Subgroups — C₂³×#13133137 #28247599

Order 13568.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133137	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₅₃	—	2	1	no	—
6	C₂³×D₂₁₂	—	2	7	no	—
7	C₂³×D₂₁₂	—	2	8	no	—
8	C₂³×D₂₁₂	C₄	1	16	yes	complemented
9	C₂³×D₂₁₂	C₂²	1	14	yes	—
10	C₂³×#88322	—	2	8	no	—
11	C₂³×#88322	C₂²	1	14	yes	complemented
12	C₂×#13133137	C₂²	1	112	yes	direct product factor
13	C₂⁴×C₄×C₅₃	—	2	1	no	—
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₅₃	—	2	7	no	—
16	C₂²×C₄⋊C₄×C₅₃	C₂²	1	14	yes	complemented
17	C₂⁴×D₅₃	—	2	16	no	—
18	C₂⁴×D₅₃	—	2	14	no	—
19	C₂⁴×D₅₃	—	4	8	no	—
20	C₂⁴×D₅₃	—	4	8	no	—
21	C₂²×D₂₁₂	—	2	112	no	—
22	C₂²×D₂₁₂	—	2	42	no	—
23	C₂²×D₂₁₂	—	2	56	no	—
24	C₂²×D₂₁₂	C₂×C₄	1	112	yes	complemented
25	C₂²×D₂₁₂	C₂³	1	28	yes	—
26	C₂²×#88322	—	2	56	no	—
27	C₂²×#88322	C₂³	1	28	yes	complemented
28	#13133137	C₂³	1	64	yes	direct product factor
29	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	—	2	14	no	—
31	C₂³×C₄×C₅₃	—	4	8	no	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₅₃	D₄	1	7	yes	—
34	C₂³×C₄×C₅₃	D₄	1	8	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	42	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂³	1	28	yes	complemented
38	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
39	C₂⁴×C₅₃	D₈	1	8	yes	—
40	C₂⁴×C₅₃	SD₁₆	1	8	yes	—
41	C₂⁴×C₅₃	C₂×D₄	1	7	yes	—
42	C₂³×D₅₃	—	2	112	no	—
43	C₂³×D₅₃	—	2	28	no	—
44	C₂³×D₅₃	—	4	128	no	—
45	C₂³×D₅₃	—	4	56	no	—
46	C₂³×D₅₃	—	4	56	no	—
47	C₂²×C₄×C₅₃	—	2	28	no	—
48	C₂²×C₄×C₅₃	—	4	56	no	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	56	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
54	C₂×D₂₁₂	—	2	336	no	—
55	C₂×D₂₁₂	—	2	28	no	—
56	C₂×D₂₁₂	—	2	56	no	—
57	C₂×D₂₁₂	C₂²×C₄	1	112	yes	complemented
58	C₂×D₂₁₂	C₂⁴	1	8	yes	—
59	C₄⋊C₄×C₅₃	—	2	28	no	—
60	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	complemented
61	C₂×#88322	—	2	56	no	—
62	C₂×#88322	C₂⁴	1	8	yes	complemented
63	C₂³×C₅₃	#102	1	64	yes	complemented
64	C₂³×C₅₃	C₂×C₂²⋊C₄	1	28	yes	—
65	C₂³×C₅₃	C₂×D₈	1	28	yes	—
66	C₂³×C₅₃	C₂×SD₁₆	1	28	yes	—
67	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
68	C₂²×D₅₃	—	2	112	no	—
69	C₂²×D₅₃	—	2	8	no	—
70	C₂²×D₅₃	—	4	56	no	—
71	C₂²×D₅₃	—	4	56	no	—
72	C₂²×D₅₃	—	4	448	no	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	4	56	no	—
75	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	84	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
77	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
78	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
79	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
80	D₂₁₂	—	2	112	no	—
81	D₂₁₂	—	2	8	no	—
82	D₂₁₂	C₂³×C₄	1	16	yes	complemented
83	#88322	—	2	8	no	—
84	C₂³×#102	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	C₂×#102	1	112	yes	complemented
86	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
87	C₂²×C₅₃	C₂²×D₈	1	14	yes	—
88	C₂²×C₅₃	C₂²×SD₁₆	1	14	yes	—
89	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
90	C₂×D₅₃	—	2	16	no	—
91	C₂×D₅₃	—	4	224	no	—
92	C₂×D₅₃	—	4	8	no	—
93	C₂×D₅₃	—	4	8	no	—
94	C₄×C₅₃	—	4	8	no	—
95	C₄×C₅₃	C₂²×C₂²⋊C₄	1	14	yes	—
96	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
97	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	—	53	1	no	—
99	C₂⁴×C₈	—	53	1	no	—
100	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
101	C₂²×#102	—	53	28	no	—
102	C₂×C₅₃	C₂²×#102	1	28	yes	complemented
103	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
105	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
106	D₅₃	—	4	16	no	—
107	C₂⁶	—	106	1	no	—
108	C₂³×D₄	—	53	16	no	—
109	C₂³×D₄	—	53	14	no	—
110	C₂³×D₄	—	106	8	no	—
111	C₂³×D₄	—	106	7	no	—
112	C₂⁴×C₄	—	2	1	no	—
113	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂³×C₈	—	53	14	no	—
115	C₂³×C₈	—	106	8	no	—
116	C₂²×C₄⋊C₄	—	2	7	no	—
117	C₂²×C₄⋊C₄	C₂×D₅₃	1	14	yes	complemented
118	C₂×#102	—	53	112	no	—
119	C₅₃	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
120	C₂⁵	—	106	14	no	—
