Subgroups — C₅₃×#9798 #27217310

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1997	C₂	1	2	yes	maximal
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1548	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	C₄	1	4	yes	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
8	C₅₃×#494	—	2	1	no	—
9	C₅₃×#494	—	2	1	no	—
10	C₅₃×#494	—	2	2	no	—
11	C₅₃×#431	C₂²	1	1	yes	characteristic
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#343	—	2	4	no	—
16	C₅₃×#335	—	2	2	no	—
17	C₅₃×#333	—	2	4	no	—
18	C₅₃×#322	—	2	1	no	—
19	C₈²×C₅₃	—	2	1	no	—
20	C₂²×Q₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×Q₁₆×C₅₃	—	2	4	no	—
22	C₂×Q₁₆×C₅₃	—	2	4	no	—
23	C₂×Q₁₆×C₅₃	—	4	8	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
37	C₂²⋊C₈×C₅₃	—	2	1	no	—
38	#97×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	—	2	1	no	—
40	C₄×C₈×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₂×Q₈×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
51	Q₁₆×C₅₃	—	4	8	no	—
52	Q₁₆×C₅₃	—	4	8	no	—
53	M₁₆×C₅₃	—	2	2	no	—
54	M₁₆×C₅₃	—	4	4	no	—
55	M₁₆×C₅₃	Q₁₆	1	4	yes	complemented
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	#41	1	1	yes	characteristic
70	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
71	Q₈×C₅₃	—	4	8	no	—
72	Q₈×C₅₃	—	8	4	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	8	4	no	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₅₃	#135	1	2	yes	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	4	no	—
87	C₈×C₅₃	—	4	2	no	—
88	C₈×C₅₃	—	8	2	no	—
89	C₈×C₅₃	#100	1	4	yes	—
90	#9798	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	#390	1	1	yes	characteristic
94	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	8	4	no	—
98	C₄×C₅₃	C₂×#100	1	2	yes	—
99	C₄×C₅₃	#467	1	2	yes	—
100	C₄×C₅₃	#473	1	2	yes	—
101	C₄×C₅₃	#494	1	1	yes	characteristic
102	C₄×C₅₃	#495	1	1	yes	characteristic
103	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
104	#1997	C₂×C₅₃	1	2	yes	—
105	#1567	C₂×C₅₃	1	1	yes	characteristic
106	#1548	C₂×C₅₃	1	2	yes	—
107	C₈×M₁₆	C₂×C₅₃	1	1	yes	characteristic
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	#1965	1	1	yes	characteristic
110	C₂×C₅₃	#1969	1	1	yes	characteristic
111	C₂×C₅₃	#1979	1	1	yes	characteristic
112	C₂²×Q₁₆	C₄×C₅₃	1	4	yes	—
113	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
114	#494	—	2	1	no	—
115	#494	—	2	1	no	—
116	#494	—	2	2	no	—
117	#431	C₂²×C₅₃	1	1	yes	characteristic
118	#423	C₂²×C₅₃	1	2	yes	—
119	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	#343	—	2	4	no	—
122	#335	—	2	2	no	—
123	#333	—	2	4	no	—
124	#322	—	2	1	no	—
125	C₈²	—	2	1	no	—
126	C₅₃	#9798	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×Q₈	C₂×C₄×C₅₃	1	4	yes	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	—	4	8	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
133	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
144	C₂²⋊C₈	—	2	1	no	—
145	#97	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
149	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
150	C₄×C₈	D₄×C₅₃	1	2	yes	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	4	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
157	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
158	Q₁₆	—	4	8	no	—
159	Q₁₆	—	4	8	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	—	4	4	no	—
162	M₁₆	Q₁₆×C₅₃	1	4	yes	complemented
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
168	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
174	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
176	C₄²	#41×C₅₃	1	1	yes	characteristic
177	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
185	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
186	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
187	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
188	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
189	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
191	C₂×C₄	#135×C₅₃	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	2	no	—
195	C₈	—	8	2	no	—
196	C₈	#100×C₅₃	1	4	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₅₃×#390	1	1	yes	characteristic
200	C₂²	C₄×Q₁₆×C₅₃	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	8	4	no	—
204	C₄	C₂×#100×C₅₃	1	2	yes	—
205	C₄	C₅₃×#467	1	2	yes	—
206	C₄	C₅₃×#473	1	2	yes	—
207	C₄	C₅₃×#494	1	1	yes	characteristic
208	C₄	C₅₃×#495	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	C₅₃×#1965	1	1	yes	characteristic
211	C₂	C₅₃×#1969	1	1	yes	characteristic
212	C₂	C₅₃×#1979	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#390
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
55. M₁₆×C₅₃	Q₁₆	semidirect
90. #9798	C₅₃	direct (N × H)
126. C₅₃	#9798	direct (N × H)
162. M₁₆	Q₁₆×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×#494	2
C₅₃×#1997	2
C₅₃×#1567	2
C₅₃×#1548	2
C₈×M₁₆×C₅₃	2
#9798	53

Subgroups — C53×#9798 #27217310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9798 #27217310