Finite Group Databasebeta

← #34181172

Subgroups — #34181172

Order 20224.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098634C211yesmaximal, characteristic
2#23763425C211yesmaximal, characteristic, complemented
3#23763385C212yesmaximal
4#23761267C212yesmaximal, complemented
5C79×#1808C211yesmaximal, characteristic, complemented
6C22×#11929607C411yescharacteristic, complemented
7C2×C4×C22⋊D79C2211yescharacteristic
8#19098634—21no—
9#19098634—21no—
10#19098634—21no—
11#19098634—21no—
12C2×#14579093—21no—
13C2×Dic79⋊C8C411yescharacteristic, complemented
14C2×#14579075—22no—
15C2×C22⋊(C79⋊C8)C2211yescharacteristic
16C2×(C2×D79)⋊C8C411yescharacteristic, complemented
17C2×#14579071C411yescharacteristic, complemented
18#19098423C2212yes—
19#19098329—22no—
20#19098326—21no—
21C23×C8×C79C2211yescharacteristic
22C2×#107×C79—21no—
23#380×C79C2212yes—
24#340×C79—21no—
25C22×C22⋊D79D411yescharacteristic
26C22×C4×D79C2×C411yescharacteristic
27C2×#11929607—21no—
28C2×#11929607—21no—
29C2×#11929607—42no—
30C22×C79⋊C8C2×C411yescharacteristic
31C22×C79⋊C8C2×C411yescharacteristic
32C4×C22⋊D79—21no—
33C4×C22⋊D79—21no—
34C4×C22⋊D79—21no—
35C4×C22⋊D79—21no—
36C2×C4⋊Dic79—22no—
37C2×Dic79⋊C4Q811yescharacteristic
38C2×C22⋊Dic79—22no—
39C2×C22⋊Dic79D411yescharacteristic
40C2×(C2×D79)⋊C4Q811yescharacteristic
41C2×C4×Dic79C2×C411yescharacteristic
42#14579093—42no—
43Dic79⋊C8—21no—
44Dic79⋊C8—21no—
45#14579075—44no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48(C2×D79)⋊C8—21no—
49(C2×D79)⋊C8—21no—
50#14579071—21no—
51#14579071—21no—
52C79⋊#95—22no—
53C23×C4×C79C2311yescharacteristic
54C22×C8×C79—42no—
55C22×C8×C79C2×C411yescharacteristic
56C22×C8×C79C2×C411yescharacteristic
57C22×C8×C79D411yescharacteristic
58C22×C8×C79D411yescharacteristic
59C22×C8×C79D411yescharacteristic
60C22×C8×C79D411yescharacteristic
61C2×C4⋊C4×C79—22no—
62C2×C22⋊C4×C79—22no—
63#107×C79—42no—
64#95×C79—22no—
65C23×D79C4⋊C411yescharacteristic
66C2×C22⋊D79—21no—
67C2×C22⋊D79—21no—
68C2×C22⋊D79—22no—
69C2×C22⋊D79—22no—
70C22×Dic79—42no—
71C22×Dic79C4⋊C411yescharacteristic
72C22×Dic79C4⋊C411yescharacteristic
73C22×Dic79C4⋊C411yescharacteristic
74C2×C4×D79—21no—
75C2×C4×D79—21no—
76C2×C4×D79—41no—
77C2×C4×D79—41no—
78#11929607—42no—
79#11929607—42no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C4⋊Dic79—42no—
85C4⋊Dic79—42no—
86Dic79⋊C4—21no—
87Dic79⋊C4—21no—
88C22⋊Dic79—21no—
89C22⋊Dic79—21no—
90C22⋊Dic79—42no—
91C22⋊Dic79—42no—
92(C2×D79)⋊C4—21no—
93(C2×D79)⋊C4—21no—
94C4×Dic79—21no—
95C4×Dic79—21no—
96C24×C79C2×D411yescharacteristic
97C22×C4×C79—41no—
98C22×C4×C79—41no—
99C22×C4×C79—42no—
100C22×C4×C79C22×C411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×Q811yescharacteristic
104C22×C4×C79#4111yescharacteristic
105C22×C4×C79#4111yescharacteristic
106C2×C8×C79—22no—
107C2×C8×C79—22no—
108C2×C8×C79—22no—
109C2×C8×C79—41no—
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C2×C8×C79C22⋊C411yescharacteristic
113C2×C8×C79C22⋊C411yescharacteristic
114C2×C8×C79C22⋊C411yescharacteristic
115C2×C8×C79C22⋊C411yescharacteristic
116C4⋊C4×C79—42no—
117C4⋊C4×C79—42no—