121	C₂⁵	—	106	16	no	—
122	C₂⁵	—	212	8	no	—
123	C₂⁵	—	212	8	no	—
124	C₂⁵	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×D₄	—	53	112	no	—
126	C₂²×D₄	—	53	28	no	—
127	C₂²×D₄	—	106	56	no	—
128	C₂²×D₄	—	106	42	no	—
129	C₂²×D₄	—	106	112	no	—
130	C₂³×C₄	—	2	14	no	—
131	C₂³×C₄	—	4	8	no	—
132	C₂³×C₄	C₄×D₅₃	1	8	yes	—
133	C₂³×C₄	D₂₁₂	1	8	yes	—
134	C₂³×C₄	C₂²⋊D₅₃	1	7	yes	—
135	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
136	C₂²×C₈	—	53	28	no	—
137	C₂²×C₈	—	106	56	no	—
138	C₂×C₄⋊C₄	—	2	42	no	—
139	C₂×C₄⋊C₄	C₂²×D₅₃	1	28	yes	complemented
140	#102	—	53	64	no	—
141	C₂⁴	—	106	112	no	—
142	C₂⁴	—	106	28	no	—
143	C₂⁴	—	212	128	no	—
144	C₂⁴	—	212	56	no	—
145	C₂⁴	—	212	56	no	—
146	C₂⁴	#11719921	1	8	yes	—
147	C₂⁴	#11719922	1	8	yes	—
148	C₂⁴	#11719924	1	8	yes	—
149	C₂⁴	C₂×C₂²⋊D₅₃	1	7	yes	—
150	C₂×D₄	—	53	112	no	—
151	C₂×D₄	—	53	8	no	—
152	C₂×D₄	—	106	336	no	—
153	C₂×D₄	—	106	28	no	—
154	C₂×D₄	—	106	56	no	—
155	C₂²×C₄	—	2	28	no	—
156	C₂²×C₄	—	4	56	no	—
157	C₂²×C₄	#11719921	1	56	yes	—
158	C₂²×C₄	C₂×C₄×D₅₃	1	28	yes	—
159	C₂²×C₄	C₂×D₂₁₂	1	28	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
161	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
162	C₂×C₈	—	53	8	no	—
163	C₂×C₈	—	106	56	no	—
164	C₄⋊C₄	—	2	28	no	—
165	C₄⋊C₄	C₂³×D₅₃	1	8	yes	complemented
166	C₂³	—	106	112	no	—
167	C₂³	—	106	8	no	—
168	C₂³	—	212	448	no	—
169	C₂³	—	212	56	no	—
170	C₂³	—	212	56	no	—
171	C₂³	#13133137	1	64	yes	direct product factor
172	C₂³	C₂×#11719921	1	28	yes	—
173	C₂³	C₂×#11719922	1	28	yes	—
174	C₂³	C₂×#11719924	1	28	yes	—
175	C₂³	C₂²×C₂²⋊D₅₃	1	7	yes	—
176	D₄	—	53	16	no	—
177	D₄	—	106	112	no	—
178	D₄	—	106	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	4	56	no	—
181	C₂×C₄	C₂×#11719921	1	84	yes	—
182	C₂×C₄	C₂²×C₄×D₅₃	1	14	yes	—
183	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
184	C₂×C₄	C₂²×D₂₁₂	1	14	yes	—
185	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
186	C₈	—	106	8	no	—
187	C₂²	—	106	16	no	—
188	C₂²	—	212	224	no	—
189	C₂²	—	212	8	no	—
190	C₂²	—	212	8	no	—
191	C₂²	C₂×#13133137	1	112	yes	direct product factor
192	C₂²	C₂²×#11719921	1	14	yes	—
193	C₂²	C₂²×#11719922	1	14	yes	—
194	C₂²	C₂²×#11719924	1	14	yes	—
195	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄	—	4	8	no	—
197	C₄	C₂²×#11719921	1	14	yes	—
198	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
199	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
200	C₂	—	212	16	no	—
201	C₂	C₂²×#13133137	1	28	yes	direct product factor
202	C₂	C₂³×#11719921	1	1	yes	characteristic
203	C₂	C₂³×#11719922	1	1	yes	characteristic
204	C₂	C₂³×#11719924	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	D₄
prime radical	C₂⁴×D₂₁₂	C₂
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#102
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×#88322	C₂	semidirect
3. C₂²×#13133137	C₂	direct (N × H)
4. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
8. C₂³×D₂₁₂	C₄	semidirect
11. C₂³×#88322	C₂²	semidirect
12. C₂×#13133137	C₂²	direct (N × H)
16. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
24. C₂²×D₂₁₂	C₂×C₄	semidirect
27. C₂²×#88322	C₂³	semidirect
28. #13133137	C₂³	direct (N × H)
37. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
57. C₂×D₂₁₂	C₂²×C₄	semidirect
60. C₄⋊C₄×C₅₃	C₂⁴	semidirect
62. C₂×#88322	C₂⁴	semidirect
63. C₂³×C₅₃	#102	semidirect
82. D₂₁₂	C₂³×C₄	semidirect
85. C₂²×C₅₃	C₂×#102	semidirect
100. C₂³×C₄⋊C₄	D₅₃	semidirect
102. C₂×C₅₃	C₂²×#102	semidirect
117. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
119. C₅₃	C₂³×#102	semidirect
139. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
165. C₄⋊C₄	C₂³×D₅₃	semidirect
171. C₂³	#13133137	direct (N × H)
191. C₂²	C₂×#13133137	direct (N × H)
201. C₂	C₂²×#13133137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁₂	2
C₂⁴×#88322	2
C₂²×#13133137	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×#102	53

Subgroups — C23×#13133137 #28247599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133137 #28247599