Finite Group Databasebeta

← #34177138

Subgroups — #34177138

Order 20224.

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

normal subgroups only

This group has 504 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#23769772C211yesmaximal, characteristic
2#23764725C211yesmaximal, characteristic, complemented
3#23764643C211yesmaximal, characteristic, complemented
4#23761222C211yesmaximal, characteristic, complemented
5#23761222C211yesmaximal, characteristic, complemented
6#23761221C211yesmaximal, characteristic
7C79×#2023C211yesmaximal, characteristic
8C2×#14579138—21no—
9C2×#14579138—21no—
10C2×#14579094—21no—
11C2×#14579085—21no—
12C2×#14579085—21no—
13#19098486—21no—
14#19098486—21no—
15#19098483—21no—
16#19098461C2211yescharacteristic
17#19098461C2211yescharacteristic
18#19098443—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×Dic158—21no—
28C23×Dic79—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31#14579138—42no—
32#14579138—42no—
33#14579138—42no—
34#14579138—42no—
35C2×C4⋊Dic79D411yescharacteristic
36C2×C4⋊Dic79D411yescharacteristic
37C2×C4⋊Dic79D411yescharacteristic
38C2×C4⋊Dic79D411yescharacteristic
39C2×Dic79⋊C4—21no—
40C2×Dic79⋊C4—21no—
41C2×Dic79⋊C4—21no—
42C2×Dic79⋊C4—21no—
43C2×C22⋊Dic79—21no—
44C2×C22⋊Dic79—21no—
45C2×C22⋊Dic79—21no—
46#14579094—42no—
47#14579085—42no—
48#14579085—42no—
49C79⋊#95—21no—
50C79⋊#95—21no—
51C79⋊#95—21no—
52C22×Q8×C79—21no—
53C22×D4×C79D411yescharacteristic
54C22×C8×C79—21no—
55C4⋊Q8×C79—42no—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79D411yescharacteristic
59C2×C22⋊C4×C79—21no—
60C2×C22⋊C4×C79—21no—
61C2×C42×C79C2311yescharacteristic
62#102×C79—42no—
63C2×Dic158—42no—
64C2×Dic158—42no—
65C2×Dic158—81no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—42no—
71C22×Dic79—42no—
72C22×Dic79—81no—
73C2×C79⋊C8—22no—
74C2×C79⋊C8—41no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C4⋊Dic79—21no—
80C4⋊Dic79—21no—
81C4⋊Dic79—41no—
82C4⋊Dic79—41no—
83C4⋊Dic79D812yes—
84C4⋊Dic79SD1612yes—
85Dic79⋊C4—42no—
86Dic79⋊C4—42no—
87Dic79⋊C4—41no—
88Dic79⋊C4—41no—
89Dic79⋊C4—41no—
90Dic79⋊C4—41no—
91C22⋊Dic79—42no—
92C22⋊Dic79—42no—
93C22⋊Dic79—81no—
94C24×C79—21no—
95C2×Q8×C79—22no—
96C2×Q8×C79—41no—
97C2×Q8×C79—41no—
98C2×Q8×C79—41no—
99C2×Q8×C79—41no—
100C2×Q8×C79—41no—
101C2×D4×C79—21no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79Q1612yes—
105C22×C4×C79—21no—
106C22×C4×C79—41no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C2×C8×C79—41no—
113C4⋊C4×C79—22no—
114C4⋊C4×C79—22no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C22⋊C4×C79—41no—
120C22⋊C4×C79—41no—
121C22⋊C4×C79—41no—
122C42×C79—41no—
123C23×C79—22no—
124C23×C79—41no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79—41no—
128C23×C79—41no—
129C23×C79—41no—
130C23×C79#12011yescharacteristic
131Q8×C79—42no—
132Q8×C79—42no—
133Q8×C79—81no—
134Q8×C79—81no—
135D4×C79—42no—
136D4×C79—81no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—42no—
147C2×C4×C79—81no—
148C2×C4×C79—81no—
149C2×C4×C79C2×D811yescharacteristic
150C2×C4×C79C2×SD1611yescharacteristic
151C2×C4×C79C2×Q1611yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79#13711yescharacteristic
155C2×Dic79—22no—
156C2×Dic79—22no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C2×Dic79—41no—
162C2×Dic79—41no—
163C2×Dic79—41no—
164C2×Dic79—41no—
165C2×Dic79—41no—
166C2×Dic79—41no—
167C2×Dic79—84no—
168Dic158—84no—
169C8×C79—81no—
170C79⋊C8—42no—
171C79⋊C8—81no—
172C22×C79—42no—
173C22×C79—42no—
174C22×C79—42no—
175C22×C79—81no—
176C22×C79—81no—
177C22×C79—81no—
178C22×C79—81no—
179C22×C79#44711yescharacteristic
180C22×C79#45011yescharacteristic
181C22×C79#45111yescharacteristic
182C22×C79#45411yescharacteristic
183C22×C79#45511yescharacteristic
184C22×C79#45611yescharacteristic
185C22×C79B(4,𝔽2)11yescharacteristic
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—82no—
193C4×C79—81no—
194Dic79—42no—
195Dic79—42no—
196Dic79—81no—
197Dic79—81no—
198#7764—791nomaximal, Sylow
199C2×C79—82no—
200C2×C79#159811yescharacteristic
201C2×C79#160211yescharacteristic
202C2×C79#160411yescharacteristic
