Subgroups — C₂³×#14556598 #33999744

Order 18688.

11422 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	C₂²×#14556598	C₂	1	56	yes	maximal, direct product factor
2	C₂³×#11924012	C₂	1	3	yes	maximal, complemented
3	C₂⁴×#10596831	C₂	1	3	yes	maximal, complemented
4	C₂³×#41×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556598	C₂²	1	448	yes	direct product factor
6	C₂²×#11924012	C₂²	1	84	yes	complemented
7	C₂³×#10596831	C₂²	1	1	yes	characteristic
8	C₂³×#10596831	C₂²	1	3	yes	—
9	C₂³×#10596831	C₂²	1	84	yes	complemented
10	C₂²×#41×C₇₃	C₄	1	28	yes	—
11	C₂²×#41×C₇₃	C₂²	1	28	yes	—
12	C₂³×Q₈×C₇₃	C₄	1	1	yes	characteristic
13	C₂³×D₄×C₇₃	C₄	1	3	yes	—
14	C₂⁴×C₄×C₇₃	C₂²	1	3	yes	—
15	#14556598	C₂³	1	512	yes	direct product factor
16	C₂×#11924012	C₂³	1	336	yes	complemented
17	C₂²×#10596831	C₂³	1	14	yes	—
18	C₂²×#10596831	C₂³	1	42	yes	—
19	C₂²×#10596831	C₂³	1	336	yes	complemented
20	C₂⁵×C₇₃	C₂×C₄	1	3	yes	—
21	C₂×#41×C₇₃	C₂×C₄	1	336	yes	—
22	C₂×#41×C₇₃	C₂³	1	112	yes	—
23	C₂²×Q₈×C₇₃	C₂×C₄	1	28	yes	—
24	C₂²×D₄×C₇₃	C₂×C₄	1	84	yes	—
25	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₇₃	C₂×C₄	1	42	yes	—
27	C₂³×C₄×C₇₃	C₂³	1	42	yes	—
28	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
29	#11924012	C₂⁴	1	192	yes	complemented
30	C₂×#10596831	C₂⁴	1	28	yes	—
31	C₂×#10596831	C₂⁴	1	84	yes	—
32	C₂×#10596831	C₂⁴	1	192	yes	complemented
33	C₂⁴×C₇₃	—	2	24	no	—
34	C₂⁴×C₇₃	C₂²×C₄	1	42	yes	—
35	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	#41×C₇₃	C₂²×C₄	1	448	yes	—
37	#41×C₇₃	C₂⁴	1	64	yes	—
38	C₂×Q₈×C₇₃	C₂²×C₄	1	112	yes	—
39	C₂×D₄×C₇₃	C₂²×C₄	1	336	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	7	yes	—
41	C₂²×C₄×C₇₃	C₂²×C₄	1	42	yes	—
42	C₂²×C₄×C₇₃	C₂²×C₄	1	252	yes	—
43	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
44	C₂²×C₄×C₇₃	C₂⁴	1	84	yes	—
45	C₂³×C₇₃	—	2	168	no	—
46	C₂³×C₇₃	#131	1	8	yes	complemented
47	C₂³×C₇₃	C₂³×C₄	1	84	yes	—
48	C₂³×C₇₃	C₂³×C₄	1	7	yes	—
49	Q₈×C₇₃	C₂³×C₄	1	64	yes	—
50	D₄×C₇₃	C₂³×C₄	1	192	yes	—
51	C₂×C₄×C₇₃	C₂³×C₄	1	21	yes	—
52	C₂×C₄×C₇₃	C₂³×C₄	1	84	yes	—
53	C₂×C₄×C₇₃	C₂³×C₄	1	168	yes	—
54	C₂×C₄×C₇₃	C₂⁵	1	7	yes	—
55	C₂×C₄×C₇₃	C₂⁵	1	24	yes	—
56	#10596831	C₂⁵	1	8	yes	—
57	#10596831	C₂⁵	1	24	yes	—
58	C₂²×C₇₃	—	2	168	no	—
59	C₂²×C₇₃	C₂×#131	1	28	yes	complemented
60	C₂²×C₇₃	C₂⁴×C₄	1	24	yes	—
61	C₂²×C₇₃	C₂⁴×C₄	1	7	yes	—
62	C₄×C₇₃	C₂⁴×C₄	1	7	yes	—
63	C₄×C₇₃	C₂⁴×C₄	1	24	yes	—
64	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
65	C₂³×#131	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	2	24	no	—
67	C₂×C₇₃	C₂²×#131	1	14	yes	complemented
68	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
69	C₂³×#41	D₇₃	1	1	yes	characteristic
70	C₂²×#131	—	73	56	no	—
71	C₂³×M₁₆	—	73	3	no	—
72	C₂⁴×C₈	—	73	3	no	—
73	C₇₃	C₂³×#131	1	1	yes	characteristic, complemented, Sylow
74	C₂²×#41	Dic₇₃	1	28	yes	—
75	C₂²×#41	C₂×D₇₃	1	28	yes	—
76	C₂³×Q₈	Dic₇₃	1	1	yes	characteristic
