Finite Group Databasebeta

← #34204244

Subgroups — #34204244

Order 20224.

512 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 103 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#23766065C211yesmaximal, characteristic
2C2×#19098352C212yesmaximal
3#23763392C211yesmaximal, characteristic
4#23761517C212yesmaximal
5C79×#1799C211yesmaximal, characteristic
6C2×#14579142C2211yescharacteristic
7C22×Dic79⋊C4C411yescharacteristic, complemented
8#19098587—21no—
9C2×#14579078C2212yes—
10#19098458—21no—
11#19098457—21no—
12#19098447—21no—
13#19098443C2211yescharacteristic
14#19098436—21no—
15#19098435C411yescharacteristic, complemented
16#19098424C411yescharacteristic, complemented
17C4×C22⋊Dic79C411yescharacteristic, complemented
18#19098352—24no—
19#19098332—21no—
20C2×C4×D4×C79C2211yescharacteristic
21C2×#99×C79C2212yes—
22#380×C79—21no—
23#342×C79—21no—
24C23×Dic79C2×C411yescharacteristic
25C23×Dic79C2×C411yescharacteristic
26C2×#11929608—22no—
27#14579142—21no—
28#14579142—21no—
29#14579142D412yes—
30#14579142Q812yes—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×Dic79⋊C4—41no—
36C2×Dic79⋊C4—41no—
37C2×C22⋊Dic79—21no—
38C2×C22⋊Dic79—21no—
39C2×C22⋊Dic79C2×C411yescharacteristic
40C2×C22⋊Dic79C2×C411yescharacteristic
41C2×C4×Dic79—21no—
42C4⋊(C79⋊C8)—41no—
43C4⋊(C79⋊C8)—41no—
44#14579078—22no—
45#14579078—22no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C79⋊#95—21no—
51C22×D4×C79C2311yescharacteristic
52C23×C4×C79C2×C411yescharacteristic
53C23×C4×C79D411yescharacteristic
54C4×D4×C79—41no—
55C4×D4×C79—41no—
56C2×C4⋊C4×C79—41no—
57C2×C4⋊C4×C79D411yescharacteristic
58C2×C22⋊C4×C79—22no—
59C2×C22⋊C4×C79C2×C411yescharacteristic
60C2×C22⋊C4×C79D411yescharacteristic
61C2×C42×C79D411yescharacteristic
62#99×C79—22no—
63#99×C79—22no—
64#95×C79—41no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—41no—
71C22×Dic79—41no—
72C22×Dic79—41no—
73C22×Dic79—81no—
74C22×Dic79C4⋊C412yescomplemented
75#11929608—42no—
76#11929608—42no—
77C2×C79⋊C8—42no—
78Dic79⋊C4—22no—
79Dic79⋊C4—22no—
80Dic79⋊C4—42no—
81Dic79⋊C4—41no—
82Dic79⋊C4—41no—
83Dic79⋊C4—81no—
84C22⋊Dic79—21no—
85C22⋊Dic79—22no—
86C22⋊Dic79—22no—
87C22⋊Dic79—41no—
88C22⋊Dic79—81no—
89C22⋊Dic79C4⋊C412yescomplemented
90C4×Dic79—81no—
91C24×C79C22×C411yescharacteristic
92C24×C79C2×D411yescharacteristic
93C2×D4×C79—41no—
94C2×D4×C79—41no—
95C2×D4×C79C2×D411yescharacteristic
96C2×D4×C79C2×Q811yescharacteristic
97C2×D4×C79#4111yescharacteristic
98C2×D4×C79#4111yescharacteristic
99C22×C4×C79—21no—
100C22×C4×C79—21no—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79—42no—
104C22×C4×C79—81no—
105C22×C4×C79—81no—
106C22×C4×C79C22⋊C411yescharacteristic
107C22×C4×C79C22⋊C411yescharacteristic
108C22×C4×C79C22⋊C411yescharacteristic
109C22×C4×C79C22⋊C411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C4⋊C4×C79—41no—
112C4⋊C4×C79—82no—
113C22⋊C4×C79—41no—
114C22⋊C4×C79—41no—
115C22⋊C4×C79—42no—
116C22⋊C4×C79—42no—
