Subgroups — C₂²×#16810730 #28248131

Order 13568.

2790 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	C₂²×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	C₂×#16810730	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133141	C₂	1	2	yes	maximal
5	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×Q₈×Dic₅₃	—	2	12	no	—
9	C₂×Q₈×Dic₅₃	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₅₃	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810730	C₂²	1	64	yes	direct product factor
15	C₂×#13133149	C₂²	1	12	yes	—
16	C₂×#13133141	C₂²	1	24	yes	—
17	C₂×#13133127	C₂²	1	12	yes	—
18	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂²×Q₁₆×C₅₃	C₄	1	16	yes	complemented
20	C₂²×Q₁₆×C₅₃	C₂²	1	12	yes	—
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₅₃	—	2	2	no	—
23	C₂³×Dic₅₃	D₄	1	2	yes	—
24	C₂²×#88322	—	2	4	no	—
25	C₂²×#88322	C₂³	1	6	yes	—
26	Q₈×Dic₅₃	—	2	48	no	—
27	Q₈×Dic₅₃	C₂³	1	32	yes	—
28	C₂×C₄⋊Dic₅₃	—	2	6	no	—
29	C₂×C₄⋊Dic₅₃	—	2	24	no	—
30	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
31	C₂×C₄×Dic₅₃	—	2	24	no	—
32	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
34	#13133149	C₂³	1	16	yes	—
35	#13133141	C₂³	1	32	yes	—
36	#13133127	C₂³	1	16	yes	—
37	C₂²×Q₈×C₅₃	—	2	6	no	—
38	C₂²×Q₈×C₅₃	—	2	8	no	—
39	C₂²×Q₈×C₅₃	C₂×C₄	1	16	yes	—
40	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
41	C₂³×C₄×C₅₃	—	2	2	no	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₅₃	C₂×C₄	1	96	yes	complemented
44	C₂×Q₁₆×C₅₃	C₂³	1	16	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	—	2	48	no	—
49	C₂×Q₈×C₅₃	—	2	12	no	—
50	C₂×Q₈×C₅₃	—	2	24	no	—
51	C₂×Q₈×C₅₃	C₂²×C₄	1	48	yes	—
52	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
53	C₂²×C₄×C₅₃	—	2	16	no	—
54	C₂²×C₄×C₅₃	—	2	12	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅₃	#41	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
59	C₂²×Dic₅₃	—	2	8	no	—
60	C₂²×Dic₅₃	—	2	12	no	—
61	C₂²×Dic₅₃	—	4	8	no	—
62	C₂²×Dic₅₃	C₂×D₄	1	12	yes	—
63	Q₁₆×C₅₃	C₂²×C₄	1	64	yes	complemented
64	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	—
65	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
66	C₄⋊Dic₅₃	—	2	12	no	—
67	C₄⋊Dic₅₃	—	2	32	no	—
68	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
69	C₄×Dic₅₃	—	2	32	no	—
70	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
71	C₄×Dic₅₃	C₂⁴	1	4	yes	—
72	C₂×#88322	—	2	12	no	—
73	C₂×#88322	C₂⁴	1	4	yes	—
74	C₂³×C₅₃	C₄×D₄	1	4	yes	—
75	C₂³×C₅₃	#137	1	8	yes	—
76	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
77	Q₈×C₅₃	—	2	48	no	—
78	Q₈×C₅₃	—	2	8	no	—
79	Q₈×C₅₃	C₂³×C₄	1	16	yes	—
80	C₂×C₄×C₅₃	—	2	48	no	—
81	C₂×C₄×C₅₃	—	2	8	no	—
82	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
84	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	24	no	—
88	C₂×Dic₅₃	—	2	8	no	—
89	C₂×Dic₅₃	—	4	24	no	—
90	C₂×Dic₅₃	C₂²×D₄	1	8	yes	—
91	C₈×C₅₃	C₂³×C₄	1	8	yes	—
92	#88322	—	2	4	no	—
93	C₂²×#441	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#441	1	16	yes	complemented
95	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
96	C₂²×C₅₃	C₂×#137	1	12	yes	—
97	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
98	C₄×C₅₃	—	2	16	no	—
99	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
100	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
102	Dic₅₃	—	2	8	no	—
103	Dic₅₃	—	4	8	no	—
104	C₂³×Q₁₆	D₅₃	1	1	yes	characteristic
105	C₂²×C₄×Q₈	—	53	2	no	—
106	C₂×#441	—	53	24	no	—
107	C₂²×#106	—	53	1	no	—
108	C₂²×#103	—	53	2	no	—
109	C₂²×#97	—	53	1	no	—
110	C₂×C₅₃	C₂×#441	1	12	yes	complemented
111	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
112	C₂×C₅₃	C₂²×#137	1	2	yes	—
113	C₂³×Q₈	C₂×D₅₃	1	2	yes	—
114	C₂²×Q₁₆	Dic₅₃	1	16	yes	complemented
115	C₂²×Q₁₆	C₂×D₅₃	1	12	yes	—
116	C₂³×C₈	—	53	1	no	—
117	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄×Q₈	—	53	24	no	—
119	C₂×C₄×Q₈	—	106	12	no	—
