Subgroups — C₂²×C₈×#88322 #27349658

Order 13568.

1342 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	2	yes	maximal
2	C₂×C₈×#88322	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₄	1	4	yes	—
5	C₂×C₄×#88322	C₄	1	12	yes	—
6	C₂×C₄×#88322	C₂²	1	12	yes	—
7	C₈×#88322	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₅₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	6	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
12	C₂²×#88322	C₈	1	32	yes	direct product factor
13	C₂²×#88322	C₂×C₄	1	24	yes	—
14	C₄×#88322	C₂×C₄	1	24	yes	—
15	C₄×#88322	C₂³	1	8	yes	—
16	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₈	1	16	yes	complemented
19	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
22	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
24	C₄×C₈×C₅₃	C₂³	1	4	yes	—
25	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₄²	1	6	yes	—
27	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
28	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
32	C₂×C₈×C₅₃	C₂×C₈	1	48	yes	complemented
33	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
34	C₄²×C₅₃	C₄²	1	6	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
36	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
37	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
38	C₂×#88322	C₂×C₈	1	96	yes	direct product factor
39	C₂×#88322	C₂²×C₄	1	16	yes	—
40	C₂³×C₅₃	C₄×C₈	1	8	yes	—
41	C₂³×C₅₃	C₄×C₈	1	4	yes	—
42	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
43	C₂×C₄×C₅₃	C₄×C₈	1	24	yes	—
44	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
45	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
46	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
47	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
49	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
50	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
51	C₈×C₅₃	C₂²×C₈	1	16	yes	complemented
52	#88322	C₂²×C₈	1	32	yes	direct product factor
53	C₂²×C₈²	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	C₈²	1	16	yes	complemented
55	C₂²×C₅₃	C₂×C₄×C₈	1	12	yes	—
56	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
57	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
58	C₄×C₅₃	C₂×C₄×C₈	1	12	yes	—
59	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
60	C₄×C₅₃	C₂³×C₈	1	4	yes	—
61	C₄×C₅₃	C₂³×C₈	1	2	yes	—
62	C₂²×C₄×C₈	—	53	2	no	—
63	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
64	C₂×C₈²	—	53	12	no	—
65	C₂×C₅₃	C₂×C₈²	1	12	yes	complemented
66	C₂×C₅₃	C₂²×C₄×C₈	1	2	yes	—
67	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
68	C₂³×C₈	—	53	4	no	—
69	C₂³×C₈	Dic₅₃	1	2	yes	—
70	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	C₂×C₄×C₈	—	53	12	no	—
72	C₂×C₄×C₈	—	53	12	no	—
73	C₂×C₄×C₈	Dic₅₃	1	6	yes	—
74	C₂×C₄×C₈	C₂×D₅₃	1	6	yes	—
75	C₈²	—	53	16	no	—
76	C₅₃	C₂²×C₈²	1	1	yes	characteristic, complemented, Sylow
77	C₂³×C₄	C₄×D₅₃	1	2	yes	—
78	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
79	C₂²×C₈	—	53	24	no	—
80	C₂²×C₈	—	53	32	no	—
81	C₂²×C₈	#88322	1	16	yes	direct product factor
82	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
83	C₂×C₄²	C₄×D₅₃	1	6	yes	—
84	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
85	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
86	C₄×C₈	—	53	24	no	—
87	C₄×C₈	—	53	8	no	—
88	C₄×C₈	C₂×Dic₅₃	1	12	yes	—
89	C₄×C₈	C₂²×D₅₃	1	4	yes	—
90	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
91	C₂²×C₄	C₈×D₅₃	1	16	yes	—
92	C₂²×C₄	C₂×#88322	1	8	yes	—
93	C₂²×C₄	C₄×Dic₅₃	1	6	yes	—
94	C₂²×C₄	C₄×Dic₅₃	1	3	yes	—
95	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
96	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
97	C₂×C₈	—	53	96	no	—
98	C₂×C₈	—	53	16	no	—
99	C₂×C₈	C₂×#88322	1	48	yes	direct product factor
100	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
101	C₄²	C₄×Dic₅₃	1	6	yes	—
102	C₄²	C₂×C₄×D₅₃	1	6	yes	—
103	C₄²	C₂²×Dic₅₃	1	3	yes	—
104	C₄²	C₂³×D₅₃	1	1	yes	characteristic
105	C₂³	C₈×Dic₅₃	1	8	yes	—
106	C₂³	C₄×#88322	1	4	yes	—
107	C₂³	C₂×C₄×Dic₅₃	1	3	yes	—
108	C₂×C₄	C₈×Dic₅₃	1	24	yes	—
109	C₂×C₄	C₄×#88322	1	12	yes	—
110	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
111	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
112	C₂×C₄	C₂×C₈×D₅₃	1	24	yes	—
113	C₂×C₄	C₂²×#88322	1	12	yes	—
114	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
115	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
116	C₈	—	53	32	no	—
117	C₈	C₂²×#88322	1	16	yes	direct product factor
118	C₂²	C₈×#88322	1	16	yes	direct product factor
119	C₂²	C₂×C₈×Dic₅₃	1	12	yes	—
120	C₂²	C₂×C₄×#88322	1	6	yes	—
121	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
122	C₄	C₂×C₈×Dic₅₃	1	12	yes	—
123	C₄	C₂×C₄×#88322	1	6	yes	—
124	C₄	C₂²×C₈×D₅₃	1	4	yes	—
125	C₄	C₂³×#88322	1	2	yes	—
126	C₂	C₂×C₈×#88322	1	12	yes	direct product factor
127	C₂	C₂²×C₈×Dic₅₃	1	2	yes	—
128	C₂	C₂²×C₄×#88322	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₈	D₅₃
commutator, hypocenter	C₅₃	C₂²×C₈²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×#88322	C₂	direct (N × H)
7. C₈×#88322	C₂²	direct (N × H)
12. C₂²×#88322	C₈	direct (N × H)
18. C₂²×C₈×C₅₃	C₈	semidirect
32. C₂×C₈×C₅₃	C₂×C₈	semidirect
38. C₂×#88322	C₂×C₈	direct (N × H)
51. C₈×C₅₃	C₂²×C₈	semidirect
52. #88322	C₂²×C₈	direct (N × H)
54. C₂²×C₅₃	C₈²	semidirect
65. C₂×C₅₃	C₂×C₈²	semidirect
76. C₅₃	C₂²×C₈²	semidirect
81. C₂²×C₈	#88322	direct (N × H)
99. C₂×C₈	C₂×#88322	direct (N × H)
117. C₈	C₂²×#88322	direct (N × H)
118. C₂²	C₈×#88322	direct (N × H)
126. C₂	C₂×C₈×#88322	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×#88322	2
C₂×C₈×#88322	2
C₂²×C₄×C₈×C₅₃	2
C₂²×C₈²	53

Subgroups — C22×C8×#88322 #27349658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×#88322 #27349658