Subgroups — #27277753

Order 13568.

422 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	#20274867	C₂	1	2	yes	maximal
2	#20272550	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	2	no	—
5	C₂×C₄⋊#88322	—	2	2	no	—
6	C₂×C₄⋊#88322	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×#88322	—	2	4	no	—
13	C₄⋊#88322	—	2	8	no	—
14	C₄⋊#88322	—	4	4	no	—
15	C₄⋊#88322	—	4	4	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₅₃	—	2	4	no	—
26	C₄×C₈×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	—	2	4	no	—
28	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	C₂×C₈	1	2	yes	—
46	C₄²×C₅₃	M₁₆	1	2	yes	—
47	C₄²×C₅₃	D₈	1	2	yes	—
48	C₄²×C₅₃	SD₁₆	1	4	yes	—
49	C₄²×C₅₃	Q₁₆	1	2	yes	—
50	C₄²×C₅₃	#41	1	4	yes	—
51	C₂×#88322	—	4	4	no	—
52	C₂×#88322	—	4	4	no	—
53	C₂×#88322	—	4	4	no	—
54	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₅₃	#106	1	2	yes	—
64	C₂×C₄×C₅₃	#106	1	2	yes	—
65	C₂×C₄×C₅₃	#107	1	2	yes	—
66	C₂×C₄×C₅₃	#107	1	2	yes	—
67	C₂×C₄×C₅₃	#108	1	2	yes	—
68	C₂×C₄×C₅₃	#108	1	2	yes	—
69	C₂×C₄×C₅₃	#117	1	2	yes	—
70	C₂×C₄×C₅₃	#124	1	2	yes	—
71	C₂×C₄×C₅₃	#125	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₈×C₅₃	—	2	4	no	—
79	C₈×C₅₃	—	2	4	no	—
80	#88322	—	8	4	no	—
81	#8784	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	#382	1	1	yes	characteristic
83	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₅₃	#423	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₅₃	—	2	2	no	—
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	—	2	2	no	—
92	C₄×C₅₃	#334	1	4	yes	—
93	C₄×C₅₃	#335	1	4	yes	—
94	C₄×C₅₃	#431	1	2	yes	—
95	C₄×C₅₃	#432	1	2	yes	—
96	C₄×C₅₃	#486	1	2	yes	—
97	C₄×C₅₃	#487	1	2	yes	—
98	#1842	—	53	2	no	—
99	C₄²×C₈	D₅₃	1	1	yes	characteristic
100	#1277	—	53	4	no	—
101	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
103	C₂×C₅₃	#1566	1	1	yes	characteristic
104	C₂×C₅₃	#1567	1	1	yes	characteristic
105	C₂×C₅₃	#1831	1	1	yes	characteristic
106	C₂×C₅₃	#1834	1	1	yes	characteristic
107	C₂×C₅₃	#1837	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	53	4	no	—
109	C₂×C₄⋊C₈	—	106	2	no	—
110	C₂×C₄⋊C₈	—	106	2	no	—
111	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
112	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
113	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
115	C₄³	C₂×D₅₃	1	1	yes	characteristic
116	C₅₃	#8784	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	106	4	no	—
118	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
119	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
120	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
121	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
122	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
123	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
127	C₄⋊C₈	—	106	8	no	—
128	C₄⋊C₈	—	212	4	no	—
129	C₄⋊C₈	—	212	4	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	#88322	1	4	yes	complemented
134	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	212	4	no	—
146	C₂×C₈	—	212	4	no	—
147	C₂×C₈	—	212	4	no	—
148	C₂×C₈	C₂×#88322	1	4	yes	—
149	C₂×C₈	#11719917	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	#11719913	1	4	yes	—
154	C₄²	D₄₂₄	1	2	yes	—
155	C₄²	Dic₂₁₂	1	2	yes	—
156	C₄²	C₂×#88322	1	2	yes	—
157	C₄²	#11719917	1	2	yes	—
158	C₄²	#11719945	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄⋊#88322	1	2	yes	—
167	C₂×C₄	C₄⋊#88322	1	2	yes	—
168	C₂×C₄	#13133149	1	2	yes	—
169	C₂×C₄	#13133149	1	2	yes	—
170	C₂×C₄	#13133151	1	2	yes	—
171	C₂×C₄	#13133151	1	2	yes	—
172	C₂×C₄	#13133153	1	2	yes	—
173	C₂×C₄	#13133153	1	2	yes	—
174	C₂×C₄	#13133176	1	2	yes	—
175	C₂×C₄	#13133205	1	2	yes	—
176	C₂×C₄	#13133209	1	2	yes	—
177	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719917	1	2	yes	—
179	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
181	C₂×C₄	C₂×#11719913	1	2	yes	—
182	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	2	4	no	—
185	C₈	—	424	4	no	—
186	C₂²	#16810489	1	1	yes	characteristic
187	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
188	C₂²	#16810647	1	1	yes	characteristic
189	C₂²	C₂×#13133149	1	1	yes	characteristic
190	C₂²	C₂×#13133151	1	1	yes	characteristic
191	C₂²	C₂×#13133153	1	1	yes	characteristic
192	C₂²	C₂×#13133153	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	#16810374	1	4	yes	—
197	C₄	#16810376	1	4	yes	—
198	C₄	#16810676	1	2	yes	—
199	C₄	#16810680	1	2	yes	—
200	C₄	#16810970	1	2	yes	—
201	C₄	#16810974	1	2	yes	—
202	C₂	C₂×#16810374	1	1	yes	characteristic
203	C₂	C₂×#16810376	1	1	yes	characteristic
204	C₂	#20273471	1	1	yes	characteristic
205	C₂	#20273474	1	1	yes	characteristic
206	C₂	#20274832	1	1	yes	characteristic
207	C₂	#20274843	1	1	yes	characteristic
208	C₂	#20274853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8784
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₈×C₅₃	C₈	semidirect
116. C₅₃	#8784	semidirect
133. C₄×C₈	#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
#20272550	2
C₄²×C₈×C₅₃	2
#8784	53