Subgroups — #27302674

Order 13568.

366 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	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	2	yes	maximal
4	#20272806	C₂	1	2	yes	maximal
5	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₅₃	—	2	1	no	—
9	C₄⋊#11719908	—	2	2	no	—
10	#16810386	—	2	1	no	—
11	C₂×#125×C₅₃	—	2	1	no	—
12	C₅₃×#499	—	2	2	no	—
13	C₅₃×#499	C₄	1	4	yes	complemented
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#106×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₄×Dic₅₃	—	4	1	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	4	no	—
22	#13133151	—	2	2	no	—
23	#13133151	Q₈	1	4	yes	—
24	#13133124	—	4	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	#125×C₅₃	—	2	4	no	—
28	#125×C₅₃	—	4	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
32	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#107×C₅₃	—	2	1	no	—
35	#107×C₅₃	—	2	1	no	—
36	#107×C₅₃	C₂×C₄	1	2	yes	—
37	#107×C₅₃	C₂×C₄	1	2	yes	—
38	#106×C₅₃	—	2	4	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₄⋊C₄×C₅₃	—	4	4	no	—
56	C₄⋊C₄×C₅₃	—	4	4	no	—
57	C₄⋊C₄×C₅₃	—	4	1	no	—
58	C₄⋊C₄×C₅₃	—	4	1	no	—
59	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₅₃	SD₁₆	1	2	yes	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄×Dic₅₃	—	8	2	no	—
64	#11719908	—	4	2	no	—
65	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	—	8	1	no	—
73	C₂×C₄×C₅₃	—	8	1	no	—
74	C₂×C₄×C₅₃	#102	1	2	yes	—
75	C₂×C₄×C₅₃	#102	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#137	1	2	yes	—
80	C₂×Dic₅₃	—	4	4	no	—
81	C₂×Dic₅₃	—	8	2	no	—
82	C₂×Dic₅₃	—	8	2	no	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	C₄×Q₈	1	2	yes	—
85	C₈×C₅₃	#124	1	1	yes	characteristic
86	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
87	#13277	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
89	C₂²×C₅₃	#474	1	2	yes	—
90	C₂²×C₅₃	#508	1	2	yes	—
91	C₂²×C₅₃	#509	1	1	yes	characteristic
92	C₂²×C₅₃	#510	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	8	2	no	—
96	C₄×C₅₃	—	8	2	no	—
97	C₄×C₅₃	#391	1	1	yes	characteristic
98	C₄×C₅₃	#419	1	1	yes	characteristic
99	C₄×C₅₃	#475	1	2	yes	—
100	Dic₅₃	—	8	4	no	—
101	C₂×#499	D₅₃	1	1	yes	characteristic
102	C₂×C₄⋊C₁₆	—	53	1	no	—
103	C₄×#107	—	53	1	no	—
104	#1380	—	53	2	no	—
105	#1380	—	53	2	no	—
106	C₂×C₅₃	#1988	1	1	yes	characteristic
107	C₂×C₅₃	#2138	1	1	yes	characteristic
108	C₂×C₅₃	#2139	1	1	yes	characteristic
109	C₂×C₅₃	#2228	1	2	yes	—
110	C₂×C₅₃	#2229	1	2	yes	—
111	C₂×#125	—	2	1	no	—
112	C₂²×C₁₆	—	53	2	no	—
113	#499	—	2	2	no	—
114	#499	Dic₅₃	1	4	yes	complemented
115	C₂×#107	—	53	2	no	—
116	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
117	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
118	C₂×#106	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	106	1	no	—
121	C₄⋊C₁₆	—	106	2	no	—
122	#340	—	106	1	no	—
123	C₅₃	#13277	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	106	2	no	—
133	C₂×C₄²	—	212	1	no	—
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₁₆	—	106	4	no	—
136	C₂×C₁₆	—	106	2	no	—
137	#107	—	2	1	no	—
138	#107	—	2	1	no	—
139	#107	—	53	4	no	—
140	#107	—	106	2	no	—
141	#107	C₂×Dic₅₃	1	2	yes	—
142	#107	C₂×Dic₅₃	1	2	yes	—
143	#106	—	2	4	no	—
144	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
145	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
146	#95	—	212	1	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	—	212	2	no	—
150	C₂²×C₄	#11719922	1	1	yes	characteristic
151	C₂²×C₄	#11719922	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
155	C₂×C₈	Q₈×D₅₃	1	2	yes	—
156	C₂×C₈	#11719949	1	2	yes	—
157	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	106	4	no	—
166	C₄⋊C₄	—	212	2	no	—
167	C₄²	—	424	2	no	—
168	C₄²	#11719923	1	2	yes	—
169	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
170	C₁₆	—	212	2	no	—
171	C₂³	C₂×#11719922	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	1	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	424	2	no	—
181	C₂×C₄	—	424	2	no	—
182	C₂×C₄	—	212	4	no	—
183	C₂×C₄	#13133138	1	2	yes	—
184	C₂×C₄	#13133138	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
186	C₂×C₄	#13133193	1	2	yes	—
187	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
188	C₂×C₄	#13133248	1	2	yes	—
189	C₈	—	2	2	no	—
190	C₈	Q₈×Dic₅₃	1	2	yes	—
191	C₈	#13133206	1	1	yes	characteristic
192	C₈	#13133215	1	1	yes	characteristic
193	C₂²	C₂×#13133138	1	1	yes	characteristic
194	C₂²	#16810900	1	2	yes	—
195	C₂²	#16811053	1	2	yes	—
196	C₂²	#16811058	1	1	yes	characteristic
197	C₂²	#16811063	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	424	4	no	—
203	C₄	#16810530	1	1	yes	characteristic
204	C₄	#16810632	1	1	yes	characteristic
205	C₄	#16810905	1	2	yes	—
206	C₂	#20275668	1	1	yes	characteristic
207	C₂	#20276422	1	1	yes	characteristic
208	C₂	#20276427	1	1	yes	characteristic
209	C₂	#20276869	1	2	yes	—
210	C₂	#20276876	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	#13277
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
13. C₅₃×#499	C₄	semidirect
114. #499	Dic₅₃	semidirect
123. C₅₃	#13277	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
C₄×#13133151	2
#20272806	2
C₂×C₅₃×#499	2
#13277	53