Subgroups — C₂×C₅₃×#3162 #27268169

Order 13568.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#533	C₂	1	2	yes	maximal
2	C₅₃×#3162	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#501	C₂	1	8	yes	maximal, complemented
4	C₂×C₅₃×#427	C₂	1	4	yes	maximal
5	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₅₃×#533	—	2	8	no	—
8	C₅₃×#533	C₂²	1	16	yes	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	4	yes	—
10	C₂×#125×C₅₃	C₂²	1	4	yes	—
11	C₂×C₂²⋊Q₈×C₅₃	—	2	4	no	—
12	C₂×C₄×Q₈×C₅₃	—	2	4	no	—
13	C₂×#117×C₅₃	C₂²	1	4	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#501	C₂²	1	64	yes	complemented
16	C₅₃×#427	C₂²	1	32	yes	—
17	C₂×#107×C₅₃	C₂²	1	8	yes	—
18	C₂×#106×C₅₃	C₂²	1	8	yes	—
19	C₂×#97×C₅₃	C₂²	1	4	yes	—
20	C₂×#97×C₅₃	C₂²	1	8	yes	—
21	C₂²×Q₈×C₅₃	—	2	4	no	—
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	Q₈	1	16	yes	complemented
25	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂³	1	8	yes	—
27	C₄⋊Q₈×C₅₃	—	2	8	no	—
28	C₄⋊Q₈×C₅₃	C₂³	1	16	yes	—
29	#125×C₅₃	—	2	8	no	—
30	#125×C₅₃	C₂³	1	16	yes	—
31	C₂²⋊Q₈×C₅₃	—	2	32	no	—
32	C₄×Q₈×C₅₃	—	2	32	no	—
33	#117×C₅₃	—	2	8	no	—
34	#117×C₅₃	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
38	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
39	C₂×C₄²×C₅₃	—	2	4	no	—
40	C₂×C₄²×C₅₃	D₄	1	4	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
44	#107×C₅₃	C₂³	1	32	yes	—
45	#106×C₅₃	C₂³	1	32	yes	—
46	#97×C₅₃	C₂³	1	16	yes	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	—	2	16	no	—
49	C₂×Q₈×C₅₃	—	4	16	no	—
50	C₂²×C₄×C₅₃	—	2	8	no	—
51	C₂²×C₄×C₅₃	—	4	4	no	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×Q₈	1	16	yes	—
60	C₂×C₈×C₅₃	C₂×Q₈	1	32	yes	complemented
61	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
62	C₄⋊C₄×C₅₃	—	2	32	no	—
63	C₄⋊C₄×C₅₃	—	2	16	no	—
64	C₄⋊C₄×C₅₃	—	2	8	no	—
65	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
66	C₂²⋊C₄×C₅₃	—	2	16	no	—
67	C₄²×C₅₃	—	2	16	no	—
68	C₄²×C₅₃	C₂×D₄	1	8	yes	—
69	C₄²×C₅₃	C₂×D₄	1	4	yes	—
70	C₄²×C₅₃	C₂⁴	1	4	yes	—
71	C₂³×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
74	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	32	no	—
76	C₂×C₄×C₅₃	—	2	8	no	—
77	C₂×C₄×C₅₃	—	2	16	no	—
78	C₂×C₄×C₅₃	—	4	8	no	—
79	C₂×C₄×C₅₃	—	4	16	no	—
80	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
81	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
87	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
88	C₈×C₅₃	C₂²×Q₈	1	16	yes	—
89	C₂×#3162	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₅₃	—	2	8	no	—
91	C₂²×C₅₃	—	2	4	no	—
92	C₂²×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
93	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
94	C₂²×C₅₃	#575	1	4	yes	—
95	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
96	C₄×C₅₃	—	2	8	no	—
97	C₄×C₅₃	—	4	16	no	—
98	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
99	C₄×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
100	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
101	C₂×#533	C₂×C₅₃	1	2	yes	—
102	#3162	C₂×C₅₃	1	16	yes	direct product factor
103	C₂×#501	C₂×C₅₃	1	8	yes	complemented
104	C₂×#427	C₂×C₅₃	1	4	yes	—
105	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
106	C₂×C₅₃	—	2	4	no	—
107	C₂×C₅₃	#3162	1	4	yes	direct product factor
108	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#575	1	2	yes	—
110	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
111	#533	—	2	8	no	—
112	#533	C₂²×C₅₃	1	16	yes	—
113	C₂×C₄⋊Q₈	C₂²×C₅₃	1	4	yes	—
114	C₂×#125	C₂²×C₅₃	1	4	yes	—
115	C₂×C₂²⋊Q₈	—	2	4	no	—
116	C₂×C₄×Q₈	—	2	4	no	—
117	C₂×#117	C₂²×C₅₃	1	4	yes	—
118	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
