Subgroups — C₂×C₄²×#88324 #28317889

Order 13568.

2046 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88324	C₂	1	6	yes	maximal
2	C₄²×#88324	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88324	C₂²	1	4	yes	—
5	C₂×C₄×#88324	C₄	1	48	yes	direct product factor
6	C₂×C₄×#88324	C₂²	1	24	yes	—
7	C₂²×C₄×Dic₅₃	C₄	1	3	yes	—
8	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
9	C₄²×Dic₅₃	C₄	1	4	yes	—
10	C₄²×Dic₅₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88324	C₂×C₄	1	48	yes	—
13	C₂²×#88324	C₂³	1	8	yes	—
14	C₄×#88324	C₂×C₄	1	96	yes	direct product factor
15	C₂³×Dic₅₃	C₂×C₄	1	3	yes	—
16	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₈	1	24	yes	complemented
18	C₂×C₄×Dic₅₃	C₂×C₄	1	24	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
21	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₅₃	C₈	1	8	yes	complemented
24	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
25	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄²	1	12	yes	—
27	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
28	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
29	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
30	C₂²×Dic₅₃	C₄²	1	12	yes	—
31	C₂²×Dic₅₃	C₂×C₈	1	24	yes	—
32	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
34	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
35	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
36	C₂×#88324	C₄²	1	64	yes	direct product factor
37	C₂×#88324	C₂²×C₄	1	48	yes	—
38	C₄²×C₅₃	C₂×C₈	1	16	yes	complemented
39	C₄×Dic₅₃	C₄²	1	24	yes	—
40	C₄×Dic₅₃	C₂×C₈	1	48	yes	complemented
41	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
42	C₂³×C₅₃	C₂×C₄²	1	6	yes	—
43	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
44	C₂³×C₅₃	C₂²×C₈	1	6	yes	—
45	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄×C₈	1	48	yes	complemented
47	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
49	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
50	C₂×Dic₅₃	C₄×C₈	1	48	yes	complemented
51	C₂×Dic₅₃	C₂×C₄²	1	6	yes	—
52	C₂×Dic₅₃	C₂×C₄²	1	16	yes	—
53	C₂×Dic₅₃	C₂×C₄²	1	12	yes	—
54	C₂×Dic₅₃	C₂²×C₈	1	24	yes	—
55	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
56	#88324	C₂×C₄²	1	32	yes	direct product factor
57	C₂×C₄²×C₈	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₄³	1	4	yes	—
59	C₂²×C₅₃	C₂×C₄×C₈	1	12	yes	—
60	C₂²×C₅₃	C₂×C₄×C₈	1	12	yes	—
61	C₂²×C₅₃	C₂²×C₄²	1	3	yes	—
62	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
63	C₂²×C₅₃	C₂³×C₈	1	3	yes	—
64	C₄×C₅₃	C₂×C₄×C₈	1	24	yes	complemented
65	Dic₅₃	C₄³	1	4	yes	—
66	Dic₅₃	C₂×C₄×C₈	1	24	yes	complemented
67	Dic₅₃	C₂²×C₄²	1	4	yes	—
68	C₂²×C₄×C₈	—	53	6	no	—
69	C₂×C₄³	—	53	1	no	—
70	C₄²×C₈	—	53	8	no	—
71	C₂×C₅₃	C₄²×C₈	1	8	yes	complemented
72	C₂×C₅₃	C₂×C₄³	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×C₄×C₈	1	3	yes	—
74	C₂×C₅₃	C₂²×C₄×C₈	1	3	yes	—
75	C₂³×C₈	—	53	4	no	—
76	C₂²×C₄²	—	53	3	no	—
77	C₂²×C₄²	—	53	3	no	—
78	C₂²×C₄²	#6157	1	1	yes	characteristic
