Finite Group Databasebeta

← #34197568

Subgroups — #34197568

Order 20224.

382 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 83 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#23772090C211yesmaximal, characteristic
2#23765019C212yesmaximal, complemented
3#23761352C212yesmaximal, complemented
4#23761336C211yesmaximal, characteristic, complemented
5C79×#1320C211yesmaximal, characteristic, complemented
6C2×#14579169C2211yescharacteristic
7C23×C79⋊C8C411yescharacteristic
8#19099074C411yescharacteristic
9M32×D79—22no—
10#19098631—22no—
11(C22⋊D79)⋊C8—22no—
12#19098591C2211yescharacteristic
13#19098531C411yescharacteristic
14#19098521C411yescharacteristic
15#19098345C2212yes—
16#19098344—22no—
17#407×C79C2211yescharacteristic
18#349×C79C2212yes—
19C2×#11929624C2×C411yescharacteristic
20#14579169—41no—
21#14579169—41no—
22C2×#11929607D412yes—
23C22×C79⋊C8—21no—
24C22×C79⋊C8—21no—
25C22×C79⋊C8—41no—
26C22×C79⋊C8—41no—
27C22×C79⋊C8C2×C411yescharacteristic
28C22×C79⋊C8C2×C411yescharacteristic
29C2×C8×D79D412yes—
30#14579144C812yes—
31#14579140—21no—
32#14579135—21no—
33#14579130C812yes—
34C4×C22⋊D79—22no—
35#14579120C2×C411yescharacteristic
36#14579101—22no—
37#14579100—42no—
38C16×D79—42no—
39Dic79⋊C8—22no—
40(C2×D79)⋊C8—22no—
41#14579071—21no—
42C8×Dic79—21no—
43C23×C4×C79C2×C411yescharacteristic
44C2×M16×C79C2×C411yescharacteristic
45C2×M16×C79C2311yescharacteristic
46M32×C79—22no—
47C22⋊C8×C79—22no—
48#11929624—41no—
49#11929624—41no—
50C2×Dic158M1611yescharacteristic
51C2×D316M1611yescharacteristic
52C2×C22⋊D79—22no—
53C2×C22⋊D79C2×C812yes—
54C2×C4×D79C22⋊C412yes—
55#11929607—42no—
56C2×C79⋊C8—21no—
57C2×C79⋊C8—21no—
58C2×C79⋊C8—21no—
59C2×C79⋊C8—41no—
60C2×C79⋊C8—42no—
61C2×C79⋊C8—41no—
62C2×C79⋊C8—41no—
63C2×C79⋊C8C22⋊C411yescharacteristic
64C2×C79⋊C8C22⋊C411yescharacteristic
65C8×D79—22no—
66C8×D79—22no—
67C4⋊Dic79—21no—
68Dic79⋊C4—22no—
69C22⋊Dic79—22no—
70C22⋊Dic79C2×C812yes—
71(C2×D79)⋊C4—22no—
72C4×Dic79—21no—
73C79⋊C16—42no—
74C24×C79C2×C811yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79C2×C811yescharacteristic
80C22×C4×C79C22×C411yescharacteristic
81M16×C79—41no—
82M16×C79—41no—
83C2×C8×C79—21no—
84C2×C8×C79C2×D412yes—
85C16×C79—42no—
86C23×C79—21no—
87C23×C79—41no—
88C23×C79—42no—
89C23×C79C22×C811yescharacteristic
90C22×D79C22⋊C812yes—
91C2×C4×C79—21no—
92C2×C4×C79—21no—
93C2×C4×C79—21no—
94C2×C4×C79—42no—
95C2×C4×C79—41no—
96C2×C4×C79—41no—
97C2×C4×C79—41no—
98C2×C4×C79C2×C22⋊C411yescharacteristic
99C2×C4×C79C2×M1611yescharacteristic
100C22⋊D79—42no—
101C22⋊D79—44no—
102C2×Dic79—22no—
103C2×Dic79C22⋊C812yes—
104D316—41no—
105C4×D79—22no—
106C4×D79—22no—
107Dic158—41no—
108C8×C79—22no—
109C8×C79—41no—
110C79⋊C8—21no—
111C79⋊C8—41no—
112C79⋊C8#9911yescharacteristic
113C79⋊C8#9911yescharacteristic
114C22×C79—22no—
115C22×C79—21no—
116C22×C79—41no—
117C22×C79—41no—
118C22×C79—42no—
119C22×C79—42no—
120C22×C79—41no—
121C22×C79C2×C22⋊C811yescharacteristic
122C2×D79—22no—
123C2×D79—42no—
124C4×C79—21no—
125C4×C79—41no—
126C4×C79C2×#9911yescharacteristic
