Subgroups — #27300294

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810449	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274779	C₂	1	2	yes	maximal
4	#20272824	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810624	—	2	2	no	—
9	#16810453	—	2	2	no	—
10	#16810449	—	2	1	no	—
11	#16810449	—	2	1	no	—
12	#16810391	—	2	2	no	—
13	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#504	—	2	2	no	—
15	C₅₃×#504	C₄	1	4	yes	complemented
16	C₂×M₃₂×C₅₃	—	2	1	no	—
17	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	—	2	1	no	—
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	#13133151	—	2	2	no	—
26	#13133141	—	4	2	no	—
27	#13133138	—	4	2	no	—
28	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
29	#131×C₅₃	—	2	1	no	—
30	#131×C₅₃	—	2	1	no	—
31	#131×C₅₃	C₂×C₄	1	2	yes	—
32	#131×C₅₃	Q₈	1	2	yes	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	M₃₂×C₅₃	—	2	1	no	—
38	M₃₂×C₅₃	—	2	4	no	—
39	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₁₆×C₅₃	—	2	1	no	—
41	C₂×C₁₆×C₅₃	—	2	4	no	—
42	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
46	#41×C₅₃	—	2	1	no	—
47	#41×C₅₃	—	2	1	no	—
48	#41×C₅₃	C₄⋊C₄	1	2	yes	—
49	#41×C₅₃	Q₁₆	1	2	yes	—
50	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
51	C₂×D₄×C₅₃	—	2	1	no	—
52	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	4	2	no	—
56	M₁₆×C₅₃	—	2	1	no	—
57	M₁₆×C₅₃	—	2	2	no	—
58	M₁₆×C₅₃	—	2	2	no	—
59	M₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	#11719917	—	4	2	no	—
72	#11719917	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₁₆×C₅₃	—	2	2	no	—
75	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	#135	1	1	yes	characteristic
79	Q₈×C₅₃	—	2	1	no	—
80	Q₈×C₅₃	#107	1	2	yes	—
81	D₄×C₅₃	—	2	2	no	—
82	D₄×C₅₃	—	2	2	no	—
83	D₄×C₅₃	—	2	1	no	—
84	D₄×C₅₃	#107	1	2	yes	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	16	2	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₈×C₅₃	#117	1	1	yes	characteristic
99	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
101	#88322	—	8	2	no	—
102	#12842	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	4	1	no	—
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	#427	1	1	yes	characteristic
109	C₂²×C₅₃	#466	1	1	yes	characteristic
110	C₂²×C₅₃	#467	1	1	yes	characteristic
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	#380	1	1	yes	characteristic
113	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
114	C₄×C₅₃	#466	1	1	yes	characteristic
115	C₄×C₅₃	#467	1	1	yes	characteristic
116	Dic₅₃	—	16	2	no	—
117	C₂×#504	D₅₃	1	1	yes	characteristic
118	C₂×#368	—	53	1	no	—
119	C₂×#366	—	53	1	no	—
120	#1816	—	53	2	no	—
121	#1387	—	53	2	no	—
122	C₂×C₅₃	—	2	1	no	—
123	C₂×C₅₃	—	2	1	no	—
124	C₂×C₅₃	—	4	2	no	—
125	C₂×C₅₃	#1808	1	1	yes	characteristic
126	C₂×C₅₃	#2221	1	1	yes	characteristic
127	C₂×C₅₃	#2222	1	1	yes	characteristic
128	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
129	#504	—	2	2	no	—
130	#504	Dic₅₃	1	4	yes	complemented
131	C₂×M₃₂	—	2	1	no	—
132	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
133	C₂²×C₁₆	—	2	1	no	—
134	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
135	C₂×#108	—	53	2	no	—
136	C₂×#107	—	53	2	no	—
137	#417	—	106	2	no	—
138	#368	—	106	2	no	—
139	#366	—	106	1	no	—
140	#366	—	106	1	no	—
141	#343	—	106	2	no	—
142	C₅₃	#12842	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	D₂₁₂	1	1	yes	characteristic
144	#131	—	2	1	no	—
145	#131	—	2	1	no	—
146	#131	Dic₁₀₆	1	2	yes	—
147	#131	C₂×Dic₅₃	1	2	yes	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	106	2	no	—
150	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	106	2	no	—
154	M₃₂	—	2	1	no	—
155	M₃₂	—	2	4	no	—
156	M₃₂	C₂×Dic₅₃	1	2	yes	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
163	#108	—	106	2	no	—
164	#108	—	106	2	no	—
165	#107	—	106	2	no	—
166	#107	—	106	2	no	—
167	#103	—	212	2	no	—
168	#102	—	212	2	no	—
169	#41	—	2	1	no	—
170	#41	—	2	1	no	—
171	#41	Dic₂₁₂	1	2	yes	—
172	#41	C₄⋊Dic₅₃	1	2	yes	—
173	C₂×Q₈	D₄₂₄	1	1	yes	characteristic
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	D₄₂₄	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	212	2	no	—
178	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
179	M₁₆	—	2	1	no	—
180	M₁₆	—	2	2	no	—
181	M₁₆	—	2	2	no	—
182	M₁₆	—	212	2	no	—
183	M₁₆	—	212	2	no	—
184	M₁₆	C₄⋊Dic₅₃	1	2	yes	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	212	2	no	—
189	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
190	C₂×C₈	#11719945	1	1	yes	characteristic
191	C₂×C₈	#11719945	1	1	yes	characteristic
192	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₁₆	—	2	2	no	—
198	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
199	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
200	C₂³	—	2	1	no	—
201	C₂³	#13133239	1	1	yes	characteristic
202	Q₈	—	2	1	no	—
203	Q₈	#13133151	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	2	2	no	—
206	D₄	—	2	1	no	—
207	D₄	#13133151	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	848	2	no	—
212	C₂×C₄	—	424	2	no	—
213	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	#13133191	1	1	yes	characteristic
215	C₂×C₄	#13133191	1	1	yes	characteristic
216	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
217	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
218	C₂×C₄	#13133239	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	424	2	no	—
221	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₈	#13133176	1	1	yes	characteristic
223	C₈	#13133196	1	1	yes	characteristic
224	C₈	#13133196	1	1	yes	characteristic
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#16810661	1	1	yes	characteristic
231	C₂²	#16810859	1	1	yes	characteristic
232	C₂²	#16810864	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	848	2	no	—
235	C₄	#16810483	1	1	yes	characteristic
236	C₄	C₂×#13133151	1	1	yes	characteristic
237	C₄	#16810859	1	1	yes	characteristic
238	C₄	#16810864	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	1	no	—
241	C₂	—	4	2	no	—
242	C₂	#20274744	1	1	yes	characteristic
243	C₂	#20276835	1	1	yes	characteristic
244	C₂	#20276840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810859
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂²×C₅₃	#466
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#12842
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16810453	C₂	semidirect
2. C₂×#16810449	C₂	semidirect
4. #20272824	C₂	semidirect
15. C₅₃×#504	C₄	semidirect
130. #504	Dic₅₃	semidirect
142. C₅₃	#12842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
C₂×#16810449	2
#20274779	2
#20272824	2
C₂×C₅₃×#504	2
#12842	53