Finite Group Databasebeta

← #34205206

Subgroups — #34205206

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098800C211yesmaximal, characteristic
2C2×#107×D79C211yesmaximal, characteristic, complemented
3C2×#19098365C211yesmaximal, characteristic, complemented
4C2×#19098363C211yesmaximal, characteristic, complemented
5C2×(C2×D79)⋊C16C211yesmaximal, characteristic, complemented
6#23761491C211yesmaximal, characteristic, complemented
7C79×#1380C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9C22×C8×D79C2211yescharacteristic
10C2×#14579136—21no—
11C2×C4⋊C4×D79—21no—
12C22×C79⋊C16C2211yescharacteristic
13#19098800—22no—
14#19098800C414yescomplemented
15C2×#14579095C2211yescharacteristic
16C2×#14579094—21no—
17#107×D79—22no—
18#107×D79—22no—
19C2×#14579081—21no—
20#19098365—22no—
21#19098363—22no—
22(C2×D79)⋊C16—22no—
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25C2×#107×C79C2211yescharacteristic
26C22×D316—21no—
27C22×D316—41no—
28C22×C4×D79—41no—
29C22×C4×D79D411yescharacteristic
30C2×D632—21no—
31C2×D632—21no—
32C2×D632—21no—
33C2×D632—21no—
34C2×D632—21no—
35C2×D632C2×C412yescomplemented
36C22×C79⋊C8D411yescharacteristic
37C2×C8×D79—21no—
38C2×C8×D79—21no—
39C2×C8×D79—21no—
40C2×C8×D79—21no—
41C2×C8×D79C2×C412yes—
42C2×C8×D79D412yes—
43#14579136—24no—
44#14579136—42no—
45C2×C4⋊Dic79—21no—
46C2×Dic79⋊C4—41no—
47C4⋊C4×D79—24no—
48C4⋊C4×D79—42no—
49C2×(C2×D79)⋊C4—41no—
50C2×C79⋊C16—21no—
51C2×C79⋊C16—21no—
52C2×C79⋊C16—21no—
53#14579095—21no—
54#14579095D412yescomplemented
55#14579094—24no—
56#14579081—24no—
57C22×C8×C79C2311yescharacteristic
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C16×C79—21no—
61C2×C16×C79—21no—
62C2×C16×C79—21no—
63#107×C79—21no—
64#107×C79—21no—
65#107×C79C2×C412yescomplemented
66#107×C79D412yescomplemented
67C23×D79—41no—
68C23×D79D811yescharacteristic
69C2×D316—22no—
70C2×D316—44no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—82no—
77C2×D316—82no—
78C22×Dic79—41no—
79C22×Dic79D811yescharacteristic
80C2×C4×D79—21no—
81C2×C4×D79—21no—
82C2×C4×D79—21no—
83C2×C4×D79—21no—
84C2×C4×D79—44no—
85C2×C4×D79—82no—
86C2×C4×D79—82no—
87C2×C4×D79C22⋊C412yes—
88C2×C4×D79SD1612yes—
89D632—22no—
90D632—22no—
91D632—41no—
92D632—41no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8C22⋊C412yes—
96C8×D79—22no—
97C8×D79—22no—
98C8×D79—22no—
99C8×D79—22no—
100C4⋊Dic79—22no—
101C4⋊Dic79—41no—
102Dic79⋊C4—44no—
103(C2×D79)⋊C4—44no—
104C79⋊C16—41no—
105C22×C4×C79—41no—
106C22×C4×C79—41no—
107C22×C4×C79C2×D411yescharacteristic
108C2×C8×C79C22×C411yescharacteristic
109C2×C8×C79C2×D411yescharacteristic
110C2×C8×C79C2×D411yescharacteristic
111C2×C8×C79C2×D411yescharacteristic
112C2×C8×C79#4111yescharacteristic
113C2×C8×C79#4111yescharacteristic
114C4⋊C4×C79—22no—
115C4⋊C4×C79—22no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C16×C79—41no—
119C23×C79C2×D811yescharacteristic
120C22×D79—21no—
121C22×D79—21no—
122C22×D79—21no—
123C22×D79—21no—
124C22×D79—42no—
