Finite Group Databasebeta

← #34187651

Subgroups — #34187651

Order 20224.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773722C211yesmaximal, characteristic
2#23763530C211yesmaximal, characteristic, complemented
3#23761965C211yesmaximal, characteristic, complemented
4#23761963C211yesmaximal, characteristic, complemented
5#23761960C211yesmaximal, characteristic
6#23761684C211yesmaximal, characteristic, complemented
7C79×#1517C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098917—21no—
10#19098917—21no—
11#19098913C411yescharacteristic, complemented
12#19098913C411yescharacteristic, complemented
13#19098909C411yescharacteristic, complemented
14#19098909C411yescharacteristic, complemented
15#19098616C2211yescharacteristic
16#19098559—21no—
17#19098559—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19#19098429C2211yescharacteristic
20#19098325—21no—
21#19098325—21no—
22#19098301—22no—
23#19098299—22no—
24#19098297—22no—
25#19098285—21no—
26C8×C79⋊C8—21no—
27#532×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29#382×C79C2211yescharacteristic
30#327×C79—22no—
31C22×C79⋊C8D411yescharacteristic
32C22×C79⋊C8D411yescharacteristic
33#14579161C2×C411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579153—21no—
36#14579140—21no—
37#14579140—21no—
38C4×Dic158—21no—
39C4×Dic158—21no—
40#14579120—21no—
41C2×C4⋊Dic79—21no—
42C4⋊(C79⋊C8)—21no—
43#14579085—21no—
44#14579085—21no—
45#14579085—21no—
46#14579085—21no—
47#14579082—21no—
48#14579082—21no—
49#14579082—21no—
50#14579082—21no—
51C22⋊(C79⋊C8)D412yes—
52C4×C79⋊C8—21no—
53C4×C79⋊C8C2×C411yescharacteristic
54C4×C79⋊C8C2×C411yescharacteristic
55C4×C79⋊C8D411yescharacteristic
56C4×C79⋊C8D411yescharacteristic
57C79⋊#95—21no—
58C79⋊#95—21no—
59C2×#41×C79—21no—
60C4⋊Q8×C79C2×C411yescharacteristic
61#127×C79C2×C411yescharacteristic
62#124×C79—21no—
63#121×C79—21no—
64#121×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C2×C4⋊C4×C79—21no—
68C2×C42×C79C2311yescharacteristic
69C4⋊C8×C79—21no—
70C22⋊C8×C79D412yes—
71C4×C8×C79—21no—
72#95×C79—21no—
73#95×C79—21no—
74C2×Dic158—21no—
75C2×Dic158—21no—
76C22×Dic79—41no—
77C2×C79⋊C8—22no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8C22⋊C411yescharacteristic
83C2×C79⋊C8C22⋊C411yescharacteristic
84C2×C79⋊C8C22⋊C411yescharacteristic
85C2×C79⋊C8C22⋊C411yescharacteristic
86C4⋊Dic79—21no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C4⋊Dic79—41no—
91C4⋊Dic79—41no—
92Dic79⋊C4—41no—
93Dic79⋊C4—41no—
94C22⋊Dic79—41no—
95C4×Dic79—41no—
96#41×C79—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100C2×Q8×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—41no—
105C22×C4×C79—41no—
106C22×C4×C79—41no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C2×C8×C79—22no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79—41no—
117C42×C79C22×C411yescharacteristic
118C42×C79C2×D411yescharacteristic
119C42×C79#4111yescharacteristic
120C42×C79#4111yescharacteristic
121C23×C79—41no—
122C23×C79#12711yescharacteristic
123Q8×C79—41no—
124Q8×C79—41no—
125D4×C79—41no—
126D4×C79—41no—
127D4×C79—81no—
128D4×C79—81no—
129D4×C79—81no—
130D4×C79—81no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—81no—
135C2×C4×C79—81no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79C2×C22⋊C411yescharacteristic
140C2×C4×C79C4×D411yescharacteristic
141C2×C4×C79C4×D411yescharacteristic
142C2×C4×C79#12111yescharacteristic
143C2×C4×C79#12111yescharacteristic
144C2×C4×C79#12411yescharacteristic
145C2×C4×C79#13511yescharacteristic
146C2×C4×C79#13511yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×C4×C79#13511yescharacteristic
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—81no—
152C2×Dic79—81no—
153C2×Dic79—81no—
154Dic158—81no—
155Dic158—81no—
156C8×C79—42no—
157C79⋊C8—21no—
158C79⋊C8—21no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—42no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—81no—
165C22×C79—81no—
166C22×C79—81no—
167C22×C79#39011yescharacteristic
168C22×C79#49511yescharacteristic
169C22×C79#49511yescharacteristic
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—81no—
175C4×C79—161no—
176C4×C79#41611yescharacteristic
177C4×C79#41611yescharacteristic
178C4×C79#48811yescharacteristic
