Subgroups — #34177624

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763145	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098285	C₂	1	2	yes	maximal
4	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₉	C₄	1	4	yes	—
9	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098317	C₄	1	8	yes	—
11	#19098285	C₄	1	8	yes	complemented
12	C₈×C₇₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₈²×C₇₉	C₄	1	4	yes	complemented
16	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
20	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
22	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
23	C₈×Dic₇₉	C₈	1	16	yes	complemented
24	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₂²×Dic₇₉	C₄²	1	4	yes	—
32	C₂×C₇₉⋊C₈	—	2	8	no	—
33	C₂×C₇₉⋊C₈	C₄²	1	8	yes	—
34	C₂×C₇₉⋊C₈	C₄²	1	4	yes	—
35	C₂×C₇₉⋊C₈	C₂×C₈	1	8	yes	—
36	C₄×Dic₇₉	—	2	8	no	—
37	C₄×Dic₇₉	C₂×C₈	1	8	yes	—
38	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
40	C₂×C₈×C₇₉	—	2	8	no	—
41	C₂×C₈×C₇₉	C₄²	1	8	yes	—
42	C₂×C₈×C₇₉	C₄²	1	4	yes	—
43	C₂×C₈×C₇₉	C₂×C₈	1	8	yes	—
44	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
45	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
50	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₇₉	C₂²×C₈	1	4	yes	—
53	C₂×C₄×C₇₉	#131	1	8	yes	—
54	C₂×Dic₇₉	—	2	8	no	—
55	C₂×Dic₇₉	C₄×C₈	1	8	yes	—
56	C₈×C₇₉	—	2	8	no	—
57	C₈×C₇₉	C₄×C₈	1	8	yes	—
58	C₇₉⋊C₈	—	2	8	no	—
59	C₇₉⋊C₈	C₄×C₈	1	8	yes	—
60	C₂²×C₇₉	C₄³	1	1	yes	characteristic
61	C₂²×C₇₉	C₂×C₄×C₈	1	2	yes	—
62	C₂²×C₇₉	#405	1	4	yes	—
63	C₄×C₇₉	C₂×C₄×C₈	1	2	yes	—
64	C₄×C₇₉	C₂×C₄×C₈	1	2	yes	—
65	C₄×C₇₉	#405	1	4	yes	—
66	C₄×C₇₉	#405	1	4	yes	—
67	Dic₇₉	—	2	8	no	—
68	#7841	—	79	1	no	maximal, Sylow
69	C₂×C₇₉	#1450	1	4	yes	—
70	C₂×C₇₉	C₄²×C₈	1	1	yes	characteristic
71	C₂×C₇₉	#1726	1	2	yes	—
72	#1744	—	79	2	no	—
73	C₄²×C₈	—	79	1	no	—
74	C₂×#322	—	79	2	no	—
75	C₂×C₈²	—	79	1	no	—
76	C₂×C₈²	D₇₉	1	1	yes	characteristic
77	C₇₉	#7841	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄×C₈	—	79	4	no	—
79	C₂×C₄×C₈	—	79	2	no	—
80	C₂×C₄×C₈	—	79	1	no	—
81	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
83	C₄³	—	79	1	no	—
84	#336	—	79	8	no	—
85	#322	—	79	8	no	—
86	C₈²	—	79	4	no	—
87	C₈²	Dic₇₉	1	4	yes	complemented
88	C₂²×C₈	—	79	2	no	—
89	C₂²×C₈	—	79	4	no	—
90	C₂²×C₈	C₄×D₇₉	1	4	yes	—
91	C₂²×C₈	C₄×D₇₉	1	2	yes	—
92	C₂×C₄²	—	79	2	no	—
93	C₂×C₄²	—	79	4	no	—
94	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
95	C₄×C₈	—	79	16	no	—
96	C₄×C₈	—	79	4	no	—
97	C₄×C₈	—	79	4	no	—
98	C₄×C₈	—	79	4	no	—
99	C₄×C₈	C₄×D₇₉	1	2	yes	—
100	C₄×C₈	C₄×D₇₉	1	4	yes	—
101	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
102	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
103	C₂²×C₄	—	79	4	no	—
104	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	79	4	no	—
108	C₂×C₈	—	79	8	no	—
109	C₂×C₈	—	79	8	no	—
110	C₂×C₈	—	158	8	no	—
111	C₂×C₈	C₄×Dic₇₉	1	8	yes	—
112	C₂×C₈	C₄×Dic₇₉	1	4	yes	—
113	C₂×C₈	C₈×D₇₉	1	8	yes	—
114	C₄²	—	79	8	no	—
115	C₄²	—	158	8	no	—
116	C₄²	C₂×C₄×D₇₉	1	2	yes	—
117	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
118	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
119	C₂³	C₄²×D₇₉	1	1	yes	characteristic
120	C₂×C₄	—	79	8	no	—
121	C₂×C₄	—	158	8	no	—
122	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
123	C₂×C₄	C₄²×D₇₉	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
126	C₂×C₄	C₂×C₈×D₇₉	1	4	yes	—
127	C₂×C₄	#14579167	1	8	yes	—
128	C₈	—	2	8	no	—
129	C₈	—	79	8	no	—
130	C₈	—	158	8	no	—
131	C₈	C₈×Dic₇₉	1	8	yes	—
132	C₂²	C₄²×Dic₇₉	1	1	yes	characteristic
133	C₂²	C₄×C₈×D₇₉	1	2	yes	—
134	C₂²	#19098517	1	4	yes	—
135	C₄	—	158	8	no	—
136	C₄	C₄×C₈×D₇₉	1	2	yes	—
137	C₄	C₂×C₈×Dic₇₉	1	2	yes	—
138	C₄	#19098517	1	4	yes	—
139	C₄	#19098518	1	4	yes	—
140	C₂	#23761682	1	4	yes	—
141	C₂	C₄×C₈×Dic₇₉	1	1	yes	characteristic
142	C₂	#23763086	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₇₉
commutator	C₂×C₇₉	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#7841
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
11. #19098285	C₄	semidirect
12. C₈×C₇₉⋊C₈	C₄	semidirect
15. C₈²×C₇₉	C₄	semidirect
23. C₈×Dic₇₉	C₈	semidirect
77. C₇₉	#7841	semidirect
87. C₈²	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763145	2
C₄×C₈×Dic₇₉	2
C₂×#19098285	2
C₂×C₈×C₇₉⋊C₈	2
C₂×C₈²×C₇₉	2
#7841	79