Finite Group Databasebeta

← #34202669

Subgroups — #34202669

Order 20224.

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

normal subgroups only

This group has 196 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#23765573C211yesmaximal, characteristic, complemented
2#23765473C211yesmaximal, characteristic
3#23765243C211yesmaximal, characteristic, complemented
4#23761468C211yesmaximal, characteristic, complemented
5#23761461C211yesmaximal, characteristic, complemented
6#23761437C211yesmaximal, characteristic, complemented
7C79×#2244C211yesmaximal, characteristic, complemented
8#19098999—21no—
9#19098979—21no—
10#19098681C2211yescharacteristic
11#19098385—21no—
12#19098384C2211yescharacteristic
13#19098384C2211yescharacteristic
14#19098384C2211yescharacteristic
15#19098376—21no—
16#19098372—21no—
17#19098346—21no—
18#499×C79C2211yescharacteristic
19#493×C79C2211yescharacteristic
20#357×C79—21no—
21#357×C79—21no—
22#346×C79C2211yescharacteristic
23#14579183D411yescharacteristic
24#14579167—21no—
25#14579105—41no—
26#14579102—41no—
27#14579101—21no—
28#14579101—21no—
29#14579101—21no—
30#14579100—41no—
31C16×D79—41no—
32#14579097D411yescharacteristic
33#14579089—21no—
34C2×D8×C79—21no—
35C2×D8×C79—21no—
36#127×C79—21no—
37#125×C79—21no—
38C2×C16×C79D411yescharacteristic
39C2×C16×C79D411yescharacteristic
40#107×C79—21no—
41#106×C79—21no—
42C4×C8×C79C2311yescharacteristic
43#11929624—21no—
44Dic316SD1611yescharacteristic
45#11929613—21no—
46D632SD1611yescharacteristic
47#11929608—21no—
48#11929607—41no—
49C8×D79—41no—
50C79⋊C16—41no—
51C79⋊C16—41no—
52C79⋊C16—41no—
53C2×D4×C79—41no—
54C2×D4×C79—81no—
55D8×C79—41no—
56D8×C79—41no—
57D8×C79—41no—
58D8×C79—41no—
59C2×C8×C79C2×D411yescharacteristic
60C2×C8×C79C2×D411yescharacteristic
61C4⋊C4×C79—41no—
62C4⋊C4×C79—81no—
63C42×C79#4111yescharacteristic
64C16×C79—21no—
65C16×C79—21no—
66C23×C79—81no—
67D4×C79—81no—
68D4×C79—81no—
69D4×C79—162no—
70C2×C4×C79—21no—
71C2×C4×C79—81no—
72C2×C4×C79#12111yescharacteristic
73C22⋊D79—41no—
74D316—21no—
75C4×D79—41no—
76Dic158—21no—
77C8×C79C2×SD1611yescharacteristic
78C8×C79#13511yescharacteristic
79C8×C79#13511yescharacteristic
80C8×C79AGL(1,ℤ/8)11yescharacteristic
81C79⋊C8—41no—
82C22×C79—162no—
83C22×C79—161no—
84C22×C79#47211yescharacteristic
85C2×D79—41no—
86C4×C79—41no—
87C4×C79—161no—
88C4×C79#46311yescharacteristic
89C4×C79#46511yescharacteristic
90Dic79—41no—
91#12465—791nomaximal, Sylow
92C2×C79—21no—
93C2×C79—321no—
94C2×C79#167711yescharacteristic
95D79—81no—
96#2244D7911yescharacteristic, complemented
97#2231—791no—
98#2213—791no—
99#2171—791no—
100#1369—791no—
101#1366—791no—
102#1357—791no—
103C79#1246511yescharacteristic, complemented, Sylow
104#509—1581no—
105#504—1581no—
106#499C2×D7911yescharacteristic
107#493C2×D7911yescharacteristic
108#443—791no—
109#365—1581no—
110#364—791no—
111#364—791no—
112#364—791no—
113#361—1581no—
114#360—1581no—
115#357—21no—
116#357—21no—
117#350—1581no—
118#346C2×D7911yescharacteristic
119#135—791no—
120C2×D8—21no—
121C2×D8—21no—
122#131—1581no—
123#127—21no—
124#125—21no—
125SD32—3161no—
126D16—3161no—
127M32—3161no—
128M32—1581no—
129M32—1581no—
130M32—1581no—
131C2×C16—3161no—
132C2×C16C22⋊D7911yescharacteristic
133C2×C16C22⋊D7911yescharacteristic
134#108—791no—
135#107—21no—
136#106—21no—
137#104—1581no—
138C4×C8C22×D7911yescharacteristic
139#41—1581no—
140C2×D4—41no—
141C2×D4—81no—
142Q16—791no—
143SD16—1581no—
144D8—41no—
145D8—41no—
146D8—41no—
147D8—41no—
148D8—791no—
149M16—3161no—
150M16—1581no—
151C2×C8—3161no—
152C2×C8D4×D7911yescharacteristic
153C2×C8C2×C22⋊D7911yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—81no—
156C42#1192962311yescharacteristic
157C16—21no—
158C16—21no—
159C16—3161no—
160C16—3161no—
161C16—3161no—
162C23—81no—
163Q8—1581no—
164D4—81no—
165D4—81no—
166D4—162no—
167D4—3161no—
168D4—1581no—
169C2×C4—21no—
170C2×C4—81no—
171C2×C4—3161no—
172C2×C4#1457913211yescharacteristic
173C8—3161no—
174C8SD16×D7911yescharacteristic
175C8#1457918111yescharacteristic
176C8#1457918211yescharacteristic
177C8#1457918611yescharacteristic
178C22—162no—
179C22—161no—
180C22—3161no—
181C22#1909883011yescharacteristic
182C4—41no—
183C4—161no—
184C4—3161no—
185C4#1909877511yescharacteristic
186C4#1909878911yescharacteristic
187C2—21no—
188C2—321no—
189C2—6321no—
190C2#2376280611yescharacteristic
subgroupG/H
centerC2#23762806
commutator, cosocleC4×C8×C79C23
FrattiniC4×C8C22×D79
FittingC79×#2244C2
socleC2×C79#1677
prime radical#23765473C2
hypercenter#2244D79
hypocenterC79#12465
LCSG ▷ C4×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #23765573C2semidirect
3. #23765243C2semidirect
4. #23761468C2semidirect
5. #23761461C2semidirect
6. #23761437C2semidirect
7. C79×#2244C2semidirect
96. #2244D79semidirect
103. C79#12465semidirect

Maximal subgroups

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

subgroupindex
#237655732
#237654732
#237652432
#237614682
#237614612
#237614372
C79×#22442
#1246579