Subgroups — C₂³×#13133304 #28330819

Order 13568.

11422 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133304	C₂	1	56	yes	maximal, direct product factor
2	C₂³×#11719936	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11719940	C₂	1	2	yes	maximal, complemented
4	C₂³×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×#88324	C₂	1	2	yes	maximal, complemented
6	C₂³×#11719947	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133304	C₂²	1	448	yes	direct product factor
8	C₂²×#11719936	C₂²	1	28	yes	complemented
9	C₂²×#11719940	C₂²	1	56	yes	complemented
10	C₂²×D₅₃⋊C₈	C₂²	1	28	yes	complemented
11	C₂³×#88324	C₂²	1	1	yes	characteristic
12	C₂³×#88324	C₂²	1	2	yes	—
13	C₂³×#88324	C₂²	1	1	yes	characteristic
14	C₂³×#88324	C₂²	1	56	yes	complemented
15	C₂²×#11719947	C₄	1	28	yes	—
16	C₂²×#11719947	C₂²	1	28	yes	—
17	C₂³×Dic₁₀₆	C₄	1	1	yes	characteristic
18	C₂³×C₂²⋊D₅₃	C₄	1	2	yes	—
19	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
20	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
21	C₂³×D₄×C₅₃	C₄	1	1	yes	characteristic
22	#13133304	C₂³	1	512	yes	direct product factor
23	C₂×#11719936	C₂³	1	112	yes	complemented
24	C₂×#11719940	C₂³	1	224	yes	complemented
25	C₂×D₅₃⋊C₈	C₂³	1	112	yes	complemented
26	C₂²×#88324	C₂³	1	14	yes	—
27	C₂²×#88324	C₂³	1	28	yes	—
28	C₂²×#88324	C₂³	1	14	yes	—
29	C₂²×#88324	C₂³	1	224	yes	complemented
30	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719947	C₂×C₄	1	336	yes	—
32	C₂×#11719947	C₂³	1	112	yes	—
33	C₂²×Dic₁₀₆	C₂×C₄	1	28	yes	—
34	C₂²×C₂²⋊D₅₃	C₂×C₄	1	56	yes	—
35	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂³×Dic₅₃	C₂×C₄	1	28	yes	—
37	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
38	C₂³×Dic₅₃	C₂³	1	28	yes	—
39	C₂²×C₄×D₅₃	C₂×C₄	1	14	yes	—
40	C₂²×C₄×D₅₃	C₂³	1	14	yes	—
41	C₂⁵×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×D₄×C₅₃	C₂×C₄	1	28	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂⁴×C₅₃	—	2	16	no	—
45	C₂⁴×C₅₃	C₂²×C₄	1	28	yes	—
46	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂³×D₅₃	—	2	8	no	—
48	C₂³×D₅₃	C₂²×C₄	1	14	yes	—
49	C₂×D₄×C₅₃	C₂²×C₄	1	112	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	14	yes	—
51	C₂×C₂²⋊D₅₃	C₂²×C₄	1	224	yes	—
52	C₂²×Dic₅₃	C₂²×C₄	1	7	yes	—
53	C₂²×Dic₅₃	C₂²×C₄	1	28	yes	—
54	C₂²×Dic₅₃	C₂²×C₄	1	168	yes	—
55	C₂²×Dic₅₃	C₂⁴	1	7	yes	—
56	C₂²×Dic₅₃	C₂⁴	1	56	yes	—
57	#11719947	C₂²×C₄	1	448	yes	—
58	#11719947	C₂⁴	1	64	yes	—
59	C₂×C₄×D₅₃	C₂²×C₄	1	84	yes	—
60	C₂×C₄×D₅₃	C₂⁴	1	28	yes	—
61	C₂×Dic₁₀₆	C₂²×C₄	1	112	yes	—
62	#11719940	C₂⁴	1	128	yes	complemented
63	C₂×#88324	C₂⁴	1	28	yes	—
64	C₂×#88324	C₂⁴	1	56	yes	—
65	C₂×#88324	C₂⁴	1	28	yes	—
66	C₂×#88324	C₂⁴	1	128	yes	complemented
