Subgroups — C₂²×#16810450 #27537115

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 164 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
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810450	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂²×#11719908	C₄	1	8	yes	complemented
7	C₂²×#11719908	C₂²	1	6	yes	—
8	C₂×#13133151	—	2	3	no	—
9	C₂×#13133151	C₂²	1	6	yes	—
10	#16810450	C₂²	1	16	yes	direct product factor
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
13	C₂×#107×C₅₃	—	2	3	no	—
14	C₂×#107×C₅₃	C₂²	1	6	yes	—
15	C₂³×Dic₅₃	—	4	1	no	—
16	C₂×C₄⋊Dic₅₃	—	2	6	no	—
17	C₂×C₄⋊Dic₅₃	—	4	3	no	—
18	C₂×#11719908	C₂×C₄	1	24	yes	complemented
19	C₂×#11719908	C₂³	1	4	yes	—
20	#13133151	—	2	6	no	—
21	#13133151	C₂³	1	4	yes	—
22	C₂³×C₄×C₅₃	—	4	1	no	—
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	D₄	1	3	yes	—
26	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
28	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
29	C₂×C₄⋊C₄×C₅₃	—	4	3	no	—
30	#107×C₅₃	—	2	6	no	—
31	#107×C₅₃	C₂³	1	4	yes	—
32	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	4	6	no	—
34	C₂²×C₄×C₅₃	—	8	4	no	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₅₃	D₈	1	3	yes	—
37	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
39	C₂²×Dic₅₃	—	4	6	no	—
40	C₂²×Dic₅₃	—	8	4	no	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	6	no	—
48	C₄⋊Dic₅₃	—	2	4	no	—
49	C₄⋊Dic₅₃	—	4	6	no	—
50	#11719908	C₂²×C₄	1	8	yes	complemented
51	C₂³×C₅₃	#107	1	4	yes	—
52	C₂³×C₅₃	SD₃₂	1	4	yes	—
53	C₂³×C₅₃	SD₃₂	1	4	yes	—
54	C₂³×C₅₃	C₂×D₈	1	3	yes	—
55	C₂×C₄×C₅₃	—	4	4	no	—
56	C₂×C₄×C₅₃	—	8	12	no	—
57	C₂×C₄×C₅₃	#107	1	12	yes	—
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
59	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
60	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
61	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×Dic₅₃	—	4	4	no	—
63	C₂×Dic₅₃	—	8	12	no	—
64	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
65	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×#367	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#367	1	16	yes	complemented
69	C₂²×C₅₃	C₂×#107	1	6	yes	—
70	C₂²×C₅₃	C₂×SD₃₂	1	6	yes	—
71	C₂²×C₅₃	C₂×SD₃₂	1	6	yes	—
72	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
73	C₄×C₅₃	—	8	4	no	—
74	C₄×C₅₃	C₂×#107	1	6	yes	—
75	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
77	Dic₅₃	—	8	4	no	—
78	C₂³×C₁₆	—	53	1	no	—
79	C₂²×#107	—	53	1	no	—
80	C₂²×#107	D₅₃	1	1	yes	characteristic
81	C₂×#367	—	53	12	no	—
82	C₂×C₅₃	C₂×#367	1	12	yes	complemented
83	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
84	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
85	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
86	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂²×C₄⋊C₄	—	2	1	no	—
88	C₂²×C₄⋊C₄	—	106	1	no	—
89	C₂²×C₁₆	—	53	8	no	—
90	C₂²×C₁₆	—	53	6	no	—
91	C₂×#107	—	2	3	no	—
92	C₂×#107	—	53	6	no	—
93	C₂×#107	—	106	3	no	—
94	C₂×#107	C₂×D₅₃	1	6	yes	—
95	#367	—	53	16	no	—
96	C₅₃	C₂²×#367	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	—	4	1	no	—
98	C₂³×C₄	—	212	1	no	—
99	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
100	C₂²×C₈	Dic₁₀₆	1	4	yes	—
101	C₂²×C₈	C₄×D₅₃	1	4	yes	—
102	C₂²×C₈	C₂²⋊D₅₃	1	3	yes	—
103	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
104	C₂×C₄⋊C₄	—	2	6	no	—
105	C₂×C₄⋊C₄	—	4	3	no	—
106	C₂×C₄⋊C₄	—	106	6	no	—
107	C₂×C₄⋊C₄	—	212	3	no	—
108	C₂×C₁₆	—	53	24	no	—
109	C₂×C₁₆	—	53	4	no	—
110	#107	—	2	6	no	—
111	#107	—	53	4	no	—
112	#107	—	106	6	no	—
113	#107	C₂²×D₅₃	1	4	yes	—
114	C₂⁴	#11719922	1	1	yes	characteristic
115	C₂²×C₄	—	4	6	no	—
116	C₂²×C₄	—	8	4	no	—
117	C₂²×C₄	—	424	4	no	—
118	C₂²×C₄	—	212	6	no	—
119	C₂²×C₄	#11719919	1	4	yes	—
120	C₂²×C₄	#11719922	1	3	yes	—
121	C₂²×C₄	#11719925	1	4	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
123	C₂×C₈	#11719919	1	12	yes	—
124	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
125	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
126	C₂×C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
127	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	6	no	—
130	C₄⋊C₄	—	106	4	no	—
131	C₄⋊C₄	—	212	6	no	—
132	C₁₆	—	53	8	no	—
133	C₂³	#13133150	1	4	yes	—
134	C₂³	#13133160	1	4	yes	—
135	C₂³	#13133161	1	4	yes	—
136	C₂³	C₂×#11719922	1	3	yes	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	12	no	—
139	C₂×C₄	—	424	12	no	—
140	C₂×C₄	—	212	4	no	—
141	C₂×C₄	#13133150	1	12	yes	—
142	C₂×C₄	C₂×#11719919	1	6	yes	—
143	C₂×C₄	C₂×#11719922	1	3	yes	—
144	C₂×C₄	C₂×#11719925	1	6	yes	—
145	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₈	C₂×#11719919	1	6	yes	—
147	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
148	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
149	C₂²	#16810450	1	16	yes	direct product factor
150	C₂²	C₂×#13133150	1	6	yes	—
151	C₂²	C₂×#13133160	1	6	yes	—
152	C₂²	C₂×#13133161	1	6	yes	—
153	C₂²	C₂²×#11719922	1	1	yes	characteristic
154	C₄	—	8	4	no	—
155	C₄	—	424	4	no	—
156	C₄	C₂×#13133150	1	6	yes	—
157	C₄	C₂²×#11719919	1	1	yes	characteristic
158	C₄	C₂²×#11719925	1	1	yes	characteristic
159	C₂	C₂×#16810450	1	12	yes	direct product factor
160	C₂	C₂²×#13133150	1	1	yes	characteristic
161	C₂	C₂²×#13133160	1	1	yes	characteristic
162	C₂	C₂²×#13133161	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂²×#367
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. C₂×#16810450	C₂	direct (N × H)
6. C₂²×#11719908	C₄	semidirect
10. #16810450	C₂²	direct (N × H)
18. C₂×#11719908	C₂×C₄	semidirect
50. #11719908	C₂²×C₄	semidirect
68. C₂²×C₅₃	#367	semidirect
82. C₂×C₅₃	C₂×#367	semidirect
96. C₅₃	C₂²×#367	semidirect
149. C₂²	#16810450	direct (N × H)
159. C₂	C₂×#16810450	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×#16810450	2
C₂²×#107×C₅₃	2
C₂²×#367	53

Subgroups — C22×#16810450 #27537115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810450 #27537115