Finite Group Databasebeta

← #34188780

Subgroups — #34188780

Order 20224.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#23774172C211yesmaximal, characteristic
2C2×#19098898C211yesmaximal, characteristic
3#23764385C211yesmaximal, characteristic
4#23763490C211yesmaximal, characteristic
5#23761972C211yesmaximal, characteristic
6#23761248C211yesmaximal, characteristic
7C79×#2502C211yesmaximal, characteristic
8C2×#14579140C2211yescharacteristic
9#19098898—22no—
10#19098898—22no—
11#19098898—21no—
12#19098896—21no—
13#19098802—22no—
14#19098719—21no—
15#19098718—21no—
16C2×#14579094—21no—
17C2×#14579084—21no—
18C2×C22⋊(C79⋊C8)C2211yescharacteristic
19C2×C22⋊(C79⋊C8)C2211yescharacteristic
20#19098423C2211yescharacteristic
21#19098329—22no—
22#19098325—22no—
23#19098300—22no—
24C2×C22⋊Q8×C79C2211yescharacteristic
25C22×C4⋊C4×C79C2211yescharacteristic
26#398×C79—22no—
27#391×C79—21no—
28#385×C79—22no—
29#384×C79—21no—
30C4×C22⋊C4×C79C2211yescharacteristic
31C22×Dic158D411yescharacteristic
32C2×#11929614—22no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8D412yes—
35#14579140—21no—
36#14579140—21no—
37#14579140—21no—
38#14579140—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79D411yescharacteristic
41C2×Dic79⋊C4—21no—
42C2×C22⋊Dic79—21no—
43C2×C22⋊Dic79—21no—
44#14579094—22no—
45#14579094—24no—
46#14579085—22no—
47#14579084—24no—
48#14579084—22no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)D412yes—
53C79⋊#95—21no—
54C22×Q8×C79D411yescharacteristic
55C23×C4×C79—21no—
56C23×C4×C79C2311yescharacteristic
57C22⋊Q8×C79—21no—
58C22⋊Q8×C79—21no—
59C22⋊Q8×C79—21no—
60C22⋊Q8×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79—21no—
66C2×C4⋊C4×C79—21no—
67C2×C4⋊C4×C79—24no—
68C2×C4⋊C4×C79D412yes—
69C2×C4⋊C4×C79D411yescharacteristic
70C2×C22⋊C4×C79—21no—
71C2×C22⋊C4×C79D412yes—
72C2×C42×C79—21no—
73#95×C79—24no—
74#95×C79—21no—
75C2×Dic158—21no—
76C2×Dic158—21no—
77C2×Dic158—42no—
78C22×Dic79—21no—
79C22×Dic79—21no—
80C22×Dic79—41no—
81#11929614—82no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—42no—
85C2×C79⋊C8—42no—
86C4⋊Dic79—21no—
87C4⋊Dic79—21no—
88C4⋊Dic79—42no—
89Dic79⋊C4—41no—
90Dic79⋊C4—41no—
91C22⋊Dic79—41no—
92C22⋊Dic79—41no—
93C22⋊Dic79—42no—
94C24×C79C2×D411yescharacteristic
95C2×Q8×C79—21no—
96C2×Q8×C79—21no—
97C2×Q8×C79—42no—
98C22×C4×C79—22no—
99C22×C4×C79—21no—
100C22×C4×C79—22no—
101C22×C4×C79—21no—
102C22×C4×C79—42no—
103C22×C4×C79—41no—
104C22×C4×C79—41no—
105C22×C4×C79—82no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79#4111yescharacteristic
112C4⋊C4×C79—22no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—22no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
120C4⋊C4×C79—84no—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—41no—
123C22⋊C4×C79—41no—
124C22⋊C4×C79Q1614yescomplemented
125C42×C79—42no—
126C23×C79—41no—
127C23×C79—41no—
128C23×C79#12011yescharacteristic
129C23×C79#12111yescharacteristic
130C23×C79#12711yescharacteristic
131C23×C79C2×Q1612yes—
132C23×C79#13712yes—
133Q8×C79—42no—
134Q8×C79—81no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—24no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79—42no—
145C2×C4×C79—44no—
146C2×C4×C79—42no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—42no—
150C2×C4×C79—82no—
151C2×C4×C79—82no—
152C2×C4×C79#12011yescharacteristic
153C2×C4×C79#12011yescharacteristic
154C2×C4×C79#12111yescharacteristic
155C2×C4×C79#12111yescharacteristic
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—81no—
159C2×Dic79—81no—
160C2×Dic79—81no—
161C2×Dic79—81no—
162C2×Dic79—81no—
163Dic158—42no—
164Dic158—42no—
165C79⋊C8—42no—
166C79⋊C8—42no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—42no—
174C22×C79—42no—
175C22×C79#39211yescharacteristic
176C22×C79#45112yes—
177C22×C79#46212yes—
178C22×C79#49411yescharacteristic
179C22×C79#49711yescharacteristic
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—44no—
184C4×C79—81no—
185C4×C79—82no—
186C4×C79—81no—
187C4×C79#45812yes—
