Finite Group Databasebeta

← #34192145

Subgroups — #34192145

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098844C211yesmaximal, characteristic
2#23764202C211yesmaximal, characteristic, complemented
3#23764194C211yesmaximal, characteristic
4#23763401C211yesmaximal, characteristic
5C2×#19098308C211yesmaximal, characteristic, complemented
6#23761284C211yesmaximal, characteristic, complemented
7C2×#334×C79C211yesmaximal, characteristic, complemented
8C2×#14579161C2211yescharacteristic
9C2×C4×D316C2211yescharacteristic
10#19098844—22no—
11#19098844—22no—
12C2×#14579095—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579086—21no—
15C2×#14579086—21no—
16C2×#14579083—21no—
17C2×#14579083—21no—
18C2×(C2×D79)⋊C8—21no—
19C2×(C2×D79)⋊C8—21no—
20#19098433C2211yescharacteristic
21#19098326—21no—
22#19098308—22no—
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×C4×C8×C79C2211yescharacteristic
26#334×C79—22no—
27C22×Dic158—21no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C22×C79⋊C8—21no—
31#14579161—21no—
32#14579161—21no—
33#14579161—21no—
34#14579161D412yes—
35C4×D316—21no—
36C4×D316—21no—
37C4×D316—21no—
38C4×D316—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79—21no—
44C2×C4⋊Dic79D411yescharacteristic
45C2×(C2×D79)⋊C4—21no—
46#14579095—24no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49#14579086—42no—
50#14579086—42no—
51#14579083—24no—
52#14579083—42no—
53(C2×D79)⋊C8—42no—
54(C2×D79)⋊C8—42no—
55C22×C8×C79—21no—
56C22×C8×C79—21no—
57C22×C8×C79D411yescharacteristic
58C22×C8×C79D411yescharacteristic
59C2×C42×C79C2311yescharacteristic
60C4⋊C8×C79—21no—
61C4⋊C8×C79—21no—
62C4⋊C8×C79—21no—
63C4⋊C8×C79D412yes—
64C4×C8×C79—21no—
65C4×C8×C79—21no—
66C23×D79—21no—
67C2×Dic158—22no—
68C2×Dic158—41no—
69C2×Dic158—41no—
70C2×Dic158—41no—
71C2×Dic158—41no—
72C2×Dic158—41no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316—21no—
76C2×D316—21no—
77C2×D316—21no—
78C2×D316SD1612yes—
79C22×Dic79—21no—
80C22×Dic79—41no—
81C22×Dic79—41no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84C2×C4×D79—41no—
85C2×C4×D79—41no—
86C2×C4×D79—41no—
87C2×C4×D79—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—22no—
92C4⋊Dic79—21no—
93C4⋊Dic79—22no—
94C4⋊Dic79—21no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—41no—
99C4⋊Dic79—41no—
100C4⋊Dic79—41no—
101C4⋊Dic79—81no—
102(C2×D79)⋊C4—41no—
103(C2×D79)⋊C4—41no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79#4111yescharacteristic
107C2×C8×C79—22no—
108C2×C8×C79—21no—
109C2×C8×C79—21no—
110C2×C8×C79—21no—
111C2×C8×C79—21no—
112C2×C8×C79—21no—
113C2×C8×C79—41no—
114C2×C8×C79—41no—
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C2×C8×C79D812yes—
119C42×C79C2×D411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79#4111yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12111yescharacteristic
124C22×D79—22no—
125C22×D79—41no—
126C22×D79—41no—
127C22×D79—41no—
128C22×D79—41no—
129C22×D79—41no—
130C22×D79—41no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79#12011yescharacteristic
134C2×C4×C79#12111yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12311yescharacteristic
137C2×C4×C79#12311yescharacteristic
138C2×C4×C79#12411yescharacteristic
139C2×C4×C79C2×D811yescharacteristic
140C2×C4×C79C2×SD1611yescharacteristic
