Subgroups — #27295650

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	#20272781	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810676	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	#16810376	—	2	1	no	—
8	#16810374	—	2	1	no	—
9	#16810374	C₂²	1	2	yes	—
10	#16810374	C₂²	1	2	yes	—
11	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
12	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
16	C₂²×#88322	—	4	1	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	4	1	no	—
20	C₂²⋊#88322	—	4	1	no	—
21	C₄×#88322	—	4	1	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	M₃₂×C₅₃	—	2	2	no	—
26	M₃₂×C₅₃	C₈	1	4	yes	complemented
27	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	2	no	—
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	2	yes	—
40	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×#88322	—	4	2	no	—
48	C₂×#88322	—	4	2	no	—
49	C₂×#88322	—	8	2	no	—
50	C₁₆×C₅₃	—	2	2	no	—
51	C₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
52	C₂³×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#108	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
66	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	#88322	—	8	2	no	—
69	#88322	—	8	2	no	—
70	#12000	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	1	no	—
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	#334	1	2	yes	—
74	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	#334	1	2	yes	—
79	C₄×C₅₃	#365	1	2	yes	—
80	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
82	C₄×M₃₂	D₅₃	1	1	yes	characteristic
83	#1564	—	53	1	no	—
84	C₂×#334	—	53	1	no	—
85	#1368	—	53	4	no	—
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
89	C₂×C₅₃	#2155	1	1	yes	characteristic
90	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
91	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
92	#431	—	106	1	no	—
93	C₂×C₄⋊C₈	—	106	1	no	—
94	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
95	#346	C₂×D₅₃	1	2	yes	—
96	C₄×C₁₆	Dic₅₃	1	2	yes	—
97	#335	—	106	1	no	—
98	#334	—	53	2	no	—
99	#334	—	53	2	no	—
100	#334	—	106	1	no	—
101	C₅₃	#12000	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	212	1	no	—
103	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
105	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	#88322	1	4	yes	complemented
108	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
110	C₄⋊C₈	—	106	2	no	—
111	C₄⋊C₈	—	106	2	no	—
112	C₄⋊C₈	—	212	1	no	—
113	C₂²⋊C₈	—	212	1	no	—
114	C₄×C₈	—	212	1	no	—
115	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
116	C₄×C₈	D₂₁₂	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₂²×C₄	#11719913	1	1	yes	characteristic
120	C₂²×C₄	#11719913	1	1	yes	characteristic
121	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	424	2	no	—
124	C₂×C₈	—	212	2	no	—
125	C₂×C₈	—	212	2	no	—
126	C₂×C₈	C₂×#88322	1	2	yes	—
127	C₂×C₈	#11719917	1	2	yes	—
128	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
131	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
132	C₄²	#11719913	1	1	yes	characteristic
133	C₄²	#11719913	1	1	yes	characteristic
134	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
135	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
136	C₁₆	—	2	2	no	—
137	C₁₆	C₂×#88322	1	4	yes	complemented
138	C₂³	#13133149	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	C₄⋊#88322	1	2	yes	—
142	C₂×C₄	#13133149	1	1	yes	characteristic
143	C₂×C₄	#13133149	1	1	yes	characteristic
144	C₂×C₄	#13133149	1	1	yes	characteristic
145	C₂×C₄	#13133153	1	2	yes	—
146	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
148	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	424	2	no	—
152	C₈	—	424	2	no	—
153	C₈	C₄⋊#88322	1	2	yes	—
154	C₈	C₂²×#88322	1	1	yes	characteristic
155	C₈	C₂×#11719917	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#16810374	1	2	yes	—
159	C₂²	C₂×#13133149	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#16810374	1	2	yes	—
164	C₄	#16810446	1	2	yes	—
165	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
166	C₄	C₂×#13133153	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#16810374	1	1	yes	characteristic
169	C₂	C₂×#16810446	1	1	yes	characteristic
170	C₂	#20276488	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#12000
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272781	C₂	semidirect
26. M₃₂×C₅₃	C₈	semidirect
51. C₁₆×C₅₃	C₂×C₈	semidirect
101. C₅₃	#12000	semidirect
107. M₃₂	#88322	semidirect
137. C₁₆	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
C₂×#16810374	2
#20272781	2
C₄×M₃₂×C₅₃	2
#12000	53