Subgroups — #27315284

Order 13568.

1710 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810493	C₂	1	2	yes	maximal
2	C₂×#16810489	C₂	1	4	yes	maximal
3	#20272945	C₂	1	8	yes	maximal
4	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	C₂²	1	8	yes	—
6	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
7	#16810493	—	2	8	no	—
8	#16810489	—	2	16	no	—
9	C₂×#13133124	C₂²	1	16	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄³×C₅₃	C₄	1	8	yes	complemented
14	C₂³×Dic₅₃	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₅₃	—	2	16	no	—
16	C₂×C₄⋊Dic₅₃	—	2	16	no	—
17	C₂×C₄⋊Dic₅₃	—	2	8	no	—
18	C₂×C₄⋊Dic₅₃	—	2	32	no	—
19	#13133124	—	2	32	no	—
20	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	16	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₅₃	D₄	1	4	yes	—
28	C₂×C₄²×C₅₃	D₄	1	8	yes	—
29	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	8	yes	—
31	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	2	16	no	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
41	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₅₃	#41	1	16	yes	—
45	C₂²×Dic₅₃	—	2	16	no	—
46	C₂²×Dic₅₃	—	2	8	no	—
47	C₂²×Dic₅₃	—	2	32	no	—
48	C₄²×C₅₃	—	2	32	no	—
49	C₄²×C₅₃	C₄⋊C₄	1	16	yes	—
50	C₄²×C₅₃	C₄⋊C₄	1	32	yes	complemented
51	C₄⋊Dic₅₃	—	4	32	no	—
52	C₄⋊Dic₅₃	—	4	16	no	—
53	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
55	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
56	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
57	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
58	C₂³×C₅₃	C₂×#41	1	8	yes	—
59	C₂×C₄×C₅₃	—	2	16	no	—
60	C₂×C₄×C₅₃	—	2	16	no	—
61	C₂×C₄×C₅₃	—	2	16	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₅₃	#117	1	16	yes	—
67	C₂×C₄×C₅₃	#124	1	16	yes	—
68	C₂×C₄×C₅₃	#125	1	16	yes	—
69	C₂×C₄×C₅₃	#127	1	4	yes	—
70	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
71	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
72	C₂×Dic₅₃	—	4	16	no	—
73	C₂×Dic₅₃	—	4	8	no	—
74	C₂×Dic₅₃	—	4	32	no	—
75	#14687	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
78	C₂²×C₅₃	C₂×#117	1	4	yes	—
79	C₂²×C₅₃	C₂×#124	1	4	yes	—
80	C₂²×C₅₃	C₂×#125	1	4	yes	—
81	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
83	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₅₃	#532	1	8	yes	—
85	C₂²×C₅₃	#533	1	8	yes	—
86	C₄×C₅₃	—	2	16	no	—
87	C₄×C₅₃	#382	1	16	yes	complemented
88	C₄×C₅₃	#384	1	8	yes	—
89	Dic₅₃	—	8	8	no	—
90	C₂×#384	—	53	2	no	—
91	C₂×#382	—	53	4	no	—
92	C₂×C₄³	D₅₃	1	1	yes	characteristic
93	C₄⋊#95	—	53	8	no	—
94	C₂×C₅₃	C₂×#382	1	2	yes	—
95	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
96	C₂×C₅₃	#2285	1	4	yes	—
97	C₂×C₅₃	#2858	1	2	yes	—
98	C₂×C₅₃	#2859	1	4	yes	—
99	C₂×C₅₃	#2862	1	2	yes	—
100	C₂²×C₄⋊C₄	—	53	8	no	—
101	C₂²×C₄⋊C₄	—	53	4	no	—
102	C₂²×C₄²	C₂×D₅₃	1	4	yes	—
103	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
104	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
105	#384	—	106	8	no	—
106	#382	—	106	16	no	—
107	C₂×#95	—	53	16	no	—
108	C₄³	Dic₅₃	1	8	yes	complemented
109	C₅₃	#14687	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₄	—	53	8	no	—
