Subgroups — C₅₃×#22538 #27230050

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#557	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₅₃	C₂	1	2	yes	maximal
3	C₅₃×#1966	C₂	1	6	yes	maximal
4	C₅₃×#1801	C₂	1	3	yes	maximal
5	C₄×#101×C₅₃	C₂	1	3	yes	maximal
6	C₅₃×#557	—	2	2	no	—
7	C₅₃×#557	—	2	6	no	—
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
9	C₂×#125×C₅₃	C₄	1	6	yes	—
10	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	6	no	—
14	C₈×Q₈×C₅₃	—	2	2	no	—
15	C₅₃×#423	C₂²	1	6	yes	—
16	C₂×#101×C₅₃	C₂²	1	6	yes	—
17	C₂×#101×C₅₃	C₂²	1	6	yes	—
18	C₄×M₁₆×C₅₃	C₂²	1	6	yes	—
19	C₅₃×#343	C₂²	1	1	yes	characteristic
20	C₅₃×#343	C₂²	1	3	yes	—
21	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
23	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
24	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
25	C₄⋊Q₈×C₅₃	—	2	6	no	—
26	C₄⋊Q₈×C₅₃	—	2	6	no	—
27	#125×C₅₃	—	2	12	no	—
28	#125×C₅₃	—	2	12	no	—
29	C₄×Q₈×C₅₃	—	2	6	no	—
30	C₄×Q₈×C₅₃	—	2	6	no	—
31	C₄×Q₈×C₅₃	—	2	6	no	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	D₄	1	2	yes	—
34	C₄×Q₈×C₅₃	D₄	1	6	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
42	C₄⋊C₈×C₅₃	—	2	6	no	—
43	C₄⋊C₈×C₅₃	—	2	6	no	—
44	#101×C₅₃	—	2	12	no	—
45	#101×C₅₃	—	4	6	no	—
46	#97×C₅₃	—	2	6	no	—
47	C₄×C₈×C₅₃	—	2	6	no	—
48	C₂×Q₈×C₅₃	—	2	4	no	—
49	C₂×Q₈×C₅₃	—	2	6	no	—
50	C₂×Q₈×C₅₃	—	2	6	no	—
51	C₂×Q₈×C₅₃	—	2	3	no	—
52	C₂×Q₈×C₅₃	—	2	3	no	—
53	C₂×Q₈×C₅₃	C₂²⋊C₄	1	3	yes	—
54	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₅₃	—	2	2	no	—
61	M₁₆×C₅₃	—	2	6	no	—
62	M₁₆×C₅₃	—	2	6	no	—
63	M₁₆×C₅₃	#41	1	4	yes	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	6	no	—
66	C₄⋊C₄×C₅₃	—	2	12	no	—
67	C₄⋊C₄×C₅₃	—	2	6	no	—
68	C₄⋊C₄×C₅₃	—	2	6	no	—
69	C₄⋊C₄×C₅₃	—	2	12	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
72	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
73	C₄²×C₅₃	—	2	6	no	—
74	C₄²×C₅₃	—	2	6	no	—
75	C₄²×C₅₃	C₂×D₄	1	6	yes	—
76	C₄²×C₅₃	C₂×D₄	1	6	yes	—
77	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	Q₈×C₅₃	—	2	6	no	—
79	Q₈×C₅₃	—	4	6	no	—
80	Q₈×C₅₃	—	4	6	no	—
81	Q₈×C₅₃	#101	1	4	yes	complemented
82	C₂×C₄×C₅₃	—	2	6	no	—
83	C₂×C₄×C₅₃	—	2	6	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	6	no	—
86	C₂×C₄×C₅₃	—	2	6	no	—
87	C₂×C₄×C₅₃	—	4	3	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
94	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	—	4	6	no	—
98	#22538	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
103	C₂²×C₅₃	#520	1	1	yes	characteristic
104	C₄×C₅₃	—	2	3	no	—
105	C₄×C₅₃	—	2	6	no	—
106	C₄×C₅₃	—	4	6	no	—
107	C₄×C₅₃	—	4	2	no	—
108	C₄×C₅₃	C₂×#101	1	6	yes	—
109	C₄×C₅₃	#548	1	2	yes	—
110	C₄×C₅₃	#550	1	2	yes	—
111	C₂×#557	C₂×C₅₃	1	1	yes	characteristic
112	Q₈×M₁₆	C₂×C₅₃	1	2	yes	—
113	#1966	C₂×C₅₃	1	6	yes	—
114	#1801	C₂×C₅₃	1	3	yes	—
115	C₄×#101	C₂×C₅₃	1	3	yes	—
116	C₂×C₅₃	—	2	1	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	#2341	1	1	yes	characteristic
119	C₂×C₅₃	C₂²×#101	1	1	yes	characteristic
120	C₂×C₅₃	#2889	1	1	yes	characteristic
121	#557	—	2	2	no	—
122	#557	—	2	6	no	—
123	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
124	C₂×#125	C₄×C₅₃	1	6	yes	—
125	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
127	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
128	#446	—	2	6	no	—
129	C₈×Q₈	—	2	2	no	—
130	#423	C₂²×C₅₃	1	6	yes	—
131	C₂×#101	C₂²×C₅₃	1	6	yes	—
