Finite Group Databasebeta

← #34184049

Subgroups — #34184049

Order 20224.

692 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 147 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#23771731C211yesmaximal, characteristic
2#23763468C211yesmaximal, characteristic, complemented
3C2×#19098296C212yesmaximal
4#23761279C212yesmaximal, complemented
5C2×#382×C79C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C2211yescharacteristic
8C2×C22⋊(C79⋊C8)C2212yes—
9#19098465C412yescomplemented
10#19098461C412yescomplemented
11#19098454—22no—
12#19098442—21no—
13#19098423C2211yescharacteristic
14#19098318—22no—
15#19098296—24no—
16C22×C4⋊C4×C79—21no—
17C22×C42×C79C2211yescharacteristic
18#382×C79—24no—
19C2×#95×C79—21no—
20C2×#95×C79C2212yes—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23C22×C79⋊C8—22no—
24#14579144—42no—
25#14579144—42no—
26#14579135—22no—
27#14579135—22no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C412yes—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—21no—
32C2×(C2×D79)⋊C4—22no—
33C2×(C2×D79)⋊C4—22no—
34C2×(C2×D79)⋊C4—21no—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C23×C4×C79—22no—
40C23×C4×C79D411yescharacteristic
41C23×C4×C79D411yescharacteristic
42C23×C4×C79C2311yescharacteristic
43C2×C4⋊C4×C79—24no—
44C2×C4⋊C4×C79—42no—
45C2×C4⋊C4×C79—42no—
46C2×C42×C79—22no—
47C2×C42×C79—22no—
48C2×C42×C79C2×C412yes—
49C2×C42×C79D412yes—
50#95×C79—24no—
51#95×C79—22no—
52#95×C79D414yes—
53C23×D79—21no—
54C2×D316—22no—
55C2×D316—42no—
56C2×C22⋊D79—41no—
57C2×C22⋊D79—41no—
58C2×C22⋊D79—41no—
59C2×C22⋊D79—41no—
60C2×C22⋊D79—41no—
61C2×C22⋊D79—41no—
62C22×Dic79—22no—
63C22×Dic79—21no—
64C2×C79⋊C8—44no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—22no—
67C4⋊Dic79—42no—
68Dic79⋊C4—44no—
69C22⋊Dic79—41no—
70C22⋊Dic79—41no—
71(C2×D79)⋊C4—44no—
72(C2×D79)⋊C4—44no—
73(C2×D79)⋊C4—42no—
74C24×C79C2×D411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—24no—
82C22×C4×C79—42no—
83C22×C4×C79—42no—
84C22×C4×C79—44no—
85C22×C4×C79—44no—
86C22×C4×C79—42no—
87C22×C4×C79—42no—
88C22×C4×C79C22⋊C412yes—
89C22×C4×C79C22⋊C412yes—
90C22×C4×C79D812yes—
91C22×C4×C79SD1612yes—
92C22×C4×C79C22×C411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D412yes—
95C22×C4×C79#4112yes—
96C4⋊C4×C79—48no—
97C42×C79—24no—
98C42×C79—24no—
99C23×C79—22no—
100C23×C79—22no—
101C23×C79C2×C22⋊C411yescharacteristic
102C23×C79#12011yescharacteristic
103C23×C79#12311yescharacteristic
104C23×C79C2×D811yescharacteristic
105C23×C79C2×SD1611yescharacteristic
106C23×C79AGL(1,ℤ/8)11yescharacteristic
107C23×C79#13711yescharacteristic
108C22×D79—41no—
109C22×D79—42no—
110C22×D79—41no—
111C22×D79—41no—
112C22×D79—42no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—22no—
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—22no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—44no—
129C2×C4×C79—44no—
130C2×C4×C79—44no—
131C2×C4×C79—84no—
132C2×C4×C79—84no—
133C2×C4×C79#10214yes—
134C2×C4×C79C4×D412yes—
135C2×C4×C79#12112yes—
136C22⋊D79—81no—
137C22⋊D79—81no—
138C22⋊D79—81no—
139C22⋊D79—81no—
140C2×Dic79—41no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—42no—
146D316—82no—
147C79⋊C8—82no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—22no—
153C22×C79—22no—
154C22×C79—22no—
155C22×C79—22no—
156C22×C79#38611yescharacteristic
157C22×C79#41012yes—
158C22×C79C2×#10211yescharacteristic
159C22×C79#41711yescharacteristic
160C22×C79#42112yes—
161C22×C79#44711yescharacteristic
162C22×C79#44811yescharacteristic
163C22×C79#48011yescharacteristic
164C22×C79#48111yescharacteristic
165C2×D79—81no—
166C2×D79—81no—
167C2×D79—81no—
168C2×D79—82no—
169C2×D79—82no—
170C4×C79—24no—
171C4×C79—22no—
172C4×C79—22no—
173C4×C79—42no—
174C4×C79—84no—
175Dic79—81no—
176Dic79—82no—
177#9156—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—21no—
