Subgroups — C₂³×#13133207 #28290758

Order 13568.

8414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133207	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic
4	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic
5	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic
6	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×#13133207	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₅₃	C₂²	1	28	yes	—
15	C₂²×#11719919	C₂²	1	28	yes	—
16	C₂²×#11719919	C₂²	1	28	yes	—
17	C₂²×#11719919	C₂²	1	28	yes	—
18	C₂²×#11719919	C₂²	1	28	yes	—
19	C₂²×C₄×Dic₅₃	C₂²	1	28	yes	—
20	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	—
24	C₂³×Dic₅₃	—	2	8	no	—
25	C₂³×Dic₅₃	—	2	8	no	—
26	C₂³×Dic₅₃	—	2	8	no	—
27	C₂³×Dic₅₃	Q₈	1	16	yes	—
28	C₂³×Dic₅₃	C₂³	1	14	yes	—
29	C₂³×Dic₅₃	C₂³	1	14	yes	—
30	C₂³×Dic₅₃	C₂³	1	14	yes	—
31	C₂³×Dic₅₃	C₂³	1	14	yes	—
32	#13133207	C₂³	1	512	yes	direct product factor
33	C₂×C₄⋊Dic₅₃	C₂³	1	112	yes	—
34	C₂×#11719919	C₂³	1	112	yes	—
35	C₂×#11719919	C₂³	1	112	yes	—
36	C₂×#11719919	C₂³	1	112	yes	—
37	C₂×#11719919	C₂³	1	112	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	112	yes	—
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	—	2	8	no	—
41	C₂³×C₄×C₅₃	—	2	8	no	—
42	C₂³×C₄×C₅₃	—	2	8	no	—
43	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	—
47	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
48	C₂⁴×C₅₃	#41	1	8	yes	—
49	C₂⁴×C₅₃	#41	1	8	yes	—
50	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
51	C₂²×C₄×C₅₃	—	2	56	no	—
52	C₂²×C₄×C₅₃	—	2	56	no	—
53	C₂²×C₄×C₅₃	—	2	56	no	—
54	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
57	C₂²×Dic₅₃	—	2	56	no	—
58	C₂²×Dic₅₃	—	2	56	no	—
59	C₂²×Dic₅₃	—	2	56	no	—
60	C₂²×Dic₅₃	C₂×Q₈	1	112	yes	—
61	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
62	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
63	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
64	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
65	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
66	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
67	#11719919	C₂⁴	1	64	yes	—
68	#11719919	C₂⁴	1	64	yes	—
69	#11719919	C₂⁴	1	64	yes	—
70	#11719919	C₂⁴	1	64	yes	—
71	C₄×Dic₅₃	C₂⁴	1	64	yes	—
72	C₂³×C₅₃	#125	1	64	yes	complemented
73	C₂³×C₅₃	C₂²×Q₈	1	28	yes	—
74	C₂³×C₅₃	C₂×#41	1	28	yes	—
75	C₂³×C₅₃	C₂×#41	1	28	yes	—
76	C₂³×C₅₃	C₂⁵	1	7	yes	—
77	C₂×C₄×C₅₃	—	2	56	no	—
78	C₂×C₄×C₅₃	—	2	56	no	—
79	C₂×C₄×C₅₃	—	2	56	no	—
80	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
81	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
82	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
83	C₂×Dic₅₃	—	2	56	no	—
84	C₂×Dic₅₃	—	2	56	no	—
85	C₂×Dic₅₃	—	2	56	no	—
86	C₂×Dic₅₃	C₂²×Q₈	1	112	yes	—
87	C₂×Dic₅₃	C₂⁵	1	8	yes	—
88	C₂×Dic₅₃	C₂⁵	1	8	yes	—
89	C₂×Dic₅₃	C₂⁵	1	8	yes	—
90	C₂×Dic₅₃	C₂⁵	1	8	yes	—
91	C₂³×#125	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	C₂×#125	1	112	yes	complemented
93	C₂²×C₅₃	C₂³×Q₈	1	14	yes	—
94	C₂²×C₅₃	C₂²×#41	1	14	yes	—
95	C₂²×C₅₃	C₂²×#41	1	14	yes	—
96	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
97	C₄×C₅₃	—	2	8	no	—
98	C₄×C₅₃	—	2	8	no	—
99	C₄×C₅₃	—	2	8	no	—
100	Dic₅₃	—	2	8	no	—
101	Dic₅₃	—	2	8	no	—
102	Dic₅₃	—	2	8	no	—
103	Dic₅₃	C₂³×Q₈	1	16	yes	—
104	C₂²×#125	—	53	56	no	—
105	C₂³×C₄⋊C₄	—	53	1	no	—
106	C₂³×C₄⋊C₄	—	53	1	no	—
107	C₂³×C₄⋊C₄	—	53	1	no	—
108	C₂³×C₄⋊C₄	—	53	1	no	—
109	C₂³×C₄⋊C₄	—	53	1	no	—
110	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
111	C₂³×C₄²	—	53	1	no	—
112	C₂×C₅₃	C₂²×#125	1	28	yes	complemented
113	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
114	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
115	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
116	C₂⁴×C₄	—	53	1	no	—
117	C₂⁴×C₄	—	53	1	no	—
118	C₂⁴×C₄	—	53	1	no	—
