Finite Group Databasebeta

← #34187656

Subgroups — #34187656

Order 20224.

408 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773722C211yesmaximal, characteristic
2#23763562C211yesmaximal, characteristic, complemented
3#23761965C211yesmaximal, characteristic, complemented
4#23761963C211yesmaximal, characteristic, complemented
5#23761960C211yesmaximal, characteristic
6#23761693C211yesmaximal, 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
16C2×#14579094—21no—
17C2×#14579092—21no—
18#19098559—21no—
19#19098559—21no—
20C2×C4×C79⋊C8C2211yescharacteristic
21#19098429C2211yescharacteristic
22#19098301—22no—
23#19098299—22no—
24#19098297—22no—
25#19098285—21no—
26#19098284—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—
42#14579094—42no—
43#14579092—42no—
44C4⋊(C79⋊C8)—21no—
45#14579085—21no—
46#14579085—21no—
47#14579085—21no—
48#14579085—21no—
49#14579082—21no—
50#14579082—21no—
51#14579082—21no—
52#14579082—21no—
53C22⋊(C79⋊C8)D412yes—
54C4×C79⋊C8—21no—
55C4×C79⋊C8C2×C411yescharacteristic
56C4×C79⋊C8C2×C411yescharacteristic
57C4×C79⋊C8D411yescharacteristic
58C4×C79⋊C8D411yescharacteristic
59C79⋊#95—21no—
60C79⋊#95—21no—
61C2×#41×C79—21no—
62C4⋊Q8×C79C2×C411yescharacteristic
63#127×C79C2×C411yescharacteristic
64#124×C79—21no—
65#121×C79—21no—
66#121×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C42×C79C2311yescharacteristic
71C4⋊C8×C79—21no—
72C22⋊C8×C79D412yes—
73C4×C8×C79—21no—
74#95×C79—21no—
75#95×C79—21no—
76C2×Dic158—21no—
77C2×Dic158—21no—
78C22×Dic79—41no—
79C2×C79⋊C8—22no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8C22⋊C411yescharacteristic
85C2×C79⋊C8C22⋊C411yescharacteristic
86C2×C79⋊C8C22⋊C411yescharacteristic
87C2×C79⋊C8C22⋊C411yescharacteristic
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92C4⋊Dic79—41no—
93C4⋊Dic79—41no—
94Dic79⋊C4—41no—
95Dic79⋊C4—41no—
96C22⋊Dic79—41no—
97C4×Dic79—41no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101#41×C79—41no—
102C2×Q8×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79—41no—
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C2×C8×C79—22no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79#4111yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79—41no—
124C23×C79#12711yescharacteristic
125Q8×C79—41no—
126Q8×C79—41no—
127D4×C79—41no—
128D4×C79—41no—
129D4×C79—81no—
130D4×C79—81no—
131D4×C79—81no—
132D4×C79—81no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79C2×C22⋊C411yescharacteristic
142C2×C4×C79C4×D411yescharacteristic
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79#12411yescharacteristic
147C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C79#13711yescharacteristic
150C2×C4×C79#13711yescharacteristic
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—81no—
154C2×Dic79—81no—
155C2×Dic79—81no—
156Dic158—81no—
157Dic158—81no—
158C8×C79—42no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—21no—
162C79⋊C8—21no—
163C79⋊C8—42no—
164C22×C79—21no—
165C22×C79—21no—
166C22×C79—81no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79#39011yescharacteristic
170C22×C79#49611yescharacteristic
171C22×C79#49711yescharacteristic
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—21no—
176C4×C79—81no—
177C4×C79—161no—
178C4×C79#41711yescharacteristic
179C4×C79#41711yescharacteristic
180C4×C79#48911yescharacteristic
181C4×C79#48911yescharacteristic
182Dic79—81no—
183Dic79—81no—
184Dic79—161no—
185#9758—791nomaximal, Sylow
186C2×C79—21no—
187C2×C79—21no—
188C2×C79—161no—
189C2×C79#195911yescharacteristic
190C2×C79#197611yescharacteristic
191C2×C79#197611yescharacteristic
192#3131—791no—
193#1846—791no—
194#1518—791no—
195#1518—791no—
196#1517—791no—
197#1517D7911yescharacteristic
198#1452—791no—
199C79#975811yescharacteristic, complemented, Sylow
200#533—791no—
