Finite Group Databasebeta

← #34187991

Subgroups — #34187991

Order 20224.

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

normal subgroups only

This group has 484 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×#19098802C211yesmaximal, characteristic
2#23763436C211yesmaximal, characteristic
3#23763178C211yesmaximal, characteristic
4#23761919C211yesmaximal, characteristic
5#23761765C211yesmaximal, characteristic
6#23761764C211yesmaximal, characteristic
7C79×#1508C211yesmaximal, characteristic
8C22×#11929614C411yescharacteristic, complemented
9C23×C79⋊C8C2211yescharacteristic
10C2×#14579140C2211yescharacteristic
11#19098802—21no—
12#19098802—21no—
13#19098802—21no—
14#19098802—21no—
15C2×#14579096—21no—
16C2×#14579094C411yescharacteristic, complemented
17C2×#14579085C411yescharacteristic, complemented
18C2×#14579084C411yescharacteristic, complemented
19#19098527C2211yescharacteristic
20C2×C22⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579072—21no—
22#19098327—21no—
23#19098318—21no—
24#19098299—22no—
25#19098289—22no—
26#19098288—22no—
27C2×C22⋊Q8×C79C2211yescharacteristic
28#407×C79C2211yescharacteristic
29C4×C22⋊C4×C79C2211yescharacteristic
30#328×C79—22no—
31C22×Dic158C2×C411yescharacteristic
32C2×#11929614—21no—
33C2×#11929614—21no—
34C2×#11929614—42no—
35C2×#11929608—21no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8—42no—
38C22×C79⋊C8C2×C411yescharacteristic
39C22×C79⋊C8C2×C411yescharacteristic
40C22×C79⋊C8D411yescharacteristic
41C22×C79⋊C8D411yescharacteristic
42C22×C79⋊C8D411yescharacteristic
43C22×C79⋊C8D411yescharacteristic
44#14579140—21no—
45#14579140—21no—
46#14579140—21no—
47#14579140—21no—
48C2×C4⋊Dic79C2×C411yescharacteristic
49C2×Dic79⋊C4—21no—
50C2×C22⋊Dic79—21no—
51#14579096—42no—
52#14579094—21no—
53#14579094—21no—
54#14579085—21no—
55#14579085—21no—
56#14579084—21no—
57#14579084—21no—
58C22⋊(C79⋊C8)—21no—
59C22⋊(C79⋊C8)—21no—
60C22⋊(C79⋊C8)—21no—
61C22⋊(C79⋊C8)D412yes—
62#14579072—42no—
63C22×Q8×C79C2×C411yescharacteristic
64C23×C4×C79C2311yescharacteristic
65C2×M16×C79—21no—
66C22⋊Q8×C79—21no—
67C22⋊Q8×C79—21no—
68C22⋊Q8×C79—21no—
69C22⋊Q8×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79C2×C411yescharacteristic
72C2×C22⋊C4×C79—21no—
73C2×C22⋊C4×C79—21no—
74C2×C42×C79—21no—
75C22⋊C8×C79—21no—
76C22⋊C8×C79—21no—
77#95×C79D412yes—
78C2×Dic158—21no—
79C2×Dic158—21no—
80C2×Dic158—41no—
81C2×Dic158—41no—
82C22×Dic79—21no—
83C22×Dic79—21no—
84#11929614—42no—
85#11929614—42no—
86#11929608—41no—
87#11929608—41no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—22no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—22no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8C22⋊C411yescharacteristic
99C2×C79⋊C8C22⋊C411yescharacteristic
100C2×C79⋊C8C22⋊C411yescharacteristic
101C2×C79⋊C8C22⋊C411yescharacteristic
102C4⋊Dic79—21no—
103C4⋊Dic79—21no—
104Dic79⋊C4—41no—
105Dic79⋊C4—41no—
106C22⋊Dic79—41no—
107C22⋊Dic79—41no—
108C24×C79#4111yescharacteristic
109C2×Q8×C79—21no—
110C2×Q8×C79—21no—
111C2×Q8×C79—41no—
112C2×Q8×C79—41no—
113C22×C4×C79—21no—
114C22×C4×C79—22no—
115C22×C4×C79—21no—
116C22×C4×C79—41no—
117C22×C4×C79—41no—
118C22×C4×C79C22×C411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79#4111yescharacteristic
124M16×C79—41no—
125M16×C79—41no—
126C2×C8×C79—41no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—41no—
130C4⋊C4×C79—41no—
131C22⋊C4×C79—42no—
132C22⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C42×C79—41no—
135C42×C79—41no—
136C23×C79—22no—
137C23×C79—22no—
138C23×C79—41no—
139C23×C79C4×D411yescharacteristic
140C23×C79#12111yescharacteristic
141C23×C79#12411yescharacteristic
142Q8×C79—41no—
143Q8×C79—41no—
144Q8×C79—81no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—42no—
155C2×C4×C79—42no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—42no—
159C2×C4×C79—42no—
160C2×C4×C79—81no—
161C2×C4×C79C2×C22⋊C411yescharacteristic
162C2×C4×C79C4×D411yescharacteristic
163C2×C4×C79#12111yescharacteristic
164C2×C4×C79#12711yescharacteristic
165C2×Dic79—41no—
166C2×Dic79—41no—
167C2×Dic79—42no—
168C2×Dic79—81no—
169Dic158—41no—
170Dic158—41no—
171Dic158—81no—
172C8×C79—81no—
173C79⋊C8—21no—
174C79⋊C8—21no—
175C79⋊C8—21no—
176C79⋊C8—42no—
177C79⋊C8—81no—
178C79⋊C8#9912yes—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—44no—
187C22×C79#39011yescharacteristic
188C22×C79#44412yes—
189C22×C79#47312yes—
