Subgroups — #27277899

Order 13568.

322 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274852	C₂	1	2	yes	maximal
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20273470	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	#16810647	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810387	—	2	4	no	—
10	#16810376	—	2	2	no	—
11	#16810373	—	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	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄×#88322	C₂×C₄	1	1	yes	characteristic
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	D₄	1	2	yes	—
29	C₄×#88322	Q₈	1	2	yes	—
30	C₂×M₁₆×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	—	2	2	no	—
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	M₁₆×C₅₃	—	4	4	no	—
42	C₂×C₈×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	#11719917	—	4	4	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	4	2	no	—
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	C₄⋊C₄	1	2	yes	—
59	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#135	1	2	yes	—
69	C₂×C₄×C₅₃	#135	1	2	yes	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	8	2	no	—
72	C₈×C₅₃	#108	1	4	yes	—
73	#88322	—	2	2	no	—
74	#88322	—	2	2	no	—
75	#88322	—	2	2	no	—
76	#88322	—	8	2	no	—
77	#88322	#108	1	4	yes	—
78	#8824	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	#384	1	1	yes	characteristic
82	C₂²×C₅₃	#499	1	2	yes	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	#427	1	2	yes	—
86	C₄×C₅₃	C₂×#108	1	2	yes	—
87	C₄×C₅₃	C₂×#108	1	2	yes	—
88	C₄×C₅₃	#495	1	1	yes	characteristic
89	C₄×C₅₃	#495	1	1	yes	characteristic
90	#1837	—	53	2	no	—
91	#1567	—	53	1	no	—
92	#1567	D₅₃	1	1	yes	characteristic
93	#1566	—	53	2	no	—
94	C₂×C₈²	—	53	1	no	—
95	C₂×C₅₃	—	2	2	no	—
96	C₂×C₅₃	#1846	1	1	yes	characteristic
97	C₂×C₅₃	#1848	1	1	yes	characteristic
98	C₂×C₅₃	#1848	1	1	yes	characteristic
99	#423	—	53	2	no	—
100	#423	C₂×D₅₃	1	2	yes	—
101	C₂×C₄×C₈	—	53	1	no	—
102	C₂×C₄×C₈	—	53	1	no	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	#341	—	106	4	no	—
105	#335	—	2	2	no	—
106	#335	—	106	2	no	—
107	#334	—	106	4	no	—
108	C₈²	—	53	1	no	—
109	C₈²	—	53	1	no	—
110	C₈²	—	53	1	no	—
111	C₈²	—	53	1	no	—
112	C₅₃	#8824	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	106	2	no	—
115	C₂²×C₈	—	53	2	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	Dic₁₀₆	1	2	yes	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	106	2	no	—
123	C₄×C₈	—	53	1	no	—
124	C₄×C₈	—	53	1	no	—
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	—	53	1	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
130	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
132	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
133	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
134	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
135	M₁₆	—	4	4	no	—
136	M₁₆	—	212	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	2	no	—
143	C₂×C₈	—	106	4	no	—
144	C₂×C₈	—	106	2	no	—
145	C₂×C₈	—	106	2	no	—
146	C₂×C₈	—	212	2	no	—
147	C₂×C₈	#11719919	1	2	yes	—
148	C₂×C₈	#11719919	1	2	yes	—
149	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
150	C₄²	D₄×D₅₃	1	1	yes	characteristic
151	C₄²	Q₈×D₅₃	1	1	yes	characteristic
152	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂³	#13133216	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
159	C₂×C₄	#13133213	1	1	yes	characteristic
160	C₂×C₄	#13133215	1	1	yes	characteristic
161	C₂×C₄	#13133216	1	1	yes	characteristic
162	C₂×C₄	#13133236	1	2	yes	—
163	C₂×C₄	#13133240	1	2	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	53	4	no	—
167	C₈	—	106	2	no	—
168	C₈	—	106	2	no	—
169	C₈	—	106	2	no	—
170	C₈	—	424	2	no	—
171	C₈	#13133152	1	4	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	#16810492	1	1	yes	characteristic
175	C₂²	#16811024	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#16810660	1	2	yes	—
179	C₄	#108×D₅₃	1	2	yes	—
180	C₄	C₂×#13133152	1	2	yes	—
181	C₄	#16811006	1	1	yes	characteristic
182	C₄	#16811008	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	#20274878	1	1	yes	characteristic
185	C₂	#20274885	1	1	yes	characteristic
186	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	#8824
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
112. C₅₃	#8824	semidirect

Maximal subgroups

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

subgroup	index
#20274852	2
#20273474	2
#20273470	2
C₂×C₈×#88322	2
C₅₃×#1567	2
#8824	53