Finite Group Databasebeta

← #34183492

Subgroups — #34183492

Order 20224.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772144C211yesmaximal, characteristic, complemented
2#23764320C211yesmaximal, characteristic, complemented
3#23761895C211yesmaximal, characteristic, complemented
4#23761799C211yesmaximal, characteristic, complemented
5#23761240C211yesmaximal, characteristic, complemented
6#23761237C211yesmaximal, characteristic, complemented
7C79×#3169C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098641—21no—
10#19098635—22no—
11D4×C79⋊C8—21no—
12#19098616C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579088—21no—
15C2×#14579088—21no—
16#19098530—21no—
17C4×#11929608C2211yescharacteristic
18C4×C4⋊Dic79C2211yescharacteristic
19#19098297—22no—
20D4⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22#532×C79C2211yescharacteristic
23#532×C79C2211yescharacteristic
24#500×C79C411yescharacteristic, complemented
25#493×C79C411yescharacteristic, complemented
26#486×C79C411yescharacteristic, complemented
27#486×C79C411yescharacteristic, complemented
28#466×C79—22no—
29C4×D8×C79—22no—
30C2×C4×C8×C79C2211yescharacteristic
31#14579168—42no—
32#14579168—42no—
33C2×#11929608—22no—
34C2×#11929608D411yescharacteristic
35C2×#11929608D411yescharacteristic
36C22×C79⋊C8—22no—
37C22×C79⋊C8—21no—
38C2×C4⋊Dic79—21no—
39C2×C4×Dic79—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)D411yescharacteristic
43C4⋊(C79⋊C8)D411yescharacteristic
44#14579088—42no—
45#14579088—42no—
46C22⋊(C79⋊C8)—22no—
47C22⋊(C79⋊C8)—22no—
48C22⋊(C79⋊C8)—21no—
49#14579072—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C79⋊#95—21no—
53C2×#41×C79—21no—
54C2×#41×C79—21no—
55C2×D8×C79—22no—
56C22×C8×C79—21no—
57C4⋊Q8×C79C2×C411yescharacteristic
58C4⋊Q8×C79C2×C411yescharacteristic
59#127×C79C2×C411yescharacteristic
60#127×C79C2×C411yescharacteristic
61#124×C79—21no—
62#124×C79—21no—
63#121×C79—21no—
64#121×C79—21no—
65#121×C79—21no—
66#121×C79D412yes—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79D412yes—
71C2×C42×C79C2311yescharacteristic
72#107×C79—22no—
73#102×C79—22no—
74#102×C79—22no—
75C4×C8×C79—21no—
76C4×C8×C79C2×C411yescharacteristic
77C4×C8×C79C2×C411yescharacteristic
78C22×Dic79—41no—
79#11929608—21no—
80#11929608—21no—
81#11929608—21no—
82#11929608—44no—
83#11929608D812yescomplemented
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—44no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C4⋊Dic79—22no—
93C4⋊Dic79—41no—
94C4×Dic79—41no—
95C4×Dic79—41no—
96#41×C79—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101#41×C79—41no—
102#41×C79—41no—
103#41×C79—41no—
104C2×Q8×C79—21no—
105C2×Q8×C79—21no—
106C2×D4×C79—22no—
107C2×D4×C79—21no—
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—21no—
111C2×D4×C79—21no—
112C2×D4×C79—41no—
113C2×D4×C79C22⋊C412yes—
114C22×C4×C79—22no—
115C22×C4×C79—41no—
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79#4111yescharacteristic
119D8×C79—44no—
120C2×C8×C79—21no—
121C2×C8×C79—21no—
122C2×C8×C79—42no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79C22⋊C412yes—
127C22⋊C4×C79—22no—
128C22⋊C4×C79—22no—
129C22⋊C4×C79—41no—
130C22⋊C4×C79—41no—
