Subgroups — C₂³×C₄⋊#88322 #28247700

Order 13568.

3870 conjugacy classes of subgroups.

The subgroup lattice has 146 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	2	yes	maximal
2	C₂²×C₄⋊#88322	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	—	2	16	no	—
5	C₂³×#88322	C₂²	1	28	yes	—
6	C₂×C₄⋊#88322	C₂²	1	112	yes	direct product factor
7	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₅₃	C₄	1	1	yes	characteristic
9	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅₃	C₄	1	14	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	14	yes	—
12	C₂²×#88322	—	2	112	no	—
13	C₂²×#88322	C₂³	1	56	yes	—
14	C₄⋊#88322	C₂³	1	64	yes	direct product factor
15	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	—	2	8	no	—
17	C₂³×C₄×C₅₃	C₈	1	16	yes	complemented
18	C₂³×C₄×C₅₃	C₂×C₄	1	14	yes	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	14	yes	—
21	C₂³×C₄×C₅₃	D₄	1	8	yes	—
22	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	84	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	28	yes	—
26	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
27	C₂⁴×C₅₃	C₂×C₈	1	8	yes	—
28	C₂⁴×C₅₃	M₁₆	1	8	yes	—
29	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
30	C₂²×C₄×C₅₃	—	2	56	no	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	56	yes	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	112	yes	complemented
33	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	28	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
37	C₂²×C₄×C₅₃	C₂×Q₈	1	28	yes	—
38	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	56	yes	—
40	C₄²×C₅₃	C₂⁴	1	8	yes	—
41	C₂×#88322	—	2	112	no	—
42	C₂×#88322	C₂⁴	1	16	yes	—
43	C₂³×C₅₃	C₄⋊C₈	1	64	yes	complemented
44	C₂³×C₅₃	C₂×C₄⋊C₄	1	28	yes	—
45	C₂³×C₅₃	C₂²×C₈	1	28	yes	—
46	C₂³×C₅₃	C₂×M₁₆	1	28	yes	—
47	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
48	C₂×C₄×C₅₃	—	2	56	no	—
49	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	84	yes	—
50	C₂×C₄×C₅₃	C₂²×C₈	1	112	yes	complemented
51	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
52	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
53	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
54	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
55	C₂×C₄×C₅₃	C₂²×Q₈	1	14	yes	—
56	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
57	#88322	—	2	16	no	—
58	C₂³×C₄⋊C₈	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₂×C₄⋊C₈	1	112	yes	complemented
60	C₂²×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
61	C₂²×C₅₃	C₂³×C₈	1	14	yes	—
62	C₂²×C₅₃	C₂²×M₁₆	1	14	yes	—
63	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
64	C₄×C₅₃	—	2	8	no	—
65	C₄×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
66	C₄×C₅₃	C₂³×C₈	1	16	yes	complemented
67	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
68	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
69	C₂⁴×C₈	—	53	2	no	—
70	C₂³×C₄²	D₅₃	1	1	yes	characteristic
71	C₂²×C₄⋊C₈	—	53	28	no	—
72	C₂×C₅₃	C₂²×C₄⋊C₈	1	28	yes	complemented
73	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
75	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
76	C₂⁴×C₄	Dic₅₃	1	1	yes	characteristic
77	C₂⁴×C₄	Dic₅₃	1	1	yes	characteristic