117C42×C79—21no—
118C42×C79—21no—
119C23×C79—21no—
120C23×C79—21no—
121C23×C79—21no—
122C23×C79—21no—
123C23×C79—41no—
124C23×C79—41no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79C2×C22⋊C411yescharacteristic
128C23×C79C2×C4⋊C411yescharacteristic
129C23×C79#11711yescharacteristic
130C23×C79#12111yescharacteristic
131C23×C79C22⋊Q811yescharacteristic
132D4×C79—41no—
133D4×C79—41no—
134D4×C79—41no—
135D4×C79—41no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—41no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—81no—
147C2×C4×C79—81no—
148C2×C4×C79—81no—
149C2×C4×C79—82no—
150C2×C4×C79—82no—
151C2×C4×C79—82no—
152C2×C4×C79#9911yescharacteristic
153C2×C4×C79#9911yescharacteristic
154C2×C4×C79#12111yescharacteristic
155C2×C4×C79C22⋊Q811yescharacteristic
156C2×Dic79—22no—
157C2×Dic79—22no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—42no—
161C2×Dic79—41no—
162C2×Dic79—41no—
163C2×Dic79—42no—
164C2×Dic79—41no—
165C2×Dic79—41no—
166C2×Dic79—41no—
167C2×Dic79—41no—
168C2×Dic79—41no—
169C2×Dic79—41no—
170C2×Dic79—82no—
171C79⋊C8—82no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—21no—
176C22×C79—21no—
177C22×C79—21no—
178C22×C79—41no—
179C22×C79—41no—
180C22×C79—41no—
181C22×C79—41no—
182C22×C79—42no—
183C22×C79—41no—
184C22×C79—41no—
185C22×C79—41no—
186C22×C79—41no—
187C22×C79—81no—
188C22×C79#38011yescharacteristic
189C22×C79C2×#9911yescharacteristic
190C22×C79#41011yescharacteristic
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—21no—
195C4×C79—41no—
196C4×C79—41no—
197C4×C79—41no—
198C4×C79—162no—
199Dic79—42no—
200Dic79—42no—
201Dic79—81no—
202Dic79—81no—
203#12796—791nomaximal, Sylow
204C2×C79—21no—
205C2×C79—21no—
206C2×C79—41no—
207C2×C79—41no—
208C2×C79—41no—
209C2×C79—41no—
210C2×C79#179911yescharacteristic
211C2×C79#212111yescharacteristic
212C2×C79#212311yescharacteristic
213#2322—791no—
214C2×#352—792no—
215#1800—791no—
216#1799D7911yescharacteristic
217#1390—792no—
218C79#1279611yescharacteristic, complemented, Sylow
219C2×#123—791no—
220C2×C4×D4C2×D7911yescharacteristic
221C22×C4⋊C4—791no—
222#423—1581no—
223C2×#100—792no—
224C2×#99C2×D7912yes—
225#389—1581no—
226#388—1581no—
227#387—1581no—
228#386—791no—
229#385—791no—
230#385—1581no—
231#381—791no—
232#380—21no—
233C4×C22⋊C4—791no—
234#352—1584no—
235#343—1581no—
236#342—21no—
237C22×D4C22×D7911yescharacteristic
238C23×C4—791no—
239C23×C4—791no—
240C23×C4C4×D7911yescharacteristic
241C23×C4C22⋊D7911yescharacteristic
242C2×M16—1582no—
243#123—792no—
244#123—792no—
245#123—1581no—
246#123—1581no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—3161no—
251C2×C4⋊C4—3161no—
252C2×C4⋊C4—1581no—
253C2×C4⋊C4—1581no—
254C2×C4⋊C4—1581no—
255C2×C4⋊C4—1581no—
256C2×C4⋊C4D31611yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—791no—
259C2×C22⋊C4—791no—
260C2×C22⋊C4—1581no—
261C2×C22⋊C4—1581no—
262C2×C22⋊C4C4×D7911yescharacteristic
263C2×C22⋊C4C22⋊D7911yescharacteristic
