Subgroups — #27295656

Order 13568.

220 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	#20272782	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×#13133157	C₄	1	1	yes	characteristic
7	C₂×#13133157	C₄	1	1	yes	characteristic
8	#16810444	—	2	1	no	—
9	#16810444	C₂²	1	2	yes	—
10	#16810399	C₂²	1	2	yes	—
11	C₄×#11719908	C₄	1	2	yes	—
12	C₂×M₃₂×C₅₃	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#364	—	2	1	no	—
15	C₅₃×#364	C₂²	1	2	yes	—
16	#13133157	—	2	1	no	—
17	#13133157	—	2	1	no	—
18	#13133157	—	2	2	no	—
19	#13133157	—	4	1	no	—
20	#13133157	C₈	1	2	yes	—
21	#13133157	C₈	1	2	yes	—
22	C₂×#11719908	—	4	1	no	—
23	C₂×#11719908	C₂×C₄	1	2	yes	—
24	C₂×#11719908	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
28	M₃₂×C₅₃	—	2	2	no	—
29	M₃₂×C₅₃	—	4	1	no	—
30	C₂×C₁₆×C₅₃	—	4	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₅₃	—	4	2	no	—
53	#11719908	—	2	2	no	—
54	#11719908	—	4	2	no	—
55	#11719908	C₂×C₈	1	4	yes	—
56	C₂³×C₅₃	#108	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	4	2	no	—
60	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
67	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
68	C₈×C₅₃	#106	1	1	yes	characteristic
69	C₈×C₅₃	#106	1	1	yes	characteristic
70	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
74	#12002	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	#334	1	1	yes	characteristic
77	C₂²×C₅₃	#334	1	1	yes	characteristic
78	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	#334	1	1	yes	characteristic
82	C₄×C₅₃	#334	1	1	yes	characteristic
83	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
85	C₂×#364	—	53	1	no	—
86	C₂×#364	D₅₃	1	1	yes	characteristic
87	C₄×M₃₂	—	53	1	no	—
88	#1369	—	53	4	no	—
89	C₂×C₅₃	—	2	1	no	—
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
93	C₂×M₃₂	—	2	1	no	—
94	C₂×M₃₂	—	53	1	no	—
95	C₂×M₃₂	—	53	1	no	—
96	C₂×M₃₂	—	106	1	no	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	#364	—	2	1	no	—
99	#364	—	53	2	no	—
100	#364	—	106	1	no	—
101	#364	C₂×D₅₃	1	2	yes	—
102	#346	—	53	2	no	—
103	C₄×C₁₆	—	53	2	no	—
104	C₅₃	#12002	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
107	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	4	1	no	—
110	M₃₂	—	53	2	no	—
111	M₃₂	—	53	2	no	—
112	M₃₂	—	106	1	no	—
113	M₃₂	—	106	1	no	—
114	M₃₂	—	106	2	no	—
115	M₃₂	—	212	1	no	—
116	C₂×C₁₆	—	4	1	no	—
117	C₂×C₁₆	—	53	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	212	1	no	—
120	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
121	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	#11719919	1	1	yes	characteristic
126	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
127	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
128	C₂×C₈	#11719912	1	1	yes	characteristic
129	C₂×C₈	#11719912	1	1	yes	characteristic
130	C₂×C₈	#11719919	1	1	yes	characteristic
131	C₂×C₈	#11719919	1	1	yes	characteristic
132	C₂×C₈	#11719923	1	1	yes	characteristic
133	C₂×C₈	#11719923	1	1	yes	characteristic
134	C₂×C₈	#11719924	1	1	yes	characteristic
135	C₂×C₈	#11719924	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄²	—	2	1	no	—
139	C₄²	#11719919	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₁₆	—	4	2	no	—
142	C₁₆	—	53	4	no	—
143	C₁₆	—	106	2	no	—
144	C₁₆	—	212	2	no	—
145	C₂³	#13133152	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	#13133146	1	1	yes	characteristic
150	C₂×C₄	#13133146	1	1	yes	characteristic
151	C₂×C₄	#13133148	1	1	yes	characteristic
152	C₂×C₄	#13133148	1	1	yes	characteristic
153	C₂×C₄	#13133152	1	1	yes	characteristic
154	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
155	C₈	#13133146	1	1	yes	characteristic
156	C₈	#13133146	1	1	yes	characteristic
157	C₈	#13133148	1	1	yes	characteristic
158	C₈	#13133148	1	1	yes	characteristic
159	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
160	C₈	C₂×#11719912	1	1	yes	characteristic
161	C₈	C₂×#11719924	1	1	yes	characteristic
162	C₈	C₂×#11719923	1	1	yes	characteristic
163	C₂²	—	4	1	no	—
164	C₂²	#16810373	1	1	yes	characteristic
165	C₂²	#16810373	1	1	yes	characteristic
166	C₂²	C₂×#13133152	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	#16810373	1	1	yes	characteristic
170	C₄	#16810373	1	1	yes	characteristic
171	C₄	C₂×#13133146	1	1	yes	characteristic
172	C₄	C₂×#13133148	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	C₂×#16810373	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133148
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	C₂×#334
prime radical	C₂³×C₅₃	#108
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12002
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #20272782	C₂	semidirect
104. C₅₃	#12002	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
C₄×#13133157	2
#20272782	2
C₂×C₅₃×#364	2
#12002	53