125C22×D79—81no—
126C22×D79—81no—
127C22×D79—81no—
128C22×D79—81no—
129C22×D79—81no—
130C22×D79—81no—
131C22×D79#10212yes—
132C22×D79D1612yes—
133C22×D79SD3212yes—
134C2×C4×C79—42no—
135C2×C4×C79—42no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79C2×C22⋊C411yescharacteristic
141C2×C4×C79#12011yescharacteristic
142C2×C4×C79#12311yescharacteristic
143C2×C4×C79C2×SD1611yescharacteristic
144C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C79#13711yescharacteristic
146C2×Dic79—21no—
147C2×Dic79—21no—
148C2×Dic79—42no—
149C2×Dic79—81no—
150C2×Dic79—81no—
151C2×Dic79#10212yes—
152D316—42no—
153D316—42no—
154D316—88no—
155D316—81no—
156D316—81no—
157C4×D79—22no—
158C4×D79—22no—
159C4×D79—22no—
160C4×D79—22no—
161C4×D79—88no—
162C8×C79C4×D411yescharacteristic
163C8×C79C4×D411yescharacteristic
164C8×C79#12111yescharacteristic
165C8×C79#12111yescharacteristic
166C79⋊C8—22no—
167C22×C79C2×#10211yescharacteristic
168C22×C79#44711yescharacteristic
169C22×C79#48011yescharacteristic
170C22×C79C2×D1611yescharacteristic
171C22×C79C2×SD3211yescharacteristic
172C22×C79#50911yescharacteristic
173C22×C79#51011yescharacteristic
174C2×D79—22no—
175C2×D79—22no—
176C2×D79—22no—
177C2×D79—22no—
178C2×D79—22no—
179C2×D79—22no—
180C2×D79—82no—
181C2×D79—82no—
182C2×D79—82no—
183C2×D79—161no—
184C2×D79—161no—
185C2×D79—161no—
186C2×D79—161no—
187C2×D79#35714yescomplemented
188C4×C79—82no—
189C4×C79—82no—
190C4×C79#38611yescharacteristic
191C4×C79#41711yescharacteristic
192C4×C79#44811yescharacteristic
193C4×C79#48111yescharacteristic
194Dic79—22no—
195Dic79—82no—
196#12991—791nomaximal, Sylow
197C2×C79#187411yescharacteristic
198C2×C79C2×#35711yescharacteristic
199C2×C79#213911yescharacteristic
200C2×C79#218411yescharacteristic
201C2×C79#218511yescharacteristic
202C2×C79#223211yescharacteristic
203C2×C79#223311yescharacteristic
204D79—24no—
205D79—162no—
206C2×#466—791no—
207C22×#107—791no—
208C2×#357—791no—
209C2×#357—791no—
210C2×C22⋊C16—791no—
211#1380—791no—
212#1380D7911yescharacteristic, complemented
213C79#1299111yescharacteristic, complemented, Sylow
214C22×D8—791no—
215C23×C8—791no—
216C2×#121—1581no—
217C22×C4⋊C4—1581no—
218C22×C16—791no—
219C22×C16C2×D7911yescharacteristic
220#466—794no—
221#466—1582no—
222C2×#107—791no—
223C2×#107—1582no—
224C2×#107—1581no—
225C2×#107—1582no—
226C2×#107C2×D7911yescharacteristic
227C2×#107C2×D7911yescharacteristic
228C2×#102—1581no—
229#357—1582no—
230#357—1582no—
231C22⋊C16—1582no—
232C22×D4—3161no—
233C22×D4—1581no—
234C23×C4—3161no—
235C23×C4—791no—
236C2×D8—792no—
237C2×D8—1581no—
238C2×D8—1581no—
239C2×D8—1581no—
240C2×D8—1581no—
241C2×D8—1581no—
242C22×C8—791no—
243C22×C8—792no—
244C22×C8—792no—
245C22×C8—1581no—
246C22×C8—1581no—
247C22×C8—1581no—
248C22×C8—1581no—
249C22×C8C22×D7911yescharacteristic
250#121—3162no—
251#121—1584no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—3162no—
255C2×C4⋊C4—3161no—
256C2×C4⋊C4—1584no—
257C2×C4⋊C4—1581no—