131C42×C79C22×C411yescharacteristic
132C42×C79C2×D411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79#4111yescharacteristic
135C23×C79—22no—
136C23×C79—41no—
137C23×C79#12111yescharacteristic
138Q8×C79—41no—
139Q8×C79—41no—
140Q8×C79—41no—
141Q8×C79—41no—
142D4×C79—42no—
143D4×C79—42no—
144D4×C79—42no—
145D4×C79—42no—
146D4×C79—41no—
147D4×C79—41no—
148D4×C79—41no—
149D4×C79—41no—
150D4×C79—41no—
151D4×C79—41no—
152D4×C79—41no—
153D4×C79—41no—
154D4×C79—81no—
155D4×C79—81no—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—22no—
159C2×C4×C79—41no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79—41no—
163C2×C4×C79—81no—
164C2×C4×C79—81no—
165C2×C4×C79C2×C22⋊C411yescharacteristic
166C2×C4×C79C4×D411yescharacteristic
167C2×C4×C79C4×D411yescharacteristic
168C2×C4×C79#12011yescharacteristic
169C2×C4×C79#12111yescharacteristic
170C2×C4×C79#12311yescharacteristic
171C2×C4×C79C2×D811yescharacteristic
172C2×C4×C79#13511yescharacteristic
173C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
175C2×Dic79—42no—
176C2×Dic79—81no—
177C2×Dic79—81no—
178C8×C79—22no—
179C8×C79—41no—
180C79⋊C8—22no—
181C79⋊C8—41no—
182C79⋊C8—41no—
183C79⋊C8—41no—
184C79⋊C8—81no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—42no—
188C22×C79—42no—
189C22×C79—42no—
190C22×C79—81no—
191C22×C79—81no—
192C22×C79—81no—
193C22×C79#38611yescharacteristic
194C22×C79#45911yescharacteristic
195C22×C79#46311yescharacteristic
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—42no—
200C4×C79—81no—
201C4×C79C4×D811yescharacteristic
202C4×C79#44211yescharacteristic
203C4×C79#44311yescharacteristic
204C4×C79#44411yescharacteristic
205C4×C79#44711yescharacteristic
206C4×C79#44911yescharacteristic
207Dic79—82no—
208#9053—791nomaximal, Sylow
209C2×C79—21no—
210C2×C79—21no—
211C2×C79—82no—
212C2×C79—161no—
213C2×C79#187611yescharacteristic
214C2×C79#187911yescharacteristic
215C2×C79#188411yescharacteristic
216#3169D7911yescharacteristic
217#2935—791no—
218#1980—791no—
219#1502—791no—
220#1480—791no—
221#1279—791no—
222#1278—791no—
223C79#905311yescharacteristic, complemented, Sylow
224C2×#131—1581no—
225#532C2×D7911yescharacteristic
226#532C2×D7911yescharacteristic
227#500Dic7911yescharacteristic, complemented
228#493Dic7911yescharacteristic, complemented
229#486Dic7911yescharacteristic, complemented
230#486Dic7911yescharacteristic, complemented
231#466—22no—
232C4×D8—22no—
233#436—1581no—
234#435—1582no—
235C8×D4—1581no—
236#431—791no—
237C2×C4⋊C8—791no—
238C2×#104—1581no—
239C2×#104—1581no—
240#408—1581no—
241C4×M16—791no—
242C2×C4×C8C2×D7911yescharacteristic
243C4×C4⋊C4—791no—
244#327—1582no—
245D4⋊C8—1581no—
246D4⋊C8—1581no—
247C2×#41—21no—
248C2×#41—21no—
249C2×D8—22no—
250#131—3162no—
251#131—3162no—
252C2×M16—791no—
253C2×M16—791no—
254C2×M16—1582no—
255C22×C8—21no—
256C22×C8—1582no—
257C22×C8—1581no—
258C4⋊Q8C2×Dic7911yescharacteristic
259C4⋊Q8C2×Dic7911yescharacteristic
260#127C2×Dic7911yescharacteristic
261#127C2×Dic7911yescharacteristic
262#124—21no—
263#124—21no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121C22⋊D7912yes—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C22⋊D7912yes—
272C2×C4⋊C4—1581no—
273C2×C42—1581no—
