Finite Group Databasebeta

← #34184481

Subgroups — #34184481

Order 20224.

578 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098765C211yesmaximal, characteristic
2C2×#19098642C211yesmaximal, characteristic
3#23763397C211yesmaximal, characteristic
4C2×#19098309C211yesmaximal, characteristic, complemented
5#23761283C211yesmaximal, characteristic, complemented
6#23761235C211yesmaximal, characteristic, complemented
7C79×#1990C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C22×#11929607—21no—
10C2×C4×D316C2211yescharacteristic
11#19098765—22no—
12#19098765C414yescomplemented
13#19098642—22no—
14#19098642—22no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×C4⋊(C79⋊C8)C2211yescharacteristic
18C2×#14579085—21no—
19C2×#14579081—21no—
20C2×(C2×D79)⋊C8—21no—
21#19098325—21no—
22#19098309—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4×C8×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26#340×C79—21no—
27#340×C79—21no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C2×#11929611—24no—
31C2×#11929611—42no—
32C2×#11929611—42no—
33C2×#11929607—24no—
34C2×#11929607—42no—
35C2×#11929607—42no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8—21no—
38C22×C79⋊C8—21no—
39C4×D316—21no—
40C4×D316—21no—
41C4×D316C2×C412yes—
42C4×D316D412yes—
43C2×C4⋊Dic79D411yescharacteristic
44C2×(C2×D79)⋊C4—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)—21no—
49C4⋊(C79⋊C8)C2×C412yes—
50C4⋊(C79⋊C8)D412yes—
51#14579085—24no—
52#14579081—42no—
53(C2×D79)⋊C8—24no—
54C22×Q8×C79—21no—
55C22×C8×C79D411yescharacteristic
56C22×C8×C79D411yescharacteristic
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79—21no—
59C4⋊Q8×C79—21no—
60C4⋊Q8×C79C2×C412yes—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79—21no—
66C2×C42×C79C2311yescharacteristic
67C4×C8×C79—21no—
68C4×C8×C79—21no—
69C23×D79—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316C22⋊C412yes—
76C22×Dic79—21no—
77C2×C4×D79—22no—
78C2×C4×D79—22no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C4×D79—41no—
83#11929611—48no—
84#11929607—48no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C4⋊Dic79—21no—
95C4⋊Dic79C22⋊C412yes—
96(C2×D79)⋊C4—22no—
97(C2×D79)⋊C4—22no—
98C2×Q8×C79—22no—
99C2×Q8×C79—41no—
100C2×Q8×C79—41no—
101C2×Q8×C79—41no—
102C2×Q8×C79—41no—
103C2×Q8×C79—41no—
104C22×C4×C79—41no—
105C22×C4×C79—41no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79#4111yescharacteristic
109C2×C8×C79—21no—
110C2×C8×C79—21no—
111C2×C8×C79—21no—
112C2×C8×C79—21no—
113C2×C8×C79—21no—
114C2×C8×C79SD1612yescomplemented
115C4⋊C4×C79—22no—
116C4⋊C4×C79—22no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
120C4⋊C4×C79—41no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—81no—
124C42×C79C22×C411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79C2×D411yescharacteristic
127C42×C79#4111yescharacteristic
128C23×C79#12111yescharacteristic
129C22×D79—22no—
130C22×D79—41no—
131C22×D79—41no—
132C22×D79—41no—
133C22×D79—41no—
134C22×D79—41no—
135C22×D79—41no—
136Q8×C79—42no—
137Q8×C79—42no—
138Q8×C79—81no—
139Q8×C79—81no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—42no—
143C2×C4×C79—81no—
144C2×C4×C79—81no—
145C2×C4×C79—81no—
146C2×C4×C79—81no—
147C2×C4×C79—81no—
148C2×C4×C79C2×C22⋊C411yescharacteristic
