171	C₄	C₄²×#6157	1	1	yes	characteristic
172	C₄	C₂×C₄×#88324	1	1	yes	characteristic
173	C₄	C₂×C₄×#88324	1	1	yes	characteristic
174	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
175	C₂	C₄²×#88324	1	1	yes	characteristic
176	C₂	#20290916	1	2	yes	—
177	C₂	#20290921	1	2	yes	—
178	C₂	#20290922	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	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₄×#346
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
6. #16811512	C₄	semidirect
7. #16811511	C₄	semidirect
8. #16811509	C₄	semidirect
56. #11719910	C₄²	semidirect
85. Dic₅₃	#346	semidirect
105. C₅₃	C₄×#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811512	2
C₂×#16811511	2
C₂×#16811509	2
C₄×C₈×Dic₅₃	2
C₄×#346	53