Subgroups — #27289828

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810388	C₂	1	2	yes	maximal
2	C₂×#16810388	C₂	1	2	yes	maximal
3	#20274409	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₂²	1	4	yes	—
6	#16810388	—	2	4	no	—
7	#16810388	C₄	1	8	yes	complemented
8	#16810378	—	2	4	no	—
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	—	2	4	no	—
13	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
14	C₂×#11719917	—	2	4	no	—
15	C₂×#11719917	—	2	4	no	—
16	C₂×#11719917	—	2	4	no	—
17	C₂×#11719917	—	2	4	no	—
18	C₂×#11719917	—	2	4	no	—
19	C₂×#11719917	C₂×C₄	1	8	yes	—
20	C₂²×#88322	—	2	4	no	—
21	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	4	yes	—
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₄×C₈×C₅₃	—	2	8	no	—
36	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	4	no	—
38	C₂²×C₄×C₅₃	—	2	8	no	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	2	yes	—
45	C₂²×C₄×C₅₃	#41	1	2	yes	—
46	C₂²×C₄×C₅₃	#41	1	2	yes	—
47	C₂²×C₄×C₅₃	#41	1	2	yes	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	8	no	—
53	C₂×C₈×C₅₃	—	2	8	no	—
54	C₂×C₈×C₅₃	C₄²	1	4	yes	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	2	4	no	—
59	#11719917	—	2	8	no	—
60	#11719917	—	2	8	no	—
61	#11719917	—	4	4	no	—
62	#11719917	—	4	4	no	—
63	C₂×#88322	—	2	8	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂×#88322	—	4	4	no	—
66	C₂³×C₅₃	—	2	4	no	—
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
69	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
70	C₂³×C₅₃	#125	1	2	yes	—
71	C₂×C₄×C₅₃	—	2	16	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	#108	1	8	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#117	1	4	yes	—
78	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	#121	1	2	yes	—
85	C₂×C₄×C₅₃	#123	1	4	yes	—
86	C₂×C₄×C₅₃	#124	1	2	yes	—
87	C₂×C₄×C₅₃	#126	1	4	yes	—
88	C₈×C₅₃	—	2	4	no	—
89	C₈×C₅₃	—	2	4	no	—
90	#88322	—	4	8	no	—
91	#10822	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	4	no	—
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₅₃	#380	1	1	yes	characteristic
97	C₂²×C₅₃	#382	1	1	yes	characteristic
98	C₂²×C₅₃	#387	1	4	yes	—
99	C₂²×C₅₃	C₂×#108	1	4	yes	—
100	C₄×C₅₃	—	2	8	no	—
101	C₄×C₅₃	#376	1	1	yes	characteristic
102	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
103	C₄×C₅₃	#381	1	1	yes	characteristic
104	C₄×C₅₃	#383	1	1	yes	characteristic
105	C₄×C₅₃	#388	1	4	yes	—
106	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
107	C₂×#341	—	53	2	no	—
108	C₂×#341	—	53	2	no	—
109	#1728	—	53	2	no	—
110	C₂×C₅₃	—	2	4	no	—
111	C₂×C₅₃	#1434	1	1	yes	characteristic
112	C₂×C₅₃	C₄×#108	1	2	yes	—
113	C₂×C₅₃	#1810	1	2	yes	—
114	C₂×C₅₃	#1837	1	2	yes	—
115	C₂²×M₁₆	—	53	4	no	—
116	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	—	2	4	no	—
120	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
121	#341	—	53	8	no	—
122	#341	—	106	4	no	—
123	#336	—	106	4	no	—
124	C₅₃	#10822	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₄	Dic₁₀₆	1	2	yes	—
126	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
127	C₂×M₁₆	—	53	8	no	—
128	C₂×M₁₆	—	106	4	no	—
129	C₂×M₁₆	—	106	4	no	—
130	C₂×M₁₆	—	106	4	no	—
131	C₂×M₁₆	—	106	4	no	—
132	C₂×M₁₆	—	106	4	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	106	4	no	—
138	C₂²×C₈	C₄×D₅₃	1	4	yes	—
139	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
140	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	D₂₁₂	1	2	yes	—
145	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
146	C₄×C₈	—	2	8	no	—
147	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
152	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
153	C₂²×C₄	#11719945	1	2	yes	—
154	C₂²×C₄	#11719945	1	2	yes	—
155	C₂²×C₄	#11719945	1	2	yes	—
156	C₂²×C₄	#11719945	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
159	M₁₆	—	106	8	no	—
160	M₁₆	—	106	8	no	—
161	M₁₆	—	212	4	no	—
162	M₁₆	—	212	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	106	8	no	—
170	C₂×C₈	—	212	4	no	—
171	C₂×C₈	—	212	4	no	—
172	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
173	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂³	C₄×Dic₁₀₆	1	2	yes	—
180	C₂³	#13133196	1	2	yes	—
181	C₂³	#13133209	1	2	yes	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	#13133153	1	8	yes	—
186	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₂×C₄	#13133174	1	4	yes	—
189	C₂×C₄	#13133176	1	1	yes	characteristic
190	C₂×C₄	#13133176	1	1	yes	characteristic
191	C₂×C₄	#13133176	1	1	yes	characteristic
192	C₂×C₄	#13133176	1	1	yes	characteristic
193	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
194	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
195	C₂×C₄	#13133191	1	2	yes	—
196	C₂×C₄	#13133200	1	4	yes	—
197	C₂×C₄	#13133205	1	2	yes	—
198	C₂×C₄	#13133211	1	4	yes	—
199	C₈	—	2	4	no	—
200	C₈	—	2	4	no	—
201	C₈	—	212	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂²	#16810483	1	1	yes	characteristic
207	C₂²	#16810489	1	1	yes	characteristic
208	C₂²	#16810507	1	4	yes	—
209	C₂²	C₂×#13133153	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#16810469	1	1	yes	characteristic
212	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₄	#16810486	1	1	yes	characteristic
214	C₄	#16810491	1	1	yes	characteristic
215	C₄	#16810514	1	4	yes	—
216	C₂	—	2	4	no	—
217	C₂	#20272942	1	1	yes	characteristic
218	C₂	C₄×#13133153	1	2	yes	—
219	C₂	#20274753	1	2	yes	—
220	C₂	#20274853	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#10822
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #16810388	C₄	semidirect
124. C₅₃	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810388	2
#20274409	2
C₂²×C₄×C₈×C₅₃	2
#10822	53