Finite Group Databasebeta

← #34180005

Subgroups — #34180005

Order 20224.

706 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#19099291C211yesmaximal, characteristic
2#23763396C211yesmaximal, characteristic
3#23762071C212yesmaximal
4#23761830C212yesmaximal
5C79×#1461C211yesmaximal, characteristic
6#19099291—24no—
7#19099291—24no—
8C2×#14579161C2211yescharacteristic
9C2×#14579159—21no—
10C2×#14579153C412yes—
11C2×#14579151—22no—
12C2×C4×Dic158C412yes—
13C2×C4×Dic158C2211yescharacteristic
14C2×#14579126—21no—
15#19098587C2212yes—
16C2×#14579080—21no—
17#19098332—21no—
18#19098308—24no—
19#19098295—24no—
20C2×C4×Q8×C79C2211yescharacteristic
21#431×C79C2212yes—
22#324×C79—22no—
23C22×Dic158—21no—
24C22×Dic158C2×C411yescharacteristic
25C2×#11929608—22no—
26#14579161—22no—
27#14579161D414yes—
28#14579159—28no—
29#14579153—24no—
30#14579153—24no—
31#14579151—216no—
32C4×Dic158—22no—
33C4×Dic158—24no—
34C4×Dic158—24no—
35C4×Dic158D414yes—
36#14579126—28no—
37C2×C4⋊Dic79—21no—
38C2×C4⋊Dic79C2×C412yes—
39C2×C4⋊Dic79C2×C411yescharacteristic
40C2×Dic79⋊C4—22no—
41C2×Dic79⋊C4—22no—
42C2×Dic79⋊C4—22no—
43C2×Dic79⋊C4—21no—
44C2×C4×Dic79—22no—
45C2×C4×Dic79—21no—
46C4⋊(C79⋊C8)—22no—
47C4⋊(C79⋊C8)—22no—
48#14579080—42no—
49C22×Q8×C79D411yescharacteristic
50C4×Q8×C79—22no—
51C4×Q8×C79—22no—
52C2×C4⋊C4×C79D412yes—
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C42×C79C2×C412yes—
55C2×C42×C79C2311yescharacteristic
56C4⋊C8×C79—22no—
57C22⋊C8×C79—22no—
58C4×C8×C79—22no—
59C2×Dic158—24no—
60C2×Dic158—21no—
61C2×Dic158—24no—
62C2×Dic158—42no—
63C2×Dic158—41no—
64C2×Dic158—41no—
65C2×Dic158C22⋊C412yes—
66C22×Dic79—21no—
67C22×Dic79—22no—
68C22×Dic79—21no—
69#11929608—42no—
70#11929608—42no—
71C2×C79⋊C8—42no—
72C4⋊Dic79—21no—
73C4⋊Dic79—22no—
74C4⋊Dic79—24no—
75C4⋊Dic79C22⋊C414yes—
76C4⋊Dic79C22⋊C412yes—
77Dic79⋊C4—28no—
78Dic79⋊C4—28no—
79Dic79⋊C4—28no—
80Dic79⋊C4—24no—
81C4×Dic79—28no—
82C4×Dic79—24no—
83C2×Q8×C79—22no—
84C2×Q8×C79—22no—
85C2×Q8×C79—21no—
86C2×Q8×C79—21no—
87C22×C4×C79C22⋊C412yes—
88C22×C4×C79C22⋊C412yes—
89C22×C4×C79C22×C411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C2×C8×C79—42no—
93C4⋊C4×C79—22no—
94C4⋊C4×C79—24no—
95C42×C79—22no—
96C42×C79—22no—
97C42×C79C2×D412yes—
98C42×C79C2×D412yes—
99C23×C79C2×C22⋊C411yescharacteristic
100Q8×C79—42no—
101Q8×C79—42no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—41no—
108C2×C4×C79—42no—
109C2×C4×C79C2×C22⋊C411yescharacteristic
110C2×C4×C79C2×C22⋊C411yescharacteristic
111C2×C4×C79#12012yes—
112C2×C4×C79#12012yes—
113C2×C4×C79#13512yes—
114C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C79#13711yescharacteristic
116C2×Dic79—24no—
117C2×Dic79—22no—
118C2×Dic79—22no—
119C2×Dic79—24no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123Dic158—24no—
124Dic158—22no—
125Dic158—44no—
126Dic158—44no—
127Dic158#10114yescomplemented
128C8×C79—82no—
129C79⋊C8—82no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79#37911yescharacteristic
133C22×C79#41611yescharacteristic
134C22×C79#41711yescharacteristic
135C4×C79—22no—
136C4×C79—21no—
137C4×C79—42no—
138C4×C79—42no—
139C4×C79C2×#10112yes—
140C4×C79#44912yes—
141C4×C79#45212yes—
142Dic79—24no—
143Dic79—42no—
144Dic79—44no—
145#8390—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79#178411yescharacteristic
149C2×C79#178811yescharacteristic
