Finite Group Databasebeta

← #34187415

Subgroups — #34187415

Order 20224.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773960C211yesmaximal, characteristic
2#23764345C211yesmaximal, characteristic, complemented
3#23764318C211yesmaximal, characteristic
4C4×#14579092C211yesmaximal, characteristic, complemented
5#23761896C211yesmaximal, characteristic
6#23761681C211yesmaximal, characteristic, complemented
7C79×#1503C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098959C411yescharacteristic, complemented
10#19098937C411yescharacteristic, complemented
11#19098913C411yescharacteristic, complemented
12#19098911C411yescharacteristic, complemented
13#19098794—21no—
14#19098794—21no—
15C4×#11929611—21no—
16C4×#11929611—21no—
17C2×#14579092—21no—
18C2×#14579089—22no—
19C4×#11929608C2211yescharacteristic
20C2×C4×C79⋊C8C2211yescharacteristic
21C4×C4⋊Dic79C2211yescharacteristic
22#19098325—21no—
23#19098322—22no—
24#19098298—22no—
25#19098285—21no—
26C8×C79⋊C8—21no—
27#533×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C4×C4⋊C4×C79C2211yescharacteristic
30#328×C79—22no—
31C2×#11929624—21no—
32C2×#11929611—21no—
33C2×#11929611—21no—
34C2×#11929608D412yes—
35C22×C79⋊C8D411yescharacteristic
36C22×C79⋊C8D411yescharacteristic
37#14579161C2×C411yescharacteristic
38#14579158C2×C411yescharacteristic
39#14579149—21no—
40#14579135—21no—
41#14579135—21no—
42C4×D316—21no—
43C4×D316—21no—
44C2×C4⋊Dic79—21no—
45C2×C4×Dic79—21no—
46#14579092—24no—
47#14579092—21no—
48#14579092—21no—
49#14579089—44no—
50#14579085—21no—
51#14579085—21no—
52#14579081—21no—
53#14579081—21no—
54#14579072—21no—
55C4×C79⋊C8—21no—
56C4×C79⋊C8C2×C411yescharacteristic
57C4×C79⋊C8C2×C411yescharacteristic
58C4×C79⋊C8D411yescharacteristic
59C4×C79⋊C8D411yescharacteristic
60C79⋊#95—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62C4⋊Q8×C79C2×C411yescharacteristic
63#125×C79—21no—
64C22⋊Q8×C79—21no—
65C22⋊Q8×C79—21no—
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68#117×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C42×C79—21no—
71C2×C42×C79C2311yescharacteristic
72C4⋊C8×C79—21no—
73C22⋊C8×C79D412yes—
74C4×C8×C79—21no—
75#95×C79—21no—
76#11929624—42no—
77#11929624—42no—
78C2×Dic158—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×C22⋊D79—41no—
83C22×Dic79—41no—
84C2×C4×D79—41no—
85#11929611—42no—
86#11929611—42no—
87#11929608—22no—
88#11929608SD1614yescomplemented
89C2×C79⋊C8—22no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8C22⋊C411yescharacteristic
93C2×C79⋊C8C22⋊C411yescharacteristic
94C2×C79⋊C8C22⋊C411yescharacteristic
95C2×C79⋊C8C22⋊C411yescharacteristic
96C4⋊Dic79—22no—
97C4⋊Dic79—21no—
98C4⋊Dic79—21no—
99C4⋊Dic79—41no—
100(C2×D79)⋊C4—41no—
101(C2×D79)⋊C4—41no—
102C4×Dic79—42no—
103C2×Q8×C79—21no—
104C2×Q8×C79—21no—
105C22×C4×C79—41no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C2×C8×C79—22no—
110C4⋊C4×C79—22no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C42×C79—41no—
120C42×C79—42no—
121C42×C79C22×C411yescharacteristic
122C42×C79C2×D411yescharacteristic
123C42×C79#4111yescharacteristic
124C42×C79#4111yescharacteristic
125C23×C79#12711yescharacteristic
126C22×D79—41no—
127Q8×C79—41no—
128Q8×C79—41no—
129Q8×C79—81no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—41no—
133C2×C4×C79—42no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79C2×C22⋊C411yescharacteristic
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79C4×D411yescharacteristic
141C2×C4×C79#12111yescharacteristic
142C2×C4×C79#12111yescharacteristic
143C2×C4×C79#12411yescharacteristic
144C2×C4×C79C2×SD1612yes—
145C2×C4×C79#13512yes—
146C22⋊D79—81no—
147C22⋊D79—81no—
148C2×Dic79—42no—
149C2×Dic79—41no—
150C2×Dic79—81no—
151C2×Dic79—81no—
152D316—41no—
153D316—41no—
154D316—42no—
155D316—81no—
156C4×D79—82no—
157Dic158—42no—
158C8×C79—42no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—21no—
162C79⋊C8—24no—
163C79⋊C8#10412yescomplemented
164C22×C79—21no—
165C22×C79—21no—
166C22×C79#39011yescharacteristic
167C22×C79#49211yescharacteristic
168C22×C79#49511yescharacteristic
169C2×D79—81no—
170C2×D79—81no—
171C2×D79—81no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—82no—
176C4×C79—81no—
177C4×C79—81no—
178C4×C79C2×#10411yescharacteristic
179C4×C79#42111yescharacteristic
180C4×C79C4×SD1612yes—
181C4×C79#46512yes—
182Dic79—82no—
183Dic79—81no—
184#9717—791nomaximal, Sylow