111	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
112	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
114	C₂×C₄⋊C₄	—	106	16	no	—
115	C₂×C₄⋊C₄	—	106	16	no	—
116	C₂×C₄⋊C₄	—	106	8	no	—
117	C₂×C₄⋊C₄	—	106	32	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	Dic₁₀₆	1	4	yes	—
120	C₂×C₄²	Dic₁₀₆	1	8	yes	—
121	C₂×C₄²	D₂₁₂	1	4	yes	—
122	C₂×C₄²	D₂₁₂	1	8	yes	—
123	C₂×C₄²	C₂×Dic₅₃	1	16	yes	—
124	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
125	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
126	#95	—	106	32	no	—
127	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	—	2	16	no	—
129	C₂²×C₄	—	106	16	no	—
130	C₂²×C₄	—	106	8	no	—
131	C₂²×C₄	—	106	32	no	—
132	C₂²×C₄	C₂×Dic₁₀₆	1	8	yes	—
133	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
134	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
135	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
136	C₂²×C₄	C₂×D₂₁₂	1	8	yes	—
137	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
138	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
139	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
140	C₂²×C₄	#11719945	1	16	yes	—
141	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
142	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
144	C₄⋊C₄	—	212	32	no	—
145	C₄⋊C₄	—	212	16	no	—
146	C₄²	—	2	32	no	—
147	C₄²	C₄⋊Dic₅₃	1	16	yes	—
148	C₄²	C₄⋊Dic₅₃	1	32	yes	complemented
149	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
150	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
151	C₂³	C₂²×D₂₁₂	1	2	yes	—
152	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
153	C₂³	C₂²×Dic₁₀₆	1	2	yes	—
154	C₂³	C₂×#11719945	1	8	yes	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	212	16	no	—
159	C₂×C₄	—	212	8	no	—
160	C₂×C₄	—	212	32	no	—
161	C₂×C₄	C₂×C₄⋊Dic₅₃	1	16	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
165	C₂×C₄	#13133176	1	16	yes	—
166	C₂×C₄	#13133205	1	16	yes	—
167	C₂×C₄	#13133209	1	16	yes	—
168	C₂×C₄	#13133214	1	4	yes	—
169	C₂×C₄	#13133217	1	8	yes	—
170	C₂×C₄	#13133217	1	4	yes	—
171	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂²	C₂²×C₄⋊Dic₅₃	1	2	yes	—
173	C₂²	C₂×#13133176	1	4	yes	—
174	C₂²	C₂×#13133205	1	4	yes	—
175	C₂²	C₂×#13133209	1	4	yes	—
176	C₂²	C₂×#13133214	1	1	yes	characteristic
177	C₂²	C₂×#13133217	1	2	yes	—
178	C₂²	C₂×#13133217	1	1	yes	characteristic
179	C₂²	#16811173	1	8	yes	—
180	C₂²	#16811180	1	8	yes	—
181	C₄	—	2	16	no	—
182	C₄	—	424	8	no	—
183	C₄	#16810489	1	16	yes	complemented
184	C₄	#16810493	1	8	yes	—
185	C₂	C₂×#16810489	1	2	yes	—
186	C₂	C₂×#16810493	1	1	yes	characteristic
187	C₂	#20277122	1	4	yes	—
188	C₂	#20283072	1	2	yes	—
189	C₂	#20283078	1	4	yes	—
190	C₂	#20283093	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#14687
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
13. C₄³×C₅₃	C₄	semidirect
50. C₄²×C₅₃	C₄⋊C₄	semidirect
87. C₄×C₅₃	#382	semidirect
108. C₄³	Dic₅₃	semidirect
109. C₅₃	#14687	semidirect
148. C₄²	C₄⋊Dic₅₃	semidirect
183. C₄	#16810489	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810493	2
C₂×#16810489	2
#20272945	2
C₂×C₄³×C₅₃	2
#14687	53