141C2×C4×C79#13511yescharacteristic
142C2×C4×C79#13511yescharacteristic
143C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C79#13711yescharacteristic
146C2×C4×C79#13711yescharacteristic
147C2×Dic79—41no—
148C2×Dic79—42no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—81no—
152C2×Dic79—81no—
153C2×Dic79—81no—
154C2×Dic79—81no—
155C2×Dic79—81no—
156D316—22no—
157D316—22no—
158D316—41no—
159D316—41no—
160C4×D79—81no—
161C4×D79—81no—
162C4×D79—81no—
163C4×D79—81no—
164Dic158—42no—
165Dic158—42no—
166Dic158—81no—
167Dic158—81no—
168C8×C79—22no—
169C8×C79—42no—
170C8×C79—41no—
171C8×C79—81no—
172C79⋊C8—81no—
173C22×C79#39411yescharacteristic
174C22×C79#45911yescharacteristic
175C22×C79#46311yescharacteristic
176C22×C79#46511yescharacteristic
177C22×C79#47011yescharacteristic
178C22×C79#47111yescharacteristic
179C22×C79#47311yescharacteristic
180C2×D79—42no—
181C2×D79—42no—
182C2×D79—42no—
183C2×D79—81no—
184C2×D79—81no—
185C2×D79—81no—
186C2×D79—81no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79#44811yescharacteristic
190C4×C79#45211yescharacteristic
191C4×C79#45911yescharacteristic
192C4×C79#46311yescharacteristic
193C4×C79#46611yescharacteristic
194C4×C79#46911yescharacteristic
195C4×C79#48311yescharacteristic
196C4×C79#48511yescharacteristic
197Dic79—82no—
198Dic79—81no—
199Dic79—161no—
200#10484—791nomaximal, Sylow
201C2×C79#166511yescharacteristic
202C2×C79#167911yescharacteristic
203C2×C79#168911yescharacteristic
204C2×C79#170311yescharacteristic
205C2×C79#203511yescharacteristic
206C2×C79#205211yescharacteristic
207C2×C79#205411yescharacteristic
208D79—82no—
209C2×#474—791no—
210#1956—791no—
211#1954—791no—
212#1802—791no—
213C2×#334D7911yescharacteristic, complemented
214C2×#332—791no—
215#1296—791no—
216C79#1048411yescharacteristic, complemented, Sylow
217C2×C4⋊Q8—791no—
218C2×C4×D4—791no—
219#474—1582no—
220#474—1582no—
221C2×#107—1581no—
222C2×C4⋊C8—791no—
223C2×C4⋊C8C2×D7911yescharacteristic
224C2×C4⋊C8C2×D7911yescharacteristic
225C2×#103—1581no—
226C2×#103—1581no—
227C2×#102—1581no—
228C2×#102—1581no—
229C2×C22⋊C8—1581no—
230C2×C22⋊C8—1581no—
231C2×C4×C8C2×D7911yescharacteristic
232#384—791no—
233#340—1581no—
234#334—22no—
235#332—1582no—
236C22×Q8—1581no—
237C22×D4—791no—
238C23×C4—1581no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—1581no—
242C22×C8D31611yescharacteristic
243C22×C8D31611yescharacteristic
244C4⋊Q8—792no—
245C4⋊Q8—1581no—
246C4⋊Q8—1581no—
247C4⋊Q8—1581no—
248C4×D4—1581no—
249C4×D4—1581no—
250C4×D4—1581no—
251C4×D4—1581no—
252C2×C4⋊C4—791no—
253C2×C4⋊C4—1581no—
254C2×C4⋊C4—1581no—
255C2×C4⋊C4—1581no—
256C2×C4⋊C4—1581no—
257C2×C4⋊C4—1581no—
258C2×C22⋊C4—1581no—
259C2×C42C22×D7911yescharacteristic
260#107—1584no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—1581no—
265C4⋊C8—1581no—
266C4⋊C8C22⋊D7912yes—
267#103—3162no—
268#103—3162no—
269#102—3162no—
270#102—1584no—
271C22⋊C8—3162no—
272C22⋊C8—3162no—
273C4×C8—21no—
274C4×C8—21no—
275C24—1581no—
276C2×Q8—3161no—
277C2×Q8—3161no—
278C2×Q8—3161no—
279C2×Q8—3161no—
280C2×Q8—3161no—
281C2×Q8—1582no—
282C2×D4—792no—
283C2×D4—1581no—
284C2×D4—1581no—
285C2×D4—1581no—
286C2×D4—1581no—
287C2×D4—1581no—
288C22×C4—3161no—
289C22×C4—3161no—
290C22×C4—3161no—
291C22×C4—3161no—
292C22×C4—3161no—
