Subgroups — C₂²×C₄×#11719917 #28247480

Order 13568.

3950 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×#11719917	C₂	1	2	yes	maximal
2	C₂×C₄×#11719917	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133128	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×#88322	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	32	yes	direct product factor
7	C₂²×#11719917	C₂²	1	24	yes	—
8	C₂³×#88322	C₂²	1	4	yes	—
9	C₄×#11719917	C₂²	1	64	yes	direct product factor
10	C₂×#13133128	C₂²	1	24	yes	complemented
11	C₂×C₄×#88322	C₂²	1	24	yes	complemented
12	C₂⁴×C₄×C₅₃	C₄	1	2	yes	—
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₄	1	12	yes	—
15	C₂²×C₄²×C₅₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₅₃	C₂²	1	12	yes	—
18	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	C₂×C₄	1	192	yes	direct product factor
20	C₂×#11719917	C₂³	1	32	yes	—
21	C₂²×#88322	C₂×C₄	1	32	yes	complemented
22	C₂²×#88322	C₂³	1	24	yes	—
23	#13133128	C₂³	1	32	yes	complemented
24	C₄×#88322	C₂³	1	32	yes	complemented
25	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	—	2	16	no	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	24	yes	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
30	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	48	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	16	yes	—
42	C₂⁴×C₅₃	—	2	8	no	—
43	C₂⁴×C₅₃	C₄²	1	8	yes	—
44	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
45	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	8	no	—
47	C₂²×C₄×C₅₃	—	2	96	no	—
48	C₂²×C₄×C₅₃	C₄²	1	48	yes	—
49	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
50	C₂²×C₄×C₅₃	M₁₆	1	16	yes	complemented
51	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	48	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	32	yes	—
63	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
64	C₄²×C₅₃	C₂⁴	1	4	yes	—
65	C₄²×C₅₃	C₂⁴	1	4	yes	—
66	#11719917	C₂²×C₄	1	128	yes	direct product factor
67	C₂×#88322	C₂²×C₄	1	96	yes	complemented
68	C₂×#88322	C₂⁴	1	16	yes	—
69	C₂³×C₅₃	—	2	24	no	—
70	C₂³×C₅₃	—	2	32	no	—
71	C₂³×C₅₃	C₂×C₄²	1	24	yes	—
72	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
73	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
74	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
75	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
76	C₂×C₄×C₅₃	—	2	24	no	—
77	C₂×C₄×C₅₃	—	2	64	no	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	48	yes	—
79	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
80	C₂×C₄×C₅₃	C₂×C₄²	1	24	yes	—
81	C₂×C₄×C₅₃	C₂×M₁₆	1	48	yes	complemented
82	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
84	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
85	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
87	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
89	#88322	C₂³×C₄	1	32	yes	complemented
90	C₂²×C₄×M₁₆	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	48	no	—
92	C₂²×C₅₃	—	2	8	no	—
93	C₂²×C₅₃	C₄×M₁₆	1	16	yes	complemented
94	C₂²×C₅₃	C₂²×C₄²	1	8	yes	—
95	C₂²×C₅₃	C₂²×C₄²	1	6	yes	—
96	C₂²×C₅₃	C₂²×M₁₆	1	12	yes	—
97	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
98	C₄×C₅₃	—	2	8	no	—
99	C₄×C₅₃	C₂²×C₄²	1	6	yes	—
100	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
101	C₄×C₅₃	C₂²×M₁₆	1	16	yes	complemented
102	C₄×C₅₃	C₂⁴×C₄	1	2	yes	—
103	C₂³×M₁₆	—	53	2	no	—
104	C₂³×C₄²	D₅₃	1	1	yes	characteristic
105	C₂×C₄×M₁₆	—	53	24	no	—
106	C₂²×#97	—	53	2	no	—
107	C₂²×C₄×C₈	—	53	2	no	—
108	C₂×C₅₃	—	2	8	no	—
109	C₂×C₅₃	C₂×C₄×M₁₆	1	12	yes	complemented
110	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
111	C₂×C₅₃	C₂³×M₁₆	1	2	yes	—
112	C₂⁴×C₄	Dic₅₃	1	2	yes	—
113	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂²×M₁₆	—	53	32	no	—
115	C₂²×M₁₆	—	53	24	no	—
116	C₂³×C₈	—	53	4	no	—
117	C₂²×C₄²	Dic₅₃	1	2	yes	—
118	C₂²×C₄²	Dic₅₃	1	12	yes	—
119	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂×D₅₃	1	12	yes	—
121	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
122	C₄×M₁₆	—	53	64	no	—
123	C₂×#97	—	53	24	no	—
124	C₂×C₄×C₈	—	53	24	no	—
125	C₅₃	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
126	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
127	C₂³×C₄	—	2	16	no	—
128	C₂³×C₄	C₄×D₅₃	1	16	yes	—
129	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
