Finite Group Databasebeta

← #34184382

Subgroups — #34184382

Order 20224.

536 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14579168C211yesmaximal, characteristic, complemented
2#23761992C211yesmaximal, characteristic, complemented
3#23761851C211yesmaximal, characteristic, complemented
4#23761839C211yesmaximal, characteristic, complemented
5#23761291C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#3106C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098587C2211yescharacteristic
10#19098519—22no—
11#19098519—21no—
12C4×#11929608—21no—
13C4×#11929608C2211yescharacteristic
14C4×#11929608C2211yescharacteristic
15C2×C4×C79⋊C8—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17#19098303—22no—
18Q8⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)—21no—
20D4⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22#532×C79C2211yescharacteristic
23C4×#41×C79C2211yescharacteristic
24#482×C79—21no—
25#477×C79C411yescharacteristic
26#476×C79C411yescharacteristic
27#460×C79C411yescharacteristic
28#459×C79C411yescharacteristic
29#449×C79—21no—
30C4×SD16×C79—21no—
31C4×D8×C79—21no—
32#431×C79C2211yescharacteristic
33#14579168—24no—
34#14579168—24no—
35C2×#11929608—22no—
36C2×#11929608—21no—
37C2×#11929608—21no—
38C2×#11929608—21no—
39C22×C79⋊C8—22no—
40C22×C79⋊C8—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43#14579072—22no—
44#14579072—21no—
45#14579072D412yes—
46C4×C79⋊C8—22no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8D411yescharacteristic
51C4×C79⋊C8D411yescharacteristic
52C2×#41×C79—21no—
53C2×#41×C79D411yescharacteristic
54C2×SD16×C79—21no—
55C2×D8×C79—21no—
56C4⋊Q8×C79C2×C411yescharacteristic
57#127×C79C2×C411yescharacteristic
58#124×C79—21no—
59#121×C79—21no—
60#121×C79—21no—
61C4×Q8×C79C2×C411yescharacteristic
62C4×Q8×C79D411yescharacteristic
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79C2×C411yescharacteristic
68C4×D4×C79D411yescharacteristic
69#117×C79—21no—
70#117×C79D411yescharacteristic
71C2×C42×C79—21no—
72C2×C42×C79C2311yescharacteristic
73#107×C79—21no—
74#106×C79—21no—
75C4⋊C8×C79C2×C411yescharacteristic
76C4⋊C8×C79C2×C411yescharacteristic
77#103×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80#102×C79—21no—
81C22⋊C8×C79—21no—
82C4×C8×C79—21no—
83#11929608—28no—
84#11929608—22no—
85#11929608—22no—
86#11929608—22no—
87#11929608—41no—
88#11929608—81no—
89C2×C79⋊C8—28no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—22no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—22no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97#41×C79—22no—
98#41×C79—22no—
99#41×C79—81no—
100#41×C79—81no—
101C2×Q8×C79—21no—
102C2×Q8×C79C22⋊C411yescharacteristic
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—21no—
107C2×D4×C79—41no—
108C2×D4×C79C22⋊C411yescharacteristic
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—41no—
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79#4111yescharacteristic
115C22×C4×C79#4111yescharacteristic
116SD16×C79—42no—
117D8×C79—42no—
118C2×C8×C79—21no—
119C2×C8×C79—21no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79C22⋊C411yescharacteristic
125C4⋊C4×C79C22⋊C411yescharacteristic
126C22⋊C4×C79—21no—
127C22⋊C4×C79—21no—
128C22⋊C4×C79—21no—
129C22⋊C4×C79—41no—
130C22⋊C4×C79—41no—
131C42×C79—21no—
132C42×C79—21no—
133C42×C79—21no—
134C42×C79C22×C411yescharacteristic
135C42×C79C2×D411yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79C2×D411yescharacteristic
138C23×C79—21no—
139C23×C79—41no—
140C23×C79#12311yescharacteristic
141Q8×C79—21no—
142Q8×C79—81no—
143Q8×C79#10412yes—
144D4×C79—24no—
145D4×C79—21no—
146D4×C79—41no—
147D4×C79—41no—
148D4×C79—41no—
149D4×C79—41no—
150D4×C79—81no—
151D4×C79—82no—
152D4×C79#10412yes—
153C2×C4×C79—24no—
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—81no—
169C2×C4×C79—81no—
170C2×C4×C79C2×C22⋊C411yescharacteristic
171C2×C4×C79C4×D411yescharacteristic
172C2×C4×C79C4×D411yescharacteristic
173C2×C4×C79#12011yescharacteristic
174C2×C4×C79#12111yescharacteristic
175C2×C4×C79#12111yescharacteristic
176C8×C79—41no—
177C8×C79—41no—
178C79⋊C8—22no—
179C79⋊C8—24no—
180C79⋊C8—22no—
181C79⋊C8—42no—
182C79⋊C8—81no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—22no—
186C22×C79—21no—
187C22×C79—41no—
188C22×C79—41no—
189C22×C79—81no—
190C22×C79—81no—
191C22×C79—81no—
192C22×C79#38611yescharacteristic
193C4×C79—21no—
194C4×C79—21no—
195C4×C79—22no—
196C4×C79—21no—
197C4×C79—41no—
198C4×C79—41no—
199C4×C79—81no—
200C4×C79C2×#10411yescharacteristic
201C4×C79C2×#10411yescharacteristic