79	C₂×C₄×C₈	—	53	48	no	—
80	C₂×C₄×C₈	—	53	24	no	—
81	C₄³	—	53	4	no	—
82	C₄³	—	53	4	no	—
83	C₅₃	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
84	C₂³×C₄	—	53	1	no	—
85	C₂³×C₄	—	53	3	no	—
86	C₂³×C₄	C₂×#6157	1	3	yes	—
87	C₂²×C₈	—	53	8	no	—
88	C₂²×C₈	—	53	48	no	—
89	C₂×C₄²	—	53	24	no	—
90	C₂×C₄²	—	53	6	no	—
91	C₂×C₄²	—	53	12	no	—
92	C₂×C₄²	—	53	6	no	—
93	C₂×C₄²	—	53	24	no	—
94	C₂×C₄²	#88324	1	8	yes	direct product factor
95	C₂×C₄²	C₂×#6157	1	4	yes	—
96	C₄×C₈	—	53	96	no	—
97	C₂⁴	C₂²×#6157	1	1	yes	characteristic
98	C₂²×C₄	—	53	12	no	—
99	C₂²×C₄	—	53	12	no	—
100	C₂²×C₄	—	53	24	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	6	no	—
103	C₂²×C₄	—	53	1	no	—
104	C₂²×C₄	D₅₃⋊C₈	1	12	yes	—
105	C₂²×C₄	C₄×#6157	1	12	yes	—
106	C₂²×C₄	C₂×#88324	1	12	yes	—
107	C₂²×C₄	C₂²×#6157	1	6	yes	—
108	C₂×C₈	—	53	64	no	—
109	C₂×C₈	—	53	48	no	—
110	C₄²	—	53	24	no	—
111	C₄²	—	53	24	no	—
112	C₄²	—	53	48	no	—
113	C₄²	C₂×#88324	1	16	yes	direct product factor
114	C₂³	C₂×C₄×#6157	1	6	yes	—
115	C₂³	C₂²×#88324	1	2	yes	—
116	C₂³	C₂×D₅₃⋊C₈	1	6	yes	—
117	C₂³	C₂³×#6157	1	1	yes	characteristic
118	C₂×C₄	—	53	16	no	—
119	C₂×C₄	—	53	48	no	—
120	C₂×C₄	—	53	6	no	—
121	C₂×C₄	—	53	12	no	—
122	C₂×C₄	—	53	6	no	—
123	C₂×C₄	—	53	24	no	—
124	C₂×C₄	C₄×#88324	1	48	yes	direct product factor
125	C₂×C₄	C₂×C₄×#6157	1	12	yes	—
126	C₂×C₄	C₂²×#88324	1	12	yes	—
127	C₂×C₄	C₂×D₅₃⋊C₈	1	12	yes	—
128	C₈	—	53	32	no	—
129	C₂²	C₄²×#6157	1	4	yes	—
130	C₂²	C₂×C₄×#88324	1	12	yes	—
131	C₂²	C₄×D₅₃⋊C₈	1	12	yes	—
132	C₂²	C₂²×C₄×#6157	1	3	yes	—
133	C₂²	C₂³×#88324	1	1	yes	characteristic
134	C₂²	C₂²×D₅₃⋊C₈	1	3	yes	—
135	C₄	—	53	4	no	—
136	C₄	—	53	4	no	—
137	C₄	—	53	24	no	—
138	C₄	C₂×C₄×#88324	1	24	yes	direct product factor
139	C₂	C₄²×#88324	1	8	yes	direct product factor
140	C₂	C₂×C₄²×#6157	1	1	yes	characteristic
141	C₂	C₂²×C₄×#88324	1	3	yes	—
142	C₂	C₂×C₄×D₅₃⋊C₈	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#6157
commutator, hypocenter	C₅₃	C₂×C₄²×C₈
Frattini	C₂³	C₂³×#6157
Fitting	C₂²×C₄²×C₅₃	C₄
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×Dic₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×#88324	C₂	direct (N × H)
5. C₂×C₄×#88324	C₄	direct (N × H)
14. C₄×#88324	C₂×C₄	direct (N × H)
17. C₂×C₄×Dic₅₃	C₈	semidirect
23. C₂×C₄²×C₅₃	C₈	semidirect
36. C₂×#88324	C₄²	direct (N × H)
38. C₄²×C₅₃	C₂×C₈	semidirect
40. C₄×Dic₅₃	C₂×C₈	semidirect
46. C₂×C₄×C₅₃	C₄×C₈	semidirect
50. C₂×Dic₅₃	C₄×C₈	semidirect
56. #88324	C₂×C₄²	direct (N × H)
64. C₄×C₅₃	C₂×C₄×C₈	semidirect
66. Dic₅₃	C₂×C₄×C₈	semidirect
71. C₂×C₅₃	C₄²×C₈	semidirect
83. C₅₃	C₂×C₄²×C₈	semidirect
94. C₂×C₄²	#88324	direct (N × H)
113. C₄²	C₂×#88324	direct (N × H)
124. C₂×C₄	C₄×#88324	direct (N × H)
138. C₄	C₂×C₄×#88324	direct (N × H)
139. C₂	C₄²×#88324	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×C₄×#88324	2
C₄²×#88324	2
C₂×C₄²×Dic₅₃	2
C₂×C₄²×C₈	53

Subgroups — C2×C42×#88324 #28317889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×#88324 #28317889