67	#11719936	C₂⁴	1	64	yes	complemented
68	D₅₃⋊C₈	C₂⁴	1	64	yes	complemented
69	C₂³×C₅₃	—	2	112	no	—
70	C₂³×C₅₃	#131	1	8	yes	complemented
71	C₂³×C₅₃	C₂³×C₄	1	56	yes	—
72	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
73	C₂²×D₅₃	—	2	56	no	—
74	C₂²×D₅₃	C₂³×C₄	1	28	yes	—
75	D₄×C₅₃	C₂³×C₄	1	64	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	28	yes	—
77	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	—
78	C₂×Dic₅₃	C₂³×C₄	1	56	yes	—
79	C₂×Dic₅₃	C₂³×C₄	1	21	yes	—
80	C₂×Dic₅₃	C₂³×C₄	1	112	yes	—
81	C₂×Dic₅₃	C₂⁵	1	7	yes	—
82	C₂×Dic₅₃	C₂⁵	1	16	yes	—
83	C₄×D₅₃	C₂³×C₄	1	56	yes	—
84	C₄×D₅₃	C₂⁵	1	8	yes	—
85	Dic₁₀₆	C₂³×C₄	1	64	yes	—
86	#88324	C₂⁵	1	16	yes	—
87	#88324	C₂⁵	1	8	yes	—
88	#88324	C₂⁵	1	8	yes	—
89	C₂³×#131	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	112	no	—
91	C₂²×C₅₃	C₂×#131	1	28	yes	complemented
92	C₂²×C₅₃	C₂⁴×C₄	1	16	yes	—
93	C₂²×C₅₃	C₂⁴×C₄	1	7	yes	—
94	C₂×D₅₃	—	2	56	no	—
95	C₂×D₅₃	C₂⁴×C₄	1	8	yes	—
96	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
97	Dic₅₃	C₂⁴×C₄	1	16	yes	—
98	Dic₅₃	C₂⁴×C₄	1	7	yes	—
99	Dic₅₃	C₂⁶	1	1	yes	characteristic
100	C₂³×#41	—	53	1	no	—
101	C₂²×#131	—	53	56	no	—
102	C₂³×M₁₆	—	53	2	no	—
103	C₂³×M₁₆	—	53	1	no	—
104	C₂⁴×C₈	—	53	2	no	—
105	C₂⁴×C₈	—	53	1	no	—
106	C₂×C₅₃	—	2	16	no	—
107	C₂×C₅₃	C₂²×#131	1	14	yes	complemented
108	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
109	D₅₃	—	2	8	no	—
110	C₂²×#41	—	53	28	no	—
111	C₂²×#41	—	53	28	no	—
112	C₂³×Q₈	—	53	1	no	—
113	C₂³×D₄	—	53	2	no	—
114	C₂³×D₄	#6157	1	1	yes	characteristic
115	C₂⁴×C₄	—	53	2	no	—
116	C₂⁴×C₄	—	53	1	no	—
117	C₂×#131	—	53	448	no	—
118	C₂²×M₁₆	—	53	56	no	—
119	C₂²×M₁₆	—	53	28	no	—
120	C₂³×C₈	—	53	1	no	—
121	C₂³×C₈	—	53	2	no	—
122	C₂³×C₈	—	53	1	no	—
123	C₂³×C₈	—	53	56	no	—
124	C₂³×C₈	—	53	28	no	—
125	C₅₃	C₂³×#131	1	1	yes	characteristic, complemented, Sylow
126	C₂⁵	—	53	1	no	—
127	C₂⁵	C₂×#6157	1	2	yes	—
128	C₂×#41	—	53	112	no	—
129	C₂×#41	—	53	336	no	—
130	C₂²×Q₈	—	53	28	no	—
131	C₂²×D₄	—	53	56	no	—
132	C₂²×D₄	C₂×#6157	1	28	yes	—
133	C₂³×C₄	—	53	28	no	—
134	C₂³×C₄	—	53	28	no	—
135	C₂³×C₄	—	53	1	no	—
136	C₂³×C₄	—	53	2	no	—
137	C₂³×C₄	—	53	14	no	—
138	C₂³×C₄	—	53	14	no	—
139	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
140	#131	—	53	512	no	—
141	C₂×M₁₆	—	53	224	no	—
142	C₂×M₁₆	—	53	112	no	—
143	C₂²×C₈	—	53	14	no	—
144	C₂²×C₈	—	53	28	no	—
145	C₂²×C₈	—	53	14	no	—
146	C₂²×C₈	—	53	224	no	—
147	C₂²×C₈	—	53	112	no	—
148	C₂⁴	—	2	16	no	—
149	C₂⁴	—	53	14	no	—
150	C₂⁴	—	106	8	no	—
151	C₂⁴	C₂²×#6157	1	28	yes	—
152	C₂⁴	C₂²×#6157	1	1	yes	characteristic
153	#41	—	53	448	no	—
154	#41	—	53	64	no	—
155	C₂×Q₈	—	53	112	no	—
156	C₂×D₄	—	53	224	no	—