202C4×C79#44311yescharacteristic
203C4×C79#44411yescharacteristic
204C4×C79#47111yescharacteristic
205C4×C79#47211yescharacteristic
206#9211—791nomaximal, Sylow
207C2×C79—41no—
208C2×C79—42no—
209C2×C79—161no—
210C2×C79#187611yescharacteristic
211C2×C79#189611yescharacteristic
212C2×C79#190711yescharacteristic
213#3106D7911yescharacteristic
214C4×#131—791no—
215#1525—791no—
216#1492—791no—
217#1489—791no—
218#1299—791no—
219#1278—791no—
220C79#921111yescharacteristic, complemented, Sylow
221C2×#131—1581no—
222#532C2×D7911yescharacteristic
223C4×#41C2×D7911yescharacteristic
224#482—21no—
225#477Dic7911yescharacteristic
226#476Dic7911yescharacteristic
227#460Dic7911yescharacteristic
228#459Dic7911yescharacteristic
229#449—21no—
230C4×SD16—21no—
231C4×D8—21no—
232#431C2×D7911yescharacteristic
233#423—791no—
234#405—1582no—
235#405—1581no—
236C4×M16—791no—
237C4×M16—791no—
238C4×M16—1581no—
239C2×C4×C8—791no—
240C2×C4×C8—1581no—
241#329—1582no—
242Q8⋊C8—1581no—
243D4⋊C8—1581no—
244D4⋊C8—1581no—
245D4⋊C8—1581no—
246C2×#41—21no—
247C2×#41C22⋊D7911yescharacteristic
248C2×SD16—21no—
249C2×D8—21no—
250#131—1584no—
251#131—1584no—
252C2×M16—1581no—
253C2×M16—1581no—
254C2×M16—1582no—
255C2×M16—1581no—
256C22×C8—1582no—
257C22×C8—1581no—
258C4⋊Q8C2×Dic7911yescharacteristic
259#127C2×Dic7911yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×Q8C2×Dic7911yescharacteristic
264C4×Q8C22⋊D7911yescharacteristic
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4C2×Dic7911yescharacteristic
270C4×D4C22⋊D7911yescharacteristic
271#117—21no—
272#117C22⋊D7911yescharacteristic
273C2×C42—21no—
274C2×C42C22×D7911yescharacteristic
275#107—21no—
276#106—21no—
277C4⋊C8—1581no—
278C4⋊C8—1581no—
279C4⋊C8C2×Dic7911yescharacteristic
280C4⋊C8C2×Dic7911yescharacteristic
281#103—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285C22⋊C8—21no—
286#97—792no—
287#97—1582no—
288#97—1581no—
289C4×C8—21no—
290C4×C8—791no—
291C4×C8—791no—
292C4×C8—1582no—
293C4×C8—1581no—
294C4×C8—1581no—
295C4×C8—1581no—
296#41—22no—
297#41—22no—
298#41—81no—
299#41—81no—
300C2×Q8—21no—
301C2×Q8C22⋊Dic7911yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4C22⋊Dic7911yescharacteristic
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4C2×C22⋊D7911yescharacteristic
313C22×C4#1192962311yescharacteristic
314C22×C4#1192962311yescharacteristic
315SD16—42no—
316D8—42no—
317M16—3161no—
318M16—6321no—
319M16—1582no—
320M16—1588no—
321M16—1582no—
322M16—1582no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—3161no—
326C2×C8—3161no—
327C2×C8—1588no—
328C2×C8—1582no—
329C2×C8—1582no—
330C2×C8—1581no—
331C2×C8—1581no—
332C2×C8—1582no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C22⋊Dic7911yescharacteristic
338C4⋊C4C22⋊Dic7911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42C22×Dic7911yescharacteristic
348C42D4×D7911yescharacteristic
349C42D4×D7911yescharacteristic
350C42C2×C22⋊D7911yescharacteristic
351C23—21no—
352C23—41no—
353C23#1457914211yescharacteristic
354Q8—21no—
355Q8—81no—
356Q8#1457908812yes—
357D4—24no—
358D4—21no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—82no—
365D4#1457908812yes—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4C2×C22⋊Dic7911yescharacteristic
384C2×C4D4×Dic7911yescharacteristic
385C2×C4D4×Dic7911yescharacteristic
386C2×C4#1457912911yescharacteristic
387C2×C4#1457913311yescharacteristic
388C2×C4#1457913311yescharacteristic
389C8—41no—
390C8—41no—
391C8—3162no—
392C8—6321no—
393C8—1582no—
394C8—1582no—
395C8—1584no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22#1909844311yescharacteristic
406C4—21no—
407C4—21no—
408C4—22no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4C2×#1457908811yescharacteristic
414C4C2×#1457908811yescharacteristic
415C4#1909868011yescharacteristic
416C4#1909868511yescharacteristic
417C4#1909882411yescharacteristic
418C4#1909883111yescharacteristic
419C2—41no—
420C2—42no—
421C2—161no—
422C2#2376372911yescharacteristic
423C2#2376384411yescharacteristic
424C2#2376391011yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3106C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3106D79
hypocenterC79#9211
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. C4×#14579168C2semidirect
2. #23761992C2semidirect
3. #23761851C2semidirect
4. #23761839C2semidirect
5. #23761291C2semidirect
6. #23761236C2semidirect
220. C79#9211semidirect

Maximal subgroups

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

subgroupindex
C4×#145791682
#237619922
#237618512
#237618392
#237612912
#237612362
C79×#31062
#921179