Subgroups — #27277895

Order 13568.

354 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274886	C₂	1	2	yes	maximal
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20273473	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133152	—	2	4	no	—
7	#16810647	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810376	—	2	2	no	—
11	#16810375	—	2	4	no	—
12	C₈×#88322	C₄	1	1	yes	characteristic
13	C₈×#88322	C₄	1	1	yes	characteristic
14	C₈×#88322	C₄	1	1	yes	characteristic
15	C₈×#88322	C₄	1	1	yes	characteristic
16	C₅₃×#423	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#335	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂²×#88322	D₄	1	2	yes	—
21	C₂²×#88322	Q₈	1	2	yes	—
22	#13133152	—	4	8	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄×#88322	C₂×C₄	1	1	yes	characteristic
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₄×#88322	C₂×C₄	1	1	yes	characteristic
28	C₄×#88322	C₂×C₄	1	1	yes	characteristic
29	C₄×#88322	D₄	1	2	yes	—
30	C₄×#88322	Q₈	1	2	yes	—
31	C₂×M₁₆×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₅₃	—	4	4	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	#11719917	—	4	4	no	—
52	C₂×#88322	—	2	4	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	C₄⋊C₄	1	2	yes	—
55	C₂×#88322	C₄⋊C₄	1	2	yes	—
56	C₂×#88322	C₄⋊C₄	1	2	yes	—
57	C₂×#88322	C₄⋊C₄	1	2	yes	—
58	C₂×#88322	D₈	1	4	yes	—
59	C₂×#88322	Q₁₆	1	4	yes	—
60	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	—	8	2	no	—
73	C₈×C₅₃	#108	1	4	yes	complemented
74	#88322	—	2	2	no	—
75	#88322	—	8	2	no	—
76	#88322	#107	1	4	yes	—
77	#88322	#107	1	4	yes	—
78	#88322	#108	1	4	yes	complemented
79	#8823	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	#384	1	1	yes	characteristic
83	C₂²×C₅₃	#500	1	2	yes	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	C₂×#107	1	2	yes	—
87	C₄×C₅₃	C₂×#108	1	2	yes	—
88	C₄×C₅₃	C₂×#108	1	2	yes	—
89	C₄×C₅₃	#493	1	1	yes	characteristic
90	C₄×C₅₃	#494	1	1	yes	characteristic
91	#1848	—	53	2	no	—
92	#1567	—	53	2	no	—
93	#1567	—	53	1	no	—
94	#1567	D₅₃	1	1	yes	characteristic
95	C₂×C₈²	—	53	1	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	#1845	1	1	yes	characteristic
98	C₂×C₅₃	#1848	1	1	yes	characteristic
99	C₂×C₅₃	#1848	1	1	yes	characteristic
100	C₂×#108	—	106	4	no	—
101	#423	—	53	2	no	—
102	#423	C₂×D₅₃	1	2	yes	—
103	C₂×C₄×C₈	—	53	1	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	#335	—	2	2	no	—
107	#335	—	106	4	no	—
108	#335	—	106	2	no	—
109	C₈²	—	53	1	no	—
110	C₈²	—	53	1	no	—
111	C₈²	—	53	1	no	—
112	C₈²	—	53	1	no	—
113	C₅₃	#8823	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	106	2	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	—	53	2	no	—
118	C₂²×C₈	Dic₁₀₆	1	2	yes	—
119	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
120	#108	—	212	8	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	106	2	no	—
124	C₄⋊C₈	—	106	2	no	—
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	—	53	1	no	—
127	C₄×C₈	—	53	1	no	—
128	C₄×C₈	—	53	1	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	53	2	no	—
131	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
134	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
135	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
137	M₁₆	—	4	4	no	—
138	M₁₆	—	212	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	2	no	—
143	C₂×C₈	—	53	2	no	—
144	C₂×C₈	—	53	4	no	—
145	C₂×C₈	—	53	4	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	106	4	no	—
148	C₂×C₈	—	212	2	no	—
149	C₂×C₈	#11719919	1	2	yes	—
150	C₂×C₈	#11719919	1	2	yes	—
151	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₄²	D₄×D₅₃	1	1	yes	characteristic
153	C₄²	Q₈×D₅₃	1	1	yes	characteristic
154	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂³	#13133216	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
161	C₂×C₄	#13133213	1	1	yes	characteristic
162	C₂×C₄	#13133215	1	1	yes	characteristic
163	C₂×C₄	#13133216	1	1	yes	characteristic
164	C₂×C₄	D₈×D₅₃	1	2	yes	—
165	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	—	53	4	no	—
169	C₈	—	53	4	no	—
170	C₈	—	53	4	no	—
171	C₈	—	106	2	no	—
172	C₈	—	424	2	no	—
173	C₈	#13133152	1	4	yes	complemented
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#16810492	1	1	yes	characteristic
177	C₂²	#16811027	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#107×D₅₃	1	2	yes	—
181	C₄	#108×D₅₃	1	2	yes	—
182	C₄	C₂×#13133152	1	2	yes	—
183	C₄	#16810998	1	1	yes	characteristic
184	C₄	#16811001	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#20274873	1	1	yes	characteristic
187	C₂	#20274885	1	1	yes	characteristic
188	C₂	#20274886	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#8823
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
73. C₈×C₅₃	#108	semidirect
78. #88322	#108	semidirect
113. C₅₃	#8823	semidirect
173. C₈	#13133152	semidirect

Maximal subgroups

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

subgroup	index
#20274886	2
#20273474	2
#20273473	2
C₂×C₈×#88322	2
C₅₃×#1567	2
#8823	53