Subgroups — #34285749

Order 20224.

1054 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11929608	C₂	1	2	yes	maximal
2	#23763152	C₂	1	8	yes	maximal, complemented
3	#23763094	C₂	1	4	yes	maximal
4	C₂×#376×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929608	C₂²	1	4	yes	—
6	C₄×#11929608	C₄	1	16	yes	complemented
7	C₂×#14579072	C₂²	1	4	yes	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
9	#19098318	C₂²	1	16	yes	—
10	C₂²×C₄²×C₇₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	#376×C₇₉	C₄	1	4	yes	—
13	#376×C₇₉	C₂²	1	4	yes	—
14	C₂×#95×C₇₉	C₄	1	4	yes	—
15	C₂×#11929608	—	2	16	no	—
16	C₂×#11929608	C₂×C₄	1	16	yes	—
17	C₂²×C₇₉⋊C₈	C₂³	1	8	yes	—
18	#14579072	C₂×C₄	1	16	yes	complemented
19	C₄×C₇₉⋊C₈	C₂×C₄	1	16	yes	complemented
20	C₂³×C₄×C₇₉	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₇₉	—	2	8	no	—
24	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
28	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
29	#95×C₇₉	C₂×C₄	1	16	yes	—
30	#11929608	—	2	32	no	—
31	C₂×C₇₉⋊C₈	—	2	16	no	—
32	C₂×C₇₉⋊C₈	C₂²×C₄	1	16	yes	—
33	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	—	2	8	no	—
35	C₂²×C₄×C₇₉	—	4	4	no	—
36	C₂²×C₄×C₇₉	—	4	8	no	—
37	C₂²×C₄×C₇₉	C₄²	1	8	yes	—
38	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
39	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₇₉	#41	1	8	yes	—
43	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
44	C₄²×C₇₉	—	2	8	no	—
45	C₄²×C₇₉	—	2	8	no	—
46	C₄²×C₇₉	C₄²	1	8	yes	—
47	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
48	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
49	C₂³×C₇₉	—	2	4	no	—
50	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
51	C₂³×C₇₉	#117	1	4	yes	—
52	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	2	8	no	—
54	C₂×C₄×C₇₉	—	2	8	no	—
55	C₂×C₄×C₇₉	—	2	16	no	—
56	C₂×C₄×C₇₉	—	4	8	no	—
57	C₂×C₄×C₇₉	—	4	16	no	—
58	C₂×C₄×C₇₉	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₇₉	#117	1	16	yes	—
61	C₂×C₄×C₇₉	#117	1	4	yes	—
62	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
64	C₇₉⋊C₈	—	2	16	no	—
65	C₂²×C₇₉	—	2	4	no	—
66	C₂²×C₇₉	—	2	8	no	—
67	C₂²×C₇₉	#376	1	4	yes	—
68	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
69	C₂²×C₇₉	C₂×#117	1	2	yes	—
70	C₂²×C₇₉	#568	1	4	yes	—
71	C₄×C₇₉	—	2	8	no	—
72	C₄×C₇₉	—	4	8	no	—
73	C₄×C₇₉	#376	1	4	yes	—
74	C₄×C₇₉	C₂×#117	1	4	yes	—
75	#20365	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	—	2	4	no	—
77	C₂×C₇₉	C₂×#376	1	1	yes	characteristic
78	C₂×C₇₉	#2873	1	2	yes	—
79	C₂×C₇₉	#2921	1	4	yes	—
80	C₂×C₄×M₁₆	—	79	2	no	—
81	C₂×#376	D₇₉	1	1	yes	characteristic
82	#1746	—	79	8	no	—
83	#1728	—	79	4	no	—
84	C₇₉	#20365	1	1	yes	characteristic, complemented, Sylow
85	C₂²×M₁₆	—	79	4	no	—
86	C₂²×C₄²	C₂×D₇₉	1	2	yes	—
87	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
88	C₄×M₁₆	—	79	16	no	—
89	C₂×#97	—	79	4	no	—
90	C₂×C₄×C₈	—	79	4	no	—
91	#376	Dic₇₉	1	4	yes	—
92	#376	C₂×D₇₉	1	4	yes	—
93	C₂×#95	Dic₇₉	1	4	yes	—
94	#336	—	79	16	no	—
95	C₂³×C₄	C₂×Dic₇₉	1	4	yes	—
96	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
97	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
98	C₂×M₁₆	—	79	16	no	—
99	C₂×M₁₆	—	158	16	no	—
100	C₂²×C₈	—	79	8	no	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	C₄×D₇₉	1	8	yes	—
103	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
104	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
105	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
106	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
107	#97	—	79	16	no	—
108	C₄×C₈	—	79	16	no	—
109	#95	C₂×Dic₇₉	1	16	yes	—
110	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	—	4	8	no	—
114	C₂²×C₄	C₄×Dic₇₉	1	8	yes	—
115	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
116	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
117	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
118	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
119	C₂²×C₄	#11929624	1	8	yes	—
120	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
121	M₁₆	—	158	32	no	—
122	C₂×C₈	—	79	16	no	—
123	C₂×C₈	—	158	16	no	—
124	C₄²	—	2	8	no	—
125	C₄²	—	2	8	no	—
126	C₄²	C₄×Dic₇₉	1	8	yes	—
127	C₄²	C₂×C₄×D₇₉	1	4	yes	—
128	C₄²	C₂×C₄×D₇₉	1	4	yes	—
129	C₂³	—	2	4	no	—
130	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
131	C₂³	#14579120	1	4	yes	—
132	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	C₂×C₄×Dic₇₉	1	8	yes	—
139	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
140	C₂×C₄	#14579118	1	16	yes	—
141	C₂×C₄	#14579120	1	4	yes	—
142	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
143	C₂×C₄	C₂×#11929624	1	4	yes	—
144	C₈	—	158	16	no	—
145	C₂²	—	2	8	no	—
146	C₂²	—	2	4	no	—
147	C₂²	#19098409	1	4	yes	—
148	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
149	C₂²	C₂×#14579120	1	2	yes	—
150	C₂²	#19099330	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	4	8	no	—
153	C₄	#19098409	1	4	yes	—
154	C₄	C₂×#14579118	1	4	yes	—
155	C₂	—	2	4	no	—
156	C₂	C₂×#19098409	1	1	yes	characteristic
157	C₂	#23771819	1	2	yes	—
158	C₂	#23772038	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×#376×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#376	D₇₉
hypocenter	C₇₉	#20365
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #23763152	C₂	semidirect
6. C₄×#11929608	C₄	semidirect
18. #14579072	C₂×C₄	semidirect
19. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
84. C₇₉	#20365	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11929608	2
#23763152	2
#23763094	2
C₂×#376×C₇₉	2
#20365	79