185C2×C79—21no—
186C2×C79—21no—
187C2×C79#196811yescharacteristic
188C2×C79#197611yescharacteristic
189C2×C79#198011yescharacteristic
190D79—161no—
191#3167—791no—
192#1986—791no—
193#1980—791no—
194C4×#106—791no—
195#1503D7911yescharacteristic, complemented
196#1502—791no—
197C8×M16—791no—
198C79#971711yescharacteristic, complemented, Sylow
199#533C2×D7911yescharacteristic
200#532—791no—
201#498—791no—
202#492—791no—
203#487—791no—
204#486—791no—
205#465—1581no—
206#465—1581no—
207C4×SD16—1581no—
208C4×SD16—1581no—
209#431C2×D7911yescharacteristic
210C2×#106—1581no—
211C2×#104—1582no—
212C4×M16—791no—
213C2×C4×C8—791no—
214C4×C4⋊C4—791no—
215C4×C4⋊C4C2×D7911yescharacteristic
216#340—1581no—
217#339—1582no—
218#328—22no—
219#327—1582no—
220#322—1581no—
221C82—1581no—
222C2×#41—1581no—
223C2×SD16—1581no—
224C2×SD16—1581no—
225C2×M16—792no—
226C22×C8—791no—
227C22×C8—791no—
228C4⋊Q8—791no—
229C4⋊Q8C4×D7911yescharacteristic
230C4⋊Q8C4×D7911yescharacteristic
231#127—791no—
232#125—21no—
233#124—1581no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—1581no—
237#121—1581no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—1581no—
241C4×D4—1581no—
242#117—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1581no—
245C2×C42—21no—
246C2×C42—1581no—
247C2×C42C22×D7911yescharacteristic
248#106—1581no—
249#106—1581no—
250#106—1584no—
251C4⋊C8—21no—
252#104—3164no—
253#103—1581no—
254#103—1581no—
255#102—1581no—
256#102—1581no—
257C22⋊C8C22⋊D7912yes—
258#97—1581no—
259C4×C8—21no—
260C4×C8—791no—
261C4×C8—791no—
262C4×C8—791no—
263C4×C8—791no—
264C4×C8—1581no—
265#95—21no—
266#95—1581no—
267#41—3162no—
268#41—3162no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—1581no—
272C2×D4—3161no—
273C2×D4—1581no—
274C2×D4—1581no—
275C2×D4—1581no—
276C22×C4—41no—
277C22×C4—3161no—
278C22×C4—3161no—
279C22×C4D4×D7911yescharacteristic
280C22×C4D4×D7911yescharacteristic
281C22×C4C2×C22⋊D7911yescharacteristic
282SD16—3162no—
283SD16—3162no—
284M16—794no—
285M16—1582no—
286C2×C8—22no—
287C2×C8—3161no—
288C2×C8—791no—
289C2×C8—791no—
290C2×C8—791no—
291C2×C8—791no—
292C2×C8—1582no—
293C2×C8—1582no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—3161no—
303C4⋊C4—1582no—
304C4⋊C4—1581no—
305C4⋊C4—1581no—
306C22⋊C4—41no—
307C22⋊C4—3161no—
308C22⋊C4—3161no—
309C42—41no—
310C42—42no—
311C42—3162no—
312C42C2×C4×D7911yescharacteristic
313C42D4×D7911yescharacteristic
314C42#1192962311yescharacteristic
315C42#1192962411yescharacteristic
316C23—3161no—
317C23#1457915711yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—81no—
321Q8—3162no—
322D4—3162no—
323D4—3161no—
324D4—3161no—
325D4—6321no—
326D4—6321no—
327D4—6321no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—3162no—
337C2×C4—3161no—
338C2×C4—6321no—
339C2×C4—6321no—
340C2×C4—6322no—
341C2×C4C22⋊C4×D7911yescharacteristic
342C2×C4(C22⋊D79)⋊C411yescharacteristic
343C2×C4C4×C22⋊D7911yescharacteristic
344C2×C4#1457913311yescharacteristic
345C2×C4#1457913411yescharacteristic
346C2×C4#1457914711yescharacteristic
347C2×C4SD16×D7912yes—
348C2×C4#1457918412yes—
349C8—42no—
350C8—792no—
351C8—1584no—
352C8—1581no—
353C8—1581no—
354C8—1581no—
355C22—21no—
356C22—21no—
357C22—6321no—
358C22—6321no—
359C22—6321no—
360C22#1909846311yescharacteristic
361C22#1909893411yescharacteristic
362C22#1909894811yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—6322no—
370C4—6321no—
371C4#104×D7911yescharacteristic
372C4#1909857811yescharacteristic
373C4#1909864812yes—
374C4#1909879512yes—
375C2—21no—
376C2—21no—
377C2—12641no—
378C2#2376425111yescharacteristic
379C2#2376429511yescharacteristic
380C2#2376431711yescharacteristic
subgroupG/H
centerC2×C4#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1503C2
socleC22×C79#390
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1503D79
hypocenterC79#9717
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #23764345C2semidirect
4. C4×#14579092C2semidirect
6. #23761681C2semidirect
7. C79×#1503C2semidirect
9. #19098959C4semidirect
10. #19098937C4semidirect
11. #19098913C4semidirect
12. #19098911C4semidirect
88. #11929608SD16semidirect
163. C79⋊C8#104semidirect
195. #1503D79semidirect
198. C79#9717semidirect

Maximal subgroups

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

subgroupindex
#237739602
#237643452
#237643182
C4×#145790922
#237618962
#237616812
C79×#15032
#971779