130	C₂³×C₄	C₂×Dic₅₃	1	24	yes	—
131	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
132	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
133	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
136	C₂×M₁₆	—	53	192	no	—
137	C₂×M₁₆	—	53	32	no	—
138	C₂²×C₈	—	53	32	no	—
139	C₂²×C₈	—	53	24	no	—
140	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
141	C₂×C₄²	C₂×Dic₅₃	1	48	yes	—
142	C₂×C₄²	C₂×Dic₅₃	1	24	yes	—
143	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
144	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
145	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
146	C₂×C₄²	C₂²×D₅₃	1	16	yes	—
147	#97	—	53	32	no	—
148	C₄×C₈	—	53	32	no	—
149	C₂⁴	—	2	8	no	—
150	C₂⁴	C₄×Dic₅₃	1	8	yes	—
151	C₂⁴	C₂²×Dic₅₃	1	6	yes	—
152	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	96	no	—
155	C₂²×C₄	#11719917	1	16	yes	direct product factor
156	C₂²×C₄	C₄×Dic₅₃	1	48	yes	—
157	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
158	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
159	C₂²×C₄	C₂×C₄×D₅₃	1	48	yes	—
160	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
161	C₂²×C₄	C₂²×Dic₅₃	1	32	yes	—
162	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
163	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
164	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
165	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
166	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
167	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
168	M₁₆	—	53	128	no	—
169	C₂×C₈	—	53	96	no	—
170	C₂×C₈	—	53	16	no	—
171	C₄²	C₂²×Dic₅₃	1	12	yes	—
172	C₄²	C₂²×Dic₅₃	1	32	yes	—
173	C₄²	C₂²×Dic₅₃	1	12	yes	—
174	C₄²	C₂³×D₅₃	1	4	yes	—
175	C₄²	C₂³×D₅₃	1	4	yes	—
176	C₂³	—	2	32	no	—
177	C₂³	—	2	24	no	—
178	C₂³	C₂×C₄×Dic₅₃	1	24	yes	—
179	C₂³	C₂×C₄×Dic₅₃	1	4	yes	—
180	C₂³	C₂×#11719917	1	8	yes	—
181	C₂³	C₂³×Dic₅₃	1	4	yes	—
182	C₂³	C₂³×Dic₅₃	1	3	yes	—
183	C₂×C₄	—	2	24	no	—
184	C₂×C₄	—	2	64	no	—
185	C₂×C₄	C₂×C₄×Dic₅₃	1	48	yes	—
186	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
187	C₂×C₄	C₂×C₄×Dic₅₃	1	24	yes	—
188	C₂×C₄	C₂×#11719917	1	48	yes	direct product factor
189	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
190	C₂×C₄	C₂²×C₄×D₅₃	1	16	yes	—
191	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
192	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
193	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
194	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
195	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
196	C₈	—	53	32	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	48	no	—
199	C₂²	C₄×#11719917	1	16	yes	direct product factor
200	C₂²	C₂²×C₄×Dic₅₃	1	8	yes	—
201	C₂²	C₂²×C₄×Dic₅₃	1	6	yes	—
202	C₂²	C₂²×#11719917	1	12	yes	—
203	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	C₂²×C₄×Dic₅₃	1	6	yes	—
206	C₄	C₂²×C₄×Dic₅₃	1	8	yes	—
207	C₄	C₂²×#11719917	1	16	yes	direct product factor
208	C₄	C₂³×C₄×D₅₃	1	2	yes	—
209	C₂	—	2	8	no	—
210	C₂	C₂×C₄×#11719917	1	12	yes	direct product factor
211	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
212	C₂	C₂³×#11719917	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11719917	C₂	direct (N × H)
3. C₂²×#13133128	C₂	semidirect
4. C₂²×C₄×#88322	C₂	semidirect
6. C₂²×#11719917	C₄	direct (N × H)
9. C₄×#11719917	C₂²	direct (N × H)
10. C₂×#13133128	C₂²	semidirect
11. C₂×C₄×#88322	C₂²	semidirect
19. C₂×#11719917	C₂×C₄	direct (N × H)
21. C₂²×#88322	C₂×C₄	semidirect
23. #13133128	C₂³	semidirect
24. C₄×#88322	C₂³	semidirect
50. C₂²×C₄×C₅₃	M₁₆	semidirect
66. #11719917	C₂²×C₄	direct (N × H)
67. C₂×#88322	C₂²×C₄	semidirect
81. C₂×C₄×C₅₃	C₂×M₁₆	semidirect
89. #88322	C₂³×C₄	semidirect
93. C₂²×C₅₃	C₄×M₁₆	semidirect
101. C₄×C₅₃	C₂²×M₁₆	semidirect
109. C₂×C₅₃	C₂×C₄×M₁₆	semidirect
125. C₅₃	C₂²×C₄×M₁₆	semidirect
155. C₂²×C₄	#11719917	direct (N × H)
188. C₂×C₄	C₂×#11719917	direct (N × H)
199. C₂²	C₄×#11719917	direct (N × H)
207. C₄	C₂²×#11719917	direct (N × H)
210. C₂	C₂×C₄×#11719917	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×#11719917	2
C₂×C₄×#11719917	2
C₂²×#13133128	2
C₂²×C₄×#88322	2
C₂³×C₄²×C₅₃	2
C₂²×C₄×M₁₆	53

Subgroups — C22×C4×#11719917 #28247480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11719917 #28247480