Subgroups — C₂²×#16810424 #27536964

Order 13568.

1470 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810424	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂²×#11719908	—	2	4	no	—
7	C₂²×#11719908	C₂²	1	6	yes	complemented
8	C₂×#13133151	—	2	3	no	—
9	C₂×#13133151	C₂²	1	6	yes	complemented
10	#16810424	C₂²	1	16	yes	direct product factor
11	C₂³×D₄×C₅₃	—	2	1	no	—
12	C₂²×D₈×C₅₃	—	2	4	no	—
13	C₂²×D₈×C₅₃	—	2	3	no	—
14	C₂²×D₈×C₅₃	C₄	1	8	yes	complemented
15	C₂²×D₈×C₅₃	C₂²	1	6	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×Dic₅₃	—	4	1	no	—
18	C₂×C₄⋊Dic₅₃	—	2	6	no	—
19	C₂×C₄⋊Dic₅₃	—	4	3	no	—
20	C₂×#11719908	—	2	12	no	—
21	C₂×#11719908	C₂³	1	4	yes	complemented
22	#13133151	—	2	6	no	—
23	#13133151	C₂³	1	4	yes	complemented
24	C₂⁵×C₅₃	—	4	1	no	—
25	C₂²×D₄×C₅₃	—	2	8	no	—
26	C₂²×D₄×C₅₃	—	2	6	no	—
27	C₂²×D₄×C₅₃	—	4	3	no	—
28	C₂²×D₄×C₅₃	—	4	4	no	—
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂×D₈×C₅₃	—	2	24	no	—
31	C₂×D₈×C₅₃	—	2	12	no	—
32	C₂×D₈×C₅₃	—	2	6	no	—
33	C₂×D₈×C₅₃	C₂×C₄	1	24	yes	complemented
34	C₂×D₈×C₅₃	C₂³	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₅₃	D₄	1	4	yes	—
37	C₂²×C₈×C₅₃	D₄	1	3	yes	—
38	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
39	C₂⁴×C₅₃	—	4	8	no	—
40	C₂⁴×C₅₃	—	4	6	no	—
41	C₂⁴×C₅₃	—	8	4	no	—
42	C₂⁴×C₅₃	—	8	4	no	—
43	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
44	C₂×D₄×C₅₃	—	2	24	no	—
45	C₂×D₄×C₅₃	—	2	4	no	—
46	C₂×D₄×C₅₃	—	4	24	no	—
47	C₂×D₄×C₅₃	—	4	6	no	—
48	C₂×D₄×C₅₃	—	4	12	no	—
49	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	D₈	1	3	yes	—
51	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
53	C₂²×Dic₅₃	—	4	6	no	—
54	C₂²×Dic₅₃	—	8	4	no	—
55	D₈×C₅₃	—	2	24	no	—
56	D₈×C₅₃	—	2	4	no	—
57	D₈×C₅₃	C₂²×C₄	1	8	yes	complemented
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	12	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
63	C₄⋊Dic₅₃	—	2	4	no	—
64	C₄⋊Dic₅₃	—	4	6	no	—
65	#11719908	—	2	4	no	—
66	C₂³×C₅₃	—	4	4	no	—
67	C₂³×C₅₃	—	4	24	no	—
68	C₂³×C₅₃	—	8	32	no	—
69	C₂³×C₅₃	—	8	12	no	—
70	C₂³×C₅₃	—	8	12	no	—
71	C₂³×C₅₃	#102	1	4	yes	—
72	C₂³×C₅₃	D₁₆	1	4	yes	—
73	C₂³×C₅₃	SD₃₂	1	4	yes	—
74	C₂³×C₅₃	C₂×D₈	1	3	yes	—
75	D₄×C₅₃	—	2	8	no	—
76	D₄×C₅₃	—	4	4	no	—
77	D₄×C₅₃	—	4	24	no	—
78	C₂×C₄×C₅₃	#102	1	12	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	8	12	no	—
85	C₈×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
86	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₂²×#357	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	4	8	no	—
90	C₂²×C₅₃	—	8	4	no	—
91	C₂²×C₅₃	—	8	4	no	—
92	C₂²×C₅₃	—	8	48	no	—
93	C₂²×C₅₃	#357	1	16	yes	complemented
94	C₂²×C₅₃	C₂×#102	1	6	yes	—
95	C₂²×C₅₃	C₂×D₁₆	1	6	yes	—
96	C₂²×C₅₃	C₂×SD₃₂	1	6	yes	—
97	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#102	1	6	yes	—
99	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
101	Dic₅₃	—	8	4	no	—
102	C₂³×D₈	D₅₃	1	1	yes	characteristic
103	C₂³×C₁₆	—	53	1	no	—
104	C₂²×#107	—	53	1	no	—
105	C₂×#357	—	53	12	no	—
106	C₂×C₅₃	—	8	8	no	—
107	C₂×C₅₃	C₂×#357	1	12	yes	complemented
108	C₂×C₅₃	C₂²×#102	1	1	yes	characteristic
109	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
110	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
111	C₂³×D₄	—	2	1	no	—
112	C₂²×D₈	—	2	4	no	—
113	C₂²×D₈	—	2	3	no	—
114	C₂²×D₈	Dic₅₃	1	8	yes	complemented
115	C₂²×D₈	C₂×D₅₃	1	6	yes	—
116	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂²×C₄⋊C₄	—	106	1	no	—
118	C₂²×C₁₆	—	53	6	no	—
119	C₂²×C₁₆	—	106	4	no	—
120	C₂×#107	—	53	6	no	—
121	C₂×#107	—	106	3	no	—
122	#357	—	53	16	no	—
123	C₅₃	C₂²×#357	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	4	1	no	—
125	C₂²×D₄	—	2	6	no	—
126	C₂²×D₄	—	2	8	no	—
127	C₂²×D₄	—	4	3	no	—
128	C₂²×D₄	—	4	4	no	—
