Finite Group Databasebeta

← #34197711

Subgroups — #34197711

Order 20224.

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

normal subgroups only

This group has 288 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#23772045C211yesmaximal, characteristic
2#23765212C211yesmaximal, characteristic, complemented
3#23761352C212yesmaximal, complemented
4#23761337C212yesmaximal
5C79×#2165C211yesmaximal, characteristic, complemented
6C4×#11929624C411yescharacteristic
7#19098625—21no—
8#19098625—21no—
9#19098614—21no—
10#19098614—21no—
11#19098591C411yescharacteristic
12#19098591C2211yescharacteristic
13#19098531C411yescharacteristic
14#19098531C2211yescharacteristic
15C2×C4×C79⋊C8C411yescharacteristic
16#19098384—21no—
17#19098345C2212yes—
18#19098341—21no—
19C4×M16×C79C2211yescharacteristic
20#364×C79—21no—
21#349×C79C2212yes—
22#347×C79—21no—
23C2×#11929624C2×C411yescharacteristic
24C22×C79⋊C8C2×C411yescharacteristic
25C22×C79⋊C8C2×C411yescharacteristic
26C4×Dic158—21no—
27C4×D316—21no—
28C4×C22⋊D79—22no—
29#14579120C2×C411yescharacteristic
30#14579118—21no—
31C42×D79—21no—
32#14579101—22no—
33Dic79⋊C8—21no—
34Dic79⋊C8—21no—
35Dic79⋊C8—21no—
36Dic79⋊C8—21no—
37(C2×D79)⋊C8—21no—
38(C2×D79)⋊C8—21no—
39(C2×D79)⋊C8—21no—
40(C2×D79)⋊C8—21no—
41C4×C79⋊C8—21no—
42C4×C79⋊C8—21no—
43C2×M16×C79C2×C411yescharacteristic
44C2×M16×C79C2311yescharacteristic
45C2×C42×C79C2×C411yescharacteristic
46M32×C79—22no—
47#97×C79D411yescharacteristic
48#97×C79D411yescharacteristic
49C4×C8×C79D411yescharacteristic
50C4×C8×C79D411yescharacteristic
51#11929624—41no—
52#11929624—41no—
53C2×Dic158M1611yescharacteristic
54C2×D316M1611yescharacteristic
55C2×C22⋊D79—21no—
56C2×C4×D79—22no—
57C2×C4×D79—21no—
58C2×C79⋊C8—21no—
59C2×C79⋊C8—21no—
60C2×C79⋊C8—41no—
61C2×C79⋊C8—41no—
62C4⋊Dic79—21no—
63Dic79⋊C4—22no—
64C22⋊Dic79M1612yes—
65(C2×D79)⋊C4—22no—
66C4×Dic79—22no—
67C4×Dic79—21no—
68C79⋊C16—42no—
69C22×C4×C79—21no—
70C22×C4×C79C22×C411yescharacteristic
71M16×C79—41no—
72M16×C79—41no—
73C2×C8×C79C22⋊C411yescharacteristic
74C2×C8×C79C22⋊C411yescharacteristic
75C2×C8×C79C2×D411yescharacteristic
76C2×C8×C79C2×D411yescharacteristic
77C42×C79—21no—
78C42×C79C22⋊C411yescharacteristic
79C42×C79C22⋊C411yescharacteristic
80C16×C79—42no—
81C23×C79C2×M1611yescharacteristic
82C22×D79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—21no—
86C2×C4×C79—21no—
87C2×C4×C79—42no—
88C2×C4×C79C2×C22⋊C411yescharacteristic
89C2×C4×C79C2×M1611yescharacteristic
90C22⋊D79—42no—
91C2×Dic79—22no—
92C2×Dic79—21no—
93D316—41no—
94C4×D79—44no—
95C4×D79—41no—
96C4×D79—41no—
97Dic158—41no—
98C8×C79—21no—
99C8×C79—21no—
100C8×C79—21no—
101C8×C79—21no—
102C79⋊C8—41no—
103C79⋊C8—41no—
104C22×C79—21no—
105C22×C79—41no—
106C22×C79#40711yescharacteristic
107C2×D79—42no—
108C2×D79—41no—
109C4×C79—22no—
110C4×C79—21no—
111C4×C79—41no—
112C4×C79#41011yescharacteristic
113C4×C79#41311yescharacteristic
114Dic79—42no—
115Dic79—41no—
116#11524—791nomaximal, Sylow
