Subgroups — C₂²×C₈×#88324 #28313734

Order 13568.

1342 conjugacy classes of subgroups.

The subgroup lattice has 148 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	2	yes	maximal
2	C₂×C₈×#88324	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88324	C₄	1	4	yes	—
5	C₂×C₄×#88324	C₄	1	12	yes	—
6	C₂×C₄×#88324	C₂²	1	12	yes	—
7	C₈×#88324	C₂²	1	16	yes	direct product factor
8	C₂³×#88322	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	6	yes	—
11	C₂×C₈×Dic₅₃	C₂²	1	6	yes	—
12	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×#88324	C₈	1	32	yes	direct product factor
14	C₂²×#88324	C₂×C₄	1	24	yes	—
15	C₄×#88324	C₂×C₄	1	24	yes	—
16	C₄×#88324	C₂³	1	8	yes	—
17	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₂²×#88322	C₈	1	8	yes	complemented
19	C₂²×#88322	C₂×C₄	1	6	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
22	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
23	C₈×Dic₅₃	C₂×C₄	1	12	yes	—
24	C₈×Dic₅₃	C₂³	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
27	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
28	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄²	1	3	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
33	C₂²×Dic₅₃	C₄²	1	6	yes	—
34	C₂²×Dic₅₃	C₂×C₈	1	16	yes	—
35	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
36	C₂×#88324	C₂×C₈	1	96	yes	direct product factor
37	C₂×#88324	C₂²×C₄	1	16	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	24	yes	complemented
39	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
40	C₄×Dic₅₃	C₄²	1	6	yes	—
41	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
42	C₄×Dic₅₃	C₂²×C₄	1	6	yes	—
43	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
44	C₂×#88322	C₂×C₈	1	24	yes	complemented
45	C₂×#88322	C₂²×C₄	1	4	yes	—
46	C₂³×C₅₃	C₄×C₈	1	8	yes	—
47	C₂³×C₅₃	C₄×C₈	1	4	yes	—
48	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
49	C₂×C₄×C₅₃	C₄×C₈	1	12	yes	—
50	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
51	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
52	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
53	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₄×C₈	1	24	yes	—
55	C₂×Dic₅₃	C₂×C₄²	1	6	yes	—
56	C₂×Dic₅₃	C₂²×C₈	1	24	yes	—
57	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
58	#88324	C₂²×C₈	1	32	yes	direct product factor
59	C₈×C₅₃	C₂²×C₈	1	8	yes	complemented
60	#88322	C₂²×C₈	1	8	yes	complemented
61	C₂²×C₈²	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	C₈²	1	16	yes	complemented
63	C₂²×C₅₃	C₂×C₄×C₈	1	12	yes	—
64	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
65	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
66	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
67	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
69	Dic₅₃	C₂×C₄×C₈	1	12	yes	—
70	Dic₅₃	C₂³×C₈	1	4	yes	—
71	C₂²×C₄×C₈	—	53	2	no	—
72	C₂²×C₄×C₈	—	53	1	no	—
73	C₂×C₈²	—	53	12	no	—
74	C₂×C₅₃	C₂×C₈²	1	12	yes	complemented
75	C₂×C₅₃	C₂²×C₄×C₈	1	2	yes	—
76	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
77	C₂³×C₈	—	53	4	no	—
78	C₂³×C₈	—	53	1	no	—
79	C₂³×C₈	#6157	1	1	yes	characteristic
80	C₂²×C₄²	—	53	1	no	—
