Subgroups — #27277914

Order 13568.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274887	C₂	1	1	yes	maximal, characteristic
2	#20274853	C₂	1	1	yes	maximal, characteristic
3	#20273474	C₂	1	1	yes	maximal, characteristic
4	#20273474	C₂	1	1	yes	maximal, characteristic
5	#20273471	C₂	1	1	yes	maximal, characteristic
6	#20273471	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133153	—	2	2	no	—
9	#16810647	C₂²	1	1	yes	characteristic
10	#16810647	C₂²	1	1	yes	characteristic
11	#16810647	C₂²	1	1	yes	characteristic
12	#16810647	C₂²	1	1	yes	characteristic
13	#16810388	—	2	2	no	—
14	#16810376	—	2	2	no	—
15	#16810376	—	2	2	no	—
16	#16810374	—	2	2	no	—
17	#16810374	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#322	C₄	1	2	yes	—
22	C₅₃×#322	C₄	1	2	yes	—
23	C₂×#11719917	—	2	1	no	—
24	C₂×#11719917	—	2	1	no	—
25	C₂×#11719917	—	2	1	no	—
26	C₂×#11719917	—	2	1	no	—
27	#13133153	—	4	4	no	—
28	C₄⋊#88322	—	2	1	no	—
29	C₄⋊#88322	—	2	1	no	—
30	C₄⋊#88322	—	2	1	no	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	—	2	1	no	—
34	C₄⋊#88322	—	2	1	no	—
35	C₄⋊#88322	—	2	1	no	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	Q₈	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	#11719917	—	4	2	no	—
63	#11719917	—	4	2	no	—
64	#11719917	—	4	2	no	—
65	#11719917	—	4	2	no	—
66	C₂×#88322	—	4	1	no	—
67	C₂×#88322	—	4	1	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂×#88322	—	4	1	no	—
70	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₅₃	#137	1	2	yes	—
81	C₈×C₅₃	—	2	4	no	—
82	C₈×C₅₃	—	2	2	no	—
83	C₈×C₅₃	—	2	4	no	—
84	C₈×C₅₃	#108	1	4	yes	complemented
85	#88322	—	8	1	no	—
86	#88322	—	8	1	no	—
87	#88322	—	8	1	no	—
88	#88322	—	8	1	no	—
89	#8828	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	#384	1	1	yes	characteristic
93	C₂²×C₅₃	#501	1	2	yes	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	#428	1	2	yes	—
97	C₄×C₅₃	C₂×#108	1	2	yes	—
98	C₄×C₅₃	#430	1	2	yes	—
99	C₄×C₅₃	#496	1	1	yes	characteristic
100	C₄×C₅₃	#497	1	1	yes	characteristic
101	#1848	—	53	1	no	—
102	#1837	—	53	1	no	—
103	#1567	—	53	1	no	—
104	#1567	—	53	1	no	—
105	#1566	—	53	1	no	—
106	#1566	—	53	1	no	—
107	C₂×#322	D₅₃	1	1	yes	characteristic
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	#1847	1	1	yes	characteristic
110	C₂×C₅₃	#1848	1	1	yes	characteristic
111	C₂×C₅₃	#1849	1	1	yes	characteristic
112	C₂×#108	—	106	2	no	—
113	#423	—	53	1	no	—
114	#423	—	53	1	no	—
115	#423	—	53	1	no	—
116	#423	—	53	1	no	—
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	#341	—	106	2	no	—
121	#335	—	106	2	no	—
122	#335	—	106	2	no	—
123	#334	—	106	2	no	—
124	#334	—	106	2	no	—
125	#322	Dic₅₃	1	2	yes	—
126	#322	Dic₅₃	1	2	yes	—
127	C₅₃	#8828	1	1	yes	characteristic, complemented, Sylow
128	C₂×M₁₆	—	106	1	no	—
129	C₂×M₁₆	—	106	1	no	—
130	C₂×M₁₆	—	106	1	no	—
131	C₂×M₁₆	—	106	1	no	—
132	C₂²×C₈	Dic₁₀₆	1	2	yes	—
133	C₂²×C₈	Dic₁₀₆	1	2	yes	—
134	C₂²×C₈	D₂₁₂	1	2	yes	—
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	#108	—	212	4	no	—
137	C₄⋊C₈	—	106	1	no	—
138	C₄⋊C₈	—	106	1	no	—
139	C₄⋊C₈	—	106	1	no	—
140	C₄⋊C₈	—	106	1	no	—
141	C₄⋊C₈	—	106	1	no	—
142	C₄⋊C₈	—	106	1	no	—
143	C₄⋊C₈	—	106	1	no	—
144	C₄⋊C₈	—	106	1	no	—
145	C₄×C₈	Dic₁₀₆	1	2	yes	—
146	C₄×C₈	D₂₁₂	1	2	yes	—
147	C₄×C₈	D₂₁₂	1	2	yes	—
148	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
149	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
151	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
155	M₁₆	—	212	2	no	—
156	M₁₆	—	212	2	no	—
157	M₁₆	—	212	2	no	—
158	M₁₆	—	212	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	212	1	no	—
164	C₂×C₈	—	212	1	no	—
165	C₂×C₈	—	212	1	no	—
166	C₂×C₈	—	212	1	no	—
167	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
168	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
169	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
170	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
171	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
172	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
173	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
174	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
175	C₂³	#13133217	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	#13133214	1	1	yes	characteristic
182	C₂×C₄	#13133217	1	1	yes	characteristic
183	C₂×C₄	#13133217	1	1	yes	characteristic
184	C₂×C₄	#13133245	1	2	yes	—
185	C₂×C₄	#13133250	1	2	yes	—
186	C₈	—	2	4	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	4	no	—
189	C₈	—	424	1	no	—
190	C₈	—	424	1	no	—
191	C₈	—	424	1	no	—
192	C₈	—	424	1	no	—
193	C₈	#13133153	1	4	yes	complemented
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#16810493	1	1	yes	characteristic
197	C₂²	#16811032	1	2	yes	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	#16810665	1	2	yes	—
201	C₄	C₂×#13133153	1	2	yes	—
202	C₄	#16810672	1	2	yes	—
203	C₄	#16811012	1	1	yes	characteristic
204	C₄	#16811017	1	1	yes	characteristic
205	C₂	—	2	2	no	—
206	C₂	#20274883	1	1	yes	characteristic
207	C₂	#20274887	1	1	yes	characteristic
208	C₂	#20274890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133214
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8828
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
84. C₈×C₅₃	#108	semidirect
127. C₅₃	#8828	semidirect
193. C₈	#13133153	semidirect

Maximal subgroups

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

subgroup	index
#20274887	2
#20274853	2
#20273474	2
#20273471	2
C₂×C₅₃×#322	2
#8828	53