Subgroups — #27295706

Order 13568.

246 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	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20272784	C₂	1	2	yes	maximal, complemented
4	#20272778	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	—	2	1	no	—
7	#16810676	C₄	1	1	yes	characteristic
8	C₄×#11719917	C₄	1	1	yes	characteristic
9	C₄⋊#11719908	—	2	1	no	—
10	C₄⋊#11719908	C₂²	1	2	yes	—
11	#16810345	C₄	1	2	yes	—
12	C₈×#88322	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₅₃	—	2	1	no	—
18	C₂×#11719917	C₈	1	1	yes	characteristic
19	C₂×#11719917	C₈	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	4	2	no	—
22	C₄⋊#88322	—	2	1	no	—
23	C₂²⋊#88322	—	2	1	no	—
24	#13133128	C₈	1	2	yes	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	1	no	—
31	C₂×C₁₆×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	4	1	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
49	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	#11719917	—	2	2	no	—
51	#11719917	C₁₆	1	4	yes	complemented
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	C₂×C₈	1	2	yes	—
54	C₂×#88322	C₂×C₈	1	2	yes	—
55	C₁₆×C₅₃	—	4	1	no	—
56	C₁₆×C₅₃	—	4	1	no	—
57	#11719908	—	4	2	no	—
58	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₈×C₅₃	#137	1	1	yes	characteristic
75	#88322	—	2	2	no	—
76	#88322	—	4	2	no	—
77	#88322	C₂×C₁₆	1	4	yes	complemented
78	#12015	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
82	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
87	C₄×C₅₃	#428	1	1	yes	characteristic
88	C₄×C₅₃	#430	1	1	yes	characteristic
89	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
90	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
91	#2162	D₅₃	1	1	yes	characteristic
92	C₂×C₄⋊C₁₆	—	53	1	no	—
93	C₈×M₁₆	—	53	1	no	—
94	#1370	—	53	2	no	—
95	#1367	—	53	2	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	#1565	1	1	yes	characteristic
98	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
99	C₂×C₅₃	#2153	1	1	yes	characteristic
100	C₂²×C₁₆	—	106	1	no	—
101	#431	—	53	1	no	—
102	C₄×M₁₆	—	53	1	no	—
103	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₄⋊C₁₆	—	53	2	no	—
105	C₄⋊C₁₆	—	106	1	no	—
106	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
107	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
108	C₂²⋊C₁₆	—	2	1	no	—
109	C₄×C₁₆	—	2	1	no	—
110	#322	—	53	2	no	—
111	C₈²	—	53	2	no	—
112	C₅₃	#12015	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	53	1	no	—
114	C₂×M₁₆	—	53	1	no	—
115	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	106	2	no	—
121	C₂×C₁₆	—	212	2	no	—
122	C₄⋊C₈	—	106	1	no	—
123	C₂²⋊C₈	—	106	1	no	—
124	#97	—	53	2	no	—
125	C₄×C₈	—	53	2	no	—
126	C₄×C₈	—	53	2	no	—
127	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
128	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
131	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
132	C₂²×C₄	#11719912	1	1	yes	characteristic
133	C₂²×C₄	#11719919	1	1	yes	characteristic
134	M₁₆	—	53	4	no	—
135	M₁₆	—	106	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	#11719919	1	1	yes	characteristic
143	C₂×C₈	#11719919	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
145	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₄²	C₈×D₅₃	1	1	yes	characteristic
147	C₄²	#11719912	1	1	yes	characteristic
148	C₄²	#11719919	1	1	yes	characteristic
149	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
150	C₁₆	—	4	1	no	—
151	C₁₆	—	4	1	no	—
152	C₁₆	—	212	2	no	—
153	C₂³	#13133146	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	#13133146	1	1	yes	characteristic
158	C₂×C₄	#13133146	1	1	yes	characteristic
159	C₂×C₄	#13133146	1	1	yes	characteristic
160	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
161	C₂×C₄	#13133156	1	2	yes	—
162	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
163	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	53	4	no	—
169	C₈	—	106	2	no	—
170	C₈	—	212	2	no	—
171	C₈	#13133242	1	1	yes	characteristic
172	C₈	#13133247	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#16810441	1	2	yes	—
176	C₂²	C₂×#13133146	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#16810441	1	2	yes	—
181	C₄	#16810666	1	1	yes	characteristic
182	C₄	#16810671	1	1	yes	characteristic
183	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
184	C₄	C₂×#13133156	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#20273468	1	1	yes	characteristic
187	C₂	C₂×#16810441	1	1	yes	characteristic
188	C₂	#20276480	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2162	D₅₃
hypocenter	C₅₃	#12015
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #20272784	C₂	semidirect
4. #20272778	C₂	semidirect
51. #11719917	C₁₆	semidirect
77. #88322	C₂×C₁₆	semidirect
112. C₅₃	#12015	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₈×#11719917	2
#20272784	2
#20272778	2
C₅₃×#2162	2
#12015	53