Subgroups — #34182040

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763560	C₂	1	2	yes	maximal
2	#23763557	C₂	1	2	yes	maximal
3	#23763528	C₂	1	2	yes	maximal
4	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579095	—	2	4	no	—
6	C₂×#14579093	—	2	4	no	—
7	#19098433	C₂²	1	4	yes	—
8	#19098326	—	2	4	no	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₈²×C₇₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₇₉	—	2	4	no	—
14	C₂×C₄⋊Dic₇₉	—	2	4	no	—
15	C₂×C₄⋊Dic₇₉	—	2	4	no	—
16	#14579095	—	4	8	no	—
17	#14579093	—	4	8	no	—
18	C₂²×C₈×C₇₉	D₄	1	2	yes	—
19	C₂²×C₈×C₇₉	D₄	1	2	yes	—
20	C₂²×C₈×C₇₉	D₄	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₉	Q₈	1	2	yes	—
26	C₄×C₈×C₇₉	Q₈	1	2	yes	—
27	C₄×C₈×C₇₉	Q₈	1	2	yes	—
28	C₂²×Dic₇₉	—	4	4	no	—
29	C₄⋊Dic₇₉	—	4	8	no	—
30	C₄⋊Dic₇₉	—	4	8	no	—
31	C₄⋊Dic₇₉	—	4	8	no	—
32	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₉	—	2	4	no	—
36	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₇₉	D₈	1	4	yes	—
40	C₂×C₈×C₇₉	SD₁₆	1	8	yes	—
41	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
42	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₇₉	#127	1	1	yes	characteristic
47	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
55	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
56	C₂×C₄×C₇₉	#135	1	4	yes	—
57	C₂×Dic₇₉	—	8	4	no	—
58	C₂×Dic₇₉	—	8	4	no	—
59	C₂×Dic₇₉	—	8	4	no	—
60	C₈×C₇₉	—	2	4	no	—
61	C₈×C₇₉	#106	1	8	yes	complemented
62	C₈×C₇₉	#107	1	8	yes	complemented
63	C₂²×C₇₉	#384	1	1	yes	characteristic
64	C₂²×C₇₉	#492	1	2	yes	—
65	C₂²×C₇₉	#493	1	1	yes	characteristic
66	C₂²×C₇₉	#494	1	1	yes	characteristic
67	C₂²×C₇₉	#495	1	2	yes	—
68	C₄×C₇₉	C₂×#106	1	2	yes	—
69	C₄×C₇₉	C₂×#107	1	2	yes	—
70	C₄×C₇₉	#427	1	2	yes	—
71	C₄×C₇₉	#498	1	2	yes	—
72	C₄×C₇₉	#499	1	2	yes	—
73	C₄×C₇₉	#500	1	2	yes	—
74	Dic₇₉	—	16	4	no	—
75	#8814	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	#1706	1	2	yes	—
77	C₂×C₇₉	#1707	1	2	yes	—
78	C₂×C₇₉	#1844	1	1	yes	characteristic
79	C₂×C₇₉	#1845	1	1	yes	characteristic
80	C₂×C₇₉	#1846	1	1	yes	characteristic
81	#1845	—	79	2	no	—
82	#1844	—	79	2	no	—
83	#1834	—	79	2	no	—
84	C₂×C₈²	D₇₉	1	1	yes	characteristic
85	C₇₉	#8814	1	1	yes	characteristic, complemented, Sylow
86	C₂×#107	—	158	4	no	—
87	C₂×#106	—	158	4	no	—
88	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
91	#384	—	79	4	no	—
92	#340	—	158	4	no	—
93	C₈²	Dic₇₉	1	4	yes	complemented
94	C₂²×C₈	D₃₁₆	1	2	yes	—
95	C₂²×C₈	D₃₁₆	1	2	yes	—
96	C₂²×C₈	D₃₁₆	1	2	yes	—
97	C₂×C₄⋊C₄	—	158	4	no	—
98	C₂×C₄⋊C₄	—	158	4	no	—
99	C₂×C₄⋊C₄	—	158	4	no	—
100	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
101	#107	—	316	8	no	—
102	#106	—	316	8	no	—
103	C₄×C₈	Dic₁₅₈	1	2	yes	—
104	C₄×C₈	Dic₁₅₈	1	2	yes	—
105	C₄×C₈	Dic₁₅₈	1	2	yes	—
106	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
107	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
108	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
109	C₂²×C₄	—	316	4	no	—
110	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
115	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
116	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
117	C₂×C₈	D₆₃₂	1	4	yes	—
118	C₂×C₈	#11929613	1	8	yes	—
119	C₂×C₈	Dic₃₁₆	1	4	yes	—
120	C₄⋊C₄	—	316	8	no	—
121	C₄⋊C₄	—	316	8	no	—
122	C₄⋊C₄	—	316	8	no	—
123	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
124	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
126	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
127	C₂³	#14579158	1	1	yes	characteristic
128	C₂×C₄	—	632	4	no	—
129	C₂×C₄	—	632	4	no	—
130	C₂×C₄	—	632	4	no	—
131	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
134	C₂×C₄	#14579161	1	1	yes	characteristic
135	C₂×C₄	#14579161	1	1	yes	characteristic
136	C₂×C₄	#14579161	1	1	yes	characteristic
137	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
138	C₂×C₄	C₂×#11929613	1	4	yes	—
139	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
140	C₂×C₄	#14579183	1	4	yes	—
141	C₈	—	2	4	no	—
142	C₈	#14579093	1	8	yes	complemented
143	C₈	#14579095	1	8	yes	complemented
144	C₂²	#19098433	1	1	yes	characteristic
145	C₂²	#19098936	1	2	yes	—
146	C₂²	#19098940	1	1	yes	characteristic
147	C₂²	#19098943	1	1	yes	characteristic
148	C₂²	#19098947	1	2	yes	—
149	C₄	—	1264	4	no	—
150	C₄	C₂×#14579093	1	2	yes	—
151	C₄	C₂×#14579095	1	2	yes	—
152	C₄	#19098601	1	2	yes	—
153	C₄	#19098961	1	2	yes	—
154	C₄	#19098965	1	2	yes	—
155	C₄	#19098968	1	2	yes	—
156	C₂	#23762992	1	2	yes	—
157	C₂	#23762997	1	2	yes	—
158	C₂	#23763557	1	1	yes	characteristic
159	C₂	#23763560	1	1	yes	characteristic
160	C₂	#23763564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579158
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#127
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#8814
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. C₈²×C₇₉	C₄	semidirect
61. C₈×C₇₉	#106	semidirect
62. C₈×C₇₉	#107	semidirect
85. C₇₉	#8814	semidirect
93. C₈²	Dic₇₉	semidirect
142. C₈	#14579093	semidirect
143. C₈	#14579095	semidirect

Maximal subgroups

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

subgroup	index
#23763560	2
#23763557	2
#23763528	2
C₂×C₈²×C₇₉	2
#8814	79