119	C₂⁴×C₄	—	53	1	no	—
120	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
121	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
122	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
123	C₂×#125	—	53	448	no	—
124	C₂²×C₄⋊C₄	—	53	28	no	—
125	C₂²×C₄⋊C₄	—	53	28	no	—
126	C₂²×C₄⋊C₄	—	53	28	no	—
127	C₂²×C₄⋊C₄	—	53	28	no	—
128	C₂²×C₄⋊C₄	—	53	28	no	—
129	C₂²×C₄⋊C₄	C₂×D₅₃	1	28	yes	—
130	C₂²×C₄²	—	53	28	no	—
131	C₅₃	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
132	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
133	C₂³×C₄	—	2	8	no	—
134	C₂³×C₄	—	2	8	no	—
135	C₂³×C₄	—	2	8	no	—
136	C₂³×C₄	—	53	16	no	—
137	C₂³×C₄	—	53	14	no	—
138	C₂³×C₄	—	53	14	no	—
139	C₂³×C₄	—	53	14	no	—
140	C₂³×C₄	—	53	14	no	—
141	C₂³×C₄	—	106	8	no	—
142	C₂³×C₄	—	106	8	no	—
143	C₂³×C₄	—	106	8	no	—
144	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
145	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
146	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
147	#125	—	53	512	no	—
148	C₂×C₄⋊C₄	—	53	112	no	—
149	C₂×C₄⋊C₄	—	53	112	no	—
150	C₂×C₄⋊C₄	—	53	112	no	—
151	C₂×C₄⋊C₄	—	53	112	no	—
152	C₂×C₄⋊C₄	—	53	112	no	—
153	C₂×C₄⋊C₄	C₂²×D₅₃	1	112	yes	—
154	C₂×C₄²	—	53	112	no	—
155	C₂⁴	#11719945	1	8	yes	—
156	C₂⁴	#11719947	1	8	yes	—
157	C₂⁴	Q₈×D₅₃	1	8	yes	—
158	C₂⁴	C₂³×D₅₃	1	7	yes	—
159	C₂²×C₄	—	2	56	no	—
160	C₂²×C₄	—	2	56	no	—
161	C₂²×C₄	—	2	56	no	—
162	C₂²×C₄	—	53	112	no	—
163	C₂²×C₄	—	53	28	no	—
164	C₂²×C₄	—	53	28	no	—
165	C₂²×C₄	—	53	28	no	—
166	C₂²×C₄	—	53	28	no	—
167	C₂²×C₄	—	106	56	no	—
168	C₂²×C₄	—	106	56	no	—
169	C₂²×C₄	—	106	56	no	—
170	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
171	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
172	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
173	C₄⋊C₄	—	53	64	no	—
174	C₄⋊C₄	—	53	64	no	—
175	C₄⋊C₄	—	53	64	no	—
176	C₄⋊C₄	—	53	64	no	—
177	C₄⋊C₄	—	53	64	no	—
178	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
179	C₄²	—	53	64	no	—
180	C₂³	#13133207	1	64	yes	direct product factor
181	C₂³	C₂×Q₈×D₅₃	1	28	yes	—
182	C₂³	C₂×#11719947	1	28	yes	—
183	C₂³	C₂×#11719945	1	28	yes	—
184	C₂³	C₂⁴×D₅₃	1	7	yes	—
185	C₂×C₄	—	2	56	no	—
186	C₂×C₄	—	2	56	no	—
187	C₂×C₄	—	2	56	no	—
188	C₂×C₄	—	53	112	no	—
189	C₂×C₄	—	53	8	no	—
190	C₂×C₄	—	53	8	no	—
191	C₂×C₄	—	53	8	no	—
192	C₂×C₄	—	53	8	no	—
193	C₂×C₄	—	106	56	no	—
194	C₂×C₄	—	106	56	no	—
195	C₂×C₄	—	106	56	no	—
196	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
197	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
198	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
199	C₂²	C₂×#13133207	1	112	yes	direct product factor
200	C₂²	C₂²×Q₈×D₅₃	1	14	yes	—
201	C₂²	C₂²×#11719947	1	14	yes	—
202	C₂²	C₂²×#11719945	1	14	yes	—
203	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	—	2	8	no	—
206	C₄	—	2	8	no	—
207	C₄	—	53	16	no	—
208	C₄	—	106	8	no	—
209	C₄	—	106	8	no	—
210	C₄	—	106	8	no	—
211	C₂	C₂²×#13133207	1	28	yes	direct product factor
212	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
213	C₂	C₂³×#11719947	1	1	yes	characteristic
214	C₂	C₂³×#11719945	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#125
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133207	C₂	direct (N × H)
13. C₂×#13133207	C₂²	direct (N × H)
32. #13133207	C₂³	direct (N × H)
72. C₂³×C₅₃	#125	semidirect
92. C₂²×C₅₃	C₂×#125	semidirect
112. C₂×C₅₃	C₂²×#125	semidirect
131. C₅₃	C₂³×#125	semidirect
180. C₂³	#13133207	direct (N × H)
199. C₂²	C₂×#13133207	direct (N × H)
211. C₂	C₂²×#13133207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133207	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719919	2
C₂³×C₄×Dic₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×#125	53

Subgroups — C23×#13133207 #28290758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133207 #28290758