Subgroups — C₄×#17988421 #33006161

Order 18688.

250 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988421	C₂	1	2	yes	maximal
2	#22608065	C₂	1	2	yes	maximal, complemented
3	C₄×#14556497	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17988421	C₄	1	8	yes	direct product factor
6	C₂×#14556497	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	#14556497	C₂×C₄	1	8	yes	complemented
15	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	C₈	1	4	yes	—
19	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
21	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₁₆	1	2	yes	—
29	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₇₃	C₁₆	1	4	yes	—
31	C₂×C₈×C₇₃	C₄²	1	2	yes	—
32	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₇₃	C₁₆	1	2	yes	—
38	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₇₃	C₄²	1	2	yes	—
41	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
42	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
43	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
44	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₇₃	C₄×C₈	1	2	yes	—
53	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
54	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
55	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
56	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₇₃	—	2	2	no	—
58	C₂²×C₇₃	C₄×C₁₆	1	2	yes	—
59	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
60	C₄×C₇₃	—	2	2	no	—
61	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
62	C₄×C₇₃	M₆₄	1	4	yes	complemented
63	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×M₆₄	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	2	2	no	—
67	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
68	C₂×C₇₃	C₂×M₆₄	1	2	yes	—
69	C₂×M₆₄	—	73	2	no	—
70	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic
71	#1397	—	73	2	no	—
72	C₄×C₃₂	—	73	2	no	—
73	C₇₃	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	Dic₇₃	1	1	yes	characteristic
75	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
76	C₂×C₄×C₈	Dic₇₃	1	1	yes	characteristic
77	M₆₄	—	73	8	no	—
78	C₂×C₃₂	—	73	4	no	—
79	C₄×C₁₆	Dic₇₃	1	1	yes	characteristic
80	C₄×C₁₆	Dic₇₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
83	C₂²×C₈	#10596831	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
85	C₂×C₄²	#10596831	1	1	yes	characteristic
86	C₂×C₁₆	#10596831	1	4	yes	—
87	C₂×C₁₆	C₄×D₇₃	1	4	yes	—
88	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
90	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
92	C₄×C₈	#10596831	1	1	yes	characteristic
93	C₄×C₈	#10596831	1	1	yes	characteristic
94	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
96	C₃₂	—	73	8	no	—
97	C₂²×C₄	#11924003	1	2	yes	—
98	C₂²×C₄	C₂×#10596831	1	1	yes	characteristic
99	C₂×C₈	#11924003	1	4	yes	—
100	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
101	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
102	C₂×C₈	C₂×#10596831	1	2	yes	—
103	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
104	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
106	C₄²	#11924003	1	2	yes	—
107	C₄²	C₂×#10596831	1	1	yes	characteristic
108	C₄²	C₂×#10596831	1	1	yes	characteristic
109	C₁₆	C₄×Dic₇₃	1	2	yes	—
110	C₁₆	C₂×#10596831	1	2	yes	—
111	C₁₆	C₂×#10596831	1	2	yes	—
112	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
113	C₂³	C₂×#11924003	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×#10596831	1	2	yes	—
116	C₂×C₄	C₂×#11924003	1	2	yes	—
117	C₂×C₄	C₂×#11924003	1	1	yes	characteristic
118	C₂×C₄	C₂×#11924003	1	2	yes	—
119	C₂×C₄	C₂×#11924003	1	2	yes	—
120	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
121	C₈	C₄×#10596831	1	2	yes	—
122	C₈	C₂×#11924003	1	2	yes	—
123	C₈	C₂×#11924003	1	2	yes	—
124	C₈	C₂×C₄×Dic₇₃	1	1	yes	characteristic
125	C₈	C₂²×#10596831	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₄×#11924003	1	2	yes	—
128	C₂²	C₂²×#11924003	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₄×#11924003	1	2	yes	—
131	C₄	#17988421	1	4	yes	direct product factor
132	C₄	C₂×C₄×#10596831	1	1	yes	characteristic
133	C₄	C₂²×#11924003	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₄×#11924003	1	1	yes	characteristic
136	C₂	C₂×#17988421	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂×D₇₃
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	C₄×M₆₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22608065	C₂	semidirect
3. C₄×#14556497	C₂	semidirect
5. #17988421	C₄	direct (N × H)
14. #14556497	C₂×C₄	semidirect
62. C₄×C₇₃	M₆₄	semidirect
73. C₇₃	C₄×M₆₄	semidirect
131. C₄	#17988421	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17988421	2
#22608065	2
C₄×#14556497	2
C₂×C₄×C₁₆×C₇₃	2
C₄×M₆₄	73

Subgroups — C4×#17988421 #33006161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17988421 #33006161