Subgroups — #27276944

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274695	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	2	yes	maximal, complemented
3	#20272550	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	C₂×C₂²⋊#88322	—	2	2	no	—
7	#16810378	—	2	2	no	—
8	#16810376	—	2	4	no	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#431	—	2	4	no	—
11	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×#88322	—	2	4	no	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄⋊#88322	—	2	4	no	—
18	C₂²⋊#88322	—	4	4	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	4	no	—
31	C₂²⋊C₈×C₅₃	—	2	4	no	—
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	D₄	1	4	yes	—
34	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	4	2	no	—
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	D₈	1	2	yes	—
46	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	—	2	1	no	—
56	C₄²×C₅₃	—	2	2	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	C₂×D₄	1	2	yes	—
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₂×#88322	—	4	4	no	—
61	C₂×#88322	—	4	4	no	—
62	C₂×#88322	—	4	4	no	—
63	C₂³×C₅₃	—	2	1	no	—
64	C₂³×C₅₃	—	2	2	no	—
65	C₂³×C₅₃	—	2	1	no	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	#107	1	2	yes	—
68	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	#107	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#121	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₅₃	#131	1	4	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	2	yes	—
90	C₈×C₅₃	—	2	4	no	—
91	C₈×C₅₃	—	4	2	no	—
92	#88322	—	8	4	no	—
93	#8599	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	—	4	1	no	—
97	C₂²×C₅₃	—	4	1	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	#380	1	1	yes	characteristic
100	C₂²×C₅₃	#424	1	2	yes	—
101	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
102	C₂²×C₅₃	#427	1	1	yes	characteristic
103	C₂²×C₅₃	#430	1	2	yes	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	1	no	—
107	C₄×C₅₃	—	4	1	no	—
108	C₄×C₅₃	#408	1	2	yes	—
109	C₄×C₅₃	#433	1	2	yes	—
110	C₄×C₅₃	#466	1	2	yes	—
111	C₄×C₅₃	#467	1	2	yes	—
112	C₂×#431	D₅₃	1	1	yes	characteristic
113	#1797	—	53	2	no	—
114	C₂×#335	—	53	2	no	—
115	#1277	—	53	2	no	—
116	C₂×C₅₃	—	4	2	no	—
117	C₂×C₅₃	#1571	1	2	yes	—
118	C₂×C₅₃	#1572	1	2	yes	—
119	C₂×C₅₃	#1798	1	1	yes	characteristic
120	C₂×C₅₃	#1808	1	1	yes	characteristic
121	C₂×C₅₃	#1811	1	1	yes	characteristic
122	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
123	#431	—	2	4	no	—
124	C₂×C₄⋊C₈	—	53	4	no	—
125	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
126	C₂×C₂²⋊C₈	—	106	2	no	—
127	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#336	—	106	2	no	—
131	#335	—	106	4	no	—
132	C₅₃	#8599	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
134	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
135	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂²×C₈	—	106	4	no	—
137	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
138	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	—	106	4	no	—
147	C₄⋊C₈	—	106	4	no	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	212	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
152	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	D₄₂₄	1	2	yes	—
161	C₂²×C₄	Dic₂₁₂	1	2	yes	—
162	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
167	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	212	4	no	—
172	C₂×C₈	—	212	4	no	—
173	C₂×C₈	—	212	4	no	—
174	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	#11719945	1	2	yes	—
180	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	1	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	1	no	—
185	C₂³	#13133151	1	2	yes	—
186	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#13133151	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₄	#13133176	1	1	yes	characteristic
202	C₂×C₄	#13133191	1	2	yes	—
203	C₂×C₄	#13133196	1	2	yes	—
204	C₂×C₄	#13133224	1	4	yes	—
205	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
207	C₂×C₄	#13133239	1	2	yes	—
208	C₈	—	2	4	no	—
209	C₈	—	4	2	no	—
210	C₈	—	424	4	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	2	no	—
216	C₂²	#16810483	1	1	yes	characteristic
217	C₂²	#16810650	1	2	yes	—
218	C₂²	C₂×#13133151	1	1	yes	characteristic
219	C₂²	#16810661	1	1	yes	characteristic
220	C₂²	#16810672	1	2	yes	—
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	1	no	—
225	C₄	#16810590	1	2	yes	—
226	C₄	#16810684	1	2	yes	—
227	C₄	#16810859	1	2	yes	—
228	C₄	#16810864	1	2	yes	—
229	C₂	—	4	2	no	—
230	C₂	#20273490	1	2	yes	—
231	C₂	#20273495	1	2	yes	—
232	C₂	#20274699	1	1	yes	characteristic
233	C₂	#20274744	1	1	yes	characteristic
234	C₂	#20274757	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#8599
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. C₂×#16810376	C₂	semidirect
3. #20272550	C₂	semidirect
132. C₅₃	#8599	semidirect

Maximal subgroups

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

subgroup	index
#20274695	2
C₂×#16810376	2
#20272550	2
C₂×C₅₃×#431	2
#8599	53