Subgroups — C₂³×#13133150 #28247736

Order 13568.

3614 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133150	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	—	2	1	no	—
6	C₂³×#88322	C₄	1	16	yes	complemented
7	C₂³×#88322	C₂²	1	14	yes	—
8	C₂²×C₄⋊Dic₅₃	—	2	7	no	—
9	C₂²×C₄⋊Dic₅₃	C₂²	1	14	yes	—
10	C₂×#13133150	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄×C₅₃	—	2	1	no	—
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₅₃	—	2	7	no	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	14	yes	—
15	C₂³×Dic₅₃	—	2	14	no	—
16	C₂³×Dic₅₃	—	4	8	no	—
17	C₂²×#88322	C₂×C₄	1	112	yes	complemented
18	C₂²×#88322	C₂³	1	28	yes	—
19	C₂×C₄⋊Dic₅₃	—	2	42	no	—
20	C₂×C₄⋊Dic₅₃	C₂³	1	28	yes	—
21	#13133150	C₂³	1	64	yes	direct product factor
22	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	—	2	14	no	—
24	C₂³×C₄×C₅₃	—	4	8	no	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₃	D₄	1	7	yes	—
27	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
29	C₂×C₄⋊C₄×C₅₃	—	2	42	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂³	1	28	yes	—
31	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
32	C₂⁴×C₅₃	D₈	1	8	yes	—
33	C₂⁴×C₅₃	Q₁₆	1	8	yes	—
34	C₂⁴×C₅₃	C₂×D₄	1	7	yes	—
35	C₂²×C₄×C₅₃	—	2	28	no	—
36	C₂²×C₄×C₅₃	—	4	56	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	56	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
40	C₂²×C₄×C₅₃	C₂×Q₈	1	28	yes	—
41	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
42	C₂²×Dic₅₃	—	2	28	no	—
43	C₂²×Dic₅₃	—	4	56	no	—
44	C₄⋊C₄×C₅₃	—	2	28	no	—
45	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
46	C₄⋊Dic₅₃	—	2	28	no	—
47	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
48	C₂×#88322	C₂²×C₄	1	112	yes	complemented
49	C₂×#88322	C₂⁴	1	8	yes	—
50	C₂³×C₅₃	#107	1	64	yes	complemented
51	C₂³×C₅₃	C₂×C₄⋊C₄	1	28	yes	—
52	C₂³×C₅₃	C₂×D₈	1	28	yes	—
53	C₂³×C₅₃	C₂×Q₁₆	1	28	yes	—
54	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
55	C₂×C₄×C₅₃	—	2	8	no	—
56	C₂×C₄×C₅₃	—	4	56	no	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	84	yes	—
58	C₂×C₄×C₅₃	C₂³×C₄	1	14	yes	—
59	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
60	C₂×C₄×C₅₃	C₂²×Q₈	1	14	yes	—
61	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
62	C₂×Dic₅₃	—	2	8	no	—
63	C₂×Dic₅₃	—	4	56	no	—
64	#88322	C₂³×C₄	1	16	yes	complemented
65	C₂³×#107	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	C₂×#107	1	112	yes	complemented
67	C₂²×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
68	C₂²×C₅₃	C₂²×D₈	1	14	yes	—
69	C₂²×C₅₃	C₂²×Q₁₆	1	14	yes	—
70	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
71	C₄×C₅₃	—	4	8	no	—
72	C₄×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
73	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
74	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
75	Dic₅₃	—	4	8	no	—
76	C₂⁴×C₈	—	53	1	no	—
77	C₂³×C₄⋊C₄	—	53	1	no	—
78	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
79	C₂²×#107	—	53	28	no	—
80	C₂×C₅₃	C₂²×#107	1	28	yes	complemented
81	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
83	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
84	C₂⁴×C₄	—	2	1	no	—
85	C₂⁴×C₄	—	106	1	no	—
86	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
87	C₂³×C₈	—	53	16	no	—
88	C₂³×C₈	—	53	14	no	—
