Finite Group Databasebeta

← #34177146

Subgroups — #34177146

Order 20224.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764728C211yesmaximal, characteristic, complemented
2#23764646C211yesmaximal, characteristic, complemented
3#23764536C211yesmaximal, characteristic
4#23761223C211yesmaximal, characteristic, complemented
5#23761223C211yesmaximal, characteristic, complemented
6#23761221C211yesmaximal, characteristic
7C79×#2727C211yesmaximal, characteristic
8C2×#14579161C2211yescharacteristic
9C2×#14579094—21no—
10C2×#14579084—21no—
11C2×#14579084—21no—
12C2×#14579082—21no—
13#19098325—21no—
14#19098325—21no—
15#19098318C2211yescharacteristic
16#19098318C2211yescharacteristic
17#19098318C2211yescharacteristic
18C2×#126×C79—21no—
19#399×C79—21no—
20#396×C79—21no—
21#394×C79—21no—
22#390×C79C2211yescharacteristic
23#389×C79—21no—
24#389×C79C2211yescharacteristic
25#389×C79C2211yescharacteristic
26#388×C79—21no—
27C22×Dic158—21no—
28C22×C79⋊C8—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31#14579161—42no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34#14579094—42no—
35#14579084—42no—
36#14579084—42no—
37#14579082—42no—
38C22×D4×C79D411yescharacteristic
39#126×C79—42no—
40#126×C79—42no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C4⋊C4×C79D411yescharacteristic
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C42×C79—21no—
57C2×C42×C79C2311yescharacteristic
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61C2×Dic158—22no—
62C2×Dic158—41no—
63C2×Dic158—41no—
64C2×Dic158—41no—
65C2×Dic158—41no—
66C2×Dic158—41no—
67C22×Dic79—41no—
68C2×C79⋊C8—22no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8—41no—
77C4⋊Dic79—22no—
78C4⋊Dic79—22no—
79C4⋊Dic79—41no—
80C4⋊Dic79—41no—
81C24×C79—21no—
82C2×D4×C79—21no—
83C2×D4×C79—21no—
84C2×D4×C79—41no—
85C2×D4×C79—41no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C4⋊C4×C79—21no—
95C4⋊C4×C79—21no—
96C4⋊C4×C79—21no—
97C4⋊C4×C79—42no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—41no—
105C4⋊C4×C79—81no—
106C4⋊C4×C79D812yes—
107C22⋊C4×C79—42no—
108C22⋊C4×C79—42no—
109C22⋊C4×C79—41no—
110C22⋊C4×C79—41no—
111C22⋊C4×C79—41no—
112C22⋊C4×C79—41no—
113C22⋊C4×C79—81no—
114C22⋊C4×C79—81no—
115C42×C79—41no—
116C42×C79—42no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79—41no—
123C23×C79—41no—
124C23×C79#12011yescharacteristic
125D4×C79—42no—
126D4×C79—81no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—22no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79C2×D811yescharacteristic
149C2×C4×C79#13511yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C79#13711yescharacteristic
153C2×C4×C79#13711yescharacteristic
154C2×Dic79—42no—
155C2×Dic79—81no—
156C2×Dic79—81no—
157Dic158—42no—
158Dic158—42no—
159Dic158—81no—
160Dic158—81no—
161C79⋊C8—42no—
162C79⋊C8—81no—
163C79⋊C8—81no—
164C22×C79—81no—
165C22×C79—81no—
166C22×C79—81no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79—81no—
170C22×C79—81no—
171C22×C79#44711yescharacteristic
172C22×C79#45211yescharacteristic
173C22×C79#45211yescharacteristic
174C22×C79#45411yescharacteristic
175C22×C79#45511yescharacteristic
176C22×C79#45611yescharacteristic
177C22×C79B(4,𝔽2)11yescharacteristic
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—42no—
185C4×C79—81no—
186C4×C79—81no—
187C4×C79—81no—
188C4×C79—81no—
189Dic79—82no—
190#7765—791nomaximal, Sylow
191C2×C79—161no—
192C2×C79#159811yescharacteristic
193C2×C79#160311yescharacteristic
