Finite Group Databasebeta

← #34184176

Subgroups — #34184176

Order 20224.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773197C211yesmaximal, characteristic
2#23761963C211yesmaximal, characteristic, complemented
3#23761898C211yesmaximal, characteristic, complemented
4#23761791C211yesmaximal, characteristic
5#23761303C211yesmaximal, characteristic, complemented
6#23761258C211yesmaximal, characteristic, complemented
7C79×#2349C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098863—22no—
10#19098853C412yescomplemented
11#19098843C412yescomplemented
12#19098616C2211yescharacteristic
13#19098616C2211yescharacteristic
14#19098603—21no—
15#19098587C2211yescharacteristic
16#19098568—21no—
17#19098299—22no—
18#19098299—21no—
19#19098299—21no—
20D4⋊(C79⋊C8)—22no—
21C2×C4×D4×C79—21no—
22C2×C4×D4×C79C2211yescharacteristic
23#389×C79—21no—
24#385×C79—22no—
25#382×C79C2211yescharacteristic
26#380×C79—21no—
27C4×C4⋊C4×C79C2211yescharacteristic
28C2×#95×C79—21no—
29C2×#11929608—21no—
30#14579161C2×C411yescharacteristic, complemented
31#14579161C2×C411yescharacteristic, complemented
32#14579153—21no—
33#14579140—21no—
34#14579140—21no—
35C4×Dic158—21no—
36C4×Dic158—21no—
37#14579120—21no—
38#14579094—22no—
39#14579092—22no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)C2×C412yescomplemented
44#14579082—22no—
45#14579082—22no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C22×D4×C79D411yescharacteristic
51C23×C4×C79—21no—
52C23×C4×C79—22no—
53C4×D4×C79—21no—
54C4×D4×C79—21no—
55C4×D4×C79—42no—
56C4×D4×C79—42no—
57C4×D4×C79C2×C412yes—
58C4×D4×C79D412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79—22no—
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C22⋊C4×C79—21no—
64C2×C22⋊C4×C79—22no—
65C2×C42×C79—21no—
66C2×C42×C79C2311yescharacteristic
67#95×C79—22no—
68#95×C79—21no—
69#95×C79D411yescharacteristic
70#95×C79D411yescharacteristic
71#95×C79D412yes—
72C2×Dic158—21no—
73C2×Dic158—21no—
74#11929608—41no—
75#11929608—41no—
76C2×C79⋊C8—22no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C4⋊Dic79—21no—
80C4⋊Dic79—21no—
81C4⋊Dic79—21no—
82C4⋊Dic79—21no—
83Dic79⋊C4—41no—
84Dic79⋊C4—41no—
85C22⋊Dic79—41no—
86C4×Dic79—41no—
87C24×C79—21no—
88C2×D4×C79—21no—
89C2×D4×C79—21no—
90C2×D4×C79—21no—
91C2×D4×C79C22⋊C412yes—
92C2×D4×C79SD1612yes—
93C2×D4×C79Q1612yes—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—22no—
97C22×C4×C79—22no—
98C22×C4×C79—22no—
99C22×C4×C79—21no—
100C22×C4×C79—44no—
101C22×C4×C79—44no—
102C22×C4×C79—82no—
103C22×C4×C79—82no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C4⋊C4×C79—21no—
108C4⋊C4×C79—42no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C4⋊C4×C79—82no—
112C4⋊C4×C79C22⋊C412yes—
113C22⋊C4×C79—22no—
114C22⋊C4×C79—22no—
115C22⋊C4×C79—44no—
116C42×C79—41no—
117C42×C79—41no—
118C42×C79C22×C411yescharacteristic
119C42×C79C2×D411yescharacteristic
120C42×C79#4111yescharacteristic
121C42×C79#4111yescharacteristic
122C23×C79—22no—
123C23×C79—22no—
124C23×C79—22no—
125C23×C79—42no—
126C23×C79—42no—
127C23×C79#12011yescharacteristic
128D4×C79—22no—
129D4×C79—22no—
130D4×C79—22no—
131D4×C79#10314yes—
132C2×C4×C79—22no—
133C2×C4×C79—22no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—42no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79—82no—
146C2×C4×C79—82no—
147C2×C4×C79—88no—
148C2×C4×C79C2×C22⋊C411yescharacteristic
149C2×C4×C79C4×D411yescharacteristic
150C2×C4×C79C4×D411yescharacteristic
151C2×C4×C79#12111yescharacteristic
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12311yescharacteristic
154C2×C4×C79C2×SD1611yescharacteristic
155C2×C4×C79C2×Q1611yescharacteristic
156C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C79#13711yescharacteristic
158C2×Dic79—41no—
159C2×Dic79—41no—
160Dic158—81no—
161Dic158—81no—
162C79⋊C8—42no—
163C79⋊C8—81no—
164C79⋊C8—81no—
165C22×C79—21no—
166C22×C79—21no—
167C22×C79—24no—
168C22×C79—44no—
169C22×C79—44no—
170C22×C79—42no—
171C22×C79—42no—
172C22×C79#38611yescharacteristic
173C22×C79#45011yescharacteristic
174C22×C79#45111yescharacteristic
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—42no—
180C4×C79—42no—
181C4×C79—81no—
182C4×C79—82no—
183C4×C79C2×#10311yescharacteristic
184C4×C79#41711yescharacteristic