132	C₂×#101	C₂²×C₅₃	1	6	yes	—
133	C₄×M₁₆	C₂²×C₅₃	1	6	yes	—
134	#343	C₂²×C₅₃	1	3	yes	—
135	#343	C₂²×C₅₃	1	1	yes	characteristic
136	C₅₃	#22538	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
138	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
139	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
140	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
141	C₄⋊Q₈	—	2	6	no	—
142	C₄⋊Q₈	—	2	6	no	—
143	#125	—	2	12	no	—
144	#125	—	2	12	no	—
145	C₄×Q₈	—	2	6	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	6	no	—
148	C₄×Q₈	—	2	6	no	—
149	C₄×Q₈	D₄×C₅₃	1	2	yes	—
150	C₄×Q₈	D₄×C₅₃	1	6	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	6	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	6	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	6	yes	—
154	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
155	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₅₃	1	6	yes	—
157	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
158	C₄⋊C₈	—	2	6	no	—
159	C₄⋊C₈	—	2	6	no	—
160	#101	—	2	12	no	—
161	#101	—	4	6	no	—
162	#97	—	2	6	no	—
163	C₄×C₈	—	2	6	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	6	no	—
166	C₂×Q₈	—	2	6	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	2	3	no	—
169	C₂×Q₈	C₂²⋊C₄×C₅₃	1	3	yes	—
170	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₅₃	1	6	yes	—
172	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
173	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
175	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	6	no	—
178	M₁₆	—	2	6	no	—
179	M₁₆	#41×C₅₃	1	4	yes	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	6	no	—
182	C₄⋊C₄	—	2	12	no	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₄⋊C₄	—	2	12	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
188	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
189	C₄²	—	2	6	no	—
190	C₄²	—	2	6	no	—
191	C₄²	C₂×D₄×C₅₃	1	6	yes	—
192	C₄²	C₂×D₄×C₅₃	1	6	yes	—
193	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
194	Q₈	—	2	6	no	—
195	Q₈	—	4	6	no	—
196	Q₈	—	4	6	no	—
197	Q₈	#101×C₅₃	1	4	yes	complemented
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	4	3	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	6	yes	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
208	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
209	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
210	C₂×C₄	H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
211	C₂×C₄	2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	—	4	6	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
217	C₂²	C₄×#41×C₅₃	1	1	yes	characteristic
218	C₂²	C₅₃×#520	1	1	yes	characteristic
219	C₄	—	2	3	no	—
220	C₄	—	2	6	no	—
221	C₄	—	4	6	no	—
222	C₄	—	4	2	no	—
223	C₄	C₂×#101×C₅₃	1	6	yes	—
224	C₄	C₅₃×#548	1	2	yes	—
225	C₄	C₅₃×#550	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	C₅₃×#2341	1	1	yes	characteristic
229	C₂	C₂²×#101×C₅₃	1	1	yes	characteristic
230	C₂	C₅₃×#2889	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
81. Q₈×C₅₃	#101	semidirect
98. #22538	C₅₃	direct (N × H)
136. C₅₃	#22538	direct (N × H)
197. Q₈	#101×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#557	2
Q₈×M₁₆×C₅₃	2
C₅₃×#1966	2
C₅₃×#1801	2
C₄×#101×C₅₃	2
#22538	53

Subgroups — C53×#22538 #27230050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#22538 #27230050