201#532C2×D7911yescharacteristic
202#488—1581no—
203#488—1581no—
204#487—791no—
205#487—791no—
206#486—791no—
207#486—791no—
208#431—791no—
209#431C2×D7911yescharacteristic
210C2×#107—1581no—
211C2×#106—1581no—
212#416—1581no—
213#416—1581no—
214C2×C4×C8—791no—
215#382—791no—
216#382C2×D7911yescharacteristic
217#328—1582no—
218#328—1582no—
219#327—22no—
220#327—1582no—
221#322—1581no—
222#322—1581no—
223C2×#41—21no—
224C22×C8—791no—
225C22×C8—791no—
226C4⋊Q8—791no—
227C4⋊Q8—791no—
228C4⋊Q8C4×D7911yescharacteristic
229#127C4×D7911yescharacteristic
230#125—1581no—
231#124—21no—
232C22⋊Q8—1581no—
233C22⋊Q8—1581no—
234#121—21no—
235#121—21no—
236C4×Q8—1581no—
237C4×Q8—1581no—
238C4×D4—21no—
239C4×D4—21no—
240#117—1581no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1581no—
243C2×C42C22×D7911yescharacteristic
244#107—3162no—
245#106—3162no—
246C4⋊C8—21no—
247C4⋊C8—1581no—
248#103—1581no—
249#103—1581no—
250#103—1581no—
251#103—1581no—
252#102—1581no—
253#102—1581no—
254#102—1581no—
255#102—1581no—
256C22⋊C8—792no—
257C22⋊C8C22⋊D7912yes—
258C4×C8—21no—
259C4×C8—791no—
260C4×C8—791no—
261C4×C8—791no—
262C4×C8—791no—
263C4×C8—1581no—
264#95—21no—
265#95—21no—
266#95—1581no—
267#95—1581no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—21no—
273C2×Q8—1581no—
274C2×Q8—1581no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—3161no—
282C22×C4D4×D7911yescharacteristic
283C22×C4D4×D7911yescharacteristic
284C22×C4C2×C22⋊D7911yescharacteristic
285C2×C8—22no—
286C2×C8—791no—
287C2×C8—791no—
288C2×C8—791no—
289C2×C8—791no—
290C2×C8—1582no—
291C2×C8—1581no—
292C2×C8—1581no—
293C2×C8—1581no—
294C2×C8—1581no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—3161no—
300C4⋊C4—3161no—
301C4⋊C4—3161no—
302C4⋊C4—3161no—
303C4⋊C4—1581no—
304C4⋊C4—1581no—
305C4⋊C4—1581no—
306C4⋊C4—1581no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—3161no—
310C42—3161no—
311C42C2×C4×D7911yescharacteristic
312C42D4×D7911yescharacteristic
313C42#1192962311yescharacteristic
314C42#1192962411yescharacteristic
315C23—41no—
316C23#1457915711yescharacteristic
317Q8—41no—
318Q8—41no—
319Q8—6321no—
320Q8—6321no—
321D4—41no—
322D4—41no—
323D4—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—3161no—
336C2×C4—3161no—
337C2×C4—6321no—
338C2×C4—6321no—
339C2×C4—6321no—
340C2×C4C22⋊C4×D7911yescharacteristic
341C2×C4(C22⋊D79)⋊C411yescharacteristic
342C2×C4C4×C22⋊D7911yescharacteristic
343C2×C4#1457913311yescharacteristic
344C2×C4#1457913411yescharacteristic
345C2×C4#1457914711yescharacteristic
346C2×C4#1457918511yescharacteristic
347C2×C4#1457918511yescharacteristic
348C2×C4#1457919011yescharacteristic
349C2×C4#1457919011yescharacteristic
350C8—42no—
351C8—3162no—
352C8—1581no—
353C8—1581no—
354C8—1581no—
355C8—1581no—
356C22—21no—
357C22—21no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#1909846311yescharacteristic
362C22#1909895111yescharacteristic
363C22#1909895411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—161no—
370C4—6321no—
371C4—6321no—
372C4—12641no—
373C4#1909856111yescharacteristic
374C4#1909856111yescharacteristic
375C4#1909892111yescharacteristic
376C4#1909892111yescharacteristic
377C2—21no—
378C2—21no—
379C2—161no—
380C2#2376421511yescharacteristic
381C2#2376429511yescharacteristic
382C2#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#9758
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. #23763562C2semidirect
3. #23761965C2semidirect
4. #23761963C2semidirect
6. #23761693C2semidirect
11. #19098913C4semidirect
12. #19098913C4semidirect
13. #19098909C4semidirect
14. #19098909C4semidirect
199. C79#9758semidirect

Maximal subgroups

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

subgroupindex
#237737222
#237635622
#237619652
#237619632
#237619602
#237616932
C79×#15172
#975879