Finite Group Databasebeta

← #34176102

Subgroups — #34176102

Order 20224.

148 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23762543C211yesmaximal, characteristic
2#23761439C211yesmaximal, characteristic, complemented
3C79×#1411C211yesmaximal, characteristic, complemented
4#19099279C411yescharacteristic
5#19098850—21no—
6#19098384—21no—
7#19098312C411yescharacteristic
8#19098292—21no—
9#494×C79C2211yescharacteristic
10#356×C79—21no—
11#14579150—41no—
12#14579136—41no—
13C4×D316—21no—
14#14579119—41no—
15#14579101—41no—
16#14579092—41no—
17C4⋊(C79⋊C8)—21no—
18#14579081—41no—
19C2×Q16×C79—41no—
20C4⋊Q8×C79C2×C411yescharacteristic
21C4⋊Q8×C79D411yescharacteristic
22#101×C79—41no—
23C4×C8×C79D411yescharacteristic
24C2×D316—21no—
25C2×D316—41no—
26C2×C4×D79—41no—
27C2×C79⋊C8—41no—
28C4⋊Dic79—21no—
29(C2×D79)⋊C4—42no—
30(C2×D79)⋊C4—41no—
31C4×Dic79—41no—
32C79⋊C16—81no—
33C2×Q8×C79—21no—
34C2×Q8×C79—21no—
35Q16×C79—81no—
36Q16×C79—81no—
37M16×C79—81no—
38C2×C8×C79—21no—
39C4⋊C4×C79—41no—
40C4⋊C4×C79—41no—
41C42×C79C22⋊C411yescharacteristic
42C22×D79—41no—
43Q8×C79—81no—
44Q8×C79—81no—
45C2×C4×C79—21no—
46C2×C4×C79—41no—
47C2×C4×C79—41no—
48C2×C4×C79#9911yescharacteristic
49C2×Dic79—41no—
50D316—41no—
51D316—41no—
52D316—161no—
53C4×D79—81no—
54C4×D79—81no—
55C8×C79—21no—
56C8×C79—21no—
57C8×C79—161no—
58C79⋊C8—81no—
59C22×C79AΣL(1,𝔽16)11yescharacteristic
60C2×D79—81no—
61C2×D79—81no—
62C2×D79—81no—
63C4×C79—21no—
64C4×C79—21no—
65C4×C79—21no—
66C4×C79—81no—
67C4×C79—81no—
68Dic79—81no—
69#7521—791nomaximal, Sylow
70C2×C79—21no—
71C2×C79#140211yescharacteristic
72D79—161no—
73#1634—791no—
74#1411D7911yescharacteristic, complemented
75#1358—791no—
76C79#752111yescharacteristic, complemented, Sylow
77#555—791no—
78#494C2×D7911yescharacteristic
79#476—1581no—
80#364—1581no—
81#356—21no—
82#333—791no—
83D4⋊C8—1581no—
84C2×Q16—41no—
85C4⋊Q8C4×D7911yescharacteristic
86C4⋊Q8C22⋊D7911yescharacteristic
87#124—3161no—
88#121—3161no—
89C4×D4—1581no—
90#117—3161no—
91M32—3161no—
92#106—3161no—
93C4⋊C8—1581no—
94#102—3161no—
95#101—41no—
96C4×C8D31611yescharacteristic
97C2×Q8—21no—
98C2×Q8—21no—
99C2×D4—3161no—
100C2×D4—1581no—
101C22×C4—3161no—
102Q16—81no—
103Q16—81no—
104M16—81no—
105C2×C8—21no—
106C2×C8—3161no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—1581no—
110C22⋊C4—3162no—
111C22⋊C4—3161no—
112C42—3161no—
113C42(C2×D79)⋊C411yescharacteristic
114C16—6321no—
115C23—3161no—
116Q8—81no—
117Q8—81no—
118D4—3161no—
119D4—3161no—
120D4—12641no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—3161no—
125C2×C4—6321no—
126C2×C4—6321no—
127C2×C4#1457907511yescharacteristic
128C8—21no—
129C8—21no—
130C8—161no—
131C8—6321no—
132C22—6321no—
133C22—6321no—
134C22—6321no—
135C22#1909834911yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—81no—
140C4—81no—
141C4—6321no—
142C2—21no—
143C2—12641no—
144C2#2376154411yescharacteristic
subgroupG/H
centerC2#23761544
commutatorC4⋊Q8×C79C2×C4
Frattini#494C2×D79
FittingC79×#1411C2
socleC2×C79#1402
prime radical#19099279C4
cosocle#494×C79C22
hypercenter#1411D79
hypocenterC79#7521
LCSG ▷ C4⋊Q8×C79 ▷ C42×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ #1411

Complements

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

NH ≅ G/Ntype
2. #23761439C2semidirect
3. C79×#1411C2semidirect
74. #1411D79semidirect
76. C79#7521semidirect

Maximal subgroups

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

subgroupindex
#237625432
#237614392
C79×#14112
#752179