81	C₂×C₄×C₈	—	53	12	no	—
82	C₂×C₄×C₈	—	53	12	no	—
83	C₂×C₄×C₈	—	53	6	no	—
84	C₂×C₄×C₈	—	53	6	no	—
85	C₈²	—	53	16	no	—
86	C₅₃	C₂²×C₈²	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₄	—	53	2	no	—
88	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
89	C₂²×C₈	—	53	24	no	—
90	C₂²×C₈	—	53	32	no	—
91	C₂²×C₈	—	53	6	no	—
92	C₂²×C₈	—	53	8	no	—
93	C₂²×C₈	#88324	1	8	yes	direct product factor
94	C₂²×C₈	C₂×#6157	1	6	yes	—
95	C₂×C₄²	—	53	6	no	—
96	C₂×C₄²	—	53	3	no	—
97	C₂×C₄²	—	53	3	no	—
98	C₄×C₈	—	53	8	no	—
99	C₄×C₈	—	53	24	no	—
100	C₄×C₈	—	53	4	no	—
101	C₄×C₈	—	53	12	no	—
102	C₂⁴	C₄×#6157	1	1	yes	characteristic
103	C₂²×C₄	—	53	16	no	—
104	C₂²×C₄	—	53	6	no	—
105	C₂²×C₄	—	53	6	no	—
106	C₂²×C₄	D₅₃⋊C₈	1	4	yes	—
107	C₂²×C₄	C₄×#6157	1	3	yes	—
108	C₂²×C₄	C₂×#88324	1	4	yes	—
109	C₂²×C₄	C₂²×#6157	1	3	yes	—
110	C₂×C₈	—	53	96	no	—
111	C₂×C₈	—	53	16	no	—
112	C₂×C₈	—	53	24	no	—
113	C₂×C₈	—	53	4	no	—
114	C₂×C₈	C₂×#88324	1	24	yes	direct product factor
115	C₂×C₈	C₂²×#6157	1	4	yes	—
116	C₄²	—	53	6	no	—
117	C₄²	—	53	6	no	—
118	C₄²	—	53	1	no	—
119	C₄²	—	53	3	no	—
120	C₂³	C₈×#6157	1	8	yes	—
121	C₂³	C₄×#88324	1	4	yes	—
122	C₂³	C₂×C₄×#6157	1	3	yes	—
123	C₂×C₄	—	53	24	no	—
124	C₂×C₄	—	53	24	no	—
125	C₂×C₄	—	53	6	no	—
126	C₂×C₄	—	53	2	no	—
127	C₂×C₄	C₄×#88324	1	12	yes	—
128	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
129	C₂×C₄	C₂²×#88324	1	6	yes	—
130	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
131	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
132	C₈	—	53	32	no	—
133	C₈	—	53	8	no	—
134	C₈	C₂²×#88324	1	8	yes	direct product factor
135	C₂²	C₈×#88324	1	16	yes	direct product factor
136	C₂²	C₂×C₈×#6157	1	12	yes	—
137	C₂²	C₂×C₄×#88324	1	6	yes	—
138	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
139	C₄	—	53	4	no	—
140	C₄	—	53	12	no	—
141	C₄	C₂×C₄×#88324	1	6	yes	—
142	C₄	C₂³×#88324	1	1	yes	characteristic
143	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
144	C₂	C₂×C₈×#88324	1	12	yes	direct product factor
145	C₂	C₂²×C₈×#6157	1	2	yes	—
146	C₂	C₂²×C₄×#88324	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	#6157
commutator, hypocenter	C₅₃	C₂²×C₈²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂³×C₈×C₅₃	C₄
socle, prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄×Dic₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×#88324	C₂	direct (N × H)
7. C₈×#88324	C₂²	direct (N × H)
13. C₂²×#88324	C₈	direct (N × H)
18. C₂²×#88322	C₈	semidirect
26. C₂²×C₈×C₅₃	C₈	semidirect
36. C₂×#88324	C₂×C₈	direct (N × H)
38. C₂×C₈×C₅₃	C₂×C₈	semidirect
44. C₂×#88322	C₂×C₈	semidirect
58. #88324	C₂²×C₈	direct (N × H)
59. C₈×C₅₃	C₂²×C₈	semidirect
60. #88322	C₂²×C₈	semidirect
62. C₂²×C₅₃	C₈²	semidirect
74. C₂×C₅₃	C₂×C₈²	semidirect
86. C₅₃	C₂²×C₈²	semidirect
93. C₂²×C₈	#88324	direct (N × H)
114. C₂×C₈	C₂×#88324	direct (N × H)
134. C₈	C₂²×#88324	direct (N × H)
135. C₂²	C₈×#88324	direct (N × H)
144. C₂	C₂×C₈×#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
C₂×C₈×#88324	2
C₂²×C₈×Dic₅₃	2
C₂²×C₈²	53

Subgroups — C22×C8×#88324 #28313734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×#88324 #28313734