Subgroups — #27302682

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
3	#20272823	C₂	1	4	yes	maximal
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₄×#11719917	—	2	1	no	—
8	C₄⋊#11719908	—	2	1	no	—
9	C₄⋊#11719908	—	2	1	no	—
10	#16810388	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₅₃×#500	—	2	1	no	—
13	C₅₃×#500	—	2	1	no	—
14	C₅₃×#500	C₄	1	2	yes	—
15	C₅₃×#500	C₄	1	2	yes	—
16	C₂×#107×C₅₃	—	2	1	no	—
17	C₂×#107×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	2	2	no	—
20	#13133157	—	2	4	no	—
21	C₂×#11719908	—	2	2	no	—
22	#13133153	—	2	2	no	—
23	#13133153	Q₈	1	4	yes	complemented
24	#13133128	—	4	1	no	—
25	C₄×#88322	—	4	1	no	—
26	C₂²×Q₈×C₅₃	—	4	1	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	4	1	no	—
32	C₄⋊Q₈×C₅₃	—	4	1	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	#107×C₅₃	—	2	2	no	—
37	#107×C₅₃	—	2	2	no	—
38	#107×C₅₃	—	2	2	no	—
39	#107×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂×Q₈×C₅₃	—	8	4	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₅₃	#41	1	2	yes	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅₃	D₈	1	1	yes	characteristic
65	C₄²×C₅₃	D₈	1	1	yes	characteristic
66	#11719917	—	2	4	no	—
67	#11719917	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	#11719908	—	4	2	no	—
70	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₅₃	—	8	4	no	—
72	Q₈×C₅₃	—	8	4	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	8	4	no	—
77	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#137	1	2	yes	—
85	C₈×C₅₃	—	2	1	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	C₄×Q₈	1	2	yes	—
88	C₈×C₅₃	#124	1	1	yes	characteristic
89	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
90	#88322	—	4	4	no	—
91	#13279	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
95	C₂²×C₅₃	#475	1	2	yes	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	8	4	no	—
99	C₄×C₅₃	#361	1	2	yes	—
100	C₄×C₅₃	#362	1	2	yes	—
101	C₄×C₅₃	#391	1	1	yes	characteristic
102	C₄×C₅₃	#419	1	1	yes	characteristic
103	C₄×C₅₃	#474	1	2	yes	—
104	C₂×#500	D₅₃	1	1	yes	characteristic
105	#2150	—	53	1	no	—
106	C₄×#108	—	53	1	no	—
107	#1387	—	53	4	no	—
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	#1988	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×M₃₂	—	53	2	no	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	Dic₅₃	1	2	yes	—
117	#500	Dic₅₃	1	2	yes	—
118	C₂×#108	—	53	2	no	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂×D₅₃	1	2	yes	—
121	C₄×M₁₆	—	106	1	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₄⋊C₁₆	—	106	1	no	—
124	C₄⋊C₁₆	—	106	1	no	—
125	#341	—	106	1	no	—
126	C₅₃	#13279	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	106	2	no	—
129	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
138	M₃₂	—	106	4	no	—
139	C₂×C₁₆	—	106	2	no	—
140	#108	—	53	4	no	—
141	#108	—	106	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×Dic₅₃	1	4	yes	—
146	#97	—	212	1	no	—
147	C₄×C₈	—	212	1	no	—
148	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
149	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#11719923	1	1	yes	characteristic
157	C₂²×C₄	#11719923	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
159	M₁₆	—	106	4	no	—
160	M₁₆	—	212	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	212	2	no	—
163	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₅₃	1	2	yes	—
166	C₂×C₈	#11719949	1	2	yes	—
167	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	#11719922	1	1	yes	characteristic
174	C₄²	#11719922	1	1	yes	characteristic
175	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₁₆	—	212	2	no	—
178	C₂³	C₂×#11719923	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#13133138	1	1	yes	characteristic
186	C₂×C₄	#13133138	1	1	yes	characteristic
187	C₂×C₄	#13133138	1	1	yes	characteristic
188	C₂×C₄	#13133138	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂×C₄	#13133193	1	2	yes	—
191	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
192	C₂×C₄	#13133248	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	212	4	no	—
196	C₈	Q₈×Dic₅₃	1	2	yes	—
197	C₈	#13133206	1	1	yes	characteristic
198	C₈	#13133215	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#13133138	1	1	yes	characteristic
202	C₂²	#16810905	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	#16810436	1	2	yes	—
207	C₄	#16810439	1	2	yes	—
208	C₄	#16810530	1	1	yes	characteristic
209	C₄	#16810632	1	1	yes	characteristic
210	C₄	#16810900	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#20275668	1	1	yes	characteristic
213	C₂	C₂×#16810436	1	1	yes	characteristic
214	C₂	C₂×#16810439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#13279
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
23. #13133153	Q₈	semidirect
126. C₅₃	#13279	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₄×#13133153	2
#20272823	2
C₂×C₅₃×#500	2
#13279	53