Subgroups — C₂³×#13133153 #28247781

Order 13568.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 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	C₂²×#13133153	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	14	no	—
5	C₂²×#11719917	C₂²	1	28	yes	—
6	C₂³×#88322	—	2	2	no	—
7	C₂×#13133153	C₂²	1	112	yes	direct product factor
8	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₄	1	14	yes	—
10	C₂³×C₈×C₅₃	C₄	1	2	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	14	yes	—
12	C₂×#11719917	—	2	84	no	—
13	C₂×#11719917	C₂³	1	56	yes	—
14	C₂²×#88322	—	2	28	no	—
15	#13133153	C₂³	1	64	yes	direct product factor
16	C₂⁵×C₅₃	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₅₃	D₄	1	7	yes	—
20	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	Q₈	1	7	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	84	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	28	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	28	yes	—
26	C₂⁴×C₅₃	—	2	8	no	—
27	C₂⁴×C₅₃	C₄⋊C₄	1	7	yes	—
28	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂⁴×C₅₃	C₂×Q₈	1	7	yes	—
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	42	yes	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	7	yes	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	7	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	7	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
37	C₂²×C₄×C₅₃	C₂×Q₈	1	21	yes	—
38	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	56	yes	—
40	C₂×C₈×C₅₃	C₂²×C₄	1	56	yes	—
41	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
42	#11719917	—	2	56	no	—
43	#11719917	C₂⁴	1	16	yes	—
44	C₂×#88322	—	2	56	no	—
45	C₂³×C₅₃	—	2	28	no	—
46	C₂³×C₅₃	—	2	28	no	—
47	C₂³×C₅₃	#108	1	8	yes	complemented
48	C₂³×C₅₃	C₂×C₄⋊C₄	1	21	yes	—
49	C₂³×C₅₃	C₂×C₄⋊C₄	1	7	yes	—
50	C₂³×C₅₃	C₂²×Q₈	1	7	yes	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	42	yes	—
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	21	yes	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	21	yes	—
54	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
55	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
56	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
57	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
58	C₂×C₄×C₅₃	C₂²×Q₈	1	7	yes	—
59	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
60	C₈×C₅₃	C₂³×C₄	1	16	yes	—
61	#88322	—	2	16	no	—
62	C₂³×#108	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	42	no	—
64	C₂²×C₅₃	—	2	14	no	—
65	C₂²×C₅₃	C₂×#108	1	28	yes	complemented
66	C₂²×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
67	C₂²×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
68	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
70	C₄×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
71	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
72	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
73	C₂³×M₁₆	—	53	2	no	—
74	C₂⁴×C₈	D₅₃	1	1	yes	characteristic
75	C₂²×#108	—	53	28	no	—
76	C₂×C₅₃	—	2	7	no	—
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	C₂²×#108	1	14	yes	complemented
79	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
80	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
81	C₂²×M₁₆	—	53	28	no	—
