Subgroups — #27275338

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#88322	C₂	1	2	yes	maximal
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810374	C₄	1	4	yes	complemented
12	#16810344	C₄	1	4	yes	complemented
13	C₂×#107×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#340	C₄	1	2	yes	—
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	D₄	1	2	yes	—
20	C₄⋊#88322	—	2	2	no	—
21	C₄⋊#88322	C₂×C₄	1	4	yes	—
22	C₄×#88322	—	2	1	no	—
23	C₄×#88322	—	4	4	no	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	—	2	2	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#107×C₅₃	C₈	1	8	yes	complemented
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	#95×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	4	no	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	C₄²	1	2	yes	—
42	C₂×C₈×C₅₃	C₄²	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₄⋊C₄×C₅₃	—	2	4	no	—
45	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
46	C₄²×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	—	4	2	no	—
48	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₅₃	#41	1	1	yes	characteristic
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	—	2	2	no	—
55	C₂×#88322	—	2	4	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	C₄⋊C₄	1	4	yes	—
60	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	8	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₃	#131	1	2	yes	—
73	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₅₃	#137	1	2	yes	—
75	C₈×C₅₃	—	2	2	no	—
76	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
77	#88322	—	2	4	no	—
78	#88322	—	4	2	no	—
79	#88322	—	4	4	no	—
80	#8284	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₅₃	C₈×D₄	1	2	yes	—
83	C₂²×C₅₃	#441	1	1	yes	characteristic
84	C₂²×C₅₃	#442	1	1	yes	characteristic
85	C₂²×C₅₃	#444	1	2	yes	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	4	8	no	—
89	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₅₃	#405	1	1	yes	characteristic
91	C₄×C₅₃	#428	1	2	yes	—
92	C₄×C₅₃	#440	1	2	yes	—
93	C₄×C₅₃	C₈×Q₈	1	2	yes	—
94	C₄×#107	D₅₃	1	1	yes	characteristic
95	C₈×C₄⋊C₄	—	53	2	no	—
96	C₂×#334	—	53	1	no	—
97	C₂×#322	—	53	1	no	—
98	#1279	—	53	2	no	—
99	C₂×C₅₃	#1579	1	2	yes	—
100	C₂×C₅₃	#1580	1	2	yes	—
101	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₅₃	#1773	1	1	yes	characteristic
103	C₂×C₅₃	#1776	1	1	yes	characteristic
104	C₂×#107	Dic₅₃	1	2	yes	—
105	C₂×C₄⋊C₈	—	53	2	no	—
106	C₂×C₄×C₈	—	53	1	no	—
107	C₂×C₄×C₈	—	53	1	no	—
108	C₂×C₄×C₈	—	106	2	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
111	#340	Dic₅₃	1	2	yes	—
112	#336	—	106	2	no	—
113	#334	—	53	4	no	—
114	#322	—	53	4	no	—
115	C₅₃	#8284	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	—	106	2	no	—
118	C₂²×C₈	—	106	1	no	—
119	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	#107	#88322	1	8	yes	complemented
125	C₄⋊C₈	—	53	4	no	—
126	C₄⋊C₈	—	106	2	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	—	53	2	no	—
130	C₄×C₈	—	106	1	no	—
131	C₄×C₈	—	212	4	no	—
132	C₄×C₈	C₄×D₅₃	1	2	yes	—
133	C₄×C₈	C₄×D₅₃	1	2	yes	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
137	C₂²×C₄	#11719947	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	53	4	no	—
141	C₂×C₈	—	106	2	no	—
142	C₂×C₈	—	106	2	no	—
143	C₂×C₈	—	106	2	no	—
144	C₂×C₈	—	106	4	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	—	212	2	no	—
147	C₂×C₈	—	212	2	no	—
148	C₂×C₈	C₂×#88322	1	4	yes	—
149	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
150	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	C₂×#88322	1	8	yes	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₄²	Q₈×D₅₃	1	1	yes	characteristic
158	C₄²	#11719949	1	1	yes	characteristic
159	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
166	C₂×C₄	#13133175	1	1	yes	characteristic
167	C₂×C₄	#13133179	1	1	yes	characteristic
168	C₂×C₄	#13133182	1	1	yes	characteristic
169	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄	C₂²×#88322	1	2	yes	—
171	C₂×C₄	#13133224	1	2	yes	—
172	C₂×C₄	#13133241	1	2	yes	—
173	C₂×C₄	#13133249	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	—	106	4	no	—
176	C₈	—	212	2	no	—
177	C₈	—	212	4	no	—
178	C₈	C₄×#88322	1	4	yes	complemented
179	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
180	C₂²	D₄×#88322	1	2	yes	—
181	C₂²	#16810730	1	1	yes	characteristic
182	C₂²	#16810735	1	1	yes	characteristic
183	C₂²	#16810745	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	8	no	—
187	C₄	C₂×C₄×#88322	1	1	yes	characteristic
188	C₄	#16810579	1	1	yes	characteristic
189	C₄	#16810664	1	2	yes	—
190	C₄	#16810722	1	2	yes	—
191	C₄	Q₈×#88322	1	2	yes	—
192	C₂	#20273534	1	2	yes	—
193	C₂	#20273539	1	2	yes	—
194	C₂	C₄⋊C₄×#88322	1	1	yes	characteristic
195	C₂	#20274601	1	1	yes	characteristic
196	C₂	#20274614	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#8284
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #16810374	C₄	semidirect
12. #16810344	C₄	semidirect
32. #107×C₅₃	C₈	semidirect
76. C₈×C₅₃	C₄×C₈	semidirect
115. C₅₃	#8284	semidirect
124. #107	#88322	semidirect
178. C₈	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#88322	2
C₂×#16810374	2
C₂×#16810344	2
#20272556	2
C₄×#107×C₅₃	2
#8284	53