264C2×C42—1581no—
265C2×C42D31611yescharacteristic
266C4⋊C8—3161no—
267C4⋊C8—3161no—
268#100—1582no—
269#100—1582no—
270#99—22no—
271#99—22no—
272#95—41no—
273#95—1581no—
274#95—1581no—
275#95—1581no—
276#95—1581no—
277#95—1581no—
278C24C2×C4×D7911yescharacteristic
279C24C2×C22⋊D7911yescharacteristic
280C2×D4—41no—
281C2×D4—41no—
282C2×D4D4×D7911yescharacteristic
283C2×D4Q8×D7911yescharacteristic
284C2×D4#1192962311yescharacteristic
285C2×D4#1192962511yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—3161no—
294C22×C4—3161no—
295C22×C4—3161no—
296C22×C4—792no—
297C22×C4—6321no—
298C22×C4—1581no—
299C22×C4—1581no—
300C22×C4—1581no—
301C22×C4—1581no—
302C22×C4—1581no—
303C22×C4(C2×D79)⋊C411yescharacteristic
304C22×C4(C2×D79)⋊C411yescharacteristic
305C22×C4(C2×D79)⋊C411yescharacteristic
306C22×C4(C2×D79)⋊C411yescharacteristic
307C22×C4C2×D31611yescharacteristic
308M16—3162no—
309M16—3162no—
310C2×C8—3162no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C4⋊C4—3162no—
314C4⋊C4—3161no—
315C4⋊C4—3161no—
316C4⋊C4—6321no—
317C4⋊C4—1582no—
318C4⋊C4—1582no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—3161no—
324C22⋊C4—792no—
325C22⋊C4—6321no—
326C22⋊C4—1581no—
327C22⋊C4—1582no—
328C22⋊C4—1582no—
329C42—21no—
330C42—21no—
331C42—6321no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23C2×(C2×D79)⋊C411yescharacteristic
341C23C4⋊C4×D7911yescharacteristic
342C23#1457911911yescharacteristic
343C23#1457913211yescharacteristic
344C23#1457913711yescharacteristic
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—3161no—
366C2×C4—3162no—
367C2×C4—3161no—
368C2×C4—3161no—
369C2×C4—3161no—
370C2×C4—3162no—
371C2×C4—3161no—
372C2×C4—3161no—
373C2×C4—3161no—
374C2×C4—3161no—
375C2×C4—3161no—
376C2×C4—3161no—
377C2×C4—6322no—
378C2×C4—1582no—
379C2×C4—1582no—
380C2×C4#1457907511yescharacteristic
381C2×C4#1457907511yescharacteristic
382C2×C4#1457913611yescharacteristic
383C2×C4#1457914111yescharacteristic
384C8—6322no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22#1909842011yescharacteristic
402C22C2×#1457907511yescharacteristic
403C22#1909853611yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—162no—
412C4—3162no—
413C4—3162no—
414C4—6321no—
415C4—6321no—
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2—41no—
421C2—41no—
422C2#2376338511yescharacteristic
423C2#2376504411yescharacteristic
424C2#2376505311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC79×#1799C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenter#1799D79
hypocenterC79#12796
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
7. C22×Dic79⋊C4C4semidirect
15. #19098435C4semidirect
16. #19098424C4semidirect
17. C4×C22⋊Dic79C4semidirect
74. C22×Dic79C4⋊C4semidirect
89. C22⋊Dic79C4⋊C4semidirect
218. C79#12796semidirect

Maximal subgroups

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

subgroupindex
#237660652
C2×#190983522
#237633922
#237615172
C79×#17992
#1279679