182C2×C79#151212yes—
183C2×C79#151712yes—
184C2×C79#187411yescharacteristic
185C2×C79#189711yescharacteristic
186C2×C79#189911yescharacteristic
187D79—161no—
188#2854—791no—
189C2×#382D7911yescharacteristic, complemented
190#1817—791no—
191C2×#327—792no—
192#1294—792no—
193C79#915611yescharacteristic, complemented, Sylow
194C2×#123—1581no—
195C2×#121—791no—
196C22×C4⋊C4—21no—
197C22×C42C2×D7911yescharacteristic
198C2×C22⋊C8—792no—
199#390—792no—
200#389—792no—
201#388—1582no—
202#386—1581no—
203#382—24no—
204#380—791no—
205C2×#95—21no—
206C2×#95C2×D7912yes—
207#336—1582no—
208#327—1584no—
209C22×D4—791no—
210C22×D4—1581no—
211C23×C4—22no—
212C23×C4C22⋊D7911yescharacteristic
213C23×C4C22⋊D7911yescharacteristic
214C23×C4C22×D7911yescharacteristic
215C22×C8—1582no—
216#123—3162no—
217#123—3162no—
218#121—1582no—
219#121—1582no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—792no—
224C2×C4⋊C4—791no—
225C2×C4⋊C4—1582no—
226C2×C22⋊C4—1581no—
227C2×C22⋊C4—1582no—
228C2×C22⋊C4—1582no—
229C2×C22⋊C4—1581no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D7912yes—
233C2×C42D31612yes—
234C22⋊C8—1582no—
235C22⋊C8—1582no—
236#95—24no—
237#95—22no—
238#95—1582no—
239#95—1581no—
240#95C22⋊D7914yes—
241C24—1581no—
242C24C2×C22⋊D7911yescharacteristic
243C2×D4—3162no—
244C2×D4—3161no—
245C2×D4—3161no—
246C2×D4—3161no—
247C2×D4—3161no—
248C2×D4—3161no—
249C2×D4—3161no—
250C2×D4—1582no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—1582no—
265C22×C4—1581no—
266C22×C4(C2×D79)⋊C412yes—
267C22×C4(C2×D79)⋊C412yes—
268C22×C4#1192960912yes—
269C22×C4#1192961112yes—
270C22×C4C2×C4×D7911yescharacteristic
271C22×C4C2×C22⋊D7912yes—
272C22×C4C2×D31611yescharacteristic
273C22×C4#1192962412yes—
274C2×C8—3164no—
275C2×C8—3162no—
276C4⋊C4—48no—
277C4⋊C4—3162no—
278C4⋊C4—3164no—
279C4⋊C4—1582no—
280C22⋊C4—3164no—
281C22⋊C4—3161no—
282C22⋊C4—3161no—
283C22⋊C4—3164no—
284C22⋊C4—3162no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—3161no—
290C23—3162no—
291C23—3161no—
292C23—3161no—
293C23—3162no—
294C23C2×(C2×D79)⋊C411yescharacteristic
295C23#1457913011yescharacteristic
296C23#1457914411yescharacteristic
297C23C2×#1192960911yescharacteristic
298C23C2×#1192961111yescharacteristic
299C23#1457918711yescharacteristic
300C23#1457919211yescharacteristic
301D4—6322no—
302D4—6321no—
303D4—6321no—
304D4—6321no—
305D4—6321no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—3161no—
327C2×C4—3162no—
328C2×C4—3162no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4—3162no—
332C2×C4#1457908114yes—
333C2×C4C4×C22⋊D7912yes—
334C2×C4#1457913512yes—
335C8—6322no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—6321no—
345C22—6322no—
346C22—6322no—
347C22—6321no—
348C22—6321no—
349C22#1909844211yescharacteristic
350C22#1909853612yes—
351C22C2×#1457908111yescharacteristic
352C22#1909856211yescharacteristic
353C22#1909858112yes—
354C22#1909869711yescharacteristic
355C22#1909870211yescharacteristic
356C22#1909887611yescharacteristic
357C22#1909888111yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—84no—
363C4—6321no—
364C4—6322no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—12641no—
370C2#2376193612yes—
371C2#2376195912yes—
372C2#2376371711yescharacteristic
373C2#2376384811yescharacteristic
374C2#2376385811yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#382×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenterC2×#382D79
hypocenterC79#9156
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. #23763468C2semidirect
4. #23761279C2semidirect
5. C2×#382×C79C2semidirect
9. #19098465C4semidirect
10. #19098461C4semidirect
189. C2×#382D79semidirect
193. C79#9156semidirect

Maximal subgroups

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

subgroupindex
#237717312
#237634682
C2×#190982962
#237612792
C2×#382×C792
#915679