203C2×C79#163711yescharacteristic
204C2×C79#163911yescharacteristic
205C2×C79#164911yescharacteristic
206C2×C79#166411yescharacteristic
207#2683—791no—
208#2057—791no—
209#2043—791no—
210#2023D7911yescharacteristic
211#1271—791no—
212#1271—791no—
213#1270—791no—
214C79#776411yescharacteristic, complemented, Sylow
215C2×C4⋊Q8C2×D7911yescharacteristic
216C2×C22⋊Q8—1581no—
217C2×C22⋊Q8—1581no—
218C2×#107—1581no—
219C2×#103—1581no—
220C2×#103—1581no—
221C2×#102—21no—
222#396—1581no—
223#396—1581no—
224#395—1581no—
225#390C2×D7911yescharacteristic
226#389—791no—
227#389—791no—
228#386—1581no—
229#340—21no—
230#340—1581no—
231#336—791no—
232#336—791no—
233#336C2×D7911yescharacteristic
234C22×Q8—21no—
235C22×Q8—1581no—
236C22×D4C22⋊D7911yescharacteristic
237C23×C4—1581no—
238C22×C8—21no—
239C22×C8—1581no—
240C22×C8—1581no—
241C4⋊Q8—42no—
242C22⋊Q8—3162no—
243C22⋊Q8—3162no—
244C22⋊Q8—3162no—
245C22⋊Q8—3162no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—791no—
249C2×C4⋊C4—791no—
250C2×C4⋊C4—791no—
251C2×C4⋊C4—791no—
252C2×C4⋊C4—1581no—
253C2×C4⋊C4—1581no—
254C2×C4⋊C4—1581no—
255C2×C4⋊C4—1581no—
256C2×C4⋊C4C22⋊D7911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—1581no—
260C2×C22⋊C4—1581no—
261C2×C22⋊C4—1581no—
262C2×C42C22×D7911yescharacteristic
263#107—3162no—
264#103—3162no—
265#103—3162no—
266#102—42no—
267#95—1581no—
268#95—1581no—
269#95—1581no—
270C24—21no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—3162no—
278C2×Q8—3162no—
279C2×Q8—6321no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4#1192961412yes—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—3162no—
287C22×C4—3162no—
288C22×C4—6321no—
289C22×C4—1581no—
290C22×C4—1581no—
291C22×C4—1581no—
292C22×C4—1581no—
293C22×C4D4×D7911yescharacteristic
294C22×C4D4×D7911yescharacteristic
295C22×C4C2×C22⋊D7911yescharacteristic
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—3161no—
300C2×C8—3161no—
301C2×C8—3161no—
302C2×C8—3161no—
303C2×C8—3161no—
304C2×C8—1582no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—3161no—
311C4⋊C4—3161no—
312C4⋊C4—3162no—
313C4⋊C4—3162no—
314C4⋊C4—3161no—
315C4⋊C4—3161no—
316C4⋊C4—3161no—
317C4⋊C4—3161no—
318C4⋊C4—792no—
319C4⋊C4—792no—
320C4⋊C4—1581no—
321C4⋊C4—1581no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—3162no—
327C22⋊C4—3162no—
328C22⋊C4—6321no—
329C42—41no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#1457912911yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342Q8—6324no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—3161no—
358C2×C4—3161no—
359C2×C4—3161no—
360C2×C4—3161no—
361C2×C4—3161no—
362C2×C4—3161no—
363C2×C4—3161no—
364C2×C4—3161no—
365C2×C4—3161no—
366C2×C4—3161no—
367C2×C4—6324no—
368C2×C4—1582no—
369C2×C4—1582no—
370C2×C4D8×D7911yescharacteristic
371C2×C4SD16×D7911yescharacteristic
372C2×C4C2×#1192961411yescharacteristic
373C2×C4#1457918511yescharacteristic
374C2×C4#1457918811yescharacteristic
375C2×C4#1457919111yescharacteristic
376C8—81no—
377C8—3162no—
378C8—6321no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1909869511yescharacteristic
387C22#1909871211yescharacteristic
388C22#1909871811yescharacteristic
389C22#1909873411yescharacteristic
390C22#1909874011yescharacteristic
391C22#1909874411yescharacteristic
392C22#1909874911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—81no—
400C4—82no—
401C4—3162no—
402C4—3162no—
403C4—6321no—
404C4—6321no—
405C2—82no—
406C2#2376233311yescharacteristic
407C2#2376235611yescharacteristic
408C2#2376236711yescharacteristic
409C2#2376256411yescharacteristic
410C2#2376257611yescharacteristic
411C2#2376264811yescharacteristic
412C2#2376273311yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2023C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2023D79
hypocenterC79#7764
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. #23761222C2semidirect
5. #23761222C2semidirect
214. C79#7764semidirect

Maximal subgroups

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

subgroupindex
#237697722
#237647252
#237646432
#237612222
#237612212
C79×#20232
#776479