Subgroups — C₅₃×#7638 #27215150

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1745	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1271	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
7	C₂×#97×C₅₃	—	2	1	no	—
8	C₂×#97×C₅₃	—	2	1	no	—
9	C₅₃×#391	—	2	1	no	—
10	C₅₃×#389	—	2	1	no	—
11	C₅₃×#389	C₂²	1	2	yes	—
12	C₅₃×#384	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
14	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₅₃×#336	—	2	1	no	—
18	C₅₃×#336	C₂²	1	2	yes	—
19	C₅₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	2	4	no	—
24	C₄×Q₈×C₅₃	—	4	4	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#97×C₅₃	—	2	4	no	—
40	#97×C₅₃	—	2	4	no	—
41	#95×C₅₃	—	2	2	no	—
42	#95×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	2	4	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	4	4	no	—
67	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	—	4	4	no	—
76	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₅₃	—	4	8	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	#101	1	2	yes	—
100	C₂×C₄×C₅₃	#104	1	2	yes	—
101	C₂×C₄×C₅₃	#104	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×C₄×C₅₃	#137	1	2	yes	—
105	C₈×C₅₃	—	4	4	no	—
106	C₈×C₅₃	—	4	4	no	—
107	#7638	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₅₃	#407	1	1	yes	characteristic
109	C₂²×C₅₃	#410	1	1	yes	characteristic
110	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
111	C₂²×C₅₃	#418	1	1	yes	characteristic
112	C₂²×C₅₃	#419	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	4	no	—
122	C₄×C₅₃	—	4	1	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
125	#1745	C₂×C₅₃	1	1	yes	characteristic
126	#1745	C₂×C₅₃	1	1	yes	characteristic
127	#1271	C₂×C₅₃	1	4	yes	—
128	C₂×C₅₃	#1467	1	1	yes	characteristic
129	C₂×C₅₃	#1490	1	1	yes	characteristic
130	C₂×C₅₃	#1491	1	1	yes	characteristic
131	C₂×C₅₃	#1505	1	1	yes	characteristic
132	C₂×C₅₃	#1506	1	1	yes	characteristic
133	C₂×C₅₃	#1526	1	1	yes	characteristic
134	C₂×C₅₃	#1556	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
136	C₂×C₄×Q₈	—	2	1	no	—
137	C₂×#97	—	2	1	no	—
138	C₂×#97	—	2	1	no	—
139	#391	—	2	1	no	—
140	#389	—	2	1	no	—
141	#389	C₂²×C₅₃	1	2	yes	—
142	#384	C₄×C₅₃	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	2	1	no	—
144	C₄×C₄⋊C₄	C₄×C₅₃	1	2	yes	—
145	C₄³	C₂²×C₅₃	1	1	yes	characteristic
146	#336	—	2	1	no	—
147	#336	—	2	1	no	—
148	#336	C₂²×C₅₃	1	2	yes	—
149	#336	C₂²×C₅₃	1	2	yes	—
150	C₅₃	#7638	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	4	no	—
155	C₄×Q₈	—	4	4	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
170	#97	—	2	4	no	—
171	#97	—	2	4	no	—
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	M₁₆×C₅₃	1	4	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	4	no	—
207	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
208	Q₈	—	4	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	#101×C₅₃	1	2	yes	—
231	C₂×C₄	#104×C₅₃	1	2	yes	—
232	C₂×C₄	#104×C₅₃	1	2	yes	—
233	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
234	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
235	C₂×C₄	#137×C₅₃	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	4	4	no	—
238	C₂²	C₅₃×#407	1	1	yes	characteristic
239	C₂²	C₅₃×#410	1	1	yes	characteristic
240	C₂²	C₂×#101×C₅₃	1	1	yes	characteristic
241	C₂²	C₅₃×#418	1	1	yes	characteristic
242	C₂²	C₅₃×#419	1	1	yes	characteristic
243	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
244	C₂²	C₂×#104×C₅₃	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	1	no	—
253	C₄	—	8	2	no	—
254	C₂	C₅₃×#1467	1	1	yes	characteristic
255	C₂	C₅₃×#1490	1	1	yes	characteristic
256	C₂	C₅₃×#1491	1	1	yes	characteristic
257	C₂	C₅₃×#1505	1	1	yes	characteristic
258	C₂	C₅₃×#1506	1	1	yes	characteristic
259	C₂	C₅₃×#1526	1	1	yes	characteristic
260	C₂	C₅₃×#1556	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
107. #7638	C₅₃	direct (N × H)
150. C₅₃	#7638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₅₃	2
C₅₃×#1745	2
C₅₃×#1271	2
#7638	53

Subgroups — C53×#7638 #27215150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7638 #27215150