Subgroups — C₈×#14556502 #32975337

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609677	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556502	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988309	C₂	1	2	yes	maximal
4	C₂×C₈×#10596831	C₂	1	2	yes	maximal
5	C₄²×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556502	C₄	1	4	yes	—
7	C₂×C₄×#10596831	C₂²	1	2	yes	—
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	#17988342	C₄	1	4	yes	—
10	#17988309	C₄	1	8	yes	complemented
11	C₈×#10596831	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₇₃	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄³×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×#10596831	C₂×C₄	1	4	yes	—
17	C₂²×#10596831	C₂×C₄	1	4	yes	—
18	#14556502	C₈	1	16	yes	direct product factor
19	C₄×#10596831	C₂×C₄	1	4	yes	—
20	C₄×#10596831	C₂×C₄	1	4	yes	—
21	C₄×#10596831	C₂×C₄	1	4	yes	—
22	C₄×#10596831	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₃	—	2	8	no	—
30	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
32	C₂×#10596831	—	2	8	no	—
33	C₂×#10596831	C₄²	1	8	yes	—
34	C₂×#10596831	C₄²	1	8	yes	—
35	C₂×#10596831	C₂×C₈	1	16	yes	—
36	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
37	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	4	no	—
41	C₂×C₈×C₇₃	C₄²	1	4	yes	—
42	C₂×C₈×C₇₃	C₄²	1	4	yes	—
43	C₂×C₈×C₇₃	M₁₆	1	8	yes	—
44	C₄²×C₇₃	—	2	4	no	—
45	C₄²×C₇₃	—	2	4	no	—
46	C₄²×C₇₃	C₂×C₈	1	8	yes	—
47	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
48	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
49	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	—	2	4	no	—
52	C₂×C₄×C₇₃	C₄×C₈	1	8	yes	—
53	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
58	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₇₃	#131	1	4	yes	—
60	C₈×C₇₃	—	2	4	no	—
61	C₈×C₇₃	#97	1	8	yes	complemented
62	#10596831	—	2	8	no	—
63	#10596831	C₄×C₈	1	16	yes	complemented
64	C₂²×C₇₃	C₄³	1	1	yes	characteristic
65	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
66	C₂²×C₇₃	C₄×M₁₆	1	2	yes	—
67	C₂²×C₇₃	#405	1	2	yes	—
68	C₄×C₇₃	—	2	8	no	—
69	C₄×C₇₃	C₂×C₄×C₈	1	4	yes	—
70	C₄×C₇₃	C₂×#97	1	2	yes	—
71	C₄×C₇₃	C₄×M₁₆	1	2	yes	—
72	C₄×C₇₃	#405	1	4	yes	—
73	C₈×#97	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	C₈×M₁₆	1	4	yes	—
75	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
76	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
77	C₂×C₇₃	#1726	1	1	yes	characteristic
78	#1744	—	73	1	no	—
79	C₄×#97	—	73	1	no	—
80	C₄²×C₈	D₇₃	1	1	yes	characteristic
81	C₂×#322	—	73	2	no	—
82	C₂×C₈²	—	73	2	no	—
83	C₇₃	C₈×#97	1	1	yes	characteristic, complemented, Sylow
84	C₂×#97	—	73	4	no	—
85	C₂×C₄×C₈	—	73	2	no	—
86	C₂×C₄×C₈	—	73	2	no	—
87	C₂×C₄×C₈	Dic₇₃	1	4	yes	—
88	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
90	C₄³	C₂×D₇₃	1	1	yes	characteristic
91	#336	—	73	4	no	—
92	#322	—	73	8	no	—
93	C₈²	—	73	8	no	—
94	C₂²×C₈	—	73	4	no	—
95	C₂²×C₈	—	73	4	no	—
96	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
97	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
98	C₂×C₄²	C₄×D₇₃	1	2	yes	—
99	C₂×C₄²	C₄×D₇₃	1	2	yes	—
100	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
101	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
102	#97	—	73	16	no	—
103	C₄×C₈	—	2	8	no	—
104	C₄×C₈	—	73	4	no	—
105	C₄×C₈	—	73	4	no	—
106	C₄×C₈	—	73	4	no	—
107	C₄×C₈	—	73	4	no	—
108	C₄×C₈	C₄×D₇₃	1	4	yes	—
109	C₄×C₈	C₄×D₇₃	1	4	yes	—
110	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	73	8	no	—
116	C₂×C₈	—	73	16	no	—
117	C₂×C₈	—	73	8	no	—
118	C₂×C₈	—	146	8	no	—
119	C₂×C₈	C₄×Dic₇₃	1	4	yes	—
120	C₂×C₈	C₄×Dic₇₃	1	4	yes	—
121	C₂×C₈	#11924012	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	C₈×D₇₃	1	8	yes	—
125	C₄²	C₂×C₄×D₇₃	1	2	yes	—
126	C₄²	C₂×C₄×D₇₃	1	2	yes	—
127	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₈×Dic₇₃	1	8	yes	—
131	C₂×C₄	C₄²×D₇₃	1	2	yes	—
132	C₂×C₄	C₄²×D₇₃	1	2	yes	—
133	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
135	C₂×C₄	C₂×C₈×D₇₃	1	4	yes	—
136	C₂×C₄	C₂×#11924012	1	4	yes	—
137	C₂×C₄	#14556597	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	73	16	no	—
140	C₈	—	146	8	no	—
141	C₈	#14556502	1	8	yes	direct product factor
142	C₂²	C₄²×Dic₇₃	1	1	yes	characteristic
143	C₂²	C₂×C₈×Dic₇₃	1	2	yes	—
144	C₂²	C₄×#11924012	1	2	yes	—
145	C₂²	#17988542	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₄×C₈×D₇₃	1	4	yes	—
148	C₄	C₂×#14556502	1	2	yes	—
149	C₄	C₄×#11924012	1	2	yes	—
150	C₄	#17988541	1	4	yes	—
151	C₂	C₈×#11924012	1	4	yes	—
152	C₂	C₄×C₈×Dic₇₃	1	1	yes	characteristic
153	C₂	C₄×#14556502	1	1	yes	characteristic
154	C₂	#22609617	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₇₃
commutator	C₂×C₇₃	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄²×C₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄²×C₈	D₇₃
hypocenter	C₇₃	C₈×#97
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
10. #17988309	C₄	semidirect
11. C₈×#10596831	C₄	semidirect
18. #14556502	C₈	direct (N × H)
61. C₈×C₇₃	#97	semidirect
63. #10596831	C₄×C₈	semidirect
83. C₇₃	C₈×#97	semidirect
141. C₈	#14556502	direct (N × H)

Maximal subgroups

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

subgroup	index
#22609677	2
C₄×#14556502	2
C₂×#17988309	2
C₂×C₈×#10596831	2
C₄²×C₈×C₇₃	2
C₈×#97	73

Subgroups — C8×#14556502 #32975337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14556502 #32975337