Subgroups — #27382823

Order 13568.

1122 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	2	yes	maximal
2	C₄⋊C₄×#88322	C₂	1	8	yes	maximal, complemented
3	C₄×C₂²⋊#88322	C₂	1	4	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₄	1	16	yes	complemented
6	C₂×C₄⋊#88322	C₂²	1	4	yes	—
7	C₂×C₂²⋊#88322	C₂²	1	4	yes	—
8	C₂×C₄×#88322	C₂²	1	8	yes	—
9	C₂×C₄×#88322	C₂²	1	4	yes	—
10	#16810378	C₂²	1	8	yes	—
11	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₅₃	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
16	C₂×#95×C₅₃	C₄	1	2	yes	—
17	C₂²×#88322	C₂³	1	8	yes	—
18	C₄⋊#88322	C₂×C₄	1	16	yes	complemented
19	C₂²⋊#88322	C₂×C₄	1	16	yes	—
20	C₄×#88322	—	4	8	no	—
21	C₄×#88322	C₂×C₄	1	16	yes	complemented
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₅₃	C₈	1	16	yes	complemented
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₅₃	—	4	4	no	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
34	#95×C₅₃	C₂×C₄	1	8	yes	—
35	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	4	4	no	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	—	4	4	no	—
41	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
42	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	C₂×C₈	1	32	yes	—
52	C₄²×C₅₃	—	2	8	no	—
53	C₄²×C₅₃	—	4	4	no	—
54	C₄²×C₅₃	C₄²	1	8	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
56	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂×#88322	—	2	8	no	—
58	C₂×#88322	—	2	8	no	—
59	C₂×#88322	C₂²×C₄	1	16	yes	—
60	C₂³×C₅₃	—	2	1	no	—
61	C₂³×C₅₃	—	2	1	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
64	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
65	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	8	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	2	8	no	—
71	C₂×C₄×C₅₃	—	2	8	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	4	16	no	—
74	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
75	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	16	yes	—
79	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
82	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
83	#88322	—	2	16	no	—
84	#20533	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	2	4	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
90	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
91	C₂²×C₅₃	#518	1	1	yes	characteristic
92	C₂²×C₅₃	#519	1	1	yes	characteristic
93	C₂²×C₅₃	C₂³×C₈	1	2	yes	—
94	C₂²×C₅₃	#568	1	2	yes	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	16	no	—
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	C₂×C₄×C₈	1	8	yes	—
99	C₄×C₅₃	#520	1	4	yes	—
100	C₂×#431	—	53	2	no	—
101	C₂×C₄×C₄⋊C₄	D₅₃	1	1	yes	characteristic
102	C₈×C₄⋊C₄	—	53	8	no	—
103	C₄×C₂²⋊C₈	—	53	4	no	—
104	C₂×C₅₃	—	2	4	no	—
105	C₂×C₅₃	#2274	1	1	yes	characteristic
106	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
107	C₂×C₅₃	#2873	1	1	yes	characteristic
108	C₂×C₅₃	#2972	1	2	yes	—
109	C₂×C₅₃	#2982	1	2	yes	—
110	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
111	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
112	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
113	#431	—	53	16	no	—
114	C₂×C₄⋊C₈	—	53	4	no	—
115	C₂×C₂²⋊C₈	—	53	4	no	—
116	C₂×C₄×C₈	—	53	4	no	—
117	C₂×C₄×C₈	—	53	8	no	—
118	C₄×C₄⋊C₄	Dic₅₃	1	4	yes	—
119	C₄×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
120	C₂×#95	Dic₅₃	1	2	yes	—
121	#336	—	53	8	no	—
122	C₅₃	#20533	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
124	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂²×C₈	—	53	8	no	—
128	C₂×C₄⋊C₄	#88322	1	16	yes	complemented
129	C₂×C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
130	C₂×C₄²	—	4	4	no	—
131	C₂×C₄²	C₄×D₅₃	1	8	yes	—
132	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
133	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
135	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
136	C₄⋊C₈	—	53	16	no	—
137	C₂²⋊C₈	—	53	16	no	—
138	C₄×C₈	—	53	16	no	—
139	C₄×C₈	—	212	8	no	—
140	#95	C₂×Dic₅₃	1	8	yes	—
141	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂×#88322	1	16	yes	—
148	C₂²×C₄	C₂×#88322	1	8	yes	—
149	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
150	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
155	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
156	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
157	C₂×C₈	—	53	16	no	—
158	C₂×C₈	—	106	8	no	—
159	C₂×C₈	—	106	8	no	—
160	C₄⋊C₄	C₂×#88322	1	32	yes	—
161	C₄²	—	2	8	no	—
162	C₄²	—	4	4	no	—
163	C₄²	C₄×Dic₅₃	1	8	yes	—
164	C₄²	C₂×C₄×D₅₃	1	4	yes	—
165	C₄²	C₂×C₄×D₅₃	1	4	yes	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
170	C₂³	C₂²×#88322	1	4	yes	—
171	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	16	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₄×#88322	1	16	yes	complemented
181	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
182	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
183	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
184	C₂×C₄	C₂²×#88322	1	16	yes	—
185	C₂×C₄	C₂²×#88322	1	8	yes	—
186	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
187	C₂×C₄	#13133262	1	2	yes	—
188	C₂×C₄	#13133265	1	2	yes	—
189	C₈	—	106	16	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	2	no	—
194	C₂²	C₂×C₄×#88322	1	4	yes	—
195	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
196	C₂²	#16811098	1	1	yes	characteristic
197	C₂²	#16811101	1	1	yes	characteristic
198	C₂²	C₂³×#88322	1	2	yes	—
199	C₂²	#16811390	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	16	no	—
202	C₄	—	2	4	no	—
203	C₄	C₂×C₄×#88322	1	8	yes	—
204	C₄	#16811104	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	#20277072	1	1	yes	characteristic
207	C₂	C₂²×C₄×#88322	1	1	yes	characteristic
208	C₂	#20283134	1	1	yes	characteristic
209	C₂	#20283739	1	2	yes	—
210	C₂	#20283796	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂×C₄×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#20533
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×#88322	C₂	semidirect
5. #16810676	C₄	semidirect
18. C₄⋊#88322	C₂×C₄	semidirect
21. C₄×#88322	C₂×C₄	semidirect
26. C₂×C₄⋊C₄×C₅₃	C₈	semidirect
74. C₂×C₄×C₅₃	C₄×C₈	semidirect
122. C₅₃	#20533	semidirect
128. C₂×C₄⋊C₄	#88322	semidirect
180. C₂×C₄	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₄⋊C₄×#88322	2
C₄×C₂²⋊#88322	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#20533	53