157	C₂×D₄	C₂²×#6157	1	112	yes	—
158	C₂²×C₄	—	53	7	no	—
159	C₂²×C₄	—	53	28	no	—
160	C₂²×C₄	—	53	168	no	—
161	C₂²×C₄	—	53	7	no	—
162	C₂²×C₄	—	53	56	no	—
163	C₂²×C₄	—	53	84	no	—
164	C₂²×C₄	—	53	28	no	—
165	C₂²×C₄	C₂²×#6157	1	14	yes	—
166	M₁₆	—	53	64	no	—
167	M₁₆	—	53	128	no	—
168	C₂×C₈	—	53	28	no	—
169	C₂×C₈	—	53	56	no	—
170	C₂×C₈	—	53	28	no	—
171	C₂×C₈	—	53	128	no	—
172	C₂×C₈	—	53	64	no	—
173	C₂³	—	2	112	no	—
174	C₂³	—	53	28	no	—
175	C₂³	—	106	56	no	—
176	C₂³	#13133304	1	8	yes	direct product factor
177	C₂³	C₂³×#6157	1	56	yes	—
178	C₂³	C₂³×#6157	1	7	yes	—
179	Q₈	—	53	64	no	—
180	D₄	—	53	128	no	—
181	D₄	C₂³×#6157	1	64	yes	—
182	C₂×C₄	—	53	21	no	—
183	C₂×C₄	—	53	56	no	—
184	C₂×C₄	—	53	112	no	—
185	C₂×C₄	—	53	7	no	—
186	C₂×C₄	—	53	16	no	—
187	C₂×C₄	—	53	56	no	—
188	C₂×C₄	—	53	8	no	—
189	C₂×C₄	C₂³×#6157	1	28	yes	—
190	C₈	—	53	8	no	—
191	C₈	—	53	16	no	—
192	C₈	—	53	8	no	—
193	C₂²	—	2	112	no	—
194	C₂²	—	53	8	no	—
195	C₂²	—	106	56	no	—
196	C₂²	C₂×#13133304	1	28	yes	direct product factor
197	C₂²	C₂⁴×#6157	1	16	yes	—
198	C₂²	C₂⁴×#6157	1	7	yes	—
199	C₄	—	53	1	no	—
200	C₄	—	53	7	no	—
201	C₄	—	53	16	no	—
202	C₄	C₂⁴×#6157	1	8	yes	—
203	C₂	—	2	16	no	—
204	C₂	—	106	8	no	—
205	C₂	C₂²×#13133304	1	14	yes	direct product factor
206	C₂	C₂⁵×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#6157
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#6157
Fitting	C₂³×D₄×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂³×#11719947	C₂
cosocle	Dic₅₃	C₂⁶
hypercenter	C₂³×D₄	#6157
hypocenter	C₅₃	C₂³×#131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133304	C₂	direct (N × H)
2. C₂³×#11719936	C₂	semidirect
3. C₂³×#11719940	C₂	semidirect
4. C₂³×D₅₃⋊C₈	C₂	semidirect
5. C₂⁴×#88324	C₂	semidirect
7. C₂×#13133304	C₂²	direct (N × H)
8. C₂²×#11719936	C₂²	semidirect
9. C₂²×#11719940	C₂²	semidirect
10. C₂²×D₅₃⋊C₈	C₂²	semidirect
14. C₂³×#88324	C₂²	semidirect
22. #13133304	C₂³	direct (N × H)
23. C₂×#11719936	C₂³	semidirect
24. C₂×#11719940	C₂³	semidirect
25. C₂×D₅₃⋊C₈	C₂³	semidirect
29. C₂²×#88324	C₂³	semidirect
62. #11719940	C₂⁴	semidirect
66. C₂×#88324	C₂⁴	semidirect
67. #11719936	C₂⁴	semidirect
68. D₅₃⋊C₈	C₂⁴	semidirect
70. C₂³×C₅₃	#131	semidirect
91. C₂²×C₅₃	C₂×#131	semidirect
107. C₂×C₅₃	C₂²×#131	semidirect
125. C₅₃	C₂³×#131	semidirect
176. C₂³	#13133304	direct (N × H)
196. C₂²	C₂×#13133304	direct (N × H)
205. C₂	C₂²×#13133304	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×#13133304	2
C₂³×#11719936	2
C₂³×#11719940	2
C₂³×D₅₃⋊C₈	2
C₂⁴×#88324	2
C₂³×#11719947	2
C₂³×#131	53

Subgroups — C23×#13133304 #28330819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133304 #28330819