Subgroups — #34204710

Order 20224.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 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	2	yes	maximal
2	C₂×#19098389	C₂	1	2	yes	maximal
3	#23763564	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579095	C₂²	1	4	yes	—
6	C₂×#14579093	—	2	2	no	—
7	#19098429	—	2	2	no	—
8	#19098391	—	2	4	no	—
9	#19098389	—	2	4	no	—
10	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₉	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₇₉	—	2	4	no	—
15	#14579095	—	2	4	no	—
16	#14579095	—	2	4	no	—
17	#14579093	—	4	4	no	—
18	C₇₉⋊#95	—	4	4	no	—
19	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
25	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
27	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
28	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₇₉	Q₈	1	2	yes	—
30	C₂²×Dic₇₉	—	4	4	no	—
31	C₄⋊Dic₇₉	—	4	4	no	—
32	C₄⋊Dic₇₉	—	4	4	no	—
33	C₂²×C₄×C₇₉	D₈	1	2	yes	—
34	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₇₉	D₈	1	2	yes	—
37	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
38	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
45	C₄²×C₇₉	Q₁₆	1	2	yes	—
46	C₁₆×C₇₉	C₄⋊C₄	1	4	yes	—
47	C₁₆×C₇₉	C₄⋊C₄	1	4	yes	—
48	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	#107	1	4	yes	—
51	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×Dic₇₉	—	8	4	no	—
58	C₂×Dic₇₉	—	8	8	no	—
59	C₈×C₇₉	#107	1	4	yes	—
60	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
65	C₂²×C₇₉	#493	1	1	yes	characteristic
66	C₂²×C₇₉	#500	1	1	yes	characteristic
67	C₂²×C₇₉	#508	1	4	yes	—
68	C₄×C₇₉	—	2	4	no	—
69	C₄×C₇₉	#384	1	1	yes	characteristic
70	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
71	C₄×C₇₉	#494	1	1	yes	characteristic
72	C₄×C₇₉	#500	1	1	yes	characteristic
73	Dic₇₉	—	16	4	no	—
74	#12907	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	#1845	1	1	yes	characteristic
76	C₂×C₇₉	#2158	1	2	yes	—
77	C₂×C₇₉	#2249	1	2	yes	—
78	C₂×C₇₉	#2253	1	2	yes	—
79	C₂×#367	—	79	2	no	—
80	C₂×#366	—	79	2	no	—
81	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
82	#1846	—	79	2	no	—
83	C₇₉	#12907	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
86	C₂×#107	—	79	4	no	—
87	C₂×#106	—	158	2	no	—
88	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
89	#382	—	158	2	no	—
90	#367	—	158	4	no	—
91	#366	—	158	4	no	—
92	C₄×C₁₆	Dic₇₉	1	4	yes	complemented
93	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	158	4	no	—
96	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
97	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
98	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
99	C₂×C₁₆	D₃₁₆	1	2	yes	—
100	C₂×C₁₆	D₃₁₆	1	2	yes	—
101	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
102	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
103	#107	—	158	4	no	—
104	#107	—	158	4	no	—
105	#106	—	316	4	no	—
106	C₄×C₈	Dic₁₅₈	1	2	yes	—
107	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
108	#95	—	316	4	no	—
109	C₂²×C₄	—	316	4	no	—
110	C₂²×C₄	D₆₃₂	1	2	yes	—
111	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
113	C₂×C₈	D₆₃₂	1	2	yes	—
114	C₂×C₈	Dic₃₁₆	1	2	yes	—
115	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
116	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
117	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
119	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
121	C₄⋊C₄	—	316	4	no	—
122	C₄⋊C₄	—	316	4	no	—
123	C₄²	C₄⋊Dic₇₉	1	2	yes	—
124	C₄²	Dic₃₁₆	1	2	yes	—
125	C₁₆	C₄⋊Dic₇₉	1	4	yes	—
126	C₁₆	C₄⋊Dic₇₉	1	4	yes	—
127	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	632	4	no	—
130	C₂×C₄	—	632	8	no	—
131	C₂×C₄	#14579095	1	4	yes	—
132	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
133	C₂×C₄	#14579158	1	1	yes	characteristic
134	C₂×C₄	#14579161	1	1	yes	characteristic
135	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
138	C₈	#14579095	1	4	yes	—
139	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
140	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
141	C₈	#14579161	1	1	yes	characteristic
142	C₈	#14579161	1	1	yes	characteristic
143	C₂²	C₂×#14579095	1	1	yes	characteristic
144	C₂²	#19098940	1	1	yes	characteristic
145	C₂²	#19098968	1	1	yes	characteristic
146	C₂²	#19098995	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	1264	4	no	—
149	C₄	#19098433	1	1	yes	characteristic
150	C₄	C₂×#14579095	1	1	yes	characteristic
151	C₄	#19098943	1	1	yes	characteristic
152	C₄	#19098968	1	1	yes	characteristic
153	C₂	#23763560	1	1	yes	characteristic
154	C₂	#23765184	1	2	yes	—
155	C₂	#23765661	1	2	yes	—
156	C₂	#23765679	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	#12907
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
13. C₄×C₁₆×C₇₉	C₄	semidirect
83. C₇₉	#12907	semidirect
92. C₄×C₁₆	Dic₇₉	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
C₂×#19098389	2
#23763564	2
C₂×C₄×C₁₆×C₇₉	2
#12907	79