77	C₂³×D₄	Dic₇₃	1	3	yes	—
78	C₂⁴×C₄	C₂×D₇₃	1	3	yes	—
79	C₂×#131	—	73	448	no	—
80	C₂²×M₁₆	—	73	84	no	—
81	C₂³×C₈	—	73	84	no	—
82	C₂³×C₈	—	73	1	no	—
83	C₂³×C₈	—	73	3	no	—
84	C₂⁵	C₂×Dic₇₃	1	3	yes	—
85	C₂×#41	C₂×Dic₇₃	1	336	yes	—
86	C₂×#41	C₂²×D₇₃	1	112	yes	—
87	C₂²×Q₈	C₂×Dic₇₃	1	28	yes	—
88	C₂²×D₄	C₂×Dic₇₃	1	84	yes	—
89	C₂³×C₄	C₂×Dic₇₃	1	3	yes	—
90	C₂³×C₄	C₂×Dic₇₃	1	42	yes	—
91	C₂³×C₄	C₂²×D₇₃	1	42	yes	—
92	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
93	#131	—	73	512	no	—
94	C₂×M₁₆	—	73	336	no	—
95	C₂²×C₈	—	73	14	no	—
96	C₂²×C₈	—	73	42	no	—
97	C₂²×C₈	—	73	336	no	—
98	C₂⁴	—	2	24	no	—
99	C₂⁴	C₂²×Dic₇₃	1	42	yes	—
100	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
101	#41	C₂²×Dic₇₃	1	448	yes	—
102	#41	C₂³×D₇₃	1	64	yes	—
103	C₂×Q₈	C₂²×Dic₇₃	1	112	yes	—
104	C₂×D₄	C₂²×Dic₇₃	1	336	yes	—
105	C₂²×C₄	C₂²×Dic₇₃	1	7	yes	—
106	C₂²×C₄	C₂²×Dic₇₃	1	42	yes	—
107	C₂²×C₄	C₂²×Dic₇₃	1	252	yes	—
108	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
109	C₂²×C₄	C₂³×D₇₃	1	84	yes	—
110	M₁₆	—	73	192	no	—
111	C₂×C₈	—	73	28	no	—
112	C₂×C₈	—	73	84	no	—
113	C₂×C₈	—	73	192	no	—
114	C₂³	—	2	168	no	—
115	C₂³	#14556598	1	8	yes	direct product factor
116	C₂³	C₂³×Dic₇₃	1	84	yes	—
117	C₂³	C₂³×Dic₇₃	1	7	yes	—
118	Q₈	C₂³×Dic₇₃	1	64	yes	—
119	D₄	C₂³×Dic₇₃	1	192	yes	—
120	C₂×C₄	C₂³×Dic₇₃	1	21	yes	—
121	C₂×C₄	C₂³×Dic₇₃	1	84	yes	—
122	C₂×C₄	C₂³×Dic₇₃	1	168	yes	—
123	C₂×C₄	C₂⁴×D₇₃	1	7	yes	—
124	C₂×C₄	C₂⁴×D₇₃	1	24	yes	—
125	C₈	—	73	8	no	—
126	C₈	—	73	24	no	—
127	C₂²	—	2	168	no	—
128	C₂²	C₂×#14556598	1	28	yes	direct product factor
129	C₂²	C₂⁴×Dic₇₃	1	24	yes	—
130	C₂²	C₂⁴×Dic₇₃	1	7	yes	—
131	C₄	C₂⁴×Dic₇₃	1	7	yes	—
132	C₄	C₂⁴×Dic₇₃	1	24	yes	—
133	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
134	C₂	—	2	24	no	—
135	C₂	C₂²×#14556598	1	14	yes	direct product factor
136	C₂	C₂⁵×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₄	C₂⁵×D₇₃
Fitting, prime radical	C₂³×#41×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₄×C₇₃	C₂⁶
hypercenter	C₂³×#41	D₇₃
hypocenter	C₇₃	C₂³×#131
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556598	C₂	direct (N × H)
2. C₂³×#11924012	C₂	semidirect
3. C₂⁴×#10596831	C₂	semidirect
5. C₂×#14556598	C₂²	direct (N × H)
6. C₂²×#11924012	C₂²	semidirect
9. C₂³×#10596831	C₂²	semidirect
15. #14556598	C₂³	direct (N × H)
16. C₂×#11924012	C₂³	semidirect
19. C₂²×#10596831	C₂³	semidirect
29. #11924012	C₂⁴	semidirect
32. C₂×#10596831	C₂⁴	semidirect
46. C₂³×C₇₃	#131	semidirect
59. C₂²×C₇₃	C₂×#131	semidirect
67. C₂×C₇₃	C₂²×#131	semidirect
73. C₇₃	C₂³×#131	semidirect
115. C₂³	#14556598	direct (N × H)
128. C₂²	C₂×#14556598	direct (N × H)
135. C₂	C₂²×#14556598	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×#14556598	2
C₂³×#11924012	2
C₂⁴×#10596831	2
C₂³×#41×C₇₃	2
C₂³×#131	73

Subgroups — C23×#14556598 #33999744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556598 #33999744