118C22⋊C4×C79—42no—
119C22⋊C4×C79—42no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79C2×C4⋊C411yescharacteristic
123C23×C79#12111yescharacteristic
124C23×C79#12111yescharacteristic
125C23×C79AGL(1,ℤ/8)12yes—
126C23×C79#13712yes—
127C22×D79—21no—
128C22×D79—22no—
129C22×D79—41no—
130C22×D79—41no—
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—82no—
141C2×C4×C79—82no—
142C2×C4×C79—82no—
143C2×C4×C79C2×C22⋊C411yescharacteristic
144C2×C4×C79#11711yescharacteristic
145C2×C4×C79C22⋊Q811yescharacteristic
146C2×C4×C79C22⋊Q811yescharacteristic
147C22⋊D79—42no—
148C22⋊D79—41no—
149C22⋊D79—41no—
150C2×Dic79—21no—
151C2×Dic79—21no—
152C2×Dic79—21no—
153C2×Dic79—22no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—82no—
157C2×Dic79—82no—
158C2×Dic79—82no—
159C4×D79—42no—
160C4×D79—41no—
161C4×D79—41no—
162C8×C79—21no—
163C8×C79—21no—
164C8×C79—21no—
165C8×C79#9912yescomplemented
166C79⋊C8—41no—
167C79⋊C8—41no—
168C22×C79—21no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—42no—
175C22×C79—42no—
176C22×C79#38011yescharacteristic
177C22×C79#42812yes—
178C22×C79#46812yes—
179C22×C79#46911yescharacteristic
180C22×C79#47011yescharacteristic
181C2×D79—41no—
182C2×D79—41no—
183C2×D79—41no—
184C2×D79—82no—
185C4×C79—21no—
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—162no—
189C4×C79C2×#9911yescharacteristic
190C4×C79#41011yescharacteristic
191Dic79—41no—
192Dic79—41no—
193Dic79—41no—
194Dic79—162no—
195#8610—791nomaximal, Sylow
196C2×C79—22no—
197C2×C79—21no—
198C2×C79—22no—
199C2×C79—21no—
200C2×C79#179911yescharacteristic
201C2×C79#180911yescharacteristic
202C2×C79#181211yescharacteristic
203D79—81no—
204C2×#435—791no—
205#1808D7911yescharacteristic, complemented
206#1807—791no—
207#1799—792no—
208#1289—792no—
209C79#861011yescharacteristic, complemented, Sylow
210C22×M16—791no—
211C23×C8C2×D7911yescharacteristic
212C2×C4×D4—791no—
213#435—1581no—
214#435—1581no—
215#435—1581no—
216#435—1581no—
217C2×#107—21no—
218C2×#106—1581no—
219C2×C4⋊C8—791no—
220C2×#99—1582no—
221C2×C22⋊C8—791no—
222C2×C22⋊C8—791no—
223C2×#97—791no—
224#380—792no—
225#380C2×D7912yes—
226#342—1582no—
227#340—21no—
228#340—1581no—
229C22×D4—791no—
230C23×C4—791no—
231C23×C4C22×D7911yescharacteristic
232C2×M16—3162no—
233C2×M16—1581no—
234C2×M16—1581no—
235C22×C8—42no—
236C22×C8—791no—
237C22×C8—791no—
238C22×C8C4×D7911yescharacteristic
239C22×C8C4×D7911yescharacteristic
240C22×C8D31611yescharacteristic
241C22×C8D31611yescharacteristic
242C22×C8C22⋊D7911yescharacteristic
243C22×C8C22⋊D7911yescharacteristic
244C4×D4—1581no—
245C4×D4—1581no—
246C4×D4—1581no—
247C4×D4—1581no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—791no—
250C2×C4⋊C4—1582no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—791no—
253C2×C22⋊C4—791no—
254C2×C22⋊C4—1582no—
255C2×C42—791no—
256#107—42no—
257#106—3162no—
258C4⋊C8—1581no—
259C4⋊C8—1581no—
260#99—3164no—
261C22⋊C8—1581no—
262C22⋊C8—1581no—
263C22⋊C8—1581no—
264C22⋊C8—1581no—
265#97—1581no—
266#97—1581no—
267#95—22no—
268#95—1582no—
269C24—791no—
270C24D4×D7911yescharacteristic
271C2×D4—1581no—
272C2×D4—1581no—