117C2×C79—21no—
118C2×C79—41no—
119C2×C79#146411yescharacteristic
120D79—81no—
121#2922—791no—
122#2165—791no—
123#2165D7911yescharacteristic, complemented
124#1328—792no—
125#1321—792no—
126C79#1152411yescharacteristic, complemented, Sylow
127C4×#41—791no—
128#433—1581no—
129#433—1581no—
130#431—1581no—
131#431—1581no—
132#424—791no—
133#424—791no—
134#408—791no—
135#408—791no—
136C4×M16C2×D7911yescharacteristic
137C2×C4×C8—791no—
138#364—21no—
139#364—1581no—
140#349—792no—
141#349C2×D7912yes—
142#347—21no—
143#347—1581no—
144C2×#41—791no—
145C2×M16C4×D7911yescharacteristic
146C2×M16C22×D7911yescharacteristic
147C22×C8—791no—
148C22×C8—791no—
149C4×Q8—1581no—
150C4×D4—1581no—
151C4×D4—1582no—
152#117—791no—
153#117—1581no—
154C2×C42—1581no—
155C2×C42C4×D7911yescharacteristic
156M32—22no—
157M32—1582no—
158C4⋊C8—1581no—
159C4⋊C8—1581no—
160C4⋊C8—1581no—
161C4⋊C8—1581no—
162C22⋊C8—1581no—
163C22⋊C8—1581no—
164C22⋊C8—1581no—
165C22⋊C8—1581no—
166#97D31611yescharacteristic
167#97C22⋊D7911yescharacteristic
168C4×C8—1581no—
169C4×C8—1581no—
170C4×C8D31611yescharacteristic
171C4×C8C22⋊D7911yescharacteristic
172#41—3161no—
173#41—3161no—
174C2×Q8—791no—
175C2×D4—791no—
176C2×D4—1581no—
177C22×C4—21no—
178C22×C4—1582no—
179C22×C4—1581no—
180C22×C4C2×C4×D7911yescharacteristic
181M16—41no—
182M16—41no—
183C2×C8—3161no—
184C2×C8—3161no—
185C2×C8—1581no—
186C2×C8—1581no—
187C2×C8(C2×D79)⋊C411yescharacteristic
188C2×C8(C2×D79)⋊C411yescharacteristic
189C2×C8C2×C22⋊D7911yescharacteristic
190C2×C8C2×D31611yescharacteristic
191C4⋊C4—1582no—
192C4⋊C4—1581no—
193C22⋊C4—792no—
194C22⋊C4—1582no—
195C42—21no—
196C42—1582no—
197C42—1581no—
198C42(C2×D79)⋊C411yescharacteristic
199C42(C2×D79)⋊C411yescharacteristic
200C16—42no—
201C16—3162no—
202C23—1581no—
203C23M16×D7911yescharacteristic
204Q8—3161no—
205D4—3161no—
206D4—3162no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—3164no—
213C2×C4—3161no—
214C2×C4—3161no—
215C2×C4—1582no—
216C2×C4—1581no—
217C2×C4C2×(C2×D79)⋊C411yescharacteristic
218C2×C4M16×D7911yescharacteristic
219C8—21no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—3161no—
224C8—3161no—
225C22—21no—
226C22—41no—
227C22—3162no—
228C22—3161no—
229C22#1909852611yescharacteristic
230C4—22no—
231C4—21no—
232C4—41no—
233C4—3162no—
234C4—3161no—
235C4#1909853611yescharacteristic
236C4#1909854511yescharacteristic
237C2—21no—
238C2—41no—
239C2—6321no—
240C2#2376173211yescharacteristic
subgroupG/H
centerC4#19098536
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#2165C2
socleC2×C79#1464
prime radicalC2×#11929624C2×C4
cosocleC2×M16×C79C23
hypercenter#2165D79
hypocenterC79#11524
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #23765212C2semidirect
3. #23761352C2semidirect
5. C79×#2165C2semidirect
123. #2165D79semidirect
126. C79#11524semidirect

Maximal subgroups

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

subgroupindex
#237720452
#237652122
#237613522
#237613372
C79×#21652
#1152479