78	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
79	C₂³×C₈	—	53	28	no	—
80	C₂³×C₈	—	106	16	no	—
81	C₂²×C₄²	Dic₅₃	1	14	yes	—
82	C₂²×C₄²	C₂×D₅₃	1	14	yes	—
83	C₂×C₄⋊C₈	—	53	112	no	—
84	C₅₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
85	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
86	C₂³×C₄	—	2	8	no	—
87	C₂³×C₄	#88322	1	16	yes	complemented
88	C₂³×C₄	Dic₁₀₆	1	8	yes	—
89	C₂³×C₄	D₂₁₂	1	8	yes	—
90	C₂³×C₄	C₂×Dic₅₃	1	14	yes	—
91	C₂³×C₄	C₂×Dic₅₃	1	14	yes	—
92	C₂³×C₄	C₂×Dic₅₃	1	7	yes	—
93	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
94	C₂²×C₈	—	53	56	no	—
95	C₂²×C₈	—	106	112	no	—
96	C₂×C₄²	C₂×Dic₅₃	1	84	yes	—
97	C₂×C₄²	C₂²×D₅₃	1	28	yes	—
98	C₄⋊C₈	—	53	64	no	—
99	C₂⁴	C₂×#88322	1	8	yes	—
100	C₂⁴	#11719917	1	8	yes	—
101	C₂⁴	C₄⋊Dic₅₃	1	8	yes	—
102	C₂⁴	C₂²×Dic₅₃	1	7	yes	—
103	C₂²×C₄	—	2	56	no	—
104	C₂²×C₄	C₂×#88322	1	112	yes	complemented
105	C₂²×C₄	C₄⋊Dic₅₃	1	56	yes	—
106	C₂²×C₄	C₂×Dic₁₀₆	1	28	yes	—
107	C₂²×C₄	C₂×D₂₁₂	1	28	yes	—
108	C₂²×C₄	C₂²×Dic₅₃	1	28	yes	—
109	C₂²×C₄	C₂²×Dic₅₃	1	21	yes	—
110	C₂²×C₄	C₂²×Dic₅₃	1	28	yes	—
111	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
112	C₂×C₈	—	53	16	no	—
113	C₂×C₈	—	106	112	no	—
114	C₄²	C₂²×Dic₅₃	1	56	yes	—
115	C₄²	C₂³×D₅₃	1	8	yes	—
116	C₂³	C₄⋊#88322	1	64	yes	direct product factor
117	C₂³	C₂×C₄⋊Dic₅₃	1	28	yes	—
118	C₂³	C₂²×#88322	1	28	yes	—
119	C₂³	C₂×#11719917	1	28	yes	—
120	C₂³	C₂³×Dic₅₃	1	7	yes	—
121	C₂×C₄	—	2	56	no	—
122	C₂×C₄	C₂×C₄⋊Dic₅₃	1	84	yes	—
123	C₂×C₄	C₂²×#88322	1	112	yes	complemented
124	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
125	C₂×C₄	C₂³×Dic₅₃	1	7	yes	—
126	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
127	C₂×C₄	C₂²×D₂₁₂	1	14	yes	—
128	C₂×C₄	C₂²×Dic₁₀₆	1	14	yes	—
129	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
130	C₈	—	106	16	no	—
131	C₂²	C₂×C₄⋊#88322	1	112	yes	direct product factor
132	C₂²	C₂²×C₄⋊Dic₅₃	1	14	yes	—
133	C₂²	C₂³×#88322	1	14	yes	—
134	C₂²	C₂²×#11719917	1	14	yes	—
135	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	C₂²×C₄⋊Dic₅₃	1	14	yes	—
138	C₄	C₂³×#88322	1	16	yes	complemented
139	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
140	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
141	C₂	C₂²×C₄⋊#88322	1	28	yes	direct product factor
142	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
143	C₂	C₂⁴×#88322	1	1	yes	characteristic
144	C₂	C₂³×#11719917	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊#88322	C₂	direct (N × H)
6. C₂×C₄⋊#88322	C₂²	direct (N × H)
14. C₄⋊#88322	C₂³	direct (N × H)
17. C₂³×C₄×C₅₃	C₈	semidirect
32. C₂²×C₄×C₅₃	C₂×C₈	semidirect
43. C₂³×C₅₃	C₄⋊C₈	semidirect
50. C₂×C₄×C₅₃	C₂²×C₈	semidirect
59. C₂²×C₅₃	C₂×C₄⋊C₈	semidirect
66. C₄×C₅₃	C₂³×C₈	semidirect
72. C₂×C₅₃	C₂²×C₄⋊C₈	semidirect
84. C₅₃	C₂³×C₄⋊C₈	semidirect
87. C₂³×C₄	#88322	semidirect
104. C₂²×C₄	C₂×#88322	semidirect
116. C₂³	C₄⋊#88322	direct (N × H)
123. C₂×C₄	C₂²×#88322	semidirect
131. C₂²	C₂×C₄⋊#88322	direct (N × H)
138. C₄	C₂³×#88322	semidirect
141. C₂	C₂²×C₄⋊#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
C₂⁴×#88322	2
C₂²×C₄⋊#88322	2
C₂³×C₄²×C₅₃	2
C₂³×C₄⋊C₈	53

Subgroups — C23×C4⋊#88322 #28247700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊#88322 #28247700