273C2×D4—1582no—
274C2×D4—1582no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—3161no—
279C22×C4—3161no—
280C22×C4—3162no—
281C22×C4—791no—
282C22×C4—791no—
283C22×C4—791no—
284C22×C4—1581no—
285C22×C4—1581no—
286C22×C4C2×C4×D7911yescharacteristic
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4C2×D31611yescharacteristic
289C22×C4Q8×D7911yescharacteristic
290C22×C4#1192962311yescharacteristic
291C22×C4#1192962511yescharacteristic
292M16—3162no—
293M16—3162no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—3161no—
301C2×C8—3161no—
302C2×C8—1581no—
303C2×C8—1581no—
304C2×C8(C2×D79)⋊C411yescharacteristic
305C2×C8(C2×D79)⋊C411yescharacteristic
306C2×C8(C2×D79)⋊C411yescharacteristic
307C2×C8(C2×D79)⋊C411yescharacteristic
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—3162no—
311C4⋊C4—3162no—
312C4⋊C4—1581no—
313C4⋊C4—1581no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—3162no—
317C22⋊C4—3162no—
318C22⋊C4—1581no—
319C22⋊C4—1581no—
320C22⋊C4—1581no—
321C22⋊C4—1581no—
322C42—1581no—
323C42—1581no—
324C23—41no—
325C23—41no—
326C23—3161no—
327C23—3161no—
328C23—1581no—
329C23—1582no—
330C23C4⋊C4×D7911yescharacteristic
331C23#1457913211yescharacteristic
332C23#1457913611yescharacteristic
333C23#1457918512yes—
334C23#1457919312yes—
335D4—3162no—
336D4—3161no—
337D4—3161no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—3162no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—3161no—
355C2×C4—6322no—
356C2×C4—6322no—
357C2×C4—6322no—
358C2×C4—1581no—
359C2×C4—1581no—
360C2×C4—1581no—
361C2×C4—1582no—
362C2×C4C2×(C2×D79)⋊C411yescharacteristic
363C2×C4#1457911911yescharacteristic
364C2×C4#1457913711yescharacteristic
365C2×C4#1457914111yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—3161no—
370C8—3161no—
371C8#1457907512yescomplemented
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—3161no—
381C22—3161no—
382C22—3161no—
383C22—6322no—
384C22#1909842011yescharacteristic
385C22#1909860412yes—
386C22#1909881012yes—
387C22#1909881311yescharacteristic
388C22#1909881811yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—162no—
393C4—3161no—
394C4—3161no—
395C4—3161no—
396C4—12642no—
397C4C2×#1457907511yescharacteristic
398C4#1909853611yescharacteristic
399C2—21no—
400C2—22no—
401C2—22no—
402C2—21no—
403C2—6321no—
404C2#2376338511yescharacteristic
405C2#2376343411yescharacteristic
406C2#2376344311yescharacteristic
subgroupG/H
centerC22#19098420
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1808C2
socleC22×C79#380
prime radicalC22×C22⋊D79D4
cosocleC23×C4×C79C23
hypercenter#1808D79
hypocenterC79#8610
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. #23763425C2semidirect
4. #23761267C2semidirect
5. C79×#1808C2semidirect
6. C22×#11929607C4semidirect
13. C2×Dic79⋊C8C4semidirect
16. C2×(C2×D79)⋊C8C4semidirect
17. C2×#14579071C4semidirect
165. C8×C79#99semidirect
205. #1808D79semidirect
209. C79#8610semidirect
371. C8#14579075semidirect

Maximal subgroups

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

subgroupindex
C2×#190986342
#237634252
#237633852
#237612672
C79×#18082
#861079