Subgroups — C₄×C₈×#10596833 #34001921

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#10596833	C₂	1	2	yes	maximal
2	C₂×C₈×#10596833	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#10596833	C₄	1	8	yes	—
5	C₂×C₄×#10596833	C₂²	1	4	yes	—
6	C₈×#10596833	C₄	1	16	yes	direct product factor
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
10	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂²×#10596833	C₂×C₄	1	8	yes	—
13	C₄×#10596833	C₈	1	32	yes	direct product factor
14	C₄×#10596833	C₂×C₄	1	16	yes	—
15	C₂²×#10596831	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
17	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
19	C₄×#10596831	C₈	1	4	yes	complemented
20	C₈×Dic₇₃	C₈	1	8	yes	complemented
21	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
22	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₈	1	4	yes	complemented
25	C₂×#10596833	C₄²	1	16	yes	—
26	C₂×#10596833	C₂×C₈	1	32	yes	—
27	C₂²×Dic₇₃	C₄²	1	2	yes	—
28	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
29	C₂×#10596831	C₄²	1	4	yes	—
30	C₂×#10596831	C₂×C₈	1	4	yes	—
31	C₂×#10596831	C₂×C₈	1	4	yes	—
32	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
33	C₄×Dic₇₃	C₂×C₈	1	16	yes	—
34	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
35	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
36	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₄²	1	4	yes	—
38	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
40	C₄²×C₇₃	C₂×C₈	1	2	yes	—
41	C₄²×C₇₃	C₂×C₈	1	2	yes	—
42	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₄×C₈	1	8	yes	—
44	C₂×C₄×C₇₃	C₄×C₈	1	4	yes	—
45	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
46	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
49	C₂×Dic₇₃	C₄×C₈	1	8	yes	—
50	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
51	C₂×Dic₇₃	C₂×C₄²	1	4	yes	—
52	C₂×Dic₇₃	C₂²×C₈	1	8	yes	—
53	#10596833	C₄×C₈	1	32	yes	direct product factor
54	C₈×C₇₃	C₄×C₈	1	8	yes	complemented
55	#10596831	C₄×C₈	1	8	yes	complemented
56	C₂²×C₇₃	C₄³	1	1	yes	characteristic
57	C₂²×C₇₃	C₂×C₄×C₈	1	4	yes	—
58	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₇₃	C₈²	1	8	yes	complemented
61	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
62	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
63	Dic₇₃	C₈²	1	8	yes	complemented
64	Dic₇₃	C₂×C₄×C₈	1	8	yes	—
65	C₄×C₈²	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	C₂×C₈²	1	2	yes	—
67	C₂×C₇₃	C₂×C₈²	1	2	yes	—
68	C₂×C₇₃	C₄²×C₈	1	2	yes	—
69	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
70	C₄²×C₈	—	73	2	no	—
71	C₄²×C₈	—	73	1	no	—
72	C₂×C₈²	—	73	4	no	—
73	C₇₃	C₄×C₈²	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×C₈	—	73	8	no	—
75	C₂×C₄×C₈	—	73	4	no	—
76	C₂×C₄×C₈	—	73	2	no	—
77	C₂×C₄×C₈	—	73	2	no	—
78	C₂×C₄×C₈	—	73	1	no	—
79	C₂×C₄×C₈	#64408	1	1	yes	characteristic
80	C₄³	—	73	1	no	—
81	C₈²	—	73	16	no	—
82	C₂²×C₈	—	73	8	no	—
83	C₂²×C₈	—	73	2	no	—
84	C₂²×C₈	C₂×#64408	1	2	yes	—
85	C₂×C₄²	—	73	4	no	—
86	C₂×C₄²	—	73	1	no	—
87	C₂×C₄²	—	73	1	no	—
88	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
89	C₄×C₈	—	73	16	no	—
90	C₄×C₈	—	73	32	no	—
91	C₄×C₈	—	73	8	no	—
92	C₄×C₈	—	73	8	no	—
93	C₄×C₈	—	73	4	no	—
94	C₄×C₈	#10596833	1	4	yes	direct product factor
95	C₂²×C₄	—	73	2	no	—
96	C₂²×C₄	—	73	2	no	—
97	C₂²×C₄	C₄×#64408	1	2	yes	—
98	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
99	C₂×C₈	—	73	16	no	—
100	C₂×C₈	—	73	32	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	—	73	4	no	—
103	C₂×C₈	—	73	4	no	—
104	C₂×C₈	C₄×#64408	1	4	yes	—
105	C₂×C₈	C₂×#10596833	1	4	yes	—
106	C₂×C₈	#11924036	1	4	yes	—
107	C₄²	—	73	4	no	—
108	C₄²	—	73	16	no	—
109	C₄²	—	73	4	no	—
110	C₄²	C₂×#10596833	1	2	yes	—
111	C₄²	#11924036	1	2	yes	—
112	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
113	C₂×C₄	—	73	4	no	—
114	C₂×C₄	—	73	8	no	—
115	C₂×C₄	—	73	8	no	—
116	C₂×C₄	—	73	4	no	—
117	C₂×C₄	C₈×#64408	1	8	yes	—
118	C₂×C₄	C₄×#10596833	1	4	yes	—
119	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
120	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
121	C₂×C₄	C₂×#11924036	1	2	yes	—
122	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
123	C₈	—	73	32	no	—
124	C₈	—	73	8	no	—
125	C₈	C₄×#10596833	1	8	yes	direct product factor
126	C₂²	C₄²×#64408	1	1	yes	characteristic
127	C₂²	C₂×C₈×#64408	1	4	yes	—
128	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
129	C₂²	C₄×#11924036	1	1	yes	characteristic
130	C₄	—	73	8	no	—
131	C₄	—	73	8	no	—
132	C₄	C₈×#10596833	1	8	yes	direct product factor
133	C₄	C₂×C₄×#10596833	1	2	yes	—
134	C₄	C₄×#11924036	1	2	yes	—
135	C₂	C₂×C₈×#10596833	1	2	yes	—
136	C₂	C₈×#11924036	1	2	yes	—
137	C₂	C₄×C₈×#64408	1	2	yes	—
138	C₂	C₄²×#10596833	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	#64408
commutator, hypocenter	C₇₃	C₄×C₈²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₈×#10596833	C₄	direct (N × H)
13. C₄×#10596833	C₈	direct (N × H)
19. C₄×#10596831	C₈	semidirect
20. C₈×Dic₇₃	C₈	semidirect
24. C₄×C₈×C₇₃	C₈	semidirect
53. #10596833	C₄×C₈	direct (N × H)
54. C₈×C₇₃	C₄×C₈	semidirect
55. #10596831	C₄×C₈	semidirect
60. C₄×C₇₃	C₈²	semidirect
63. Dic₇₃	C₈²	semidirect
73. C₇₃	C₄×C₈²	semidirect
94. C₄×C₈	#10596833	direct (N × H)
125. C₈	C₄×#10596833	direct (N × H)
132. C₄	C₈×#10596833	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄²×#10596833	2
C₂×C₈×#10596833	2
C₄×C₈×Dic₇₃	2
C₄×C₈²	73

Subgroups — C4×C8×#10596833 #34001921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×#10596833 #34001921