120	C₂²×C₄⋊C₄	—	53	2	no	—
121	C₂²×C₄⋊C₄	—	106	2	no	—
122	C₂²×C₄²	—	53	1	no	—
123	C₂²×C₄²	—	106	2	no	—
124	#441	—	53	64	no	—
125	C₂×#106	—	53	12	no	—
126	C₂×#103	—	53	24	no	—
127	C₂×#97	—	53	12	no	—
128	C₅₃	C₂²×#441	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	6	no	—
130	C₂²×Q₈	—	2	8	no	—
131	C₂²×Q₈	C₂×Dic₅₃	1	16	yes	—
132	C₂²×Q₈	C₂²×D₅₃	1	12	yes	—
133	C₂³×C₄	—	2	2	no	—
134	C₂³×C₄	—	53	2	no	—
135	C₂³×C₄	—	106	2	no	—
136	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
137	C₂×Q₁₆	C₂×Dic₅₃	1	96	yes	complemented
138	C₂×Q₁₆	C₂²×D₅₃	1	16	yes	—
139	C₂²×C₈	—	53	6	no	—
140	C₂²×C₈	—	106	4	no	—
141	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
142	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
143	C₄×Q₈	—	53	32	no	—
144	C₄×Q₈	—	106	48	no	—
145	C₂×C₄⋊C₄	—	53	12	no	—
146	C₂×C₄⋊C₄	—	106	24	no	—
147	C₂×C₄⋊C₄	—	106	6	no	—
148	C₂×C₄²	—	53	6	no	—
149	C₂×C₄²	—	53	6	no	—
150	C₂×C₄²	—	106	24	no	—
151	#106	—	53	16	no	—
152	#103	—	53	32	no	—
153	#97	—	53	16	no	—
154	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
155	C₂×Q₈	—	2	48	no	—
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	2	24	no	—
158	C₂×Q₈	C₂²×Dic₅₃	1	48	yes	—
159	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
160	C₂²×C₄	—	2	16	no	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	53	12	no	—
163	C₂²×C₄	—	106	8	no	—
164	C₂²×C₄	—	106	12	no	—
165	C₂²×C₄	—	212	8	no	—
166	C₂²×C₄	D₄×D₅₃	1	3	yes	—
167	C₂²×C₄	#11719947	1	4	yes	—
168	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
169	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
170	Q₁₆	C₂²×Dic₅₃	1	64	yes	complemented
171	C₂×C₈	—	53	4	no	—
172	C₂×C₈	—	106	12	no	—
173	C₂×C₈	C₂²×Dic₅₃	1	24	yes	—
174	C₂×C₈	C₂³×D₅₃	1	4	yes	—
175	C₄⋊C₄	—	53	8	no	—
176	C₄⋊C₄	—	106	32	no	—
177	C₄⋊C₄	—	106	12	no	—
178	C₄²	—	53	12	no	—
179	C₄²	—	53	4	no	—
180	C₄²	—	106	32	no	—
181	C₂³	D₄×Dic₅₃	1	4	yes	—
182	C₂³	#13133249	1	8	yes	—
183	C₂³	C₂×D₄×D₅₃	1	3	yes	—
184	Q₈	—	2	48	no	—
185	Q₈	—	2	8	no	—
186	Q₈	C₂³×Dic₅₃	1	16	yes	—
187	C₂×C₄	—	2	48	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	53	8	no	—
190	C₂×C₄	—	106	24	no	—
191	C₂×C₄	—	106	8	no	—
192	C₂×C₄	—	212	24	no	—
193	C₂×C₄	D₄×Dic₅₃	1	12	yes	—
194	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
195	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
196	C₂×C₄	C₂×#11719947	1	6	yes	—
197	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
198	C₈	—	106	4	no	—
199	C₈	C₂³×Dic₅₃	1	8	yes	—
200	C₂²	#16810730	1	16	yes	direct product factor
201	C₂²	C₂×D₄×Dic₅₃	1	6	yes	—
202	C₂²	C₂×#13133249	1	12	yes	—
203	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
204	C₄	—	2	16	no	—
205	C₄	—	106	8	no	—
206	C₄	—	212	8	no	—
207	C₄	C₂×D₄×Dic₅₃	1	6	yes	—
208	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
209	C₄	C₂²×#11719947	1	1	yes	characteristic
210	C₂	C₂×#16810730	1	12	yes	direct product factor
211	C₂	C₂²×D₄×Dic₅₃	1	1	yes	characteristic
212	C₂	C₂²×#13133249	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×Q₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×Q₁₆	D₅₃
hypocenter	C₅₃	C₂²×#441
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810730	C₂	direct (N × H)
14. #16810730	C₂²	direct (N × H)
19. C₂²×Q₁₆×C₅₃	C₄	semidirect
43. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
63. Q₁₆×C₅₃	C₂²×C₄	semidirect
94. C₂²×C₅₃	#441	semidirect
110. C₂×C₅₃	C₂×#441	semidirect
114. C₂²×Q₁₆	Dic₅₃	semidirect
128. C₅₃	C₂²×#441	semidirect
137. C₂×Q₁₆	C₂×Dic₅₃	semidirect
170. Q₁₆	C₂²×Dic₅₃	semidirect
200. C₂²	#16810730	direct (N × H)
210. C₂	C₂×#16810730	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×Q₈×Dic₅₃	2
C₂×#16810730	2
C₂²×#13133149	2
C₂²×#13133141	2
C₂²×#13133127	2
C₂³×Q₁₆×C₅₃	2
C₂²×#441	53

Subgroups — C22×#16810730 #28248131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810730 #28248131