Subgroups — #27382204

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	2	yes	maximal
2	#20283134	C₂	1	2	yes	maximal
3	#20274621	C₂	1	4	yes	maximal, complemented
4	C₄×#13133153	C₂	1	4	yes	maximal, complemented
5	C₂×#16810388	C₂	1	2	yes	maximal
6	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	4	yes	—
8	#16810672	C₄	1	16	yes	complemented
9	C₂×#13133153	C₂²	1	8	yes	—
10	#16810579	—	2	8	no	—
11	C₄×#11719917	C₂²	1	8	yes	—
12	#16810388	C₂²	1	4	yes	—
13	#16810388	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#405	C₄	1	8	yes	—
18	C₂×#97×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	—	2	8	no	—
22	C₂×#11719917	C₂×C₄	1	16	yes	—
23	C₂×#11719917	C₂³	1	8	yes	—
24	C₂²×#88322	—	2	4	no	—
25	#13133153	C₂×C₄	1	32	yes	—
26	#13133128	—	2	8	no	—
27	C₄×#88322	—	2	8	no	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	#117×C₅₃	C₂×C₄	1	4	yes	—
37	#117×C₅₃	D₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
39	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
41	#97×C₅₃	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
43	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₅₃	C₄²	1	16	yes	—
52	C₂×C₈×C₅₃	—	2	8	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
59	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
60	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
62	C₄²×C₅₃	C₂×D₄	1	4	yes	—
63	#11719917	—	2	16	no	—
64	#11719917	—	2	8	no	—
65	#11719917	C₂²×C₄	1	16	yes	—
66	C₂×#88322	—	2	16	no	—
67	C₂×#88322	—	2	8	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
71	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₅₃	#117	1	4	yes	—
77	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
78	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
79	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	C₂×C₄²	1	8	yes	—
86	#88322	—	2	16	no	—
87	#20459	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	4	no	—
90	C₂²×C₅₃	—	4	4	no	—
91	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
92	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
94	C₄×C₅₃	—	2	8	no	—
95	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
96	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
98	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
99	C₂×#430	—	53	2	no	—
100	#2873	—	53	2	no	—
101	C₂×#405	D₅₃	1	1	yes	characteristic
102	#1778	—	53	4	no	—
103	C₄×#108	—	53	4	no	—
104	C₂×#341	—	53	2	no	—
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₅₃	#2916	1	2	yes	—
109	C₂²×M₁₆	—	53	4	no	—
110	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
111	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
112	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
113	#430	—	53	16	no	—
114	C₂×#108	—	53	8	no	—
115	#405	—	106	8	no	—
116	#405	Dic₅₃	1	8	yes	—
117	C₄×M₁₆	—	53	8	no	—
118	C₂×#97	C₂×D₅₃	1	2	yes	—
119	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
120	#341	—	53	4	no	—
121	#341	—	53	4	no	—
122	C₅₃	#20459	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×M₁₆	—	53	16	no	—
125	C₂×M₁₆	—	53	8	no	—
126	C₂×M₁₆	—	106	4	no	—
127	C₂×M₁₆	—	106	8	no	—
128	C₂×M₁₆	C₄×D₅₃	1	8	yes	—
129	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	106	4	no	—
132	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
133	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
136	#117	D₂₁₂	1	4	yes	—
137	#117	C₂×Dic₅₃	1	4	yes	—
138	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
139	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
140	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
141	#108	—	53	32	no	—
142	#97	—	106	8	no	—
143	#97	C₂×Dic₅₃	1	8	yes	—
144	C₄×C₈	—	106	8	no	—
145	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
146	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
148	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
150	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
151	C₂²×C₄	#11719945	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
153	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
154	M₁₆	—	53	16	no	—
155	M₁₆	—	106	16	no	—
156	M₁₆	—	106	8	no	—
157	M₁₆	C₄×Dic₅₃	1	16	yes	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	106	16	no	—
162	C₂×C₈	—	106	8	no	—
163	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
164	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
165	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
166	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
167	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
168	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
169	C₄²	C₂×D₂₁₂	1	4	yes	—
170	C₄²	C₂²×Dic₅₃	1	4	yes	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
173	C₂³	C₄×Dic₁₀₆	1	4	yes	—
174	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
179	C₂×C₄	#13133176	1	4	yes	—
180	C₂×C₄	C₄×D₂₁₂	1	8	yes	—
181	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
182	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
183	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719945	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	106	16	no	—
189	C₈	C₂×C₄×Dic₅₃	1	8	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₄×C₄⋊Dic₅₃	1	4	yes	—
194	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
195	C₂²	C₂×C₄×Dic₁₀₆	1	2	yes	—
196	C₄	—	2	8	no	—
197	C₄	C₄×C₄⋊Dic₅₃	1	4	yes	—
198	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
199	C₄	C₂×#13133176	1	1	yes	characteristic
200	C₄	C₂×C₄×D₂₁₂	1	2	yes	—
201	C₂	—	2	2	no	—
202	C₂	—	2	4	no	—
203	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
204	C₂	#20283326	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#20459
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #20274621	C₂	semidirect
4. C₄×#13133153	C₂	semidirect
8. #16810672	C₄	semidirect
122. C₅₃	#20459	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20283134	2
#20274621	2
C₄×#13133153	2
C₂×#16810388	2
C₂×C₅₃×#405	2
#20459	53