127C4×C79#41311yescharacteristic
128Dic79—22no—
129Dic79—42no—
130#11483—791nomaximal, Sylow
131C2×C79—21no—
132C2×C79—42no—
133C2×C79—41no—
134C2×C79#145911yescharacteristic
135D79—42no—
136#2929—791no—
137#2115—792no—
138#1328—792no—
139#1320—791no—
140#1320D7911yescharacteristic, complemented
141C79#1148311yescharacteristic, complemented, Sylow
142C2×#131—791no—
143C23×C8—791no—
144#525—791no—
145C2×M32—1582no—
146#435—1582no—
147C8×D4—1582no—
148#424—791no—
149#408—791no—
150#407C2×D7911yescharacteristic
151#405—791no—
152#349—792no—
153#349—1582no—
154#349C2×D7912yes—
155C2×#41—791no—
156C23×C4C4×D7911yescharacteristic
157#131—3161no—
158#131—3161no—
159C2×M16—792no—
160C2×M16C4×D7911yescharacteristic
161C2×M16C22×D7911yescharacteristic
162C22×C8—3161no—
163C22×C8—3161no—
164C22×C8—791no—
165C22×C8—791no—
166C22×C8—792no—
167C22×C8—1581no—
168C22×C8—1581no—
169#123—792no—
170C22⋊Q8—1581no—
171#121—1581no—
172#120—792no—
173C4×D4—1582no—
174#117—791no—
175M32—22no—
176M32—3162no—
177M32—1582no—
178C2×C16—3162no—
179C4⋊C8—1582no—
180C22⋊C8—22no—
181C22⋊C8—1582no—
182#97—1581no—
183C4×C8—1581no—
184C24C8×D7911yescharacteristic
185#41—3161no—
186#41—3161no—
187C2×Q8—791no—
188C2×D4—792no—
189C2×D4—791no—
190C2×D4—1582no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—792no—
196C22×C4C8×D7911yescharacteristic
197C22×C4C2×C4×D7911yescharacteristic
198M16—41no—
199M16—41no—
200M16—3162no—
201C2×C8—21no—
202C2×C8—3161no—
203C2×C8—3161no—
204C2×C8—3162no—
205C2×C8—3161no—
206C2×C8—791no—
207C2×C8—791no—
208C2×C8—1582no—
209C2×C8—1582no—
210C2×C8—1581no—
211C2×C8—1581no—
212C2×C8—1581no—
213C2×C8D4×D7912yes—
214C4⋊C4—1582no—
215C4⋊C4—1581no—
216C22⋊C4—792no—
217C22⋊C4—1582no—
218C22⋊C4—1582no—
219C42—1581no—
220C16—42no—
221C16—3162no—
222C23—21no—
223C23—41no—
224C23—42no—
225C23—792no—
226C23C2×C8×D7911yescharacteristic
227Q8—3161no—
228D4—3161no—
229D4—3164no—
230D4—3162no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—792no—
239C2×C4—1582no—
240C2×C4—1582no—
241C2×C4—1582no—
242C2×C4C22⋊C4×D7911yescharacteristic
243C2×C4M16×D7911yescharacteristic
244C8—22no—
245C8—41no—
246C8—3161no—
247C8—791no—
248C8—791no—
249C8—1581no—
250C22—22no—
251C22—21no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—3162no—
258C22—1582no—
259C22C22⋊C8×D7911yescharacteristic
260C4—21no—
261C4—41no—
262C4—3162no—
263C4—1582no—
264C4#99×D7911yescharacteristic
265C4#1909854411yescharacteristic
266C2—21no—
267C2—42no—
268C2—41no—
269C2—3162no—
270C2#2376171511yescharacteristic
subgroupG/H
centerC4#99×D79
commutatorC23×C79C22×C8
FrattiniC2×M16C22×D79
FittingC79×#1320C2
socleC2×C79#1459
prime radical#19099074C4
cosocleC2×M16×C79C23
hypercenter#1320D79
hypocenterC79#11483
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #23765019C2semidirect
3. #23761352C2semidirect
4. #23761336C2semidirect
5. C79×#1320C2semidirect
140. #1320D79semidirect
141. C79#11483semidirect

Maximal subgroups

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

subgroupindex
#237720902
#237650192
#237613522
#237613362
C79×#13202
#1148379