Finite Group Databasebeta

← #34177144

Subgroups — #34177144

Order 20224.

446 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 81 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#23770359C211yesmaximal, characteristic
2#23764725C211yesmaximal, characteristic, complemented
3#23764643C211yesmaximal, characteristic, complemented
4#23761224C211yesmaximal, characteristic, complemented
5#23761222C211yesmaximal, characteristic, complemented
6#23761221C211yesmaximal, characteristic
7C79×#2023C211yesmaximal, characteristic
8C2×#14579154—21no—
9C2×#14579094—21no—
10C2×#14579085—21no—
11C2×#14579085—21no—
12#19098499—21no—
13#19098486—21no—
14#19098477—21no—
15#19098461C2211yescharacteristic
16#19098461C2211yescharacteristic
17#19098458—21no—
18#19098456—21no—
19#19098325—21no—
20#19098318C2211yescharacteristic
21#19098318C2211yescharacteristic
22C2×C4⋊Q8×C79C2211yescharacteristic
23C2×#102×C79—21no—
24#390×C79C2211yescharacteristic
25#340×C79—21no—
26#336×C79C2211yescharacteristic
27C22×C79⋊C8—21no—
28C22×C79⋊C8—21no—
29#14579154—42no—
30#14579154—42no—
31C2×C4⋊Dic79D411yescharacteristic
32C2×C4⋊Dic79D411yescharacteristic
33C2×C4⋊Dic79D411yescharacteristic
34C2×C4⋊Dic79D411yescharacteristic
35C2×Dic79⋊C4—21no—
36C2×Dic79⋊C4—21no—
37C2×Dic79⋊C4—21no—
38C2×Dic79⋊C4—21no—
39C2×C22⋊Dic79—21no—
40C2×C22⋊Dic79—21no—
41C2×C22⋊Dic79—21no—
42C2×C22⋊Dic79—21no—
43C2×C4×Dic79—21no—
44#14579094—42no—
45#14579085—42no—
46#14579085—42no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C22×Q8×C79—21no—
51C22×D4×C79D411yescharacteristic
52C22×C8×C79—21no—
53C4⋊Q8×C79—42no—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79—21no—
56C2×C4⋊C4×C79D411yescharacteristic
57C2×C22⋊C4×C79—21no—
58C2×C22⋊C4×C79—21no—
59C2×C42×C79C2311yescharacteristic
60#102×C79—42no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C2×C79⋊C8—22no—
66C2×C79⋊C8—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73C4⋊Dic79—21no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79D812yes—
77Dic79⋊C4—42no—
78Dic79⋊C4—41no—
79Dic79⋊C4—41no—
80Dic79⋊C4—41no—
81Dic79⋊C4—41no—
82Dic79⋊C4—81no—
83C22⋊Dic79—42no—
84C22⋊Dic79—42no—
85C22⋊Dic79—81no—
86C22⋊Dic79—81no—
87C4×Dic79—42no—
88C24×C79—21no—
89C2×Q8×C79—22no—
90C2×Q8×C79—41no—
91C2×Q8×C79—41no—
92C2×Q8×C79—41no—
93C2×Q8×C79—41no—
94C2×Q8×C79—41no—
95C2×D4×C79—21no—
96C2×D4×C79—21no—
97C2×D4×C79—41no—
98C2×D4×C79—41no—
99C22×C4×C79—21no—
100C22×C4×C79—41no—
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C2×C8×C79—41no—
105C2×C8×C79—41no—
106C2×C8×C79—41no—
107C4⋊C4×C79—22no—
108C4⋊C4×C79—22no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C4⋊C4×C79—41no—
112C22⋊C4×C79—41no—
113C22⋊C4×C79—41no—
114C22⋊C4×C79—41no—
115C22⋊C4×C79—41no—
116C42×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79—41no—
123C23×C79—41no—
124C23×C79#12011yescharacteristic
125Q8×C79—42no—
126Q8×C79—42no—
127Q8×C79—81no—
128Q8×C79—81no—
129D4×C79—42no—
130D4×C79—81no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—42no—
141C2×C4×C79—81no—
142C2×C4×C79—81no—
143C2×C4×C79C2×D811yescharacteristic
144C2×C4×C79#13511yescharacteristic
145C2×C4×C79#13511yescharacteristic
146C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C79#13711yescharacteristic
148C2×C4×C79#13711yescharacteristic
149C2×Dic79—22no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C8×C79—81no—
162C79⋊C8—42no—
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—82no—
185C4×C79—81no—
186Dic79—42no—
187Dic79—81no—
188Dic79—81no—
189Dic79—81no—
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#2727—791no—
200#2057—791no—
201#2043—791no—
