Subgroups — #34175967

Order 20224.

178 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098397	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#2108	C₂	1	1	yes	maximal, characteristic
4	#19098397	—	2	1	no	—
5	#19098397	C₄	1	2	yes	complemented
6	C₂×C₇₉⋊C₃₂	—	2	1	no	—
7	C₂×C₇₉⋊C₃₂	—	2	1	no	—
8	C₂×C₇₉⋊C₃₂	—	2	1	no	—
9	C₂×C₇₉⋊C₃₂	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₇₉	C₄	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
12	#405×C₇₉	C₄	1	1	yes	characteristic
13	#346×C₇₉	—	2	1	no	—
14	C₄×C₁₆×C₇₉	—	2	1	no	—
15	C₇₉⋊C₃₂	—	2	2	no	—
16	C₇₉⋊C₃₂	—	2	2	no	—
17	C₂×M₁₆×C₇₉	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
19	#117×C₇₉	C₈	1	1	yes	characteristic
20	M₃₂×C₇₉	—	2	1	no	—
21	M₃₂×C₇₉	C₈	1	2	yes	—
22	C₂×C₁₆×C₇₉	—	2	1	no	—
23	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	Q₈	1	1	yes	characteristic
29	#97×C₇₉	—	2	1	no	—
30	C₄×C₈×C₇₉	—	2	1	no	—
31	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₇₉	—	2	1	no	—
33	M₁₆×C₇₉	C₁₆	1	2	yes	—
34	C₂×C₈×C₇₉	—	2	1	no	—
35	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₇₉	C₁₆	1	2	yes	—
42	C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₄²×C₇₉	—	2	1	no	—
44	C₁₆×C₇₉	—	2	2	no	—
45	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆×C₇₉	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₇₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₇₉	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₉	M₃₂	1	1	yes	characteristic
57	C₈×C₇₉	—	2	2	no	—
58	C₈×C₇₉	#95	1	1	yes	characteristic
59	C₈×C₇₉	#97	1	1	yes	characteristic
60	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#346	1	1	yes	characteristic
64	C₂²×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
65	C₂²×C₇₉	C₄⋊C₁₆	1	1	yes	characteristic
66	C₄×C₇₉	—	4	2	no	—
67	C₄×C₇₉	#336	1	1	yes	characteristic
68	C₄×C₇₉	C₄×C₁₆	1	1	yes	characteristic
69	C₄×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₄×C₇₉	C₄⋊C₁₆	1	1	yes	characteristic
71	#7470	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	—	2	1	no	—
73	C₂×C₇₉	—	2	1	no	—
74	C₂×C₇₉	#1374	1	1	yes	characteristic
75	C₂×C₇₉	#1401	1	2	yes	—
76	C₂×M₆₄	—	79	1	no	—
77	C₂²×C₃₂	—	79	1	no	—
78	#2108	D₇₉	1	1	yes	characteristic
79	C₇₉	#7470	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	Dic₇₉	1	1	yes	characteristic
81	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
82	#405	Dic₇₉	1	1	yes	characteristic
83	M₆₄	—	79	2	no	—
84	M₆₄	—	158	1	no	—
85	C₂×C₃₂	—	79	2	no	—
86	C₂×C₃₂	—	158	1	no	—
87	C₂×C₃₂	—	158	1	no	—
88	C₂×C₃₂	—	158	1	no	—
89	#346	—	2	1	no	—
90	C₄×C₁₆	—	2	1	no	—
91	C₂×M₁₆	C₇₉⋊C₈	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
93	#117	C₇₉⋊C₈	1	1	yes	characteristic
94	M₃₂	—	2	1	no	—
95	M₃₂	C₇₉⋊C₈	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	Dic₁₅₈	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
100	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
101	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₃₂	—	158	2	no	—
106	C₃₂	—	158	2	no	—
107	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	C₇₉⋊C₁₆	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
113	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
114	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
115	C₂×C₈	#11929608	1	1	yes	characteristic
116	C₂×C₈	#11929608	1	1	yes	characteristic
117	C₄⋊C₄	C₇₉⋊C₁₆	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
122	C₁₆	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
123	C₁₆	Dic₇₉⋊C₄	1	1	yes	characteristic
124	C₁₆	Dic₇₉⋊C₄	1	1	yes	characteristic
125	C₂³	#14579101	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
129	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
130	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
131	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
132	C₂×C₄	#14579101	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	C₇₉⋊#95	1	1	yes	characteristic
135	C₈	#14579072	1	1	yes	characteristic
136	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
137	C₈	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#19098339	1	1	yes	characteristic
140	C₂²	C₂²⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
141	C₂²	C₄⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	#19098318	1	1	yes	characteristic
144	C₄	C₄×C₇₉⋊C₁₆	1	1	yes	characteristic
145	C₄	C₂²⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
146	C₄	C₄⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#23761478	1	1	yes	characteristic
150	C₂	#23761542	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₄×C₁₆
Frattini	C₂²×C₁₆	C₂×D₇₉
Fitting	C₇₉×#2108	C₂
socle	C₂²×C₇₉	C₂²⋊C₁₆
prime radical	C₂³×C₇₉	M₃₂
cosocle	C₂²×C₁₆×C₇₉	C₂²
hypercenter	#2108	D₇₉
hypocenter	C₇₉	#7470
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
5. #19098397	C₄	semidirect
9. C₂×C₇₉⋊C₃₂	C₄	semidirect
79. C₇₉	#7470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098397	2
C₂²×C₇₉⋊C₃₂	2
C₇₉×#2108	2
#7470	79