194C2×C79#160611yescharacteristic
195C2×C79#163811yescharacteristic
196C2×C79#164111yescharacteristic
197C2×C79#164711yescharacteristic
198C2×C79#166411yescharacteristic
199#2727D7911yescharacteristic
200#2057—791no—
201#2043—791no—
202#2023—791no—
203#1271—791no—
204#1271—791no—
205#1270—791no—
206C79#776511yescharacteristic, complemented, Sylow
207C2×C4⋊Q8—791no—
208C2×#126—21no—
209C2×#107—1581no—
210C2×#103—1581no—
211C2×#103—1581no—
212C2×#102—1581no—
213#399—21no—
214#396—21no—
215#394—21no—
216#390C2×D7911yescharacteristic
217#389—21no—
218#389C2×D7911yescharacteristic
219#389C2×D7911yescharacteristic
220#388—21no—
221#340—1581no—
222#340—1581no—
223#336—791no—
224#336—791no—
225#336—791no—
226C22×Q8—1581no—
227C22×D4C22⋊D7911yescharacteristic
228C22×C8—1581no—
229C22×C8—1581no—
230C22×C8—1581no—
231C4⋊Q8—3162no—
232#126—42no—
233#126—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1581no—
239C2×C4⋊C4—1581no—
240C2×C4⋊C4C22⋊D7911yescharacteristic
241C2×C4⋊C4C22⋊D7911yescharacteristic
242C2×C4⋊C4C22⋊D7911yescharacteristic
243C2×C4⋊C4C22⋊D7911yescharacteristic
244C2×C4⋊C4C22⋊D7911yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42C22×D7911yescharacteristic
253#107—3162no—
254#103—3162no—
255#103—3162no—
256#102—3162no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×Q8—3161no—
262C2×Q8—3161no—
263C2×Q8—3161no—
264C2×Q8—3161no—
265C2×Q8—3161no—
266C2×Q8—1582no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—3161no—
277C22×C4C2×C22⋊D7911yescharacteristic
278C22×C4C2×C22⋊D7911yescharacteristic
279C22×C4C2×C22⋊D7911yescharacteristic
280C2×C8—3161no—
281C2×C8—3161no—
282C2×C8—3161no—
283C2×C8—3161no—
284C2×C8—3161no—
285C2×C8—3161no—
286C2×C8—3161no—
287C2×C8—3161no—
288C2×C8—1582no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4—3161no—
302C4⋊C4—3161no—
303C4⋊C4—1582no—
304C4⋊C4—1582no—
305C4⋊C4#1192960912yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1457913011yescharacteristic
324Q8—3162no—
325Q8—3162no—
326Q8—6321no—
327Q8—6321no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—3162no—
352C2×C4—6321no—
353C2×C4—6321no—
354C2×C4C2×#1192960911yescharacteristic
355C2×C4#1457918111yescharacteristic
356C2×C4#1457918111yescharacteristic
357C2×C4#1457918711yescharacteristic
358C2×C4#1457919111yescharacteristic
359C2×C4#1457919111yescharacteristic
360C8—3162no—
361C8—6321no—
362C8—6321no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1909869711yescharacteristic
371C22#1909872511yescharacteristic
372C22#1909872511yescharacteristic
373C22#1909873611yescharacteristic
374C22#1909874111yescharacteristic
375C22#1909874611yescharacteristic
376C22#1909875011yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—6322no—
389C2—161no—
390C2#2376233511yescharacteristic
391C2#2376236311yescharacteristic
392C2#2376238011yescharacteristic
393C2#2376257211yescharacteristic
394C2#2376259311yescharacteristic
395C2#2376263511yescharacteristic
396C2#2376273411yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2727C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2727D79
hypocenterC79#7765
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #23764728C2semidirect
2. #23764646C2semidirect
4. #23761223C2semidirect
5. #23761223C2semidirect
206. C79#7765semidirect

Maximal subgroups

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

subgroupindex
#237647282
#237646462
#237645362
#237612232
#237612212
C79×#27272
#776579