Subgroups — #27295484

Order 13568.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20272764	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	#16810647	—	2	1	no	—
6	C₄⋊#11719908	C₂²	1	2	yes	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₄×#11719908	—	2	1	no	—
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₅₃	—	2	1	no	—
12	C₂²×Q₁₆×C₅₃	C₄	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₅₃×#494	—	2	1	no	—
15	C₅₃×#494	—	2	1	no	—
16	C₅₃×#494	C₄	1	2	yes	—
17	C₅₃×#494	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	4	1	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	4	1	no	—
23	C₂²×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₁₆×C₅₃	—	2	2	no	—
25	C₂×Q₁₆×C₅₃	—	2	2	no	—
26	C₂×Q₁₆×C₅₃	—	2	2	no	—
27	C₂×Q₁₆×C₅₃	—	4	4	no	—
28	C₂×Q₁₆×C₅₃	—	4	4	no	—
29	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	2	2	no	—
34	C₄⋊Q₈×C₅₃	—	4	1	no	—
35	C₄⋊Q₈×C₅₃	—	4	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	4	4	no	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₅₃	—	4	4	no	—
52	Q₁₆×C₅₃	—	4	4	no	—
53	Q₁₆×C₅₃	—	4	4	no	—
54	Q₁₆×C₅₃	—	4	4	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₃	D₈	1	1	yes	characteristic
67	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	#11719917	—	8	2	no	—
70	C₂×#88322	—	4	2	no	—
71	#11719908	—	4	2	no	—
72	C₂³×C₅₃	#102	1	1	yes	characteristic
73	Q₈×C₅₃	—	4	4	no	—
74	Q₈×C₅₃	—	4	4	no	—
75	Q₈×C₅₃	—	8	4	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	8	4	no	—
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
85	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₅₃	—	2	1	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	#100	1	2	yes	—
92	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
93	#88322	—	8	2	no	—
94	#11962	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	#358	1	2	yes	—
98	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	8	4	no	—
102	C₄×C₅₃	#358	1	2	yes	—
103	C₄×C₅₃	#362	1	2	yes	—
104	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
105	C₄×C₅₃	#418	1	1	yes	characteristic
106	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
107	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
108	C₂×#494	D₅₃	1	1	yes	characteristic
109	#2162	—	53	1	no	—
110	#1567	—	53	1	no	—
111	#1362	—	53	4	no	—
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	#1545	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
116	C₂²×Q₁₆	—	2	1	no	—
117	C₂²×Q₁₆	Dic₅₃	1	2	yes	—
118	C₂×C₄⋊Q₈	—	2	1	no	—
119	#494	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	Dic₅₃	1	2	yes	—
122	#494	C₂×D₅₃	1	2	yes	—
123	#423	—	106	1	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₄⋊C₁₆	—	53	2	no	—
126	C₂²⋊C₁₆	—	106	1	no	—
127	C₄×C₁₆	—	106	1	no	—
128	#335	—	53	2	no	—
129	#335	—	106	1	no	—
130	C₅₃	#11962	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
138	C₂×M₁₆	—	212	1	no	—
139	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	4	1	no	—
144	C₄⋊Q₈	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₁₆	—	106	2	no	—
149	C₄⋊C₈	—	106	2	no	—
150	C₄⋊C₈	—	212	1	no	—
151	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	#11719922	1	1	yes	characteristic
161	C₂²×C₄	#11719923	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	424	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	212	2	no	—
171	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	#11719922	1	1	yes	characteristic
180	C₄²	#11719923	1	1	yes	characteristic
181	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₁₆	—	212	2	no	—
184	C₂³	#13133138	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#13133138	1	1	yes	characteristic
194	C₂×C₄	#13133138	1	1	yes	characteristic
195	C₂×C₄	#13133138	1	1	yes	characteristic
196	C₂×C₄	#13133160	1	2	yes	—
197	C₂×C₄	#13133163	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
200	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	424	2	no	—
204	C₈	#13133134	1	2	yes	—
205	C₈	#13133243	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16810427	1	2	yes	—
209	C₂²	C₂×#13133138	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#16810427	1	2	yes	—
214	C₄	#16810439	1	2	yes	—
215	C₄	C₂×#13133134	1	1	yes	characteristic
216	C₄	#16810628	1	1	yes	characteristic
217	C₄	C₂×#13133160	1	1	yes	characteristic
218	C₄	C₂×#13133163	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#20273383	1	1	yes	characteristic
221	C₂	C₂×#16810427	1	1	yes	characteristic
222	C₂	C₂×#16810439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#11962
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #20272764	C₂	semidirect
130. C₅₃	#11962	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
#20273474	2
#20272764	2
C₂×C₅₃×#494	2
#11962	53