Subgroups — C₅₃×#32693 #27240205

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2546	C₂	1	2	yes	maximal
2	C₅₃×#2087	C₂	1	2	yes	maximal
3	C₅₃×#2057	C₂	1	4	yes	maximal
4	C₅₃×#1985	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1944	C₂	1	2	yes	maximal
6	C₅₃×#1919	C₂	1	4	yes	maximal
7	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#107×C₅₃	C₂²	1	2	yes	—
11	C₂×#107×C₅₃	C₂²	1	2	yes	—
12	C₂×#106×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#398	—	2	2	no	—
16	C₅₃×#398	—	2	2	no	—
17	C₅₃×#391	—	2	2	no	—
18	C₅₃×#389	C₂²	1	4	yes	—
19	C₅₃×#389	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#340	C₂²	1	2	yes	—
22	C₅₃×#340	C₂²	1	4	yes	—
23	C₅₃×#340	C₂²	1	4	yes	—
24	C₅₃×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₅₃	—	2	2	no	—
26	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	4	8	no	—
30	C₄⋊Q₈×C₅₃	—	4	2	no	—
31	C₄⋊Q₈×C₅₃	—	4	2	no	—
32	C₄×Q₈×C₅₃	—	4	1	no	—
33	C₄×Q₈×C₅₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#107×C₅₃	—	4	2	no	—
45	#107×C₅₃	—	4	2	no	—
46	#106×C₅₃	Q₈	1	8	yes	complemented
47	C₄⋊C₈×C₅₃	—	4	1	no	—
48	#97×C₅₃	—	4	1	no	—
49	#95×C₅₃	—	2	2	no	—
50	#95×C₅₃	—	2	4	no	—
51	#95×C₅₃	—	2	4	no	—
52	C₂×Q₈×C₅₃	—	2	4	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	4	no	—
55	C₂×Q₈×C₅₃	—	4	4	no	—
56	C₂×Q₈×C₅₃	—	4	4	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	—	2	4	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	2	4	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	4	2	no	—
73	C₄⋊C₄×C₅₃	—	4	1	no	—
74	C₄⋊C₄×C₅₃	—	4	4	no	—
75	C₄⋊C₄×C₅₃	—	4	8	no	—
76	C₄⋊C₄×C₅₃	—	4	8	no	—
77	C₄⋊C₄×C₅₃	C₂×Q₈	1	8	yes	—
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	2	2	no	—
82	C₄²×C₅₃	—	4	4	no	—
83	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₅₃	—	4	4	no	—
85	Q₈×C₅₃	—	8	8	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	8	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
101	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
105	C₈×C₅₃	—	2	4	no	—
106	C₈×C₅₃	—	4	2	no	—
107	#32693	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	#534	1	1	yes	characteristic
109	C₂²×C₅₃	D₄×Q₈	1	2	yes	—
110	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
111	C₂²×C₅₃	#578	1	2	yes	—
112	C₄×C₅₃	—	2	4	no	—
113	C₄×C₅₃	—	4	8	no	—
114	C₄×C₅₃	—	4	1	no	—
115	C₄×C₅₃	—	4	1	no	—
116	C₄×C₅₃	—	4	2	no	—
117	C₄×C₅₃	—	8	4	no	—
118	C₄×C₅₃	#544	1	2	yes	—
119	C₄×C₅₃	Q₈²	1	1	yes	characteristic
120	C₄×C₅₃	#563	1	1	yes	characteristic
121	#2546	C₂×C₅₃	1	2	yes	—
122	#2087	C₂×C₅₃	1	2	yes	—
123	#2057	C₂×C₅₃	1	4	yes	—
124	#1985	C₂×C₅₃	1	1	yes	characteristic
125	#1944	C₂×C₅₃	1	2	yes	—
126	#1919	C₂×C₅₃	1	4	yes	—
127	C₂×C₅₃	#2734	1	1	yes	characteristic
128	C₂×C₅₃	#3192	1	1	yes	characteristic
129	C₂×C₅₃	#3202	1	1	yes	characteristic
130	C₂×C₅₃	#3386	1	4	yes	—
131	C₂×C₄⋊Q₈	—	2	2	no	—
132	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
133	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
134	C₂×#107	C₂²×C₅₃	1	2	yes	—
135	C₂×#107	C₂²×C₅₃	1	2	yes	—
136	C₂×#106	C₂²×C₅₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
138	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
139	#398	—	2	2	no	—
140	#398	—	2	2	no	—
141	#391	—	2	2	no	—
142	#389	C₂²×C₅₃	1	4	yes	—
143	#389	C₂²×C₅₃	1	4	yes	—
144	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
145	#340	C₂²×C₅₃	1	4	yes	—
146	#340	C₂²×C₅₃	1	4	yes	—
147	#340	C₂²×C₅₃	1	2	yes	—
148	#336	C₂²×C₅₃	1	4	yes	—
149	C₅₃	#32693	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
153	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₄⋊Q₈	—	4	8	no	—
157	C₄×Q₈	—	4	1	no	—
158	C₄×Q₈	—	4	1	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
163	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
165	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
168	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	#107	—	4	2	no	—
170	#107	—	4	2	no	—
171	#106	Q₈×C₅₃	1	8	yes	complemented
172	C₄⋊C₈	—	4	1	no	—
173	#97	—	4	1	no	—
174	#95	—	2	2	no	—
175	#95	—	2	4	no	—
176	#95	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	4	4	no	—
181	C₂×Q₈	—	4	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	8	no	—
201	C₄⋊C₄	—	4	8	no	—
202	C₄⋊C₄	C₂×Q₈×C₅₃	1	8	yes	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	4	no	—
208	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
209	Q₈	—	4	4	no	—
210	Q₈	—	8	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
226	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
227	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
228	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
229	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	4	2	no	—
232	C₂²	C₅₃×#534	1	1	yes	characteristic
233	C₂²	D₄×Q₈×C₅₃	1	2	yes	—
234	C₂²	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
235	C₂²	C₅₃×#578	1	2	yes	—
236	C₄	—	2	4	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	8	no	—
241	C₄	—	8	4	no	—
242	C₄	C₅₃×#544	1	2	yes	—
243	C₄	Q₈²×C₅₃	1	1	yes	characteristic
244	C₄	C₅₃×#563	1	1	yes	characteristic
245	C₂	C₅₃×#2734	1	1	yes	characteristic
246	C₂	C₅₃×#3192	1	1	yes	characteristic
247	C₂	C₅₃×#3202	1	1	yes	characteristic
248	C₂	C₅₃×#3386	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
46. #106×C₅₃	Q₈	semidirect
107. #32693	C₅₃	direct (N × H)
149. C₅₃	#32693	direct (N × H)
171. #106	Q₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2546	2
C₅₃×#2087	2
C₅₃×#2057	2
C₅₃×#1985	2
C₅₃×#1944	2
C₅₃×#1919	2
#32693	53

Subgroups — C53×#32693 #27240205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#32693 #27240205