293C22×C4—3161no—
294C22×C4—3161no—
295C22×C4—3161no—
296C22×C4—1581no—
297C22×C4D4×D7911yescharacteristic
298C22×C4C2×D31611yescharacteristic
299C22×C4#1192962511yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—3161no—
312C2×C8—3161no—
313C2×C8—3161no—
314C2×C8D63212yes—
315C4⋊C4—3161no—
316C4⋊C4—3161no—
317C4⋊C4—3161no—
318C4⋊C4—3161no—
319C4⋊C4—3161no—
320C4⋊C4—3161no—
321C4⋊C4—6321no—
322C4⋊C4—1582no—
323C4⋊C4—1581no—
324C4⋊C4—1581no—
325C4⋊C4—1582no—
326C22⋊C4—3161no—
327C22⋊C4—3161no—
328C42D4×D7911yescharacteristic
329C42C2×C22⋊D7911yescharacteristic
330C42#1192962411yescharacteristic
331C42#1192962511yescharacteristic
332C23—3161no—
333C23—3161no—
334C23—3161no—
335C23—3161no—
336C23—3161no—
337C23—3161no—
338C23—1582no—
339C23#1457913611yescharacteristic
340Q8—3162no—
341Q8—3162no—
342Q8—6321no—
343Q8—6321no—
344D4—3161no—
345D4—3161no—
346D4—1582no—
347D4—1582no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—3161no—
351C2×C4—3162no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—6321no—
355C2×C4—6321no—
356C2×C4—6321no—
357C2×C4—6321no—
358C2×C4—6321no—
359C2×C4—6321no—
360C2×C4—6321no—
361C2×C4—6321no—
362C2×C4—6321no—
363C2×C4#1457912911yescharacteristic
364C2×C4#1457913511yescharacteristic
365C2×C4#1457913611yescharacteristic
366C2×C4#1457914611yescharacteristic
367C2×C4#1457914611yescharacteristic
368C2×C4#1457915011yescharacteristic
369C2×C4C2×D63211yescharacteristic
370C2×C4SD16×D7911yescharacteristic
371C2×C4#1457918211yescharacteristic
372C2×C4#1457918311yescharacteristic
373C2×C4#1457918911yescharacteristic
374C2×C4#1457918911yescharacteristic
375C2×C4#1457919011yescharacteristic
376C2×C4#1457919011yescharacteristic
377C8—22no—
378C8—42no—
379C8—41no—
380C8—81no—
381C8—6321no—
382C22—3162no—
383C22—3162no—
384C22—3162no—
385C22—6321no—
386C22—6321no—
387C22—6321no—
388C22—6321no—
389C22#1909848011yescharacteristic
390C22#1909875911yescharacteristic
391C22#1909877911yescharacteristic
392C22#1909879311yescharacteristic
393C22#1909882211yescharacteristic
394C22#1909882711yescharacteristic
395C22#1909883911yescharacteristic
396C4—21no—
397C4—21no—
398C4—6321no—
399C4—6322no—
400C4—12641no—
401C4#1909870011yescharacteristic
402C4#1909872211yescharacteristic
403C4#1909875911yescharacteristic
404C4#1909877911yescharacteristic
405C4#1909879911yescharacteristic
406C4#1909881611yescharacteristic
407C4#1909889511yescharacteristic
408C4#1909890711yescharacteristic
409C2—6322no—
410C2#2376274211yescharacteristic
411C2#2376282411yescharacteristic
412C2#2376289011yescharacteristic
413C2#2376297611yescharacteristic
414C2#2376459811yescharacteristic
415C2#2376470111yescharacteristic
416C2#2376471111yescharacteristic
subgroupG/H
centerC23#14579136
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#334×C79C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenterC2×#334D79
hypocenterC79#10484
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #23764202C2semidirect
5. C2×#19098308C2semidirect
6. #23761284C2semidirect
7. C2×#334×C79C2semidirect
213. C2×#334D79semidirect
216. C79#10484semidirect

Maximal subgroups

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

subgroupindex
C2×#190988442
#237642022
#237641942
#237634012
C2×#190983082
#237612842
C2×#334×C792
#1048479