Finite Group Databasebeta

← #34183231

Subgroups — #34183231

Order 20224.

398 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 95 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#23773748C211yesmaximal, characteristic
2#23772506C211yesmaximal, characteristic, complemented
3#23761870C212yesmaximal
4#23761300C212yesmaximal, complemented
5C79×#1544C211yesmaximal, characteristic, complemented
6#19099113C2211yescharacteristic
7#19098933C412yes—
8#19098929C412yes—
9#19098925—22no—
10#19098694—24no—
11#19098624—22no—
12#19098620C2212yes—
13#19098557—22no—
14C2×#14579072C2211yescharacteristic
15#19098292—24no—
16#533×C79C2211yescharacteristic
17#431×C79C2212yes—
18#332×C79—22no—
19C2×#11929624—21no—
20C22×C79⋊C8—21no—
21#14579161C2×C411yescharacteristic
22#14579158C2×C411yescharacteristic
23#14579149—21no—
24#14579135—22no—
25C4×D316—22no—
26C4⋊(C79⋊C8)—24no—
27C4⋊(C79⋊C8)D414yes—
28#14579084—24no—
29#14579081—24no—
30C22⋊(C79⋊C8)—22no—
31#14579072—22no—
32#14579072—21no—
33#14579072—22no—
34#14579072C2×C412yes—
35C4×C79⋊C8—22no—
36C4⋊Q8×C79D412yes—
37#125×C79C2×C412yes—
38C22⋊Q8×C79—22no—
39C4×Q8×C79—22no—
40#117×C79D412yes—
41C2×C42×C79C2311yescharacteristic
42C4⋊C8×C79—22no—
43C22⋊C8×C79—22no—
44C4×C8×C79—22no—
45#11929624—44no—
46C2×Dic158—21no—
47C2×D316—21no—
48C2×D316—22no—
49C2×C22⋊D79—41no—
50C2×C4×D79—41no—
51C2×C79⋊C8—22no—
52C2×C79⋊C8—24no—
53C2×C79⋊C8—42no—
54C4⋊Dic79—22no—
55(C2×D79)⋊C4—42no—
56C2×Q8×C79—22no—
57C22×C4×C79C2×D411yescharacteristic
58C22×C4×C79#4112yes—
59C2×C8×C79—42no—
60C4⋊C4×C79—24no—
61C4⋊C4×C79—22no—
62C4⋊C4×C79C22⋊C414yes—
63C22⋊C4×C79—22no—
64C42×C79—22no—
65C42×C79C22×C411yescharacteristic
66C42×C79C2×D411yescharacteristic
67C42×C79C2×D412yes—
68C23×C79#12311yescharacteristic
69C22×D79—41no—
70Q8×C79—82no—
71C2×C4×C79—22no—
72C2×C4×C79—24no—
73C2×C4×C79—42no—
74C2×C4×C79C2×C22⋊C411yescharacteristic
75C2×C4×C79C4×D412yes—
76C2×C4×C79#12011yescharacteristic
77C2×C4×C79#12112yes—
78C22⋊D79—82no—
79C2×Dic79—41no—
80D316—44no—
81D316—42no—
82C4×D79—82no—
83Dic158—42no—
84C8×C79—82no—
85C79⋊C8—42no—
86C79⋊C8—44no—
87C22×C79—22no—
88C22×C79—21no—
89C22×C79#38611yescharacteristic
90C2×D79—82no—
91C2×D79—81no—
92C4×C79—22no—
93C4×C79—22no—
94C4×C79—21no—
95C4×C79—44no—
96C4×C79#45412yes—
97Dic79—81no—
98#9008—791nomaximal, Sylow
99C2×C79—41no—
100C2×C79#188112yes—
101C2×C79#188811yescharacteristic
102D79—161no—
103#3135—791no—
104#2987—791no—
105#1544D7911yescharacteristic, complemented
106#1496—792no—
107#1302—792no—
108C79#900811yescharacteristic, complemented, Sylow
109#533C2×D7911yescharacteristic
110#532—791no—
111#491—792no—
112#490—792no—
113#489—1582no—
114#446—1584no—
115#433—1582no—
116#432—792no—
117#431C2×D7912yes—
118#416—1582no—
119C2×#97—791no—
120#332—22no—
121D4⋊C8—1584no—
122C2×#41—1581no—
123C22×C8—1581no—
124C4⋊Q8—791no—
125C4⋊Q8C22⋊D7912yes—
126#127—791no—
127#125C4×D7912yes—
128#124—1581no—
129C22⋊Q8—22no—
130#121—1582no—
131C4×Q8—22no—
132C4×D4—1582no—
133#117D31612yes—
134C2×C42C22×D7911yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—794no—
137C4⋊C8—1584no—
138#103—1584no—
139#102—1584no—
140C22⋊C8—22no—
141C22⋊C8—1582no—
142#97—792no—
143#97—1582no—
144#97—1582no—
145#97—1581no—
146C4×C8—22no—
147C4×C8—1582no—
148#41—3164no—
149C2×Q8—22no—
150C2×Q8—1581no—
151C2×D4—3161no—
152C2×D4—1581no—
153C2×D4—1582no—
154C22×C4—3161no—
155C22×C4C2×D31611yescharacteristic
156C22×C4#1192962312yes—
157C2×C8—42no—
158C2×C8—3162no—
159C2×C8—1582no—
160C2×C8—1584no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—1582no—
164C4⋊C4(C2×D79)⋊C414yes—
165C22⋊C4—22no—
166C22⋊C4—3162no—
167C42—22no—
168C42C2×C4×D7911yescharacteristic
169C42D4×D7912yes—
170C42C2×C22⋊D7911yescharacteristic
171C23—3161no—
172C23#1457914511yescharacteristic
173Q8—82no—
174Q8—3162no—
175D4—3162no—
176D4—3164no—
177D4—6322no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—3161no—
182C2×C4—6322no—
183C2×C4C2×(C2×D79)⋊C411yescharacteristic
184C2×C4(C22⋊D79)⋊C412yes—
185C2×C4#1457913111yescharacteristic
186C2×C4#1457913312yes—
187C8—82no—
188C8—3162no—
189C8—3164no—
190C22—22no—
191C22—21no—
192C22—6322no—
193C22—6321no—
194C22#1909844011yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—6321no—
200C4#1909873812yes—
201C2—41no—
202C2—12641no—
203C2#2376376012yes—
204C2#2376380111yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1544C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1544D79
hypocenterC79#9008
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
2. #23772506C2semidirect
4. #23761300C2semidirect
5. C79×#1544C2semidirect
105. #1544D79semidirect
108. C79#9008semidirect

Maximal subgroups

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

subgroupindex
#237737482
#237725062
#237618702
#237613002
C79×#15442
#900879