Finite Group Databasebeta

← #34184335

Subgroups — #34184335

Order 20224.

622 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 127 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#23769142C211yesmaximal, characteristic
2#23766018C211yesmaximal, characteristic, complemented
3#23761969C212yesmaximal
4#23761268C212yesmaximal, complemented
5C2×#324×C79C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C2211yescharacteristic
8C2×#14579120—21no—
9#19098493C412yescomplemented
10#19098485—22no—
11#19098480C412yescomplemented
12#19098462—24no—
13#19098442—21no—
14#19098423—21no—
15#19098423C2211yescharacteristic
16#19098423C2212yes—
17#19098298—24no—
18C22×C4⋊C4×C79C2211yescharacteristic
19C2×C22⋊C8×C79C2212yes—
20#324×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23#14579144—42no—
24#14579144—42no—
25#14579135—22no—
26#14579135—22no—
27#14579120—42no—
28#14579120—42no—
29C2×C4⋊Dic79—22no—
30C2×C4⋊Dic79—22no—
31C2×C4⋊Dic79—22no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79C2×C411yescharacteristic
34C2×Dic79⋊C4—22no—
35C2×C22⋊Dic79—22no—
36C2×C22⋊Dic79—21no—
37C2×(C2×D79)⋊C4—22no—
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—22no—
40C2×C4×Dic79—22no—
41C79⋊#95—22no—
42C79⋊#95—22no—
43C79⋊#95C2×C412yes—
44C79⋊#95D414yes—
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C23×C4×C79C2311yescharacteristic
48C22×C8×C79—22no—
49C2×C4⋊C4×C79—22no—
50C2×C4⋊C4×C79—22no—
51C2×C4⋊C4×C79C2×C412yes—
52C2×C4⋊C4×C79D412yes—
53C22⋊C8×C79—22no—
54C22⋊C8×C79—22no—
55C23×D79—21no—
56C2×D316—22no—
57C2×D316—81no—
58C2×C22⋊D79—41no—
59C2×C22⋊D79—41no—
60C2×C22⋊D79—41no—
61C2×C22⋊D79—41no—
62C2×C22⋊D79—41no—
63C2×C22⋊D79—41no—
64C22×Dic79—21no—
65C22×Dic79—22no—
66C22×Dic79—24no—
67C4⋊Dic79—22no—
68C4⋊Dic79—44no—
69C4⋊Dic79—81no—
70C4⋊Dic79—82no—
71C4⋊Dic79—82no—
72Dic79⋊C4—44no—
73C22⋊Dic79—42no—
74C22⋊Dic79—41no—
75C22⋊Dic79—41no—
76C22⋊Dic79—42no—
77(C2×D79)⋊C4—44no—
78(C2×D79)⋊C4—42no—
79(C2×D79)⋊C4—82no—
80C4×Dic79—44no—
81C24×C79C2×D411yescharacteristic
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—81no—
89C22×C4×C79—81no—
90C22×C4×C79C22⋊C412yes—
91C22×C4×C79C22⋊C412yes—
92C22×C4×C79D812yes—
93C22×C4×C79SD1612yes—
94C22×C4×C79C22×C411yescharacteristic
95C22×C4×C79C2×D412yes—
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79#4112yes—
98C2×C8×C79—42no—
99C2×C8×C79—82no—
100C4⋊C4×C79—24no—
101C4⋊C4×C79—24no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79C2×C22⋊C411yescharacteristic
105C23×C79#12011yescharacteristic
106C23×C79#12311yescharacteristic
107C23×C79C2×D811yescharacteristic
108C23×C79C2×SD1611yescharacteristic
109C23×C79AGL(1,ℤ/8)11yescharacteristic
110C23×C79#13711yescharacteristic
111C22×D79—42no—
112C22×D79—42no—
113C22×D79—41no—
114C22×D79—41no—
115C22×D79—41no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—22no—
123C2×C4×C79—42no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—81no—
130C2×C4×C79#10214yes—
131C2×C4×C79C4×D412yes—
132C2×C4×C79#12112yes—
133C22⋊D79—81no—
134C22⋊D79—81no—
135C22⋊D79—81no—
136C22⋊D79—81no—
137C2×Dic79—42no—
138C2×Dic79—44no—
139C2×Dic79—42no—
140C2×Dic79—44no—
141C2×Dic79—41no—
142C2×Dic79—42no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—44no—
146D316—82no—
147C8×C79—82no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79#38611yescharacteristic
159C22×C79C2×#10211yescharacteristic
160C22×C79#41711yescharacteristic
161C22×C79#44711yescharacteristic
162C22×C79#44811yescharacteristic
163C22×C79#48011yescharacteristic
164C22×C79#48111yescharacteristic
165C2×D79—82no—
166C2×D79—82no—
167C2×D79—81no—
168C2×D79—81no—
169C2×D79—81no—
170C4×C79—24no—
171C4×C79—22no—
172C4×C79—22no—
173C4×C79—42no—
174Dic79—84no—
175Dic79—82no—
176Dic79—81no—
177#9204—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—21no—
182C2×C79—21no—
183C2×C79—21no—
184C2×C79#187411yescharacteristic
