Subgroups — #27302610

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₈×Dic₁₀₆	—	2	2	no	—
8	C₈×Dic₁₀₆	—	2	2	no	—
9	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
10	C₂×#13133146	C₄	1	2	yes	—
11	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
12	C₄×#11719908	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	—	2	2	no	—
16	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₀₆	—	2	2	no	—
19	C₄×Dic₁₀₆	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
21	C₂×#11719919	C₈	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	C₂×#11719908	Q₈	1	4	yes	complemented
26	C₄⋊#88322	—	2	2	no	—
27	#13133146	—	2	4	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	—	2	2	no	—
34	C₂×C₁₆×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
41	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
42	C₂×Dic₁₀₆	—	2	2	no	—
43	C₂×Dic₁₀₆	—	2	2	no	—
44	C₂×Dic₁₀₆	C₁₆	1	4	yes	complemented
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₅₃	#41	1	2	yes	—
51	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	#11719919	—	2	4	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	2	4	no	—
59	C₁₆×C₅₃	—	4	2	no	—
60	#11719908	—	2	4	no	—
61	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#131	1	2	yes	—
71	C₂×Dic₅₃	—	2	2	no	—
72	C₂×Dic₅₃	—	2	2	no	—
73	C₂×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	2	4	no	—
75	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
76	Dic₁₀₆	—	2	8	no	—
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₅₃	#124	1	1	yes	characteristic
80	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	#88322	—	4	2	no	—
82	#13263	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
84	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
85	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
87	C₂²×C₅₃	#504	1	2	yes	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	C₂²⋊C₁₆	1	4	yes	—
90	C₄×C₅₃	#391	1	1	yes	characteristic
91	C₄×C₅₃	#408	1	1	yes	characteristic
92	C₄×C₅₃	#446	1	2	yes	—
93	Dic₅₃	—	2	4	no	—
94	Dic₅₃	—	4	2	no	—
95	C₂×C₈×Q₈	—	53	1	no	—
96	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
97	C₂×C₄×C₁₆	—	53	1	no	—
98	#1374	—	53	4	no	—
99	C₂×C₅₃	#1984	1	1	yes	characteristic
100	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
101	C₂×C₅₃	#2113	1	1	yes	characteristic
102	C₂×C₅₃	Q₈×C₁₆	1	2	yes	—
103	C₂×C₅₃	#2183	1	2	yes	—
104	C₂×C₄×Q₈	—	53	1	no	—
105	C₂²×C₁₆	—	53	2	no	—
106	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
107	C₈×Q₈	—	106	2	no	—
108	C₈×Q₈	—	106	2	no	—
109	C₂×C₄⋊C₈	—	53	2	no	—
110	C₂×C₄⋊C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	2	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₄⋊C₁₆	—	2	2	no	—
114	C₄×C₁₆	—	106	2	no	—
115	C₅₃	#13263	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	53	1	no	—
117	C₂²×C₈	—	53	2	no	—
118	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₄×Q₈	—	106	2	no	—
121	C₄×Q₈	—	106	2	no	—
122	C₂×C₄⋊C₄	—	53	2	no	—
123	C₂×C₄⋊C₄	—	53	1	no	—
124	C₂×C₄²	—	53	2	no	—
125	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₁₆	—	2	4	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	53	4	no	—
129	C₂×C₁₆	—	106	2	no	—
130	C₂×C₁₆	—	106	2	no	—
131	C₄⋊C₈	—	106	4	no	—
132	C₄⋊C₈	—	106	2	no	—
133	C₄×C₈	—	106	2	no	—
134	C₄×C₈	—	106	2	no	—
135	C₄×C₈	D₂₁₂	1	2	yes	—
136	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
137	C₂×Q₈	—	53	4	no	—
138	C₂×Q₈	—	106	2	no	—
139	C₂×Q₈	—	106	2	no	—
140	C₂²×C₄	—	53	2	no	—
141	C₂²×C₄	—	53	2	no	—
142	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	106	2	no	—
147	C₂×C₈	—	106	4	no	—
148	C₂×C₈	#11719921	1	2	yes	—
149	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂×C₈	#11719947	1	2	yes	—
151	C₂×C₈	Q₈×D₅₃	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₄⋊C₄	—	106	4	no	—
154	C₄⋊C₄	—	106	2	no	—
155	C₄²	—	106	2	no	—
156	C₄²	—	106	2	no	—
157	C₄²	#11719912	1	2	yes	—
158	C₄²	#11719921	1	2	yes	—
159	C₁₆	—	4	2	no	—
160	C₁₆	—	106	4	no	—
161	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
162	Q₈	—	106	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	53	4	no	—
165	C₂×C₄	—	106	2	no	—
166	C₂×C₄	—	106	2	no	—
167	C₂×C₄	—	106	4	no	—
168	C₂×C₄	—	106	2	no	—
169	C₂×C₄	#13133129	1	2	yes	—
170	C₂×C₄	#13133129	1	2	yes	—
171	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
172	C₂×C₄	#13133156	1	2	yes	—
173	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
174	C₂×C₄	#13133182	1	2	yes	—
175	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
176	C₂×C₄	#13133225	1	2	yes	—
177	C₈	—	2	2	no	—
178	C₈	—	212	2	no	—
179	C₈	#13133197	1	2	yes	—
180	C₈	#13133204	1	1	yes	characteristic
181	C₈	#13133215	1	1	yes	characteristic
182	C₂²	C₂×#13133129	1	1	yes	characteristic
183	C₂²	#16810747	1	2	yes	—
184	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
185	C₂²	C₂×#13133156	1	1	yes	characteristic
186	C₂²	#16811041	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	106	4	no	—
189	C₄	—	212	2	no	—
190	C₄	#16810402	1	4	yes	—
191	C₄	#16810528	1	1	yes	characteristic
192	C₄	#16810591	1	1	yes	characteristic
193	C₄	#16810750	1	2	yes	—
194	C₂	#20275650	1	1	yes	characteristic
195	C₂	C₂×#16810402	1	1	yes	characteristic
196	C₂	#20276330	1	1	yes	characteristic
197	C₂	#20276617	1	2	yes	—
198	C₂	#20276620	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#13263
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. C₂×#11719908	Q₈	semidirect
44. C₂×Dic₁₀₆	C₁₆	semidirect
115. C₅₃	#13263	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
C₂×C₄×#11719908	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#13263	53