179C4×C79#48811yescharacteristic
180Dic79—81no—
181Dic79—81no—
182Dic79—161no—
183#9757—791nomaximal, Sylow
184C2×C79—21no—
185C2×C79—21no—
186C2×C79—161no—
187C2×C79#195811yescharacteristic
188C2×C79#197611yescharacteristic
189C2×C79#197611yescharacteristic
190#3131—791no—
191#1835—791no—
192#1518—791no—
193#1518—791no—
194#1517—791no—
195#1517D7911yescharacteristic
196#1450—791no—
197C79#975711yescharacteristic, complemented, Sylow
198#533—791no—
199#532C2×D7911yescharacteristic
200#488—1581no—
201#488—1581no—
202#487—791no—
203#487—791no—
204#486—791no—
205#486—791no—
206#431—791no—
207#431C2×D7911yescharacteristic
208#416—1581no—
209#416—1581no—
210C2×C4×C8—791no—
211#382—791no—
212#382C2×D7911yescharacteristic
213#340—1581no—
214#340—1581no—
215#328—1582no—
216#328—1582no—
217#327—22no—
218#327—1582no—
219#322—1581no—
220C82—1581no—
221C2×#41—21no—
222C22×C8—791no—
223C22×C8—791no—
224C4⋊Q8—791no—
225C4⋊Q8—791no—
226C4⋊Q8C4×D7911yescharacteristic
227#127C4×D7911yescharacteristic
228#125—1581no—
229#124—21no—
230C22⋊Q8—1581no—
231C22⋊Q8—1581no—
232#121—21no—
233#121—21no—
234C4×Q8—1581no—
235C4×Q8—1581no—
236C4×D4—21no—
237C4×D4—21no—
238#117—1581no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—1581no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—1581no—
244#103—1581no—
245#103—1581no—
246#103—1581no—
247#103—1581no—
248#102—1581no—
249#102—1581no—
250#102—1581no—
251#102—1581no—
252C22⋊C8—792no—
253C22⋊C8C22⋊D7912yes—
254C4×C8—21no—
255C4×C8—791no—
256C4×C8—791no—
257C4×C8—791no—
258C4×C8—791no—
259C4×C8—1581no—
260#95—21no—
261#95—21no—
262#95—1581no—
263#95—1581no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—1581no—
270C2×Q8—1581no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—3161no—
278C22×C4D4×D7911yescharacteristic
279C22×C4D4×D7911yescharacteristic
280C22×C4C2×C22⋊D7911yescharacteristic
281C2×C8—22no—
282C2×C8—791no—
283C2×C8—791no—
284C2×C8—791no—
285C2×C8—791no—
286C2×C8—1582no—
287C2×C8—1581no—
288C2×C8—1581no—
289C2×C8—1581no—
290C2×C8—1581no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—3161no—
296C4⋊C4—3161no—
297C4⋊C4—3161no—
298C4⋊C4—3161no—
299C4⋊C4—1581no—
300C4⋊C4—1581no—
301C4⋊C4—1581no—
302C4⋊C4—1581no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—3161no—
306C42—3161no—
307C42C2×C4×D7911yescharacteristic
308C42D4×D7911yescharacteristic
309C42#1192962311yescharacteristic
310C42#1192962411yescharacteristic
311C23—41no—
312C23#1457915711yescharacteristic
313Q8—41no—
314Q8—41no—
315Q8—6321no—
316Q8—6321no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4—6321no—
334C2×C4—6321no—
335C2×C4—6321no—
336C2×C4C22⋊C4×D7911yescharacteristic
337C2×C4(C22⋊D79)⋊C411yescharacteristic
338C2×C4C4×C22⋊D7911yescharacteristic
339C2×C4#1457913311yescharacteristic
340C2×C4#1457913411yescharacteristic
341C2×C4#1457914711yescharacteristic
342C2×C4#1457918011yescharacteristic
343C2×C4#1457918011yescharacteristic
344C2×C4#1457918211yescharacteristic
345C2×C4#1457918211yescharacteristic
346C8—42no—
347C8—3162no—
348C8—1581no—
349C8—1581no—
350C8—1581no—
351C8—1581no—
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#1909846311yescharacteristic
358C22#1909894411yescharacteristic
359C22#1909894611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—161no—
366C4—6321no—
367C4—6321no—
368C4—12641no—
369C4#1909855611yescharacteristic
370C4#1909855611yescharacteristic
371C4#1909891611yescharacteristic
372C4#1909891611yescharacteristic
373C2—21no—
374C2—21no—
375C2—161no—
376C2#2376421011yescharacteristic
377C2#2376429511yescharacteristic
378C2#2376429711yescharacteristic
subgroupG/H
centerC22#19098463
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1517C2
socleC22×C79#390
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#1517D79
hypocenterC79#9757
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #23763530C2semidirect
3. #23761965C2semidirect
4. #23761963C2semidirect
6. #23761684C2semidirect
11. #19098913C4semidirect
12. #19098913C4semidirect
13. #19098909C4semidirect
14. #19098909C4semidirect
197. C79#9757semidirect

Maximal subgroups

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

subgroupindex
#237737222
#237635302
#237619652
#237619632
#237619602
#237616842
C79×#15172
#975779