149C2×C4×C79C4×D411yescharacteristic
150C2×C4×C79C4×D411yescharacteristic
151C2×C4×C79#12011yescharacteristic
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12311yescharacteristic
154C2×C4×C79C2×SD1611yescharacteristic
155C2×C4×C79#13511yescharacteristic
156C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C79#13711yescharacteristic
161C2×C4×C79#13711yescharacteristic
162C2×Dic79—22no—
163C2×Dic79—41no—
164C2×Dic79—41no—
165D316—22no—
166D316—22no—
167D316—41no—
168D316—41no—
169C4×D79—42no—
170C4×D79—42no—
171C4×D79—42no—
172C4×D79—42no—
173C8×C79—22no—
174C8×C79—41no—
175C79⋊C8—42no—
176C79⋊C8—42no—
177C79⋊C8—81no—
178C22×C79#38611yescharacteristic
179C22×C79#46111yescharacteristic
180C22×C79#46311yescharacteristic
181C22×C79#46811yescharacteristic
182C22×C79#46911yescharacteristic
183C22×C79#47111yescharacteristic
184C22×C79#47311yescharacteristic
185C2×D79—42no—
186C2×D79—42no—
187C2×D79—42no—
188C2×D79—81no—
189C2×D79—81no—
190C2×D79—81no—
191C2×D79—81no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—82no—
195C4×C79—161no—
196C4×C79#41711yescharacteristic
197C4×C79#41811yescharacteristic
198C4×C79C4×SD1611yescharacteristic
199C4×C79#44011yescharacteristic
200C4×C79#45011yescharacteristic
201C4×C79#45211yescharacteristic
202C4×C79#46811yescharacteristic
203C4×C79#46911yescharacteristic
204Dic79—42no—
205#9225—791nomaximal, Sylow
206C2×C79#169111yescharacteristic
207C2×C79#169311yescharacteristic
208C2×C79#169911yescharacteristic
209C2×C79#170111yescharacteristic
210C2×C79#187711yescharacteristic
211C2×C79#189511yescharacteristic
212C2×C79#190811yescharacteristic
213D79—82no—
214C2×#461—791no—
215C2×#436—791no—
216#1990D7911yescharacteristic, complemented
217#1802—791no—
218C2×#332—791no—
219#1296—791no—
220#1277—791no—
221C79#922511yescharacteristic, complemented, Sylow
222C22×SD16—1581no—
223C22×M16—1581no—
224C2×C4⋊Q8C2×D7911yescharacteristic
225C2×C4×D4—791no—
226#461—794no—
227#461—1582no—
228#436—1582no—
229#436—1582no—
230C2×C4⋊C8—791no—
231C2×C4⋊C8—791no—
232C2×C4⋊C8—791no—
233C2×#103—1581no—
234C2×#102—1581no—
235C2×C22⋊C8—1581no—
236C2×C4×C8C2×D7911yescharacteristic
237#384C2×D7911yescharacteristic
238#340—21no—
239#340—21no—
240#340—1581no—
241#332—1582no—
242C22×Q8—21no—
243C22×D4—791no—
244C23×C4—1581no—
245C2×SD16—3162no—
246C2×SD16—3162no—
247C2×SD16—1584no—
248C2×M16—3162no—
249C2×M16—3162no—
250C2×M16—1584no—
251C22×C8—1581no—
252C22×C8—1581no—
253C22×C8—1581no—
254C22×C8C22⋊D7911yescharacteristic
255C22×C8C22⋊D7911yescharacteristic
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C4×D7912yes—
260C4×D4—792no—
261C4×D4—792no—
262C4×D4—1581no—
263C4×D4—1581no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—791no—
270C2×C22⋊C4—1581no—
271C2×C42C22×D7911yescharacteristic
272C4⋊C8—792no—
273C4⋊C8—792no—
274C4⋊C8—1581no—
275C4⋊C8—1581no—
276C4⋊C8—1581no—
277C4⋊C8—1581no—
278#103—1584no—
279#102—3162no—
280C22⋊C8—1584no—
281C4×C8—21no—
282C4×C8—21no—
283C24—1581no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—792no—
291C2×D4—1581no—
292C2×D4—1581no—
293C2×D4—1581no—
294C2×D4—1581no—
295C2×D4—1581no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—3161no—
299C22×C4—3161no—
300C22×C4—3161no—
301C22×C4—3161no—
302C22×C4—1581no—