129	C₂³×C₄	—	212	1	no	—
130	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂×D₈	—	2	6	no	—
132	C₂×D₈	—	2	12	no	—
133	C₂×D₈	—	2	24	no	—
134	C₂×D₈	C₂×Dic₅₃	1	24	yes	complemented
135	C₂×D₈	C₂²×D₅₃	1	4	yes	—
136	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
137	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
138	C₂²×C₈	C₂²⋊D₅₃	1	3	yes	—
139	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
140	C₂×C₄⋊C₄	—	106	6	no	—
141	C₂×C₄⋊C₄	—	212	3	no	—
142	C₂×C₁₆	—	53	4	no	—
143	C₂×C₁₆	—	106	12	no	—
144	#107	—	53	4	no	—
145	#107	—	106	6	no	—
146	C₂⁴	—	4	8	no	—
147	C₂⁴	—	4	6	no	—
148	C₂⁴	—	8	4	no	—
149	C₂⁴	—	8	4	no	—
150	C₂⁴	#11719922	1	1	yes	characteristic
151	C₂×D₄	—	2	24	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	24	no	—
154	C₂×D₄	—	4	6	no	—
155	C₂×D₄	—	4	12	no	—
156	C₂²×C₄	—	424	4	no	—
157	C₂²×C₄	—	212	6	no	—
158	C₂²×C₄	#11719922	1	3	yes	—
159	C₂²×C₄	#11719923	1	4	yes	—
160	C₂²×C₄	C₂²⋊Dic₅₃	1	4	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
162	D₈	—	2	24	no	—
163	D₈	—	2	4	no	—
164	D₈	C₂²×Dic₅₃	1	8	yes	complemented
165	C₂×C₈	C₂²⋊Dic₅₃	1	12	yes	—
166	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
167	C₂×C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
168	C₂×C₈	C₂×C₂²⋊D₅₃	1	6	yes	—
169	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
170	C₄⋊C₄	—	106	4	no	—
171	C₄⋊C₄	—	212	6	no	—
172	C₁₆	—	106	4	no	—
173	C₂³	—	4	24	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	8	12	no	—
176	C₂³	—	8	12	no	—
177	C₂³	—	8	32	no	—
178	C₂³	#13133138	1	4	yes	—
179	C₂³	#13133158	1	4	yes	—
180	C₂³	#13133161	1	4	yes	—
181	C₂³	C₂×#11719922	1	3	yes	—
182	D₄	—	2	8	no	—
183	D₄	—	4	24	no	—
184	D₄	—	4	4	no	—
185	C₂×C₄	—	424	12	no	—
186	C₂×C₄	—	212	4	no	—
187	C₂×C₄	#13133138	1	12	yes	—
188	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	6	yes	—
189	C₂×C₄	C₂×#11719922	1	3	yes	—
190	C₂×C₄	C₂×#11719923	1	6	yes	—
191	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₈	C₂×C₂²⋊Dic₅₃	1	6	yes	—
193	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
194	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂²	—	4	8	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	48	no	—
199	C₂²	#16810424	1	16	yes	direct product factor
200	C₂²	C₂×#13133138	1	6	yes	—
201	C₂²	C₂×#13133158	1	6	yes	—
202	C₂²	C₂×#13133161	1	6	yes	—
203	C₂²	C₂²×#11719922	1	1	yes	characteristic
204	C₄	—	424	4	no	—
205	C₄	C₂×#13133138	1	6	yes	—
206	C₄	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
207	C₄	C₂²×#11719923	1	1	yes	characteristic
208	C₂	—	8	8	no	—
209	C₂	C₂×#16810424	1	12	yes	direct product factor
210	C₂	C₂²×#13133138	1	1	yes	characteristic
211	C₂	C₂²×#13133158	1	1	yes	characteristic
212	C₂	C₂²×#13133161	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂³×D₈×C₅₃	C₂
socle	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×D₈	D₅₃
hypocenter	C₅₃	C₂²×#357
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719908	C₂	semidirect
2. C₂²×#13133151	C₂	semidirect
3. C₂×#16810424	C₂	direct (N × H)
7. C₂²×#11719908	C₂²	semidirect
9. C₂×#13133151	C₂²	semidirect
10. #16810424	C₂²	direct (N × H)
14. C₂²×D₈×C₅₃	C₄	semidirect
21. C₂×#11719908	C₂³	semidirect
23. #13133151	C₂³	semidirect
33. C₂×D₈×C₅₃	C₂×C₄	semidirect
57. D₈×C₅₃	C₂²×C₄	semidirect
93. C₂²×C₅₃	#357	semidirect
107. C₂×C₅₃	C₂×#357	semidirect
114. C₂²×D₈	Dic₅₃	semidirect
123. C₅₃	C₂²×#357	semidirect
134. C₂×D₈	C₂×Dic₅₃	semidirect
164. D₈	C₂²×Dic₅₃	semidirect
199. C₂²	#16810424	direct (N × H)
209. C₂	C₂×#16810424	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂²×#13133151	2
C₂×#16810424	2
C₂³×D₈×C₅₃	2
C₂²×#357	53

Subgroups — C22×#16810424 #27536964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810424 #27536964