Finite Group Databasebeta

← #34205043

Subgroups — #34205043

Order 20224.

504 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 99 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×#19098353C211yesmaximal, characteristic
2#23763814C211yesmaximal, characteristic
3#23763680C211yesmaximal, characteristic
4#23763609C211yesmaximal, characteristic
5#23761525C211yesmaximal, characteristic
6#23761518C211yesmaximal, characteristic
7C79×#2376C211yesmaximal, characteristic
8#19099038C2211yescharacteristic
9#19098603—21no—
10C2×#14579078C2211yescharacteristic
11C2×#14579076C2211yescharacteristic
12C2×#14579076C2211yescharacteristic
13#19098436—21no—
14#19098426—21no—
15#19098353C414yescomplemented
16#19098329—21no—
17#19098329—21no—
18#19098323—21no—
19C2×#123×C79C2211yescharacteristic
20#390×C79C2211yescharacteristic
21#389×C79—21no—
22#388×C79—21no—
23#388×C79—21no—
24#387×C79—21no—
25#386×C79C2211yescharacteristic
26#385×C79—21no—
27#380×C79—21no—
28C2×#11929608—21no—
29D4×Dic79—41no—
30D4×Dic79—41no—
31#14579120—21no—
32C2×Dic79⋊C4—41no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37#14579094—41no—
38#14579092—41no—
39#14579078—21no—
40#14579078C2×C412yes—
41#14579076—21no—
42#14579076—21no—
43#14579076—21no—
44#14579076C2×C412yes—
45C79⋊#95—41no—
46C79⋊#95—41no—
47C79⋊#95—41no—
48C22×D4×C79C2311yescharacteristic
49C23×C4×C79D411yescharacteristic
50C23×C4×C79D411yescharacteristic
51#123×C79—21no—
52#123×C79—21no—
53#123×C79C2×C412yes—
54#123×C79Q812yes—
55C2×C4⋊C4×C79—21no—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79D411yescharacteristic
60C2×C22⋊C4×C79—21no—
61C2×C22⋊C4×C79—21no—
62C2×C22⋊C4×C79—21no—
63C2×C22⋊C4×C79—21no—
64C2×C22⋊C4×C79D411yescharacteristic
65C2×C22⋊C4×C79D411yescharacteristic
66C2×C42×C79—21no—
67C2×C42×C79D411yescharacteristic
68#95×C79—21no—
69#95×C79—21no—
70#95×C79—21no—
71#95×C79—21no—
72#95×C79—21no—
73C22×Dic79—21no—
74C22×Dic79—21no—
75C22×Dic79—21no—
76C22×Dic79—21no—
77C22×Dic79—41no—
78C22×Dic79—41no—
79#11929608—22no—
80#11929608—41no—
81C2×C79⋊C8—41no—
82C4⋊Dic79—21no—
83C4⋊Dic79—21no—
84Dic79⋊C4—82no—
85C22⋊Dic79—22no—
86C22⋊Dic79—41no—
87C22⋊Dic79—41no—
88C22⋊Dic79—41no—
89C22⋊Dic79—41no—
90C22⋊Dic79—41no—
91C22⋊Dic79—41no—
92C4×Dic79—41no—
93C24×C79C2×D411yescharacteristic
94C24×C79C2×D411yescharacteristic
95C2×D4×C79—41no—
96C2×D4×C79—41no—
97C2×D4×C79C22×C411yescharacteristic
98C2×D4×C79C2×Q811yescharacteristic
99C2×D4×C79#4111yescharacteristic
100C2×D4×C79#4111yescharacteristic
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—21no—
108C22×C4×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79—41no—
111C22×C4×C79—81no—
112C22×C4×C79C4⋊C412yescomplemented
113C22×C4×C79C2×D411yescharacteristic
114C4⋊C4×C79—22no—
115C4⋊C4×C79—22no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—81no—
118C4⋊C4×C79—81no—
119C4⋊C4×C79—81no—
120C22⋊C4×C79—22no—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—21no—
123C22⋊C4×C79—41no—
124C22⋊C4×C79—81no—
125C22⋊C4×C79—81no—
126C22⋊C4×C79—81no—
127C22⋊C4×C79C4⋊C412yescomplemented
128C42×C79—41no—
129C42×C79—81no—
130C23×C79—21no—
131C23×C79—21no—
132C23×C79—21no—
133C23×C79—21no—
134C23×C79—41no—
135C23×C79—41no—
136C23×C79—41no—
137C23×C79—41no—
138C23×C79C2×C4⋊C411yescharacteristic
139C23×C79C4×D411yescharacteristic
140C23×C79#12011yescharacteristic
141C23×C79C22⋊Q811yescharacteristic
142C23×C79#12311yescharacteristic
143D4×C79—41no—
144D4×C79—41no—
145D4×C79—41no—
146D4×C79—41no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—22no—
150C2×C4×C79—22no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—42no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—42no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—81no—
170C2×C4×C79—81no—
171C2×C4×C79C4×D411yescharacteristic
172C2×C4×C79C22⋊Q811yescharacteristic
173C2×Dic79—42no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176C2×Dic79—81no—
177C2×Dic79—81no—
178C2×Dic79—81no—
179C2×Dic79—81no—
180C2×Dic79—81no—
181C2×Dic79—81no—
182C2×Dic79—81no—
183C2×Dic79—81no—
184C2×Dic79—81no—
185C79⋊C8—42no—
186C22×C79—21no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79—41no—
193C22×C79—41no—
194C22×C79—41no—
195C22×C79—41no—
196C22×C79—41no—
197C22×C79—41no—
198C22×C79—41no—
199C22×C79—41no—
200C22×C79—81no—
201C22×C79—81no—
202C22×C79#38511yescharacteristic
203C22×C79B(4,𝔽2)11yescharacteristic