303C22×C4—1582no—
304C22×C4—1582no—
305C22×C4D4×D7911yescharacteristic
306C22×C4C2×C22⋊D7911yescharacteristic
307C22×C4#1192962311yescharacteristic
308SD16—3168no—
309M16—3168no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—3161no—
316C2×C8—3161no—
317C2×C8—3161no—
318C2×C8—3161no—
319C2×C8—3161no—
320C2×C8—3161no—
321C2×C8—3161no—
322C2×C8—1582no—
323C2×C8—1582no—
324C2×C8#1192961112yescomplemented
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—792no—
335C4⋊C4—1581no—
336C22⋊C4—1582no—
337C22⋊C4—1582no—
338C42C2×C4×D7911yescharacteristic
339C42D4×D7911yescharacteristic
340C42D4×D7911yescharacteristic
341C42#1192962411yescharacteristic
342C23—3161no—
343C23—3161no—
344C23—3161no—
345C23—3161no—
346C23—3161no—
347C23—3161no—
348C23—1582no—
349C23#1457913211yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354D4—3161no—
355D4—3161no—
356D4—1582no—
357D4—1582no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—3161no—
367C2×C4—3161no—
368C2×C4—3162no—
369C2×C4—3162no—
370C2×C4—3162no—
371C2×C4—3162no—
372C2×C4—1582no—
373C2×C4C22⋊C4×D7911yescharacteristic
374C2×C4(C22⋊D79)⋊C411yescharacteristic
375C2×C4C4×C22⋊D7911yescharacteristic
376C2×C4#1457912911yescharacteristic
377C2×C4#1457913411yescharacteristic
378C2×C4#1457914311yescharacteristic
379C2×C4C2×#1192961111yescharacteristic
380C2×C4#1457918111yescharacteristic
381C2×C4#1457918511yescharacteristic
382C2×C4#1457918511yescharacteristic
383C2×C4#1457918611yescharacteristic
384C2×C4#1457918811yescharacteristic
385C2×C4#1457919011yescharacteristic
386C2×C4#1457919211yescharacteristic
387C8—22no—
388C8—41no—
389C8—3162no—
390C8—3162no—
391C8—6321no—
392C22—3162no—
393C22—3162no—
394C22—3162no—
395C22—6321no—
396C22—6321no—
397C22—6321no—
398C22—6321no—
399C22#1909843911yescharacteristic
400C22#1909876511yescharacteristic
401C22#1909877511yescharacteristic
402C22#1909880611yescharacteristic
403C22#1909881311yescharacteristic
404C22#1909882311yescharacteristic
405C22#1909883511yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—161no—
410C4—3162no—
411C4#1909856111yescharacteristic
412C4#1909856611yescharacteristic
413C4C4×#1192961111yescharacteristic
414C4#1909866111yescharacteristic
415C4#1909871311yescharacteristic
416C4#1909872311yescharacteristic
417C4#1909880811yescharacteristic
418C4#1909881511yescharacteristic
419C2—6322no—
420C2#2376289611yescharacteristic
421C2#2376291011yescharacteristic
422C2#2376294411yescharacteristic
423C2#2376295811yescharacteristic
424C2#2376373711yescharacteristic
425C2#2376383511yescharacteristic
426C2#2376391811yescharacteristic
subgroupG/H
centerC23#14579132
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1990C2
socleC23×C79#121
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1990D79
hypocenterC79#9225
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#19098309C2semidirect
5. #23761283C2semidirect
6. #23761235C2semidirect
7. C79×#1990C2semidirect
12. #19098765C4semidirect
114. C2×C8×C79SD16semidirect
216. #1990D79semidirect
221. C79#9225semidirect
324. C2×C8#11929611semidirect

Maximal subgroups

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

subgroupindex
C2×#190987652
C2×#190986422
#237633972
C2×#190983092
#237612832
#237612352
C79×#19902
#922579