82	C₂²×M₁₆	—	106	14	no	—
83	C₂³×C₈	—	106	2	no	—
84	C₂³×C₈	Dic₅₃	1	2	yes	—
85	C₂³×C₈	Dic₅₃	1	14	yes	—
86	C₂³×C₈	C₂×D₅₃	1	14	yes	—
87	C₂×#108	—	53	112	no	—
88	C₅₃	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
89	C₂⁵	Dic₁₀₆	1	1	yes	characteristic
90	C₂³×C₄	Dic₁₀₆	1	7	yes	—
91	C₂³×C₄	D₂₁₂	1	7	yes	—
92	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
93	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
94	C₂³×C₄	C₂×Dic₅₃	1	7	yes	—
95	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
96	C₂×M₁₆	—	53	56	no	—
97	C₂×M₁₆	—	106	84	no	—
98	C₂²×C₈	—	106	28	no	—
99	C₂²×C₈	C₂×Dic₅₃	1	84	yes	—
100	C₂²×C₈	C₂×Dic₅₃	1	28	yes	—
101	C₂²×C₈	C₂²×D₅₃	1	28	yes	—
102	#108	—	53	64	no	—
103	C₂⁴	—	2	8	no	—
104	C₂⁴	C₄⋊Dic₅₃	1	7	yes	—
105	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
106	C₂⁴	C₂×Dic₁₀₆	1	7	yes	—
107	C₂²×C₄	C₄⋊Dic₅₃	1	42	yes	—
108	C₂²×C₄	C₄⋊Dic₅₃	1	7	yes	—
109	C₂²×C₄	C₄⋊Dic₅₃	1	7	yes	—
110	C₂²×C₄	C₂×Dic₁₀₆	1	21	yes	—
111	C₂²×C₄	C₂×D₂₁₂	1	7	yes	—
112	C₂²×C₄	C₂×D₂₁₂	1	21	yes	—
113	C₂²×C₄	C₂²×Dic₅₃	1	21	yes	—
114	C₂²×C₄	C₂²×Dic₅₃	1	7	yes	—
115	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
116	M₁₆	—	53	16	no	—
117	M₁₆	—	106	56	no	—
118	C₂×C₈	—	106	56	no	—
119	C₂×C₈	C₂²×Dic₅₃	1	56	yes	—
120	C₂×C₈	C₂²×Dic₅₃	1	56	yes	—
121	C₂×C₈	C₂³×D₅₃	1	8	yes	—
122	C₂³	—	2	28	no	—
123	C₂³	—	2	28	no	—
124	C₂³	#13133153	1	8	yes	direct product factor
125	C₂³	C₂×C₄⋊Dic₅₃	1	21	yes	—
126	C₂³	C₂×C₄⋊Dic₅₃	1	7	yes	—
127	C₂³	C₂²×Dic₁₀₆	1	7	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₅₃	1	42	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₅₃	1	21	yes	—
130	C₂×C₄	C₂×C₄⋊Dic₅₃	1	21	yes	—
131	C₂×C₄	C₂³×Dic₅₃	1	7	yes	—
132	C₂×C₄	C₂³×Dic₅₃	1	7	yes	—
133	C₂×C₄	C₂²×D₂₁₂	1	7	yes	—
134	C₂×C₄	C₂²×D₂₁₂	1	7	yes	—
135	C₂×C₄	C₂²×Dic₁₀₆	1	7	yes	—
136	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
137	C₈	—	106	16	no	—
138	C₈	C₂³×Dic₅₃	1	16	yes	—
139	C₂²	—	2	42	no	—
140	C₂²	—	2	14	no	—
141	C₂²	C₂×#13133153	1	28	yes	direct product factor
142	C₂²	C₂²×C₄⋊Dic₅₃	1	7	yes	—
143	C₂²	C₂²×C₄⋊Dic₅₃	1	7	yes	—
144	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
145	C₄	C₂²×C₄⋊Dic₅₃	1	7	yes	—
146	C₄	C₂²×C₄⋊Dic₅₃	1	7	yes	—
147	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
148	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
149	C₂	—	2	7	no	—
150	C₂	—	2	1	no	—
151	C₂	C₂²×#13133153	1	14	yes	direct product factor
152	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₁₂
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂⁴×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₄⋊C₄
prime radical	C₂⁵×C₅₃	Q₈
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁴×C₈	D₅₃
hypocenter	C₅₃	C₂³×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#13133153	C₂	direct (N × H)
7. C₂×#13133153	C₂²	direct (N × H)
15. #13133153	C₂³	direct (N × H)
47. C₂³×C₅₃	#108	semidirect
65. C₂²×C₅₃	C₂×#108	semidirect
78. C₂×C₅₃	C₂²×#108	semidirect
88. C₅₃	C₂³×#108	semidirect
124. C₂³	#13133153	direct (N × H)
141. C₂²	C₂×#13133153	direct (N × H)
151. C₂	C₂²×#13133153	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
C₂²×#13133153	2
C₂⁴×C₈×C₅₃	2
C₂³×#108	53

Subgroups — C23×#13133153 #28247781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133153 #28247781