Finite Group Databasebeta

← #34175898

Subgroups — #34175898

Order 20224.

198 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 41 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#23764745C211yesmaximal, characteristic
2#23761346C211yesmaximal, characteristic
3C79×#1296C211yesmaximal, characteristic
4#19099194C411yescharacteristic
5#19098603—21no—
6#19098332C411yescharacteristic
7#19098323—21no—
8#19098289—21no—
9C2×C4⋊C8×C79—21no—
10#384×C79C2211yescharacteristic
11C2×#11929608—21no—
12#14579160—41no—
13#14579152—41no—
14#14579140—41no—
15#14579120—21no—
16#14579094—41no—
17#14579092—41no—
18C22⋊(C79⋊C8)—41no—
19C22×C8×C79—41no—
20C2×C4⋊C4×C79—41no—
21C2×C4⋊C4×C79C2×C411yescharacteristic
22C2×C4⋊C4×C79D411yescharacteristic
23C2×C42×C79D411yescharacteristic
24C4⋊C8×C79—22no—
25C4⋊C8×C79—41no—
26C2×Dic158—41no—
27#11929608—22no—
28#11929608—41no—
29C2×C79⋊C8—41no—
30C2×C79⋊C8—81no—
31C4⋊Dic79—21no—
32C4⋊Dic79—21no—
33C4⋊Dic79—41no—
34Dic79⋊C4—42no—
35C22⋊Dic79—41no—
36C4×Dic79—41no—
37C22×C4×C79—21no—
38C22×C4×C79—21no—
39C22×C4×C79—21no—
40C22×C4×C79C22⋊C411yescharacteristic
41C2×C8×C79—42no—
42C2×C8×C79—81no—
43C2×C8×C79—81no—
44C4⋊C4×C79—41no—
45C4⋊C4×C79—41no—
46C4⋊C4×C79—81no—
47C4⋊C4×C79—81no—
48C42×C79—21no—
49C42×C79SD1611yescharacteristic
50C42×C79Q1611yescharacteristic
51C23×C79#9911yescharacteristic
52C2×C4×C79—21no—
53C2×C4×C79—21no—
54C2×C4×C79—21no—
55C2×C4×C79—21no—
56C2×C4×C79—21no—
57C2×C4×C79—21no—
58C2×C4×C79—21no—
59C2×C4×C79—21no—
60C2×C4×C79—41no—
61C2×C4×C79—41no—
62C2×C4×C79—81no—
63C2×C4×C79—81no—
64C2×C4×C79#10311yescharacteristic
65C2×C4×C79#10411yescharacteristic
66C2×Dic79—41no—
67C2×Dic79—41no—
68Dic158—161no—
69C8×C79—82no—
70C79⋊C8—42no—
71C79⋊C8—161no—
72C22×C79—21no—
73C22×C79—21no—
74C22×C79#32811yescharacteristic
75C22×C79AΣL(1,𝔽16)11yescharacteristic
76C22×C79#35611yescharacteristic
77C4×C79—21no—
78C4×C79—21no—
79C4×C79—21no—
80C4×C79—21no—
81C4×C79—21no—
82C4×C79—21no—
83C4×C79—81no—
84C4×C79—81no—
85Dic79—81no—
86Dic79—81no—
87#7443—791nomaximal, Sylow
88C2×C79—21no—
89C2×C79—21no—
90C2×C79#135411yescharacteristic
91C2×C79#140211yescharacteristic
92C2×C79#140511yescharacteristic
93#2060—791no—
94#1325—791no—
95#1296D7911yescharacteristic
96C79#744311yescharacteristic, complemented, Sylow
97#544—791no—
98#428—1581no—
99C2×C4⋊C8—21no—
100#384C2×D7911yescharacteristic
101#343—791no—
102#339—1581no—
103#324—1581no—
104C2×M16—1581no—
105C22×C8—41no—
106C4⋊Q8—3161no—
107#125—3161no—
108C22⋊Q8—3161no—
109#117—1581no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C4×D7911yescharacteristic
112C2×C4⋊C4D31611yescharacteristic
113C2×C42C22⋊D7911yescharacteristic
114#107—3161no—
115#106—3161no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118C22⋊C8—3161no—
119C2×Q8—3161no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4(C2×D79)⋊C411yescharacteristic
124M16—3161no—
125M16—1582no—
126C2×C8—42no—
127C2×C8—81no—
128C2×C8—81no—
129C2×C8—3161no—
130C2×C8—6321no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—81no—
134C4⋊C4—81no—
135C4⋊C4—3161no—
136C4⋊C4—3162no—
137C4⋊C4—1581no—
138C4⋊C4—1581no—
139C22⋊C4—3161no—
140C42—21no—
141C42—3161no—
142C42#1192961211yescharacteristic
143C42#1192961411yescharacteristic
144C23#1457907511yescharacteristic
145Q8—12641no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—3161no—
159C2×C4—3161no—
160C2×C4#1457908411yescharacteristic
161C2×C4#1457908911yescharacteristic
162C8—82no—
163C8—3162no—
164C8—12641no—
165C22—21no—
166C22—21no—
167C22#1909829911yescharacteristic
168C22#1909835111yescharacteristic
169C22#1909836111yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—81no—
177C4—81no—
178C4—6321no—
179C4—6321no—
180C2—21no—
181C2—21no—
182C2#2376142911yescharacteristic
183C2#2376154611yescharacteristic
184C2#2376155511yescharacteristic
subgroupG/H
centerC22#19098351
commutatorC2×C4⋊C4×C79C2×C4
Frattini#384C2×D79
FittingC79×#1296C2
socleC22×C79AΣL(1,𝔽16)
prime radicalC23×C79#99
cosocle#384×C79C22
hypercenter#1296D79
hypocenterC79#7443
LCSG ▷ C2×C4⋊C4×C79 ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #1296

Complements

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

NH ≅ G/Ntype
96. C79#7443semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237613462
C79×#12962
#744379