185C4×C79#48311yescharacteristic
186C4×C79#48411yescharacteristic
187Dic79—81no—
188Dic79—81no—
189#9175—791nomaximal, Sylow
190C2×C79—21no—
191C2×C79—21no—
192C2×C79—44no—
193C2×C79#187211yescharacteristic
194C2×C79#190411yescharacteristic
195C2×C79#190611yescharacteristic
196#3077—791no—
197#2349D7911yescharacteristic
198#1518—791no—
199#1503—791no—
200#1478—791no—
201#1303—791no—
202#1286—791no—
203C79#917511yescharacteristic, complemented, Sylow
204#533—791no—
205C2×C4×D4—21no—
206C2×C4×D4C2×D7911yescharacteristic
207#478—1582no—
208#476—792no—
209#474—792no—
210#431—791no—
211#431—791no—
212#428—1581no—
213#423—791no—
214#418—1581no—
215#389—21no—
216#385—22no—
217#382C2×D7911yescharacteristic
218#380—21no—
219C4×C4⋊C4C2×D7911yescharacteristic
220C2×#95—21no—
221#328—1581no—
222#328—1581no—
223#328—1582no—
224D4⋊C8—1582no—
225C22×D4C22⋊D7911yescharacteristic
226C23×C4—21no—
227C23×C4—22no—
228C2×M16—1581no—
229C4⋊Q8—791no—
230C4⋊Q8—791no—
231#125—1581no—
232C22⋊Q8—1581no—
233C22⋊Q8—1581no—
234C4×Q8—1581no—
235C4×Q8—1581no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4C4×D7912yes—
241C4×D4D31612yes—
242#117—1581no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C22⋊D7911yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C42—21no—
250C2×C42C22×D7911yescharacteristic
251#107—1582no—
252#106—1582no—
253C4⋊C8—792no—
254C4⋊C8—1581no—
255C4⋊C8—1581no—
256C4⋊C8—1581no—
257#102—1582no—
258#102—1582no—
259C22⋊C8—1581no—
260C22⋊C8—1581no—
261C4×C8—1581no—
262C4×C8—1581no—
263#95—22no—
264#95—21no—
265#95C22⋊D7911yescharacteristic
266#95C22⋊D7911yescharacteristic
267#95C22⋊D7912yes—
268C24—21no—
269C2×Q8—1581no—
270C2×Q8—1581no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4(C2×D79)⋊C412yes—
275C2×D4#1192961212yes—
276C2×D4#1192961412yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4C2×C22⋊D7911yescharacteristic
289C22×C4C2×C22⋊D7911yescharacteristic
290M16—3161no—
291M16—3161no—
292C2×C8—3161no—
293C2×C8—3161no—
294C2×C8—1582no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—3161no—
301C4⋊C4—3161no—
302C4⋊C4—1581no—
303C4⋊C4—1581no—
304C4⋊C4—1581no—
305C4⋊C4—1581no—
306C4⋊C4(C2×D79)⋊C412yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—44no—
310C22⋊C4—3161no—
311C42—41no—
312C42—41no—
313C42—3161no—
314C42C2×C4×D7911yescharacteristic
315C42C2×D31611yescharacteristic
316C42#1192962411yescharacteristic
317C42#1192962411yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1457913011yescharacteristic
324Q8—6321no—
325Q8—6321no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1457908414yes—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—88no—
346C2×C4—3161no—
347C2×C4—3161no—
348C2×C4C2×(C2×D79)⋊C411yescharacteristic
349C2×C4C4×C22⋊D7911yescharacteristic
350C2×C4C4×C22⋊D7911yescharacteristic
351C2×C4#1457913511yescharacteristic
352C2×C4#1457913511yescharacteristic
353C2×C4#1457914411yescharacteristic
354C2×C4C2×#1192961211yescharacteristic
355C2×C4C2×#1192961411yescharacteristic
356C2×C4#1457918711yescharacteristic
357C2×C4#1457919211yescharacteristic
358C8—3162no—
359C8—6321no—
360C8—6321no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1909844211yescharacteristic
369C22#1909871411yescharacteristic
370C22#1909871911yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4—81no—
379C4—6321no—
380C4—6321no—
381C4C2×#1457908411yescharacteristic
382C4#1909856211yescharacteristic
383C4#1909889311yescharacteristic
384C4#1909889811yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2376370711yescharacteristic
389C2#2376388711yescharacteristic
390C2#2376390111yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2349C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#2349D79
hypocenterC79#9175
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #23761963C2semidirect
3. #23761898C2semidirect
5. #23761303C2semidirect
6. #23761258C2semidirect
10. #19098853C4semidirect
11. #19098843C4semidirect
30. #14579161C2×C4semidirect
31. #14579161C2×C4semidirect
43. C4⋊(C79⋊C8)C2×C4semidirect
203. C79#9175semidirect

Maximal subgroups

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

subgroupindex
#237731972
#237619632
#237618982
#237617912
#237613032
#237612582
C79×#23492
#917579