Subgroups — #27295715

Order 13568.

232 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic
2	#20276471	C₂	1	1	yes	maximal, characteristic
3	#20272785	C₂	1	2	yes	maximal, complemented
4	#20272779	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	1	yes	characteristic
9	C₄⋊#11719908	C₂²	1	1	yes	characteristic
10	C₄⋊#11719908	C₂²	1	2	yes	—
11	C₂²⋊#11719908	—	2	1	no	—
12	#16810399	—	2	1	no	—
13	C₅₃×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#322	C₄	1	2	yes	—
17	C₈²×C₅₃	C₂²	1	2	yes	—
18	#13133157	—	4	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×M₁₆×C₅₃	C₈	1	2	yes	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₅₃	—	2	1	no	—
27	C₂²⋊C₈×C₅₃	—	2	1	no	—
28	#97×C₅₃	C₈	1	1	yes	characteristic
29	#97×C₅₃	C₈	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
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	M₁₆×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	4	1	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	#11719908	—	4	1	no	—
55	#11719908	—	4	2	no	—
56	#11719908	—	4	1	no	—
57	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	2	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×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	2	1	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	—	4	2	no	—
73	C₈×C₅₃	M₃₂	1	4	yes	complemented
74	C₈×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₈×C₅₃	#137	1	1	yes	characteristic
76	#12017	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	#428	1	1	yes	characteristic
86	C₄×C₅₃	#430	1	1	yes	characteristic
87	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
88	#2163	—	53	1	no	—
89	#2150	—	53	1	no	—
90	C₈×M₁₆	D₅₃	1	1	yes	characteristic
91	#1370	—	53	2	no	—
92	#1367	—	53	2	no	—
93	C₂×C₅₃	—	4	1	no	—
94	C₂×C₅₃	#1565	1	1	yes	characteristic
95	C₂×C₅₃	#2150	1	1	yes	characteristic
96	C₂×C₅₃	#2153	1	1	yes	characteristic
97	C₂×M₃₂	—	106	1	no	—
98	#431	Dic₅₃	1	1	yes	characteristic
99	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₄⋊C₁₆	—	53	1	no	—
102	C₄⋊C₁₆	—	53	2	no	—
103	C₄⋊C₁₆	—	53	1	no	—
104	C₄⋊C₁₆	—	106	1	no	—
105	C₂²⋊C₁₆	—	106	1	no	—
106	#346	—	106	1	no	—
107	#322	Dic₅₃	1	2	yes	—
108	C₈²	C₂×D₅₃	1	2	yes	—
109	C₅₃	#12017	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	#88322	1	2	yes	—
111	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
112	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
114	M₃₂	—	212	2	no	—
115	C₂×C₁₆	—	106	1	no	—
116	C₂×C₁₆	—	106	1	no	—
117	C₂×C₁₆	—	106	2	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₂²⋊C₈	—	2	1	no	—
120	#97	#88322	1	1	yes	characteristic
121	#97	#88322	1	1	yes	characteristic
122	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
123	C₄×C₈	D₂₁₂	1	1	yes	characteristic
124	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
125	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
126	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
129	C₂²×C₄	#11719917	1	1	yes	characteristic
130	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
131	M₁₆	—	2	4	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	C₂×#88322	1	2	yes	—
137	C₂×C₈	C₂×#88322	1	2	yes	—
138	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
142	C₄²	C₂×#88322	1	1	yes	characteristic
143	C₄²	#11719917	1	1	yes	characteristic
144	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
145	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
146	C₁₆	—	212	2	no	—
147	C₁₆	—	212	1	no	—
148	C₁₆	—	212	1	no	—
149	C₂³	C₄⋊#88322	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
155	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
156	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
159	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	4	2	no	—
165	C₈	#13133157	1	4	yes	complemented
166	C₈	#13133245	1	1	yes	characteristic
167	C₈	#13133250	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#16810665	1	1	yes	characteristic
177	C₄	#16810672	1	1	yes	characteristic
178	C₄	C₂×#13133157	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#20273467	1	1	yes	characteristic
181	C₂	#20276471	1	1	yes	characteristic
182	C₂	#20276481	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#12017
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20272785	C₂	semidirect
4. #20272779	C₂	semidirect
73. C₈×C₅₃	M₃₂	semidirect
109. C₅₃	#12017	semidirect
165. C₈	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
#20276471	2
#20272785	2
#20272779	2
C₈×M₁₆×C₅₃	2
#12017	53