Finite Group Databasebeta

← #34192413

Subgroups — #34192413

Order 20224.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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#23763540C211yesmaximal, characteristic, complemented
3#23762123C212yesmaximal, complemented
4#23762012C212yesmaximal, complemented
5C79×#1307C211yesmaximal, characteristic, complemented
6#19099113C2212yes—
7#19098854—22no—
8#19098816—22no—
9#19098759—22no—
10#19098587C2211yescharacteristic
11C2×#14579072C2211yescharacteristic
12#19098327—22no—
13#19098310—22no—
14#19098306—22no—
15#19098302—24no—
16#432×C79C2212yes—
17#423×C79C2211yescharacteristic
18C2×#11929624—22no—
19C2×D632—22no—
20C2×#11929608—21no—
21C22×C79⋊C8—21no—
22#14579161D412yes—
23#14579158D412yes—
24#14579149—22no—
25#14579135—22no—
26#14579135—22no—
27C4×D316—22no—
28C4×D316—22no—
29#14579093—22no—
30C4⋊(C79⋊C8)—22no—
31#14579083—24no—
32#14579072—21no—
33#14579072D412yes—
34C2×M16×C79D412yes—
35C2×C42×C79C2311yescharacteristic
36C4⋊C8×C79—22no—
37C4⋊C8×C79—22no—
38C22⋊C8×C79—22no—
39#97×C79—22no—
40#11929624—84no—
41C2×Dic158—22no—
42C2×D316—22no—
43C2×D316—22no—
44C2×D316—22no—
45C2×C22⋊D79—42no—
46C2×C4×D79—42no—
47D632—82no—
48#11929608—42no—
49C2×C79⋊C8—22no—
50C2×C79⋊C8—41no—
51C2×C79⋊C8—41no—
52C2×C79⋊C8—41no—
53C4⋊Dic79—22no—
54C4⋊Dic79—22no—
55(C2×D79)⋊C4—42no—
56(C2×D79)⋊C4—42no—
57C22×C4×C79C2×D411yescharacteristic
58C22×C4×C79#4112yes—
59M16×C79—42no—
60C2×C8×C79—22no—
61C2×C8×C79—42no—
62C42×C79C2×D412yes—
63C42×C79C2×D411yescharacteristic
64C42×C79#4111yescharacteristic
65C23×C79#12411yescharacteristic
66C22×D79—42no—
67C2×C4×C79—21no—
68C2×C4×C79—21no—
69C2×C4×C79—42no—
70C2×C4×C79#12011yescharacteristic
71C2×C4×C79#12112yes—
72C2×C4×C79#12111yescharacteristic
73C2×C4×C79#12312yes—
74C22⋊D79—84no—
75C2×Dic79—42no—
76D316—42no—
77D316—42no—
78D316—82no—
79D316—82no—
80C4×D79—84no—
81Dic158—82no—
82C8×C79—42no—
83C8×C79—82no—
84C79⋊C8—42no—
85C79⋊C8—81no—
86C22×C79—22no—
87C22×C79—21no—
88C22×C79#39411yescharacteristic
89C2×D79—82no—
90C2×D79—82no—
91C2×D79—82no—
92C4×C79—22no—
93C4×C79—22no—
94C4×C79—21no—
95C4×C79#45411yescharacteristic
96C4×C79#45511yescharacteristic
97Dic79—82no—
98#10525—791nomaximal, Sylow
99C2×C79—41no—
100C2×C79#204011yescharacteristic
101C2×C79#204812yes—
102D79—162no—
103#3253—791no—
104#1838—791no—
105#1557—792no—
106#1530—792no—
107#1307D7911yescharacteristic, complemented
108C79#1052511yescharacteristic, complemented, Sylow
109#532—792no—
110#476—1582no—
111#469—1582no—
112#459—1582no—
113#432C2×D7912yes—
114#423—791no—
115#423C2×D7911yescharacteristic
116C2×#97—791no—
117#341—1582no—
118#332—1582no—
119#331—1582no—
120#329—1584no—
121C2×#41—1582no—
122C2×D8—1582no—
123C2×M16—1581no—
124C2×M16D31612yes—
125C22×C8—1581no—
126C4⋊Q8—792no—
127#127—792no—
128#124—1582no—
129#121—1582no—
130#121—1582no—
131C4×D4—1582no—
132C4×D4—1582no—
133C2×C42C22×D7911yescharacteristic
134#106—1582no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—1582no—
138#102—1584no—
139C22⋊C8—22no—
140#97—22no—
141#97—792no—
142#97—1581no—
143#41—6324no—
144C2×Q8—1582no—
145C2×D4—3162no—
146C2×D4—1582no—
147C2×D4—1582no—
148C2×D4—1582no—
149C22×C4—3162no—
150C22×C4C2×D31611yescharacteristic
151C22×C4#1192962412yes—
152D8—6322no—
153M16—42no—
154M16—3162no—
155C2×C8—22no—
156C2×C8—42no—
157C2×C8—3161no—
158C2×C8—3161no—
159C2×C8—3161no—
160C2×C8—1582no—
161C4⋊C4—1582no—
162C4⋊C4—1582no—
163C22⋊C4—3162no—
164C22⋊C4—3162no—
165C42D4×D7912yes—
166C42D4×D7911yescharacteristic
167C42#1192962511yescharacteristic
168C23—3162no—
169C23#1457914911yescharacteristic
170Q8—6322no—
171D4—3162no—
172D4—3162no—
173D4—6322no—
174D4—6322no—
175D4—6324no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—3162no—
180C2×C4—6324no—
181C2×C4#1457913111yescharacteristic
182C2×C4#1457913412yes—
183C2×C4#1457913611yescharacteristic
184C2×C4#1457914612yes—
185C8—42no—
186C8—82no—
187C8—3162no—
188C8—6321no—
189C22—21no—
190C22—22no—
191C22—6322no—
192C22—6322no—
193C22—6322no—
194C22#1909848111yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—6322no—
199C4#1909873811yescharacteristic
200C4#1909874311yescharacteristic
201C2—41no—
202C2—12642no—
203C2#2376463011yescharacteristic
204C2#2376467612yes—
subgroupG/H
centerC22#19098481
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1307C2
socleC22×C79#394
prime radical#23774781C2
hypercenter#1307D79
hypocenterC79#10525
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #23763540C2semidirect
3. #23762123C2semidirect
4. #23762012C2semidirect
5. C79×#1307C2semidirect
107. #1307D79semidirect
108. C79#10525semidirect

Maximal subgroups

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

subgroupindex
#237747812
#237635402
#237621232
#237620122
C79×#13072
#1052579