Subgroups — #27302098

Order 13568.

338 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×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11719908	C₂	1	2	yes	maximal
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
4	#20272793	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₄⋊#11719908	—	2	4	no	—
10	C₄⋊#11719908	—	2	2	no	—
11	#16810399	—	2	2	no	—
12	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
16	C₅₃×#336	C₄	1	2	yes	—
17	C₂×#11719908	—	2	1	no	—
18	C₂×#11719908	—	2	1	no	—
19	C₂×#11719908	—	2	1	no	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	D₄	1	2	yes	—
25	C₂×#11719908	Q₈	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	#95×C₅₃	C₈	1	1	yes	characteristic
38	#95×C₅₃	C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅₃	M₁₆	1	2	yes	—
61	#11719908	—	2	2	no	—
62	#11719908	—	2	2	no	—
63	#11719908	—	4	2	no	—
64	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	4	2	no	—
83	C₈×C₅₃	#121	1	1	yes	characteristic
84	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₈×C₅₃	#125	1	1	yes	characteristic
86	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
87	#13160	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
92	C₂²×C₅₃	#504	1	2	yes	—
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	#389	1	1	yes	characteristic
98	C₄×C₅₃	#424	1	1	yes	characteristic
99	C₄×C₅₃	#436	1	1	yes	characteristic
100	C₄×C₅₃	#446	1	1	yes	characteristic
101	C₂×C₄⋊C₁₆	—	53	2	no	—
102	C₂×C₄⋊C₁₆	—	53	1	no	—
103	C₂×#346	—	53	1	no	—
104	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
105	#1374	—	53	2	no	—
106	C₂×C₅₃	#1939	1	1	yes	characteristic
107	C₂×C₅₃	#2150	1	1	yes	characteristic
108	C₂×C₅₃	#2151	1	1	yes	characteristic
109	C₂×C₅₃	#2168	1	2	yes	—
110	C₂×C₅₃	#2183	1	2	yes	—
111	C₂²×C₁₆	—	53	1	no	—
112	C₂²×C₁₆	—	53	1	no	—
113	C₂²×C₁₆	—	53	2	no	—
114	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	106	4	no	—
119	C₄⋊C₁₆	—	106	2	no	—
120	#346	—	106	2	no	—
121	#336	Dic₅₃	1	2	yes	—
122	C₅₃	#13160	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
124	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	#88322	1	2	yes	—
127	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
128	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
129	C₂×C₁₆	—	53	2	no	—
130	C₂×C₁₆	—	53	2	no	—
131	C₂×C₁₆	—	106	1	no	—
132	C₂×C₁₆	—	106	1	no	—
133	C₂×C₁₆	—	106	1	no	—
134	C₂×C₁₆	—	106	1	no	—
135	C₂×C₁₆	—	106	2	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	Dic₁₀₆	1	2	yes	—
142	C₄×C₈	D₂₁₂	1	2	yes	—
143	#95	#88322	1	1	yes	characteristic
144	#95	#88322	1	1	yes	characteristic
145	C₂²×C₄	C₂×#88322	1	2	yes	—
146	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
147	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
148	C₂²×C₄	C₂×#88322	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
156	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
157	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
158	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
159	C₂×C₈	#11719947	1	1	yes	characteristic
160	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
161	C₂×C₈	#11719949	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#11719917	1	2	yes	—
166	C₄²	C₄⋊Dic₅₃	1	2	yes	—
167	C₁₆	—	106	2	no	—
168	C₁₆	—	106	2	no	—
169	C₁₆	—	212	2	no	—
170	C₂³	C₂²×#88322	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₄⋊#88322	1	2	yes	—
179	C₂×C₄	C₄⋊#88322	1	2	yes	—
180	C₂×C₄	#13133157	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
185	C₂×C₄	#13133224	1	1	yes	characteristic
186	C₂×C₄	#13133224	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#13133192	1	1	yes	characteristic
190	C₈	#13133197	1	1	yes	characteristic
191	C₈	#13133208	1	1	yes	characteristic
192	C₈	#13133216	1	1	yes	characteristic
193	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
194	C₂²	D₄×#88322	1	1	yes	characteristic
195	C₂²	Q₈×#88322	1	1	yes	characteristic
196	C₂²	C₂×#13133157	1	2	yes	—
197	C₂²	#16811040	1	2	yes	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	#16810522	1	1	yes	characteristic
203	C₄	#16810650	1	1	yes	characteristic
204	C₄	#16810701	1	1	yes	characteristic
205	C₄	#16810754	1	1	yes	characteristic
206	C₂	#20275434	1	1	yes	characteristic
207	C₂	#20276471	1	1	yes	characteristic
208	C₂	#20276474	1	1	yes	characteristic
209	C₂	#20276541	1	2	yes	—
210	C₂	#20276624	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#13160
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
122. C₅₃	#13160	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810399	2
#20272793	2
C₈×C₄⋊C₄×C₅₃	2
#13160	53