Subgroups — #34204706

Order 20224.

374 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×#19098391	C₂	1	4	yes	maximal
2	#23763560	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579095	—	2	2	no	—
5	C₂×#14579095	C₂²	1	4	yes	—
6	#19098433	—	2	2	no	—
7	#19098391	—	2	8	no	—
8	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₇₉	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₇₉	—	2	4	no	—
12	C₂×C₄⋊Dic₇₉	—	4	4	no	—
13	#14579095	—	2	8	no	—
14	#14579095	—	4	4	no	—
15	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
20	C₂×C₁₆×C₇₉	Q₈	1	4	yes	—
21	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₉	Q₈	1	2	yes	—
23	C₂²×Dic₇₉	—	4	4	no	—
24	C₄⋊Dic₇₉	—	4	4	no	—
25	C₄⋊Dic₇₉	—	4	4	no	—
26	C₄⋊Dic₇₉	—	8	8	no	—
27	C₂²×C₄×C₇₉	D₈	1	2	yes	—
28	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₇₉	D₈	1	2	yes	—
31	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
32	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
34	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
36	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
37	C₄²×C₇₉	Q₁₆	1	2	yes	—
38	C₁₆×C₇₉	C₄⋊C₄	1	8	yes	complemented
39	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₇₉	#107	1	4	yes	—
41	C₂×C₄×C₇₉	SD₃₂	1	8	yes	—
42	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×Dic₇₉	—	8	4	no	—
49	C₂×Dic₇₉	—	8	8	no	—
50	C₈×C₇₉	#107	1	4	yes	—
51	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
52	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
54	C₂²×C₇₉	#493	1	1	yes	characteristic
55	C₂²×C₇₉	#500	1	1	yes	characteristic
56	C₂²×C₇₉	C₂×SD₃₂	1	4	yes	—
57	C₄×C₇₉	#367	1	8	yes	complemented
58	C₄×C₇₉	#384	1	1	yes	characteristic
59	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
60	C₄×C₇₉	#494	1	1	yes	characteristic
61	C₄×C₇₉	#500	1	1	yes	characteristic
62	Dic₇₉	—	16	4	no	—
63	#12906	—	79	1	no	maximal, Sylow
64	C₂×C₇₉	#1845	1	1	yes	characteristic
65	C₂×C₇₉	C₂×#367	1	2	yes	—
66	C₂×C₇₉	#2248	1	2	yes	—
67	C₂×C₇₉	#2254	1	2	yes	—
68	C₂×#367	—	79	4	no	—
69	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
70	#1845	—	79	2	no	—
71	C₇₉	#12906	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
73	C₂×#107	—	79	4	no	—
74	C₂×#107	—	158	2	no	—
75	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
76	#384	—	158	2	no	—
77	#367	—	158	8	no	—
78	C₄×C₁₆	Dic₇₉	1	4	yes	complemented
79	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
81	C₂×C₄⋊C₄	—	316	4	no	—
82	C₂×C₄⋊C₄	—	158	4	no	—
83	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
84	C₂×C₁₆	Dic₁₅₈	1	4	yes	—
85	C₂×C₁₆	D₃₁₆	1	4	yes	—
86	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
87	#107	—	316	4	no	—
88	#107	—	158	8	no	—
89	C₄×C₈	Dic₁₅₈	1	2	yes	—
90	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
91	C₂²×C₄	—	316	4	no	—
92	C₂²×C₄	D₆₃₂	1	2	yes	—
93	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
95	C₂×C₈	D₆₃₂	1	2	yes	—
96	C₂×C₈	Dic₃₁₆	1	2	yes	—
97	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
98	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
99	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
100	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
101	C₄⋊C₄	—	316	4	no	—
102	C₄⋊C₄	—	316	4	no	—
103	C₄⋊C₄	—	632	8	no	—
104	C₄²	C₄⋊Dic₇₉	1	2	yes	—
105	C₄²	Dic₃₁₆	1	2	yes	—
106	C₁₆	C₄⋊Dic₇₉	1	8	yes	complemented
107	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
108	C₂×C₄	—	632	4	no	—
109	C₂×C₄	—	632	8	no	—
110	C₂×C₄	#14579095	1	4	yes	—
111	C₂×C₄	#14579106	1	8	yes	—
112	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
113	C₂×C₄	#14579158	1	1	yes	characteristic
114	C₂×C₄	#14579161	1	1	yes	characteristic
115	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
118	C₈	#14579095	1	4	yes	—
119	C₈	C₂×C₄⋊Dic₇₉	1	2	yes	—
120	C₈	#14579161	1	2	yes	—
121	C₂²	C₂×#14579095	1	1	yes	characteristic
122	C₂²	#19098940	1	1	yes	characteristic
123	C₂²	#19098968	1	1	yes	characteristic
124	C₂²	C₂×#14579106	1	4	yes	—
125	C₄	—	1264	4	no	—
126	C₄	#19098391	1	8	yes	complemented
127	C₄	#19098433	1	1	yes	characteristic
128	C₄	C₂×#14579095	1	1	yes	characteristic
129	C₄	#19098943	1	1	yes	characteristic
130	C₄	#19098968	1	1	yes	characteristic
131	C₂	#23763560	1	1	yes	characteristic
132	C₂	C₂×#19098391	1	2	yes	—
133	C₂	#23765656	1	2	yes	—
134	C₂	#23765682	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#12906
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₇₉	C₄	semidirect
38. C₁₆×C₇₉	C₄⋊C₄	semidirect
57. C₄×C₇₉	#367	semidirect
71. C₇₉	#12906	semidirect
78. C₄×C₁₆	Dic₇₉	semidirect
106. C₁₆	C₄⋊Dic₇₉	semidirect
126. C₄	#19098391	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098391	2
#23763560	2
C₂×C₄×C₁₆×C₇₉	2
#12906	79