Finite Group Databasebeta

← #34202513

Subgroups — #34202513

Order 20224.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#23765652C211yesmaximal, characteristic, complemented
2#23765482C211yesmaximal, characteristic
3#23765478C211yesmaximal, characteristic
4#23765236C211yesmaximal, characteristic
5#23761475C211yesmaximal, characteristic, complemented
6#23761438C211yesmaximal, characteristic, complemented
7C79×#1358C211yesmaximal, characteristic, complemented
8#19099006—21no—
9#19099004—21no—
10C2×#14579107—21no—
11C2×#14579107—21no—
12#19098981—21no—
13#19098943C2211yescharacteristic
14#19098681C2211yescharacteristic
15#19098384C2211yescharacteristic
16#19098380—21no—
17#19098379—21no—
18#19098348—21no—
19C4×C79⋊C16C2211yescharacteristic
20#494×C79C2211yescharacteristic
21#364×C79C2211yescharacteristic
22#364×C79C2211yescharacteristic
23#14579183D411yescharacteristic
24C2×Dic316—21no—
25C2×Dic316—21no—
26#14579167—21no—
27#14579161—21no—
28Dic632—41no—
29#14579107—42no—
30#14579107—41no—
31#14579107—41no—
32#14579107—41no—
33#14579106—41no—
34#14579104—41no—
35#14579101—21no—
36#14579100—41no—
37C2×C79⋊C16D411yescharacteristic
38C2×C79⋊C16D411yescharacteristic
39C16×D79—41no—
40#14579097D411yescharacteristic
41#14579089—21no—
42C2×Q16×C79—21no—
43C2×Q16×C79—21no—
44C4⋊Q8×C79—21no—
45M32×C79—21no—
46M32×C79—21no—
47C4×C8×C79C2311yescharacteristic
48#11929624—21no—
49C2×Dic158—41no—
50Dic316—41no—
51Dic316—41no—
52Dic316—41no—
53Dic316—41no—
54Dic316D811yescharacteristic
55#11929613—21no—
56D632D811yescharacteristic
57#11929608—21no—
58#11929607—41no—
59C8×D79—41no—
60C4⋊Dic79—81no—
61C79⋊C16—21no—
62C79⋊C16—41no—
63C79⋊C16D811yescharacteristic
64C79⋊C16D811yescharacteristic
65C2×Q8×C79—41no—
66Q16×C79—41no—
67Q16×C79—41no—
68Q16×C79—41no—
69Q16×C79—41no—
70C2×C8×C79C2×D411yescharacteristic
71C2×C8×C79C2×D411yescharacteristic
72C4⋊C4×C79—81no—
73C42×C79#4111yescharacteristic
74C16×C79—41no—
75C16×C79—41no—
76Q8×C79—81no—
77Q8×C79—81no—
78C2×C4×C79—21no—
79C2×C4×C79—81no—
80C2×C4×C79#12111yescharacteristic
81C22⋊D79—41no—
82C2×Dic79—81no—
83D316—21no—
84C4×D79—41no—
85Dic158—21no—
86Dic158—81no—
87Dic158—81no—
88C8×C79C2×D811yescharacteristic
89C8×C79C2×D811yescharacteristic
90C8×C79#13511yescharacteristic
91C8×C79AGL(1,ℤ/8)11yescharacteristic
92C79⋊C8—41no—
93C22×C79#47211yescharacteristic
94C2×D79—41no—
95C4×C79—22no—
96C4×C79—161no—
97C4×C79#45911yescharacteristic
98C4×C79#46611yescharacteristic
99Dic79—41no—
100Dic79—161no—
101#12437—791nomaximal, Sylow
102C2×C79—21no—
103C2×C79#166711yescharacteristic
104D79—81no—
105#2247—791no—
106#2214—791no—
107#2214—791no—
108#2170—791no—
109#1373—791no—
110#1358—791no—
111#1358D7911yescharacteristic, complemented
112C79#1243711yescharacteristic, complemented, Sylow
113#510—1581no—
114#510—1581no—
115C2×Q32—1581no—
116C2×Q32—1581no—
117#504—1581no—
118#494—791no—
119#494C2×D7911yescharacteristic
120#443—791no—
121#364—791no—
122#364C2×D7911yescharacteristic
123#364C2×D7911yescharacteristic
124#362—1581no—
125#362—1581no—
126#350—1581no—
127C4×C16—791no—
128#135—791no—
129C2×Q16—21no—
130C2×Q16—21no—
131C2×Q16—1581no—
132C2×Q16—1581no—
133#131—1581no—
134C4⋊Q8—21no—
135C4⋊Q8—1581no—
136Q32—3161no—
137Q32—3161no—
138Q32—3161no—
139Q32—3161no—
140Q32—3162no—
141SD32—3161no—
142SD32—3161no—
143M32—21no—
144M32—21no—
145M32—3161no—
146M32—1581no—
147C2×C16—3161no—
148C2×C16—791no—
149C2×C16—791no—
150#108—791no—
151#104—1581no—
152C4×C8C22×D7911yescharacteristic
153#41—1581no—
154C2×Q8—41no—
155C2×Q8—3161no—
156Q16—41no—
157Q16—41no—
158Q16—41no—
159Q16—41no—
160Q16—3161no—
161Q16—3161no—
162Q16—3161no—
163Q16—3161no—
164Q16—791no—
165SD16—1581no—
166D8—791no—
167M16—3161no—
168M16—1581no—
169C2×C8—3161no—
170C2×C8D4×D7911yescharacteristic
171C2×C8D4×D7911yescharacteristic
172C4⋊C4—81no—
173C4⋊C4—6321no—
174C42#1192962411yescharacteristic
175C16—41no—
176C16—41no—
177C16—3161no—
178C16—791no—
179C16—791no—
180C16—1581no—
181Q8—81no—
182Q8—81no—
183Q8—6321no—
184Q8—6321no—
185Q8—1581no—
186D4—3161no—
187D4—1581no—
188C2×C4—21no—
189C2×C4—81no—
190C2×C4—3161no—
191C2×C4—6321no—
192C2×C4#1457913411yescharacteristic
193C8—3161no—
194C8D8×D7911yescharacteristic
195C8D8×D7911yescharacteristic
196C8#1457918211yescharacteristic
197C8#1457918811yescharacteristic
198C22—3161no—
199C22#1909883211yescharacteristic
200C4—22no—
201C4—161no—
202C4—3161no—
203C4—12641no—
204C4#1909875711yescharacteristic
205C4#1909879811yescharacteristic
206C2—21no—
207C2—6321no—
208C2#2376275411yescharacteristic
subgroupG/H
centerC2#23762754
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
FittingC79×#1358C2
socleC2×C79#1667
prime radical#14579183D4
hypercenter#1358D79
hypocenterC79#12437
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
1. #23765652C2semidirect
5. #23761475C2semidirect
6. #23761438C2semidirect
7. C79×#1358C2semidirect
111. #1358D79semidirect
112. C79#12437semidirect

Maximal subgroups

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

subgroupindex
#237656522
#237654822
#237654782
#237652362
#237614752
#237614382
C79×#13582
#1243779