150C2×C79#179611yescharacteristic
151C2×#557—791no—
152#1801—791no—
153#1546—792no—
154#1487—792no—
155#1461D7911yescharacteristic
156C79#839011yescharacteristic, complemented, Sylow
157#557—1584no—
158#557—1584no—
159C2×C4⋊Q8—791no—
160C2×C4⋊Q8—1581no—
161C2×#125—792no—
162C2×#125—1582no—
163C2×C4×Q8—791no—
164C2×C4×Q8—792no—
165C2×C4×Q8—1581no—
166C2×C4×Q8C2×D7911yescharacteristic
167#431C2×D7912yes—
168#423—792no—
169C2×#101—1581no—
170#343—1581no—
171#332—1584no—
172Q8⋊C8—1584no—
173#324—22no—
174C22×Q8—791no—
175C22×Q8—1581no—
176C22×Q8D31611yescharacteristic
177C2×M16—1582no—
178C4⋊Q8—794no—
179C4⋊Q8—1588no—
180C4⋊Q8—1582no—
181#125—1584no—
182#125—1584no—
183#125—15816no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—794no—
187C4×Q8—1584no—
188C4×Q8—1584no—
189C4×Q8—1588no—
190C4×Q8—1582no—
191C2×C4⋊C4—791no—
192C2×C4⋊C4—792no—
193C2×C4⋊C4—1582no—
194C2×C4⋊C4—1582no—
195C2×C4⋊C4—1582no—
196C2×C4⋊C4—1581no—
197C2×C4⋊C4—1581no—
198C2×C4⋊C4D31611yescharacteristic
199C2×C4⋊C4C22⋊D7912yes—
200C2×C42—1582no—
201C2×C42—1581no—
202C2×C42C4×D7912yes—
203C2×C42C22×D7911yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—1582no—
206C4⋊C8—1582no—
207#101—3162no—
208C22⋊C8—22no—
209C4×C8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—3162no—
215C2×Q8—3161no—
216C2×Q8—3161no—
217C2×Q8—792no—
218C2×Q8—1584no—
219C2×Q8—1581no—
220C2×Q8—1584no—
221C22×C4—1581no—
222C22×C4—1582no—
223C22×C4—1581no—
224C22×C4(C2×D79)⋊C412yes—
225C22×C4(C2×D79)⋊C412yes—
226C22×C4C2×C4×D7911yescharacteristic
227C22×C4C2×C22⋊D7911yescharacteristic
228C22×C4C2×D31611yescharacteristic
229M16—3162no—
230M16—3162no—
231C2×C8—42no—
232C2×C8—3162no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—792no—
236C4⋊C4—794no—
237C4⋊C4—1588no—
238C4⋊C4—1588no—
239C4⋊C4—1588no—
240C4⋊C4—1581no—
241C4⋊C4—1582no—
242C4⋊C4—1584no—
243C4⋊C4—1584no—
244C42—22no—
245C42—22no—
246C42—1588no—
247C42—1584no—
248C42D4×D7912yes—
249C42D4×D7912yes—
250C23C2×(C2×D79)⋊C411yescharacteristic
251Q8—42no—
252Q8—42no—
253Q8—3164no—
254Q8—3164no—
255Q8—794no—
256Q8—1584no—
257Q8—1582no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—3162no—
266C2×C4—3162no—
267C2×C4—3162no—
268C2×C4—1582no—
269C2×C4—1584no—
270C2×C4—1584no—
271C2×C4—1582no—
272C2×C4C22⋊C4×D7911yescharacteristic
273C2×C4C22⋊C4×D7911yescharacteristic
274C2×C4#1457912912yes—
275C2×C4#1457913112yes—
276C2×C4#1457918312yes—
277C2×C4#1457918911yescharacteristic
278C2×C4#1457919411yescharacteristic
279C8—82no—
280C8—6322no—
281C22—21no—
282C22—21no—
283C22#1909841811yescharacteristic
284C22#1909856011yescharacteristic
285C22#1909856511yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—3164no—
291C4—3162no—
292C4—1584no—
293C4#101×D7912yes—
294C4#1909871012yes—
295C4#1909872712yes—
296C2—21no—
297C2—21no—
298C2#2376332211yescharacteristic
299C2#2376334111yescharacteristic
300C2#2376337311yescharacteristic
subgroupG/H
centerC22#19098418
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1461C2
socleC22×C79#379
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1461D79
hypocenterC79#8390
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
127. Dic158#101semidirect
156. C79#8390semidirect

Maximal subgroups

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

subgroupindex
C2×#190992912
#237633962
#237620712
#237618302
C79×#14612
#839079