119	#501	C₂²×C₅₃	1	64	yes	complemented
120	#427	C₂²×C₅₃	1	32	yes	—
121	C₂×#107	C₂²×C₅₃	1	8	yes	—
122	C₂×#106	C₂²×C₅₃	1	8	yes	—
123	C₂×#97	C₂²×C₅₃	1	4	yes	—
124	C₂×#97	C₂²×C₅₃	1	8	yes	—
125	C₅₃	C₂×#3162	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×Q₈	—	2	4	no	—
127	C₂³×C₄	D₄×C₅₃	1	2	yes	—
128	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
129	C₂²×C₈	Q₈×C₅₃	1	16	yes	complemented
130	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
131	C₂²×C₈	C₂³×C₅₃	1	8	yes	—
132	C₄⋊Q₈	—	2	8	no	—
133	C₄⋊Q₈	C₂³×C₅₃	1	16	yes	—
134	#125	—	2	8	no	—
135	#125	C₂³×C₅₃	1	16	yes	—
136	C₂²⋊Q₈	—	2	32	no	—
137	C₄×Q₈	—	2	32	no	—
138	#117	—	2	8	no	—
139	#117	C₂³×C₅₃	1	16	yes	—
140	C₂×C₄⋊C₄	—	2	8	no	—
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
143	C₂×C₂²⋊C₄	—	2	4	no	—
144	C₂×C₄²	—	2	4	no	—
145	C₂×C₄²	D₄×C₅₃	1	2	yes	—
146	C₂×C₄²	D₄×C₅₃	1	4	yes	—
147	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
148	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
149	#107	C₂³×C₅₃	1	32	yes	—
150	#106	C₂³×C₅₃	1	32	yes	—
151	#97	C₂³×C₅₃	1	16	yes	—
152	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂×Q₈	—	2	16	no	—
154	C₂×Q₈	—	4	16	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
158	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
159	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
160	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
161	C₂²×C₄	C₂×Q₈×C₅₃	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
164	C₂×C₈	C₂×Q₈×C₅₃	1	16	yes	—
165	C₂×C₈	C₂×Q₈×C₅₃	1	32	yes	complemented
166	C₂×C₈	C₂⁴×C₅₃	1	8	yes	—
167	C₄⋊C₄	—	2	32	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₄⋊C₄	—	2	8	no	—
170	C₄⋊C₄	C₂⁴×C₅₃	1	16	yes	—
171	C₂²⋊C₄	—	2	16	no	—
172	C₄²	—	2	16	no	—
173	C₄²	C₂×D₄×C₅₃	1	8	yes	—
174	C₄²	C₂×D₄×C₅₃	1	4	yes	—
175	C₄²	C₂⁴×C₅₃	1	4	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄⋊Q₈×C₅₃	1	4	yes	—
178	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
179	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
180	Q₈	—	4	32	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	—	4	16	no	—
185	C₂×C₄	C₄⋊Q₈×C₅₃	1	8	yes	—
186	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
187	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
188	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
189	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
191	C₂×C₄	C₂²×Q₈×C₅₃	1	8	yes	—
192	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
193	C₈	C₂²×Q₈×C₅₃	1	16	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×C₄⋊Q₈×C₅₃	1	4	yes	—
197	C₂²	C₂×C₄⋊Q₈×C₅₃	1	2	yes	—
198	C₂²	C₅₃×#575	1	4	yes	—
199	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	4	16	no	—
202	C₄	C₂×C₄⋊Q₈×C₅₃	1	4	yes	—
203	C₄	C₂×C₄⋊Q₈×C₅₃	1	2	yes	—
204	C₄	C₂³×Q₈×C₅₃	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₅₃×#3162	1	4	yes	direct product factor
207	C₂	C₂²×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
208	C₂	C₂×C₅₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#3162	C₂	direct (N × H)
3. C₂×C₅₃×#501	C₂	semidirect
15. C₅₃×#501	C₂²	semidirect
24. C₂²×C₈×C₅₃	Q₈	semidirect
60. C₂×C₈×C₅₃	C₂×Q₈	semidirect
89. C₂×#3162	C₅₃	direct (N × H)
102. #3162	C₂×C₅₃	direct (N × H)
103. C₂×#501	C₂×C₅₃	semidirect
107. C₂×C₅₃	#3162	direct (N × H)
119. #501	C₂²×C₅₃	semidirect
125. C₅₃	C₂×#3162	direct (N × H)
129. C₂²×C₈	Q₈×C₅₃	semidirect
165. C₂×C₈	C₂×Q₈×C₅₃	semidirect
206. C₂	C₅₃×#3162	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#533	2
C₅₃×#3162	2
C₂×C₅₃×#501	2
C₂×C₅₃×#427	2
C₂²×#97×C₅₃	2
C₂×#3162	53

Subgroups — C2×C53×#3162 #27268169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#3162 #27268169