Subgroups — #27384328

Order 13568.

778 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	#20283279	C₂	1	2	yes	maximal
2	#20274856	C₂	1	4	yes	maximal, complemented
3	#20274853	C₂	1	4	yes	maximal, complemented
4	#20274442	C₂	1	2	yes	maximal
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	#16810650	—	2	8	no	—
9	#16810647	C₂²	1	8	yes	—
10	#16810388	C₂²	1	4	yes	—
11	#16810388	C₂²	1	4	yes	—
12	#16810388	C₂²	1	8	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	8	yes	—
17	C₂×#97×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	8	no	—
21	C₂×#11719917	—	2	8	no	—
22	C₂×#11719917	C₂³	1	8	yes	—
23	C₂²×#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	16	no	—
25	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₅₃	—	2	4	no	—
27	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
33	#117×C₅₃	C₂×C₄	1	4	yes	—
34	#117×C₅₃	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	#97×C₅₃	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
40	C₂⁴×C₅₃	C₂×Q₈	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×Q₈	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	#41	1	4	yes	—
46	C₂²×C₄×C₅₃	#41	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	M₁₆×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	—	2	8	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
56	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
57	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
58	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
59	C₄²×C₅₃	C₂×D₄	1	4	yes	—
60	#11719917	—	2	8	no	—
61	#11719917	—	2	8	no	—
62	#11719917	—	4	8	no	—
63	C₂×#88322	—	2	8	no	—
64	C₂×#88322	—	4	8	no	—
65	C₂³×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₅₃	#125	1	4	yes	—
68	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	4	4	no	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₅₃	#117	1	4	yes	—
73	C₂×C₄×C₅₃	#117	1	4	yes	—
74	C₂×C₄×C₅₃	#124	1	8	yes	—
75	C₂×C₄×C₅₃	#125	1	4	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
80	C₈×C₅₃	—	2	2	no	—
81	C₈×C₅₃	—	2	2	no	—
82	C₈×C₅₃	—	2	4	no	—
83	#88322	—	4	8	no	—
84	#20768	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	4	4	no	—
88	C₂²×C₅₃	#382	1	4	yes	—
89	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×#125	1	2	yes	—
91	C₄×C₅₃	—	2	8	no	—
92	C₄×C₅₃	#382	1	4	yes	—
93	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
94	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
95	C₄×C₅₃	C₂×#124	1	2	yes	—
96	#2905	—	53	2	no	—
97	C₂×#405	D₅₃	1	1	yes	characteristic
98	#1838	—	53	4	no	—
99	#1837	—	53	4	no	—
100	#1738	—	53	2	no	—
101	C₂×#341	—	53	2	no	—
102	C₂×C₅₃	—	2	2	no	—
103	C₂×C₅₃	—	2	4	no	—
104	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
105	C₂×C₅₃	#2916	1	2	yes	—
106	C₂²×M₁₆	—	53	4	no	—
107	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
108	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
110	#424	—	106	8	no	—
111	#423	—	53	8	no	—
112	#405	Dic₅₃	1	8	yes	—
113	C₂×#97	C₂×D₅₃	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
115	#341	—	53	4	no	—
116	#341	—	53	8	no	—
117	#341	—	53	4	no	—
118	C₅₃	#20768	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	53	8	no	—
122	C₂×M₁₆	—	106	4	no	—
123	C₂×M₁₆	—	106	8	no	—
124	C₂×M₁₆	—	106	8	no	—
125	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	106	4	no	—
128	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
129	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
132	#117	D₂₁₂	1	4	yes	—
133	#117	C₂×Dic₅₃	1	4	yes	—
134	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
135	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
136	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
137	C₄⋊C₈	—	106	16	no	—
138	#97	C₂×Dic₅₃	1	8	yes	—
139	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
140	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
142	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
143	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
144	C₂²×C₄	#11719945	1	4	yes	—
145	C₂²×C₄	#11719945	1	4	yes	—
146	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
147	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
148	M₁₆	—	2	8	no	—
149	M₁₆	—	106	8	no	—
150	M₁₆	—	106	8	no	—
151	M₁₆	—	212	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	106	8	no	—
157	C₂×C₈	—	212	8	no	—
158	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
159	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
160	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
161	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
162	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
163	C₄²	C₂×D₂₁₂	1	4	yes	—
164	C₄²	C₂²×Dic₅₃	1	4	yes	—
165	C₂³	—	4	2	no	—
166	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
167	C₂³	#13133209	1	4	yes	—
168	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
172	C₂×C₄	#13133176	1	4	yes	—
173	C₂×C₄	#13133176	1	4	yes	—
174	C₂×C₄	#13133205	1	8	yes	—
175	C₂×C₄	#13133209	1	4	yes	—
176	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
178	C₂×C₄	C₂×#11719945	1	2	yes	—
179	C₂×C₄	C₂×#11719945	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	212	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	#16810489	1	4	yes	—
188	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
189	C₂²	C₂×#13133209	1	2	yes	—
190	C₄	—	2	8	no	—
191	C₄	#16810489	1	4	yes	—
192	C₄	C₂×#13133176	1	1	yes	characteristic
193	C₄	C₂×#13133176	1	1	yes	characteristic
194	C₄	C₂×#13133205	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂×#16810489	1	1	yes	characteristic
198	C₂	#20283326	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
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	#20768
LCS	G ▷ C₂×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
2. #20274856	C₂	semidirect
3. #20274853	C₂	semidirect
118. C₅₃	#20768	semidirect

Maximal subgroups

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

subgroup	index
#20283279	2
#20274856	2
#20274853	2
#20274442	2
C₂×#16810388	2
C₂×C₅₃×#405	2
#20768	53