Subgroups — #27537676

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11719908	C₂	1	8	yes	maximal, complemented
2	#20276514	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	8	yes	complemented
6	C₄⋊#11719908	C₂²	1	4	yes	complemented
7	C₂²⋊#11719908	C₂²	1	4	yes	—
8	C₂²⋊#11719908	C₂²	1	8	yes	complemented
9	C₄×#11719908	C₂²	1	4	yes	complemented
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
12	C₈×D₄×C₅₃	C₄	1	4	yes	—
13	C₈×D₄×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719908	—	4	8	no	—
18	C₂×#11719908	C₂³	1	8	yes	—
19	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₅₃	—	4	4	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₄×D₄×C₅₃	C₈	1	4	yes	—
28	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×D₄×C₅₃	C₁₆	1	8	yes	complemented
38	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	—	4	4	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₂×C₈×C₅₃	—	4	1	no	—
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
58	C₄²×C₅₃	C₂×C₈	1	2	yes	—
59	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
60	#11719908	—	2	8	no	—
61	C₂³×C₅₃	—	4	2	no	—
62	C₂³×C₅₃	—	4	2	no	—
63	C₂³×C₅₃	C₂×C₁₆	1	8	yes	complemented
64	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
66	D₄×C₅₃	C₂×C₁₆	1	16	yes	complemented
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	1	no	—
74	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	complemented
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	4	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
84	#33486	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	2	4	no	—
89	C₂²×C₅₃	—	4	8	no	—
90	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
91	C₂²×C₅₃	C₂²×C₁₆	1	8	yes	—
92	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
97	C₄×C₅₃	#518	1	1	yes	characteristic
98	C₄×C₅₃	#568	1	1	yes	characteristic
99	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
100	D₄×C₁₆	—	53	8	no	—
101	#2162	—	53	2	no	—
102	C₂×C₂²⋊C₁₆	—	53	4	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	2	8	no	—
105	C₂×C₅₃	#2972	1	1	yes	characteristic
106	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
107	C₂×C₅₃	#3407	1	1	yes	characteristic
108	C₂³×C₈	C₂×D₅₃	1	2	yes	—
109	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
110	C₂²×C₁₆	—	53	8	no	—
111	C₈×D₄	Dic₅₃	1	4	yes	—
112	C₈×D₄	C₂×D₅₃	1	4	yes	—
113	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
114	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₄⋊C₁₆	—	53	4	no	—
117	C₂²⋊C₁₆	—	53	4	no	—
118	C₂²⋊C₁₆	—	53	8	no	—
119	C₄×C₁₆	—	53	4	no	—
120	C₅₃	#33486	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	#88322	1	1	yes	characteristic
122	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
123	C₂²×C₈	—	4	4	no	—
124	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
125	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
126	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
129	C₄×D₄	#88322	1	4	yes	—
130	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
131	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	#88322	1	2	yes	—
133	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×C₁₆	—	53	8	no	—
135	C₂×C₁₆	—	212	8	no	—
136	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
137	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
138	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
139	C₄×C₈	C₂²×D₅₃	1	2	yes	—
140	C₂⁴	C₂×#88322	1	2	yes	—
141	C₂×D₄	#11719908	1	8	yes	complemented
142	C₂×D₄	C₂×#88322	1	4	yes	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
146	C₂²×C₄	C₂×#88322	1	2	yes	—
147	C₂²×C₄	C₂×#88322	1	4	yes	—
148	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
158	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
159	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	C₂×#88322	1	4	yes	—
161	C₂²⋊C₄	C₂×#88322	1	8	yes	—
162	C₄²	C₂×#88322	1	2	yes	—
163	C₄²	C₂²×Dic₅₃	1	2	yes	—
164	C₁₆	—	106	8	no	—
165	C₂³	—	4	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂×#11719908	1	8	yes	complemented
168	C₂³	C₂²×#88322	1	4	yes	—
169	C₂³	C₂²×#88322	1	1	yes	characteristic
170	D₄	C₂×#11719908	1	16	yes	complemented
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×#11719908	1	4	yes	complemented
179	C₂×C₄	C₂²×#88322	1	2	yes	—
180	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
181	C₂×C₄	C₂²×#88322	1	4	yes	—
182	C₂×C₄	C₂²×#88322	1	2	yes	—
183	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	#13133262	1	2	yes	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	C₂²×#11719908	1	2	yes	—
194	C₂²	C₂²×#11719908	1	8	yes	—
195	C₂²	C₂³×#88322	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂²×#11719908	1	4	yes	—
200	C₄	#16811098	1	1	yes	characteristic
201	C₄	#16811390	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	2	8	no	—
204	C₂	#20283739	1	1	yes	characteristic
205	C₂	C₂³×#11719908	1	1	yes	characteristic
206	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#33486
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. D₄×#11719908	C₂	semidirect
2. #20276514	C₂	semidirect
3. C₂×C₂²⋊#11719908	C₂	semidirect
5. C₂²×#11719908	C₂²	semidirect
6. C₄⋊#11719908	C₂²	semidirect
8. C₂²⋊#11719908	C₂²	semidirect
9. C₄×#11719908	C₂²	semidirect
37. C₂×D₄×C₅₃	C₁₆	semidirect
63. C₂³×C₅₃	C₂×C₁₆	semidirect
66. D₄×C₅₃	C₂×C₁₆	semidirect
74. C₂×C₄×C₅₃	C₂×C₁₆	semidirect
120. C₅₃	#33486	semidirect
141. C₂×D₄	#11719908	semidirect
167. C₂³	C₂×#11719908	semidirect
170. D₄	C₂×#11719908	semidirect
178. C₂×C₄	C₂×#11719908	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11719908	2
#20276514	2
C₂×C₂²⋊#11719908	2
C₂×C₈×D₄×C₅₃	2
#33486	53