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—82no—
194C4×C79—81no—
195C4×C79—81no—
196C4×C79C2×#9911yescharacteristic
197C4×C79#41011yescharacteristic
198Dic79—81no—
199Dic79—81no—
200#9817—791nomaximal, Sylow
201C2×C79—22no—
202C2×C79—22no—
203C2×C79—41no—
204C2×C79#196411yescharacteristic
205C2×C79#197511yescharacteristic
206C2×C79#197911yescharacteristic
207C2×#467—791no—
208#1810—791no—
209#1752—791no—
210#1508—791no—
211#1508D7911yescharacteristic
212#1472—791no—
213#1472—791no—
214C79#981711yescharacteristic, complemented, Sylow
215C22×Q16—791no—
216C23×C8—791no—
217C2×C22⋊Q8—791no—
218C2×C22⋊Q8C2×D7911yescharacteristic
219#467—1581no—
220#467—1581no—
221#467—1581no—
222#467—1581no—
223C2×#108—1581no—
224C2×#107—791no—
225C2×#103—791no—
226C2×#103—791no—
227#407—791no—
228#407C2×D7911yescharacteristic
229C2×C22⋊C8—791no—
230C2×#97—1581no—
231C4×C22⋊C4C2×D7911yescharacteristic
232#341—1581no—
233#336—1581no—
234#328—22no—
235#328—1582no—
236#324—1582no—
237#324—1582no—
238C22×Q8—791no—
239C22×Q8C4×D7911yescharacteristic
240C23×C4C22×D7911yescharacteristic
241C2×Q16—3162no—
242C2×Q16—1581no—
243C2×Q16—1581no—
244C2×M16—21no—
245C2×M16—1581no—
246C22×C8—3162no—
247C22×C8—791no—
248C22×C8—791no—
249C22×C8—791no—
250C22×C8—791no—
251C22×C8—791no—
252C22×C8—791no—
253C22×C8—1581no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—1581no—
259C22⋊Q8—1581no—
260C22⋊Q8—1581no—
261C22⋊Q8—1581no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—791no—
264C2×C4⋊C4—1581no—
265C2×C4⋊C4C4×D7911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1581no—
269C2×C42—21no—
270#108—3162no—
271#107—1581no—
272#107—1581no—
273#103—1581no—
274#103—1581no—
275#103—1581no—
276#103—1581no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—792no—
280C22⋊C8—1581no—
281C22⋊C8—1581no—
282C22⋊C8—1581no—
283#97—3162no—
284#95C22⋊D7912yes—
285C24#1192962311yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—3161no—
291C2×Q8—3161no—
292C2×Q8—1581no—
293C2×Q8—1581no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—1581no—
300C22×C4—1581no—
301C22×C4C2×C4×D7911yescharacteristic
302C22×C4D4×D7911yescharacteristic
303C22×C4D4×D7911yescharacteristic
304C22×C4D4×D7911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306C22×C4#1192962411yescharacteristic
307Q16—3162no—
308Q16—3162no—
309M16—41no—
310M16—41no—
311M16—3161no—
312M16—3161no—
313C2×C8—41no—
314C2×C8—3161no—
315C2×C8—3161no—
316C2×C8—3161no—
317C2×C8—3161no—
318C2×C8—3161no—
319C2×C8—3161no—
320C2×C8—791no—
321C2×C8—791no—
322C2×C8—791no—
323C2×C8—791no—
324C2×C8—1582no—
325C2×C8—1582no—
326C2×C8—1582no—
327C2×C8—1582no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—3161no—
333C4⋊C4—3161no—
334C4⋊C4—1581no—
335C4⋊C4—1581no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—3161no—
340C22⋊C4—3161no—
341C42—41no—
342C42—41no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23(C22⋊D79)⋊C411yescharacteristic
347C23#1457913311yescharacteristic
348C23#1457914711yescharacteristic
349Q8—41no—
350Q8—41no—
351Q8—81no—
352Q8—3161no—
353Q8—3161no—
354Q8—6321no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—3162no—
372C2×C4—3161no—
373C2×C4—3161no—
374C2×C4—6321no—
375C2×C4C22⋊C4×D7911yescharacteristic
376C2×C4C4×C22⋊D7911yescharacteristic
377C2×C4#1457913411yescharacteristic
378C2×C4#1457915711yescharacteristic
379C8—81no—
380C8—3162no—
381C8—792no—
382C8—6321no—
383C8—1581no—
384C8—1581no—
385C8—1581no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22#1909846311yescharacteristic
395C22#1909868212yes—
396C22#1909883612yes—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—81no—
402C4—81no—
403C4—6321no—
404C4—6321no—
405C4#99×D7911yescharacteristic
406C4#1909853511yescharacteristic
407C2—22no—
408C2—22no—
409C2—41no—
410C2#2376423411yescharacteristic
411C2#2376429011yescharacteristic
412C2#2376431411yescharacteristic
subgroupG/H
centerC22#19098463
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1508C2
socleC22×C79#390
prime radicalC24×C79#41
cosocleC23×C4×C79C23
hypercenter#1508D79
hypocenterC79#9817
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
8. C22×#11929614C4semidirect
16. C2×#14579094C4semidirect
17. C2×#14579085C4semidirect
18. C2×#14579084C4semidirect
214. C79#9817semidirect

Maximal subgroups

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

subgroupindex
C2×#190988022
#237634362
#237631782
#237619192
#237617652
#237617642
C79×#15082
#981779