202#2023D7911yescharacteristic
203#1271—791no—
204#1271—791no—
205#1270—791no—
206C79#776511yescharacteristic, complemented, Sylow
207C2×C4⋊Q8C2×D7911yescharacteristic
208C2×#126—1581no—
209C2×#107—1581no—
210C2×#103—1581no—
211C2×#103—1581no—
212C2×#102—21no—
213#399—1581no—
214#396—1581no—
215#394—1581no—
216#390C2×D7911yescharacteristic
217#389—791no—
218#389—791no—
219#389—1581no—
220#388—1581no—
221#340—21no—
222#340—1581no—
223#336—791no—
224#336—791no—
225#336C2×D7911yescharacteristic
226C22×Q8—21no—
227C22×D4C22⋊D7911yescharacteristic
228C22×C8—21no—
229C22×C8—1581no—
230C22×C8—1581no—
231C4⋊Q8—42no—
232#126—3162no—
233#126—3162no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—791no—
237C2×C4⋊C4—791no—
238C2×C4⋊C4—791no—
239C2×C4⋊C4—791no—
240C2×C4⋊C4—1581no—
241C2×C4⋊C4—1581no—
242C2×C4⋊C4—1581no—
243C2×C4⋊C4—1581no—
244C2×C4⋊C4C22⋊D7911yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1581no—
248C2×C22⋊C4—1581no—
249C2×C22⋊C4—1581no—
250C2×C22⋊C4—1581no—
251C2×C42—1581no—
252C2×C42C22×D7911yescharacteristic
253#107—3162no—
254#103—3162no—
255#103—3162no—
256#102—42no—
257#95—1581no—
258#95—1581no—
259#95—1581no—
260C24—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—1581no—
274C22×C4—1581no—
275C22×C4—1581no—
276C22×C4—1581no—
277C22×C4D4×D7911yescharacteristic
278C22×C4D4×D7911yescharacteristic
279C22×C4C2×C22⋊D7911yescharacteristic
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—3161no—
284C2×C8—3161no—
285C2×C8—3161no—
286C2×C8—3161no—
287C2×C8—3161no—
288C2×C8—1582no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—3161no—
295C4⋊C4—3161no—
296C4⋊C4—3162no—
297C4⋊C4—3161no—
298C4⋊C4—3161no—
299C4⋊C4—3161no—
300C4⋊C4—3161no—
301C4⋊C4—792no—
302C4⋊C4—6321no—
303C4⋊C4—1581no—
304C4⋊C4—1581no—
305C4⋊C4—1581no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—3162no—
311C22⋊C4—3162no—
312C22⋊C4—6321no—
313C22⋊C4—6321no—
314C42—41no—
315C42—3162no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1457912911yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—3161no—
343C2×C4—3161no—
344C2×C4—3161no—
345C2×C4—3161no—
346C2×C4—3161no—
347C2×C4—3161no—
348C2×C4—3161no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—1582no—
354C2×C4D8×D7911yescharacteristic
355C2×C4#1457918111yescharacteristic
356C2×C4#1457918211yescharacteristic
357C2×C4#1457918511yescharacteristic
358C2×C4#1457919011yescharacteristic
359C2×C4#1457919211yescharacteristic
360C8—81no—
361C8—3162no—
362C8—6321no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1909869511yescharacteristic
371C22#1909872211yescharacteristic
372C22#1909872311yescharacteristic
373C22#1909873411yescharacteristic
374C22#1909874011yescharacteristic
375C22#1909874411yescharacteristic
376C22#1909874911yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—81no—
385C4—3162no—
386C4—6321no—
387C4—6321no—
388C4—6321no—
389C2—161no—
390C2#2376233311yescharacteristic
391C2#2376236011yescharacteristic
392C2#2376237811yescharacteristic
393C2#2376257011yescharacteristic
394C2#2376259011yescharacteristic
395C2#2376263411yescharacteristic
396C2#2376273311yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2023C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2023D79
hypocenterC79#7765
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
2. #23764725C2semidirect
3. #23764643C2semidirect
4. #23761224C2semidirect
5. #23761222C2semidirect
206. C79#7765semidirect

Maximal subgroups

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

subgroupindex
#237703592
#237647252
#237646432
#237612242
#237612222
#237612212
C79×#20232
#776579