Subgroups — C₂×#23761466 #34199821

Order 20224.

254 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098314	C₂	1	2	yes	maximal
2	#23761466	C₂	1	4	yes	maximal, direct product factor
3	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
5	#19098314	—	2	2	no	—
6	#19098314	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	#346×C₇₉	C₄	1	2	yes	—
11	#346×C₇₉	C₂²	1	2	yes	—
12	C₂²×C₇₉⋊C₈	—	4	2	no	—
13	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
15	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	—	2	2	no	—
19	C₂×C₁₆×C₇₉	C₈	1	4	yes	complemented
20	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₇₉⋊C₈	—	4	4	no	—
27	C₂×C₇₉⋊C₈	—	8	4	no	—
28	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
30	C₂×C₈×C₇₉	—	2	2	no	—
31	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
35	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₇₉	SD₁₆	1	2	yes	—
39	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₉	—	2	2	no	—
41	C₁₆×C₇₉	C₂×C₈	1	4	yes	complemented
42	C₂³×C₇₉	#106	1	1	yes	characteristic
43	C₂×C₄×C₇₉	—	2	4	no	—
44	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
45	C₂×C₄×C₇₉	#106	1	2	yes	—
46	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
47	C₂×C₄×C₇₉	#108	1	2	yes	—
48	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
50	C₈×C₇₉	—	2	1	no	—
51	C₈×C₇₉	—	2	1	no	—
52	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
53	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₇₉	C₂×M₁₆	1	1	yes	characteristic
55	C₇₉⋊C₈	—	8	4	no	—
56	C₂²×C₇₉	#334	1	2	yes	—
57	C₂²×C₇₉	#365	1	4	yes	—
58	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
59	C₄×C₇₉	—	2	2	no	—
60	C₄×C₇₉	—	2	2	no	—
61	C₄×C₇₉	#334	1	2	yes	—
62	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
64	C₂×#1368	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	#1368	1	4	yes	complemented
66	C₂×C₇₉	C₂×#334	1	1	yes	characteristic
67	C₂×C₇₉	C₂×#365	1	2	yes	—
68	C₂×#346	D₇₉	1	1	yes	characteristic
69	C₂×#334	—	79	2	no	—
70	#1368	—	79	4	no	—
71	C₇₉	C₂×#1368	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
73	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
74	C₂×C₄⋊C₈	—	158	2	no	—
75	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
76	#346	Dic₇₉	1	2	yes	—
77	#346	C₂×D₇₉	1	2	yes	—
78	#334	—	79	4	no	—
79	#334	—	158	2	no	—
80	C₂²×C₈	—	316	2	no	—
81	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
83	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₇₉⋊C₈	1	4	yes	complemented
86	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
87	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
88	C₄⋊C₈	—	316	2	no	—
89	C₄⋊C₈	—	158	4	no	—
90	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
91	C₄×C₈	D₃₁₆	1	1	yes	characteristic
92	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
94	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
95	C₂²×C₄	#11929613	1	2	yes	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	316	4	no	—
98	C₂×C₈	—	632	4	no	—
99	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
100	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
101	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
102	C₂×C₈	#11929608	1	2	yes	—
103	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
105	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
106	C₄²	#11929613	1	2	yes	—
107	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
108	C₁₆	—	2	2	no	—
109	C₁₆	C₂×C₇₉⋊C₈	1	4	yes	complemented
110	C₂³	#14579093	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
113	C₂×C₄	#14579093	1	2	yes	—
114	C₂×C₄	#14579093	1	1	yes	characteristic
115	C₂×C₄	#14579097	1	2	yes	—
116	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
117	C₂×C₄	C₂×#11929613	1	2	yes	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	—	632	4	no	—
121	C₈	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
122	C₈	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
123	C₈	C₂×#11929608	1	1	yes	characteristic
124	C₂²	#19098314	1	2	yes	—
125	C₂²	#19098386	1	4	yes	—
126	C₂²	C₂×#14579093	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#19098314	1	2	yes	—
130	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
131	C₄	C₂×#14579097	1	1	yes	characteristic
132	C₂	#23761466	1	4	yes	direct product factor
133	C₂	C₂×#19098314	1	1	yes	characteristic
134	C₂	C₂×#19098386	1	2	yes	—

	subgroup	G/H
center	C₂³	#14579093
commutator	C₈×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	C₂×#1368
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #23761466	C₂	direct (N × H)
19. C₂×C₁₆×C₇₉	C₈	semidirect
41. C₁₆×C₇₉	C₂×C₈	semidirect
65. C₂×C₇₉	#1368	semidirect
71. C₇₉	C₂×#1368	semidirect
85. C₂×C₁₆	C₇₉⋊C₈	semidirect
109. C₁₆	C₂×C₇₉⋊C₈	semidirect
132. C₂	#23761466	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×#19098314	2
#23761466	2
C₂×#346×C₇₉	2
C₂×#1368	79

Subgroups — C2×#23761466 #34199821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761466 #34199821