204C22×C79#45811yescharacteristic
205C4×C79—21no—
206C4×C79—21no—
207C4×C79—42no—
208C4×C79—42no—
209C4×C79—81no—
210C4×C79—81no—
211C4×C79—81no—
212C4×C79—81no—
213Dic79—82no—
214Dic79—81no—
215Dic79—81no—
216Dic79—161no—
217#12961—791nomaximal, Sylow
218C2×C79—21no—
219C2×C79—21no—
220C2×C79—41no—
221C2×C79—41no—
222C2×C79—41no—
223C2×C79—41no—
224C2×C79#186711yescharacteristic
225C2×C79#220011yescharacteristic
226C2×C79#220111yescharacteristic
227#2376D7911yescharacteristic
228C2×#352—791no—
229#1890—791no—
230#1867—791no—
231#1856—791no—
232#1393—791no—
233#1390—791no—
234C79#1296111yescharacteristic, complemented, Sylow
235C2×#123C2×D7911yescharacteristic
236#518—791no—
237#428—1581no—
238C2×#100—791no—
239C2×#99—791no—
240C2×#99—791no—
241#390C2×D7911yescharacteristic
242#389—21no—
243#388—21no—
244#388—21no—
245#387—21no—
246#386C2×D7911yescharacteristic
247#385—21no—
248#385—1581no—
249#381—1581no—
250#380—21no—
251#352—794no—
252#342—1581no—
253#342—1581no—
254#339—1581no—
255C22×D4C22×D7911yescharacteristic
256C23×C4C22⋊D7911yescharacteristic
257C23×C4C22⋊D7911yescharacteristic
258C2×M16—1581no—
259#123—21no—
260#123—21no—
261#123Dic15812yes—
262#123C4×D7912yes—
263C4×D4—3161no—
264C4×D4—3161no—
265#117—1581no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—3161no—
271C2×C4⋊C4C22⋊D7911yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1581no—
277C2×C22⋊C4—1581no—
278C2×C22⋊C4—1581no—
279C2×C22⋊C4—1581no—
280C2×C22⋊C4C22⋊D7911yescharacteristic
281C2×C22⋊C4C22⋊D7911yescharacteristic
282C2×C42—21no—
283C2×C42C22⋊D7911yescharacteristic
284#107—3161no—
285#106—3161no—
286#100—792no—
287#100—1581no—
288#99—792no—
289#99—1581no—
290#99—1581no—
291#99—1581no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—3161no—
298#95—3161no—
299#95—3161no—
300C24C2×C22⋊D7911yescharacteristic
301C24C2×C22⋊D7911yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C2×C4×D7911yescharacteristic
305C2×D4C2×Dic15811yescharacteristic
306C2×D4#1192962411yescharacteristic
307C2×D4#1192962411yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—81no—
319C22×C4—3161no—
320C22×C4—3161no—
321C22×C4—1581no—
322C22×C4—1581no—
323C22×C4—1581no—
324C22×C4—1581no—
325C22×C4Dic79⋊C412yescomplemented
326C22×C4C2×C22⋊D7911yescharacteristic
327M16—3161no—
328M16—1582no—
329C2×C8—3161no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—6322no—
337C4⋊C4—1581no—
338C4⋊C4—1581no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—21no—
342C22⋊C4—41no—
343C22⋊C4—81no—
344C22⋊C4—81no—
345C22⋊C4—81no—
346C22⋊C4—3161no—
347C22⋊C4—3161no—
348C22⋊C4—3161no—
349C22⋊C4—3161no—
350C22⋊C4—3161no—
351C22⋊C4—3161no—
352C22⋊C4—1582no—
353C22⋊C4Dic79⋊C412yescomplemented
354C42—41no—
355C42—81no—
356C42—3161no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×Dic79⋊C411yescharacteristic
366C23C4×C22⋊D7911yescharacteristic
367C23#1457913011yescharacteristic
368C23#1457914011yescharacteristic
369C23#1457914411yescharacteristic
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—3162no—
399C2×C4—3161no—
400C2×C4—3161no—
401C2×C4—6321no—
402C2×C4—6321no—
403C2×C4—6321no—
404C2×C4—6321no—
405C2×C4—6321no—
406C2×C4—6321no—
407C2×C4—6321no—
408C2×C4—6321no—
409C2×C4—6321no—
410C2×C4C4×C22⋊D7911yescharacteristic
411C2×C4#1457914011yescharacteristic
412C8—3162no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22#1909843611yescharacteristic
430C22#1909875011yescharacteristic
431C22#1909875311yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—6322no—
441C4—6321no—
442C4—6321no—
443C4—12641no—
444C2—21no—
445C2—21no—
446C2—41no—
447C2—41no—
448C2—41no—
449C2—41no—
450C2#2376368111yescharacteristic
451C2#2376540711yescharacteristic
452C2#2376541311yescharacteristic
subgroupG/H
centerC22#19098750
commutatorC2×D4×C79C22×C4
FrattiniC22×D4C22×D79
FittingC79×#2376C2
socleC22×C79B(4,𝔽2)
prime radical, cosocleC22×D4×C79C23
hypercenter#2376D79
hypocenterC79#12961
LCSG ▷ C2×D4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2376

Complements

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

NH ≅ G/Ntype
15. #19098353C4semidirect
112. C22×C4×C79C4⋊C4semidirect
127. C22⋊C4×C79C4⋊C4semidirect
234. C79#12961semidirect
325. C22×C4Dic79⋊C4semidirect
353. C22⋊C4Dic79⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#190983532
#237638142
#237636802
#237636092
#237615252
#237615182
C79×#23762
#1296179