Subgroups — #27277841

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	2	yes	maximal
4	C₂×#16810374	C₂	1	2	yes	maximal
5	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810376	—	2	4	no	—
12	#16810374	—	2	4	no	—
13	C₂×#97×C₅₃	C₄	1	2	yes	—
14	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×#88322	—	2	1	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	4	4	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	#97×C₅₃	—	2	4	no	—
34	#97×C₅₃	—	2	2	no	—
35	#97×C₅₃	C₈	1	4	yes	complemented
36	C₄×C₈×C₅₃	D₄	1	4	yes	—
37	C₄×C₈×C₅₃	Q₈	1	4	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	4	2	no	—
53	C₄²×C₅₃	C₂×C₈	1	2	yes	—
54	C₄²×C₅₃	M₁₆	1	2	yes	—
55	C₄²×C₅₃	C₂×D₄	1	2	yes	—
56	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	1	no	—
59	C₂×#88322	—	4	1	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₅₃	#137	1	2	yes	—
80	C₈×C₅₃	—	2	4	no	—
81	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
82	#88322	—	8	2	no	—
83	#88322	—	8	1	no	—
84	#88322	—	8	1	no	—
85	#8808	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#384	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₅₃	#423	1	1	yes	characteristic
89	C₂²×C₅₃	#428	1	2	yes	—
90	C₂²×C₅₃	#430	1	2	yes	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	4	2	no	—
94	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
95	C₄×C₅₃	#423	1	2	yes	—
96	C₄×C₅₃	#496	1	1	yes	characteristic
97	C₄×C₅₃	#497	1	1	yes	characteristic
98	C₄×C₅₃	#501	1	2	yes	—
99	#1842	—	53	1	no	—
100	#1831	—	53	1	no	—
101	C₄×#97	D₅₃	1	1	yes	characteristic
102	C₂×#335	—	53	2	no	—
103	C₂×#334	—	53	2	no	—
104	C₂×C₅₃	#1565	1	2	yes	—
105	C₂×C₅₃	#1569	1	2	yes	—
106	C₂×C₅₃	#1842	1	1	yes	characteristic
107	C₂×C₅₃	#1847	1	1	yes	characteristic
108	C₂×C₅₃	#1849	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	53	1	no	—
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₂×C₄⋊C₈	—	53	1	no	—
112	C₂×C₄⋊C₈	—	106	2	no	—
113	C₂×#97	Dic₅₃	1	2	yes	—
114	C₂×#97	Dic₅₃	1	1	yes	characteristic
115	C₂×#97	Dic₅₃	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
117	C₄³	C₂×D₅₃	1	1	yes	characteristic
118	#336	—	106	2	no	—
119	#335	—	106	4	no	—
120	#334	—	106	4	no	—
121	C₅₃	#8808	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	106	1	no	—
123	C₂²×C₈	—	106	1	no	—
124	C₂²×C₈	—	106	2	no	—
125	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
126	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
127	C₂×C₄²	Dic₁₀₆	1	2	yes	—
128	C₂×C₄²	D₂₁₂	1	2	yes	—
129	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
132	C₄⋊C₈	—	106	2	no	—
133	C₄⋊C₈	—	106	2	no	—
134	C₄⋊C₈	—	106	2	no	—
135	C₄⋊C₈	—	106	2	no	—
136	C₄⋊C₈	—	212	4	no	—
137	#97	—	2	4	no	—
138	#97	—	2	2	no	—
139	#97	#88322	1	4	yes	complemented
140	C₄×C₈	Dic₁₀₆	1	4	yes	—
141	C₄×C₈	D₂₁₂	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
143	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	212	2	no	—
150	C₂×C₈	—	212	1	no	—
151	C₂×C₈	—	212	1	no	—
152	C₂×C₈	—	212	4	no	—
153	C₂×C₈	—	212	2	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	C₂×#88322	1	4	yes	—
156	C₂×C₈	#11719917	1	4	yes	—
157	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
158	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×#88322	1	2	yes	—
164	C₄²	#11719917	1	2	yes	—
165	C₄²	C₂×Dic₁₀₆	1	2	yes	—
166	C₄²	C₂×D₂₁₂	1	2	yes	—
167	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₄⋊#88322	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
176	C₂×C₄	#13133214	1	1	yes	characteristic
177	C₂×C₄	#13133217	1	2	yes	—
178	C₂×C₄	#13133217	1	1	yes	characteristic
179	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
180	C₂×C₄	C₂×#11719917	1	2	yes	—
181	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
182	C₂×C₄	#13133245	1	2	yes	—
183	C₂×C₄	#13133250	1	2	yes	—
184	C₈	—	2	4	no	—
185	C₈	—	424	2	no	—
186	C₈	—	424	1	no	—
187	C₈	—	424	1	no	—
188	C₈	C₄⋊#88322	1	8	yes	complemented
189	C₂²	#16810493	1	1	yes	characteristic
190	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
191	C₂²	#16810647	1	1	yes	characteristic
192	C₂²	#16810665	1	2	yes	—
193	C₂²	#16810672	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₄⋊#88322	1	2	yes	—
198	C₄	#16810647	1	2	yes	—
199	C₄	#16811012	1	1	yes	characteristic
200	C₄	#16811017	1	1	yes	characteristic
201	C₄	#16811032	1	2	yes	—
202	C₂	#20273467	1	2	yes	—
203	C₂	#20273482	1	2	yes	—
204	C₂	#20274867	1	1	yes	characteristic
205	C₂	#20274883	1	1	yes	characteristic
206	C₂	#20274890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#97×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#8808
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
35. #97×C₅₃	C₈	semidirect
81. C₈×C₅₃	C₄⋊C₈	semidirect
121. C₅₃	#8808	semidirect
139. #97	#88322	semidirect
188. 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
#20274867	2
#20274832	2
C₂×#16810376	2
C₂×#16810374	2
C₄×#97×C₅₃	2
#8808	53