89	C₂²×C₄⋊C₄	—	2	7	no	—
90	C₂²×C₄⋊C₄	—	53	14	no	—
91	C₂²×C₄⋊C₄	—	106	7	no	—
92	C₂²×C₄⋊C₄	C₂×D₅₃	1	14	yes	—
93	C₂×#107	—	53	112	no	—
94	C₅₃	C₂³×#107	1	1	yes	characteristic, complemented, Sylow
95	C₂⁵	C₂²⋊D₅₃	1	1	yes	characteristic
96	C₂³×C₄	—	2	14	no	—
97	C₂³×C₄	—	4	8	no	—
98	C₂³×C₄	—	106	14	no	—
99	C₂³×C₄	—	212	8	no	—
100	C₂³×C₄	Dic₁₀₆	1	8	yes	—
101	C₂³×C₄	C₄×D₅₃	1	8	yes	—
102	C₂³×C₄	C₂²⋊D₅₃	1	7	yes	—
103	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
104	C₂²×C₈	—	53	112	no	—
105	C₂²×C₈	—	53	28	no	—
106	C₂×C₄⋊C₄	—	2	42	no	—
107	C₂×C₄⋊C₄	—	53	28	no	—
108	C₂×C₄⋊C₄	—	106	42	no	—
109	C₂×C₄⋊C₄	C₂²×D₅₃	1	28	yes	—
110	#107	—	53	64	no	—
111	C₂⁴	#11719919	1	8	yes	—
112	C₂⁴	#11719922	1	8	yes	—
113	C₂⁴	#11719925	1	8	yes	—
114	C₂⁴	C₂×C₂²⋊D₅₃	1	7	yes	—
115	C₂²×C₄	—	2	28	no	—
116	C₂²×C₄	—	4	56	no	—
117	C₂²×C₄	—	106	28	no	—
118	C₂²×C₄	—	212	56	no	—
119	C₂²×C₄	#11719919	1	56	yes	—
120	C₂²×C₄	C₂×Dic₁₀₆	1	28	yes	—
121	C₂²×C₄	C₂×C₄×D₅₃	1	28	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
123	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
124	C₂×C₈	—	53	112	no	—
125	C₂×C₈	—	53	8	no	—
126	C₄⋊C₄	—	2	28	no	—
127	C₄⋊C₄	—	53	8	no	—
128	C₄⋊C₄	—	106	28	no	—
129	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
130	C₂³	#13133150	1	64	yes	direct product factor
131	C₂³	C₂×#11719919	1	28	yes	—
132	C₂³	C₂×#11719922	1	28	yes	—
133	C₂³	C₂×#11719925	1	28	yes	—
134	C₂³	C₂²×C₂²⋊D₅₃	1	7	yes	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	4	56	no	—
137	C₂×C₄	—	106	8	no	—
138	C₂×C₄	—	212	56	no	—
139	C₂×C₄	C₂×#11719919	1	84	yes	—
140	C₂×C₄	C₂²×C₄×D₅₃	1	14	yes	—
141	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
142	C₂×C₄	C₂²×Dic₁₀₆	1	14	yes	—
143	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
144	C₈	—	53	16	no	—
145	C₂²	C₂×#13133150	1	112	yes	direct product factor
146	C₂²	C₂²×#11719919	1	14	yes	—
147	C₂²	C₂²×#11719922	1	14	yes	—
148	C₂²	C₂²×#11719925	1	14	yes	—
149	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
150	C₄	—	4	8	no	—
151	C₄	—	212	8	no	—
152	C₄	C₂²×#11719919	1	14	yes	—
153	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
154	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
155	C₂	C₂²×#13133150	1	28	yes	direct product factor
156	C₂	C₂³×#11719919	1	1	yes	characteristic
157	C₂	C₂³×#11719922	1	1	yes	characteristic
158	C₂	C₂³×#11719925	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, prime radical	C₂⁵×C₅₃	D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#107
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
3. C₂²×#13133150	C₂	direct (N × H)
6. C₂³×#88322	C₄	semidirect
10. C₂×#13133150	C₂²	direct (N × H)
17. C₂²×#88322	C₂×C₄	semidirect
21. #13133150	C₂³	direct (N × H)
48. C₂×#88322	C₂²×C₄	semidirect
50. C₂³×C₅₃	#107	semidirect
64. #88322	C₂³×C₄	semidirect
66. C₂²×C₅₃	C₂×#107	semidirect
80. C₂×C₅₃	C₂²×#107	semidirect
94. C₅₃	C₂³×#107	semidirect
130. C₂³	#13133150	direct (N × H)
145. C₂²	C₂×#13133150	direct (N × H)
155. C₂	C₂²×#13133150	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×#88322	2
C₂³×C₄⋊Dic₅₃	2
C₂²×#13133150	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×#107	53

Subgroups — C23×#13133150 #28247736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133150 #28247736