Subgroups — C₅₃×#7176 #27214688

Order 13568.

246 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₅₃	—	2	1	no	—
5	C₂×#108×C₅₃	C₄	1	2	yes	—
6	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
7	C₄×M₁₆×C₅₃	—	2	1	no	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#346	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₃	—	2	1	no	—
11	C₅₃×#341	—	2	1	no	—
12	C₅₃×#335	—	2	1	no	—
13	C₅₃×#335	C₄	1	2	yes	—
14	C₂×M₁₆×C₅₃	—	2	2	no	—
15	C₂²×C₈×C₅₃	—	4	1	no	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
19	M₃₂×C₅₃	—	2	4	no	—
20	C₂×C₁₆×C₅₃	—	2	2	no	—
21	#108×C₅₃	—	2	2	no	—
22	#108×C₅₃	C₈	1	4	yes	complemented
23	C₄⋊C₈×C₅₃	—	2	2	no	—
24	C₄⋊C₈×C₅₃	—	4	1	no	—
25	#97×C₅₃	—	4	1	no	—
26	C₄×C₈×C₅₃	—	4	1	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₅₃	—	2	4	no	—
35	M₁₆×C₅₃	—	4	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	8	2	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	D₈	1	1	yes	characteristic
49	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₁₆×C₅₃	—	2	4	no	—
51	C₂³×C₅₃	#103	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#104	1	2	yes	—
61	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	1	no	—
65	C₈×C₅₃	—	4	4	no	—
66	C₈×C₅₃	—	8	2	no	—
67	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
68	C₈×C₅₃	#97	1	1	yes	characteristic
69	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
70	#7176	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₃	—	2	1	no	—
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
74	C₂²×C₅₃	#340	1	1	yes	characteristic
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
79	C₄×C₅₃	#336	1	1	yes	characteristic
80	C₄×C₅₃	#339	1	1	yes	characteristic
81	C₄×C₅₃	#361	1	1	yes	characteristic
82	C₄×C₅₃	#362	1	1	yes	characteristic
83	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic
84	C₄×#108	C₂×C₅₃	1	1	yes	characteristic
85	C₂×#335	C₂×C₅₃	1	1	yes	characteristic
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	#1279	1	1	yes	characteristic
88	C₂×C₅₃	#1387	1	1	yes	characteristic
89	C₂×C₅₃	#1388	1	1	yes	characteristic
90	C₂×M₃₂	—	2	1	no	—
91	C₂×#108	C₄×C₅₃	1	2	yes	—
92	C₂×C₄⋊C₈	—	2	1	no	—
93	C₄×M₁₆	—	2	1	no	—
94	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
95	#346	C₄×C₅₃	1	2	yes	—
96	C₄×C₁₆	—	2	1	no	—
97	#341	—	2	1	no	—
98	#335	—	2	1	no	—
99	#335	C₄×C₅₃	1	2	yes	—
100	C₅₃	#7176	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
104	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
105	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
106	M₃₂	—	2	4	no	—
107	C₂×C₁₆	—	2	2	no	—
108	#108	—	2	2	no	—
109	#108	C₈×C₅₃	1	4	yes	complemented
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	#97	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
115	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
116	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₅₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
119	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
120	C₂²×C₄	Q₁₆×C₅₃	1	1	yes	characteristic
121	M₁₆	—	2	4	no	—
122	M₁₆	—	4	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	8	2	no	—
127	C₂×C₈	C₄²×C₅₃	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₈×C₅₃	1	2	yes	—
132	C₂×C₈	M₁₆×C₅₃	1	2	yes	—
133	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
134	C₄²	C₄⋊C₄×C₅₃	1	1	yes	characteristic
135	C₄²	D₈×C₅₃	1	1	yes	characteristic
136	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₂³	#103×C₅₃	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	#95×C₅₃	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
144	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
145	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
146	C₂×C₄	#103×C₅₃	1	1	yes	characteristic
147	C₂×C₄	#104×C₅₃	1	2	yes	—
148	C₂×C₄	#106×C₅₃	1	1	yes	characteristic
149	C₂×C₄	#107×C₅₃	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	4	4	no	—
153	C₈	—	8	2	no	—
154	C₈	C₄×C₈×C₅₃	1	1	yes	characteristic
155	C₈	#97×C₅₃	1	1	yes	characteristic
156	C₈	C₄⋊C₈×C₅₃	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	C₅₃×Q₈⋊C₈	1	2	yes	—
160	C₂²	C₅₃×#340	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₅₃×D₄⋊C₈	1	2	yes	—
165	C₄	C₅₃×#336	1	1	yes	characteristic
166	C₄	C₅₃×#339	1	1	yes	characteristic
167	C₄	C₅₃×#361	1	1	yes	characteristic
168	C₄	C₅₃×#362	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	C₅₃×#1279	1	1	yes	characteristic
171	C₂	C₅₃×#1387	1	1	yes	characteristic
172	C₂	C₅₃×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	#102
commutator	C₈	C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅₃
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#103
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
22. #108×C₅₃	C₈	semidirect
70. #7176	C₅₃	direct (N × H)
100. C₅₃	#7176	direct (N × H)
109. #108	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₄×#108×C₅₃	2
C₂×C₅₃×#335	2
#7176	53

Subgroups — C53×#7176 #27214688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7176 #27214688