Finite Group Databasebeta

← #34192407

Subgroups — #34192407

Order 20224.

386 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 75 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#23774781C211yesmaximal, characteristic
2#23762120C212yesmaximal, complemented
3#23762010C212yesmaximal, complemented
4#23761311C211yesmaximal, characteristic, complemented
5C79×#1849C211yesmaximal, characteristic, complemented
6#19099113C2212yes—
7#19098854—22no—
8#19098816—22no—
9#19098759—22no—
10#19098620C2212yes—
11#19098308—22no—
12#19098306—22no—
13#19098302—24no—
14C2×#108×C79—22no—
15#423×C79C2211yescharacteristic
16#423×C79C2211yescharacteristic
17C2×#97×C79C2211yescharacteristic
18C2×#11929624—22no—
19C2×D632—22no—
20#14579161D412yes—
21#14579158D412yes—
22#14579149—22no—
23#14579135—22no—
24#14579135—22no—
25C4×D316—22no—
26C4×D316—22no—
27#14579093—22no—
28C4⋊(C79⋊C8)—22no—
29#14579083—24no—
30C22⋊(C79⋊C8)—22no—
31#14579072—22no—
32C2×M16×C79—21no—
33C2×M16×C79D412yes—
34C22×C8×C79—21no—
35C2×C42×C79C2311yescharacteristic
36#108×C79—44no—
37C4⋊C8×C79—22no—
38C4⋊C8×C79—22no—
39#97×C79—21no—
40#97×C79D412yes—
41#11929624—84no—
42C2×Dic158—22no—
43C2×D316—22no—
44C2×D316—22no—
45C2×D316—22no—
46C2×C22⋊D79—42no—
47C2×C4×D79—42no—
48D632—82no—
49C2×C79⋊C8—42no—
50C4⋊Dic79—22no—
51C4⋊Dic79—22no—
52(C2×D79)⋊C4—42no—
53(C2×D79)⋊C4—42no—
54C22×C4×C79C2×D411yescharacteristic
55C22×C4×C79#4112yes—
56M16×C79—42no—
57M16×C79—42no—
58C2×C8×C79—22no—
59C2×C8×C79—22no—
60C2×C8×C79—22no—
61C2×C8×C79—22no—
62C2×C8×C79—41no—
63C42×C79C2×D412yes—
64C42×C79C2×D411yescharacteristic
65C42×C79#4111yescharacteristic
66C23×C79#12411yescharacteristic
67C22×D79—42no—
68C2×C4×C79—21no—
69C2×C4×C79—21no—
70C2×C4×C79—42no—
71C2×C4×C79#12011yescharacteristic
72C2×C4×C79#12111yescharacteristic
73C2×C4×C79#12112yes—
74C2×C4×C79#12312yes—
75C22⋊D79—84no—
76C2×Dic79—42no—
77D316—42no—
78D316—42no—
79D316—82no—
80D316—82no—
81C4×D79—84no—
82Dic158—82no—
83C8×C79—24no—
84C8×C79—42no—
85C8×C79—81no—
86C79⋊C8—82no—
87C22×C79—22no—
88C22×C79—21no—
89C22×C79#39411yescharacteristic
90C2×D79—82no—
91C2×D79—82no—
92C2×D79—82no—
93C4×C79—22no—
94C4×C79—22no—
95C4×C79—21no—
96C4×C79#45311yescharacteristic
97C4×C79#45611yescharacteristic
98Dic79—82no—
99#10524—791nomaximal, Sylow
100C2×C79—41no—
101C2×C79#203911yescharacteristic
102C2×C79#204812yes—
103D79—162no—
104#3253—791no—
105#1849D7911yescharacteristic, complemented
106#1557—792no—
107#1530—792no—
108#1307—791no—
109C79#1052411yescharacteristic, complemented, Sylow
110#532—792no—
111#476—1582no—
112#469—1582no—
113#459—1582no—
114#432—792no—
115C2×#108—22no—
116#423C2×D7911yescharacteristic
117#423C2×D7911yescharacteristic
118C2×#97C2×D7911yescharacteristic
119#332—1582no—
120#331—1582no—
121#329—1584no—
122C2×#41—1582no—
123C2×D8—1582no—
124C2×M16—21no—
125C2×M16C22⋊D7912yes—
126C22×C8—21no—
127C4⋊Q8—792no—
128#127—792no—
129#124—1582no—
130#121—1582no—
131#121—1582no—
132C4×D4—1582no—
133C4×D4—1582no—
134C2×C42C22×D7911yescharacteristic
135#108—44no—
136#106—1582no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—1582no—
140#102—1584no—
141C22⋊C8—1582no—
142#97—21no—
143#97—1582no—
144#97D31612yes—
145#41—6324no—
146C2×Q8—1582no—
147C2×D4—3162no—
148C2×D4—1582no—
149C2×D4—1582no—
150C2×D4—1582no—
151C22×C4—3162no—
152C22×C4C2×C22⋊D7911yescharacteristic
153C22×C4#1192962512yes—
154D8—6322no—
155M16—42no—
156M16—42no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—3162no—
163C4⋊C4—1582no—
164C4⋊C4—1582no—
165C22⋊C4—3162no—
166C22⋊C4—3162no—
167C42D4×D7912yes—
168C42C2×D31611yescharacteristic
169C42#1192962411yescharacteristic
170C23—3162no—
171C23#1457915011yescharacteristic
172Q8—6322no—
173D4—3162no—
174D4—3162no—
175D4—6322no—
176D4—6322no—
177D4—6324no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—3162no—
182C2×C4—6324no—
183C2×C4#1457912911yescharacteristic
184C2×C4#1457913511yescharacteristic
185C2×C4#1457913612yes—
186C2×C4#1457914612yes—
187C8—24no—
188C8—42no—
189C8—81no—
190C8—6322no—
191C22—21no—
192C22—22no—
193C22—6322no—
194C22—6322no—
195C22—6322no—
196C22#1909848011yescharacteristic
197C4—21no—
198C4—22no—
199C4—22no—
200C4—6322no—
201C4#1909873011yescharacteristic
202C4#1909874511yescharacteristic
203C2—41no—
204C2—12642no—
205C2#2376462411yescharacteristic
206C2#2376467512yes—
subgroupG/H
centerC22#19098480
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1849C2
socleC22×C79#394
prime radical#23774781C2
hypercenter#1849D79
hypocenterC79#10524
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1849

Complements

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

NH ≅ G/Ntype
2. #23762120C2semidirect
3. #23762010C2semidirect
4. #23761311C2semidirect
5. C79×#1849C2semidirect
105. #1849D79semidirect
109. C79#10524semidirect

Maximal subgroups

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

subgroupindex
#237747812
#237621202
#237620102
#237613112
C79×#18492
#1052479