185C2×C79#191211yescharacteristic
186C2×C79#191311yescharacteristic
187D79—161no—
188#2618—791no—
189#2316—791no—
190#1519—792no—
191C2×#324D7911yescharacteristic, complemented
192#1289—792no—
193C79#920411yescharacteristic, complemented, Sylow
194C2×#123—1581no—
195C2×#121—791no—
196C2×#117—1581no—
197C22×C4⋊C4C2×D7911yescharacteristic
198C2×C22⋊C8C2×D7912yes—
199#398—792no—
200#396—1582no—
201#394—792no—
202#389—1584no—
203#386—1581no—
204#380—791no—
205#380—792no—
206#380—1581no—
207#327—1584no—
208#324—22no—
209C22×D4—791no—
210C22×D4—1581no—
211C23×C4D31611yescharacteristic
212C23×C4D31611yescharacteristic
213C23×C4C22×D7911yescharacteristic
214C22×C8—22no—
215#123—3162no—
216#123—3162no—
217#121—1582no—
218#121—1582no—
219#117—3162no—
220#117—3162no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—791no—
224C2×C4⋊C4—1582no—
225C2×C4⋊C4—1582no—
226C2×C4⋊C4—1582no—
227C2×C4⋊C4—1582no—
228C2×C4⋊C4—1581no—
229C2×C4⋊C4C4×D7912yes—
230C2×C4⋊C4C22⋊D7912yes—
231C2×C22⋊C4—1582no—
232C2×C22⋊C4—1581no—
233C2×C22⋊C4—1582no—
234C2×C22⋊C4—1581no—
235C2×C22⋊C4—1582no—
236C2×C42—1582no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#95—792no—
240#95—794no—
241#95—1582no—
242#95—1582no—
243C24—1581no—
244C24C2×D31611yescharacteristic
245C2×D4—3161no—
246C2×D4—3161no—
247C2×D4—3161no—
248C2×D4—3161no—
249C2×D4—3161no—
250C2×D4—3161no—
251C2×D4—6321no—
252C2×D4—1582no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—1582no—
262C22×C4—1581no—
263C22×C4—1584no—
264C22×C4(C2×D79)⋊C412yes—
265C22×C4(C2×D79)⋊C412yes—
266C22×C4D63212yes—
267C22×C4#1192961312yes—
268C22×C4C2×C4×D7911yescharacteristic
269C22×C4D4×D7912yes—
270C22×C4C2×C22⋊D7911yescharacteristic
271C22×C4#1192962312yes—
272C2×C8—42no—
273C2×C8—82no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—3164no—
277C4⋊C4—3164no—
278C4⋊C4—6322no—
279C4⋊C4—6321no—
280C4⋊C4—6322no—
281C4⋊C4—1582no—
282C22⋊C4—3162no—
283C22⋊C4—3161no—
284C22⋊C4—3161no—
285C22⋊C4—3162no—
286C22⋊C4—3164no—
287C22⋊C4—3162no—
288C22⋊C4—6322no—
289C42—3164no—
290C23—41no—
291C23—41no—
292C23—3162no—
293C23—3162no—
294C23—3161no—
295C23—3161no—
296C23—3161no—
297C23C2×(C2×D79)⋊C411yescharacteristic
298C23#1457913111yescharacteristic
299C23#1457914511yescharacteristic
300C23C2×D63211yescharacteristic
301C23C2×#1192961311yescharacteristic
302C23#1457918911yescharacteristic
303C23#1457919411yescharacteristic
304D4—6322no—
305D4—6321no—
306D4—6321no—
307D4—6321no—
308D4—6321no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—3162no—
324C2×C4—3164no—
325C2×C4—3164no—
326C2×C4—3162no—
327C2×C4—3161no—
328C2×C4—3164no—
329C2×C4—3162no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4#1457908314yes—
333C2×C4(C22⋊D79)⋊C412yes—
334C2×C4#1457913312yes—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—6322no—
347C22—6322no—
348C22—6321no—
349C22—6321no—
350C22—6321no—
351C22#1909844011yescharacteristic
352C22C2×#1457908311yescharacteristic
353C22#1909856511yescharacteristic
354C22#1909869911yescharacteristic
355C22#1909870411yescharacteristic
356C22#1909887711yescharacteristic
357C22#1909888211yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—6324no—
363C4—6322no—
364C4—6321no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—12641no—
372C2#2376371511yescharacteristic
373C2#2376393511yescharacteristic
374C2#2376394011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#324×C79C2
socleC22×C79#386
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenterC2×#324D79
hypocenterC79#9204
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #23766018C2semidirect
4. #23761268C2semidirect
5. C2×#324×C79C2semidirect
9. #19098493C4semidirect
11. #19098480C4semidirect
191. C2×#324D79semidirect
193. C79#9204semidirect

Maximal subgroups

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

subgroupindex
#237691422
#237660182
#237619692
#237612682
C2×#324×C792
#920479