274C2×C42C22×D7911yescharacteristic
275#107—22no—
276C4⋊C8—791no—
277C4⋊C8—791no—
278C4⋊C8—1581no—
279C4⋊C8—1581no—
280#104—3162no—
281#104—3162no—
282#102—22no—
283#102—22no—
284C22⋊C8—1581no—
285C22⋊C8—1582no—
286C22⋊C8—1582no—
287#97—1581no—
288C4×C8—21no—
289C4×C8—1581no—
290C4×C8—1581no—
291C4×C8C2×Dic7911yescharacteristic
292C4×C8C2×Dic7911yescharacteristic
293#95—1581no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4C22⋊Dic7912yes—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—3161no—
315C22×C4D4×D7911yescharacteristic
316C22×C4C2×C22⋊D7911yescharacteristic
317C22×C4#1192962311yescharacteristic
318D8—44no—
319M16—3164no—
320M16—792no—
321M16—1581no—
322M16—1581no—
323M16—1581no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—42no—
327C2×C8—3161no—
328C2×C8—3161no—
329C2×C8—3164no—
330C2×C8—3161no—
331C2×C8—1581no—
332C2×C8—1581no—
333C2×C8—1581no—
334C2×C8—1581no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—3161no—
339C4⋊C4—1582no—
340C4⋊C4C22⋊Dic7912yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C42—3161no—
346C42—3161no—
347C42C22×Dic7911yescharacteristic
348C42D4×D7911yescharacteristic
349C42C2×C22⋊D7911yescharacteristic
350C42#1192962311yescharacteristic
351C23—22no—
352C23—41no—
353C23#1457913311yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—41no—
357Q8—41no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—3162no—
382C2×C4—6321no—
383C2×C4—6321no—
384C2×C4C2×C22⋊Dic7911yescharacteristic
385C2×C4D4×Dic7911yescharacteristic
386C2×C4D4×Dic7911yescharacteristic
387C2×C4#1457912911yescharacteristic
388C2×C4#1457913311yescharacteristic
389C2×C4#1457914211yescharacteristic
390C2×C4D8×D7911yescharacteristic
391C2×C4#1457918011yescharacteristic
392C2×C4#1457918511yescharacteristic
393C2×C4#1457918511yescharacteristic
394C8—22no—
395C8—41no—
396C8—3161no—
397C8—3161no—
398C8—3161no—
399C8—6321no—
400C8—1582no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#1909844311yescharacteristic
410C22#1909875611yescharacteristic
411C22#1909877611yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—6322no—
418C4D8×Dic7911yescharacteristic
419C4#1909867511yescharacteristic
420C4#1909868011yescharacteristic
421C4#1909868511yescharacteristic
422C4#1909869511yescharacteristic
423C4#1909870511yescharacteristic
424C2—21no—
425C2—21no—
426C2—82no—
427C2—161no—
428C2#2376372911yescharacteristic
429C2#2376375011yescharacteristic
430C2#2376378411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3169C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3169D79
hypocenterC79#9053
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #23772144C2semidirect
2. #23764320C2semidirect
3. #23761895C2semidirect
4. #23761799C2semidirect
5. #23761240C2semidirect
6. #23761237C2semidirect
24. #500×C79C4semidirect
25. #493×C79C4semidirect
26. #486×C79C4semidirect
27. #486×C79C4semidirect
83. #11929608D8semidirect
223. C79#9053semidirect
227. #500Dic79semidirect
228. #493Dic79semidirect
229. #486Dic79semidirect
230. #486Dic79semidirect

Maximal subgroups

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

subgroupindex
#237721442
#237643202
#237618952
#237617992
#237612402
#237612372
C79×#31692
#905379