188Dic79—81no—
189Dic79—81no—
190Dic79—161no—
191#9959—791nomaximal, Sylow
192C2×C79—21no—
193C2×C79—22no—
194C2×C79—22no—
195C2×C79—21no—
196C2×C79#200811yescharacteristic
197C2×C79#201611yescharacteristic
198C2×C79#202211yescharacteristic
199#3189—791no—
200C2×#484—791no—
201#2502D7911yescharacteristic
202#1996—791no—
203#1824—791no—
204#1520—791no—
205#1282—791no—
206C79#995911yescharacteristic, complemented, Sylow
207C2×C22⋊Q8—791no—
208C2×C22⋊Q8C2×D7911yescharacteristic
209C22×C4⋊C4C2×D7911yescharacteristic
210#484—1582no—
211#484—1581no—
212#484—1582no—
213#484—1581no—
214#467—1582no—
215#451—1581no—
216#451—1581no—
217C2×#107—1581no—
218C2×#103—1581no—
219C2×C22⋊C8—791no—
220C2×C22⋊C8—791no—
221#398—22no—
222#391—21no—
223#385—22no—
224#384—21no—
225#380—791no—
226C4×C22⋊C4C2×D7911yescharacteristic
227#342—1582no—
228#340—1582no—
229#328—1582no—
230C22×Q8—791no—
231C22×Q8C22⋊D7911yescharacteristic
232C23×C4—21no—
233C23×C4C22×D7911yescharacteristic
234C2×Q16—1582no—
235C22×C8—792no—
236C22×C8—1581no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—1581no—
242C22⋊Q8—1581no—
243C22⋊Q8—1581no—
244C22⋊Q8—1581no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—791no—
253C2×C4⋊C4—1581no—
254C2×C4⋊C4—1581no—
255C2×C4⋊C4C22⋊D7912yes—
256C2×C4⋊C4C22⋊D7911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1581no—
259C2×C22⋊C4—1581no—
260C2×C22⋊C4C22⋊D7912yes—
261C2×C42—21no—
262#107—1584no—
263#107—1582no—
264#103—1584no—
265#103—1582no—
266#103—1582no—
267C22⋊C8—792no—
268C22⋊C8—1581no—
269C22⋊C8—1581no—
270C22⋊C8—1581no—
271#95—24no—
272#95—21no—
273#95—1581no—
274C24C2×C22⋊D7911yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—3162no—
279C2×Q8—1581no—
280C2×Q8—1581no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—3161no—
290C22×C4—1581no—
291C22×C4—1581no—
292C22×C4D4×D7911yescharacteristic
293C22×C4D4×D7911yescharacteristic
294C22×C4D4×D7911yescharacteristic
295C22×C4C2×C22⋊D7911yescharacteristic
296C22×C4C2×C22⋊D7911yescharacteristic
297C22×C4#1192962311yescharacteristic
298Q16—6322no—
299C2×C8—3162no—
300C2×C8—3162no—
301C2×C8—1582no—
302C2×C8—1582no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—84no—
312C4⋊C4—3162no—
313C4⋊C4—3161no—
314C4⋊C4—3161no—
315C4⋊C4—1581no—
316C4⋊C4—1581no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—3161no—
321C22⋊C4—3161no—
322C22⋊C4—3162no—
323C22⋊C4#1192961414yescomplemented
324C42—42no—
325C23—41no—
326C23—41no—
327C23#1457913011yescharacteristic
328C23#1457913211yescharacteristic
329C23#1457915711yescharacteristic
330C23C2×#1192961412yes—
331C23#1457919212yes—
332Q8—42no—
333Q8—81no—
334Q8—3162no—
335Q8—3162no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—3161no—
354C2×C4—3161no—
355C2×C4—6321no—
356C2×C4—6321no—
357C2×C4—6321no—
358C2×C4—6321no—
359C2×C4—6321no—
360C2×C4#1457912911yescharacteristic
361C2×C4#1457912911yescharacteristic
362C2×C4#1457913311yescharacteristic
363C2×C4#1457913311yescharacteristic
364C8—3162no—
365C8—3162no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1909847111yescharacteristic
375C22#1909871912yes—
376C22#1909877012yes—
377C22#1909894211yescharacteristic
378C22#1909895511yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—81no—
384C4—82no—
385C4—81no—
386C4—6321no—
387C4—6321no—
388C4—12641no—
389C4#1909875212yes—
390C2—22no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2#2376445311yescharacteristic
395C2#2376450011yescharacteristic
396C2#2376452811yescharacteristic
subgroupG/H
centerC22#19098471
commutator, cosocleC23×C4×C79C23
FrattiniC23×C4C22×D79
FittingC79×#2502C2
socleC22×C79#392
prime radicalC24×C79C2×D4
hypercenter#2502D79
hypocenterC79#9959
LCSG ▷ C23×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
124. C22⋊C4×C79Q16semidirect
206. C79#9959semidirect
323. C22⋊C4#11929614semidirect

Maximal subgroups

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

subgroupindex
#237741722
C2×#190988982
#237643852
#237634902
#237619722
#237612482
C79×#25022
#995979