Subgroups — #27277830

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	2	yes	maximal
2	C₂×#16810374	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	4	no	—
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	#16810374	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×#88322	—	2	4	no	—
13	C₄⋊#88322	—	2	8	no	—
14	C₄⋊#88322	—	4	8	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	D₄	1	2	yes	—
20	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₅₃	—	2	4	no	—
23	C₄×C₈×C₅₃	—	2	2	no	—
24	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
25	C₄×C₈×C₅₃	D₄	1	4	yes	—
26	C₄×C₈×C₅₃	Q₈	1	4	yes	—
27	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
38	C₄²×C₅₃	—	2	4	no	—
39	C₄²×C₅₃	—	2	2	no	—
40	C₄²×C₅₃	C₂×C₈	1	2	yes	—
41	C₄²×C₅₃	M₁₆	1	2	yes	—
42	C₄²×C₅₃	SD₁₆	1	8	yes	—
43	C₄²×C₅₃	C₂×D₄	1	2	yes	—
44	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×#88322	—	4	4	no	—
46	C₂×#88322	—	4	8	no	—
47	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₅₃	#106	1	4	yes	—
53	C₂×C₄×C₅₃	#106	1	4	yes	—
54	C₂×C₄×C₅₃	#108	1	4	yes	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
63	C₈×C₅₃	—	2	4	no	—
64	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
65	#88322	—	8	4	no	—
66	#8805	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	#384	1	1	yes	characteristic
68	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₃	#423	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×#106	1	2	yes	—
71	C₂²×C₅₃	C₂×#108	1	2	yes	—
72	C₄×C₅₃	—	2	4	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	#334	1	8	yes	complemented
75	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
76	C₄×C₅₃	#423	1	2	yes	—
77	C₄×C₅₃	#492	1	2	yes	—
78	C₄×C₅₃	#498	1	2	yes	—
79	#1842	—	53	2	no	—
80	C₄²×C₈	D₅₃	1	1	yes	characteristic
81	C₂×#334	—	53	4	no	—
82	C₂×C₅₃	C₂×#334	1	2	yes	—
83	C₂×C₅₃	#1566	1	2	yes	—
84	C₂×C₅₃	#1842	1	1	yes	characteristic
85	C₂×C₅₃	#1844	1	1	yes	characteristic
86	C₂×C₅₃	#1848	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	53	4	no	—
88	C₂×C₄⋊C₈	—	106	4	no	—
89	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
90	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
91	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
93	C₄³	C₂×D₅₃	1	1	yes	characteristic
94	#334	—	106	8	no	—
95	C₅₃	#8805	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	106	4	no	—
97	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
98	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
99	C₂×C₄²	Dic₁₀₆	1	2	yes	—
100	C₂×C₄²	D₂₁₂	1	2	yes	—
101	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
104	C₄⋊C₈	—	106	8	no	—
105	C₄⋊C₈	—	212	8	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	#88322	1	4	yes	complemented
109	C₄×C₈	Dic₁₀₆	1	4	yes	—
110	C₄×C₈	D₂₁₂	1	4	yes	—
111	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
112	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
113	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
115	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	212	4	no	—
119	C₂×C₈	—	212	8	no	—
120	C₂×C₈	C₂×#88322	1	4	yes	—
121	C₂×C₈	#11719917	1	4	yes	—
122	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
123	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	#11719913	1	8	yes	—
127	C₄²	C₂×#88322	1	2	yes	—
128	C₄²	#11719917	1	2	yes	—
129	C₄²	C₂×Dic₁₀₆	1	2	yes	—
130	C₄²	C₂×D₂₁₂	1	2	yes	—
131	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊#88322	1	4	yes	—
136	C₂×C₄	#13133149	1	4	yes	—
137	C₂×C₄	#13133149	1	4	yes	—
138	C₂×C₄	#13133153	1	4	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
140	C₂×C₄	#13133214	1	1	yes	characteristic
141	C₂×C₄	#13133217	1	2	yes	—
142	C₂×C₄	#13133217	1	1	yes	characteristic
143	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
144	C₂×C₄	C₂×#11719917	1	2	yes	—
145	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
146	C₂×C₄	C₂×#11719913	1	4	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	424	4	no	—
149	C₈	C₄⋊#88322	1	8	yes	complemented
150	C₂²	#16810493	1	1	yes	characteristic
151	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
152	C₂²	#16810647	1	1	yes	characteristic
153	C₂²	C₂×#13133149	1	2	yes	—
154	C₂²	C₂×#13133153	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	2	no	—
157	C₄	#16810374	1	8	yes	complemented
158	C₄	C₂×C₄⋊#88322	1	2	yes	—
159	C₄	#16810647	1	2	yes	—
160	C₄	#16810996	1	2	yes	—
161	C₄	#16811021	1	2	yes	—
162	C₂	C₂×#16810374	1	2	yes	—
163	C₂	#20273471	1	2	yes	—
164	C₂	#20274867	1	1	yes	characteristic
165	C₂	#20274872	1	1	yes	characteristic
166	C₂	#20274887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8805
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₅₃	C₈	semidirect
64. C₈×C₅₃	C₄⋊C₈	semidirect
74. C₄×C₅₃	#334	semidirect
95. C₅₃	#8805	semidirect
108. C₄×C₈	#88322	semidirect
149. C₈	C₄⋊#88322	semidirect
157. C₄	#16810374	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₂×#16810374	2
C₄²×C₈×C₅₃	2
#8805	53