258C2×C22⋊C4—3161no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—1581no—
263C2×C16—1581no—
264C2×C16—1581no—
265#107—21no—
266#107—21no—
267#107—792no—
268#107—1584no—
269#107—1581no—
270#107C4×D7912yescomplemented
271#107D31612yescomplemented
272#102—1584no—
273C24—3161no—
274C24—791no—
275C2×D4—3164no—
276C2×D4—3161no—
277C2×D4—3161no—
278C2×D4—3161no—
279C2×D4—3161no—
280C2×D4—3161no—
281C2×D4—6322no—
282C2×D4—6322no—
283C2×D4—1582no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—3164no—
287C22×C4—3161no—
288C22×C4—791no—
289C22×C4—792no—
290C22×C4—792no—
291C22×C4—6322no—
292C22×C4—6322no—
293C22×C4—1581no—
294C22×C4—1581no—
295C22×C4—1581no—
296C22×C4—1581no—
297C22×C4D4×D7911yescharacteristic
298D8—3161no—
299D8—3161no—
300D8—1582no—
301D8—1582no—
302C2×C8—792no—
303C2×C8—1581no—
304C2×C8—1581no—
305C2×C8—1582no—
306C2×C8—1582no—
307C2×C8—1582no—
308C2×C8—1582no—
309C2×C8C2×C4×D7911yescharacteristic
310C2×C8D4×D7911yescharacteristic
311C2×C8D4×D7911yescharacteristic
312C2×C8C2×D31611yescharacteristic
313C2×C8#1192962411yescharacteristic
314C2×C8#1192962511yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—3164no—
320C4⋊C4—3161no—
321C4⋊C4—1582no—
322C22⋊C4—3164no—
323C16—41no—
324C16—3161no—
325C23—3162no—
326C23—792no—
327C23—792no—
328C23—792no—
329C23—6321no—
330C23—6321no—
331C23—6321no—
332C23—6321no—
333C23—6321no—
334C23—6321no—
335C23—1581no—
336C23—1581no—
337C23—1581no—
338C23—1581no—
339C23D8×D7911yescharacteristic
340D4—3162no—
341D4—3162no—
342D4—6328no—
343D4—6321no—
344D4—6321no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—3162no—
352C2×C4—792no—
353C2×C4—6328no—
354C2×C4—6321no—
355C2×C4—6321no—
356C2×C4—1581no—
357C2×C4—1581no—
358C2×C4—1582no—
359C2×C4—1582no—
360C2×C4—1582no—
361C2×C4—1582no—
362C2×C4C22⋊C4×D7911yescharacteristic
363C2×C4#1457913111yescharacteristic
364C2×C4#1457914611yescharacteristic
365C2×C4SD16×D7911yescharacteristic
366C2×C4#1457918811yescharacteristic
367C2×C4#1457919311yescharacteristic
368C8—1582no—
369C8#1457912311yescharacteristic
370C8C4×D31611yescharacteristic
371C8#1457913411yescharacteristic
372C8#1457913611yescharacteristic
373C22—794no—
374C22—6322no—
375C22—6322no—
376C22—6322no—
377C22—1582no—
378C22—1582no—
379C22—1582no—
380C22—1582no—
381C22—1582no—
382C22—1582no—
383C22—12641no—
384C22—12641no—
385C22—12641no—
386C22—12641no—
387C22#102×D7911yescharacteristic
388C22#1909869811yescharacteristic
389C22#1909887811yescharacteristic
390C22D16×D7911yescharacteristic
391C22SD32×D7911yescharacteristic
392C22#1909900011yescharacteristic
393C22#1909900511yescharacteristic
394C4—82no—
395C4—82no—
396C4—6322no—
397C4—1582no—
398C4#1909844111yescharacteristic
399C4#1909856311yescharacteristic
400C4#1909870311yescharacteristic
401C4#1909888311yescharacteristic
402C2—1584no—
403C2—12642no—
404C2#2376371611yescharacteristic
405C2#357×D7911yescharacteristic
406C2#2376511111yescharacteristic
407C2#2376531311yescharacteristic
408C2#2376531811yescharacteristic
409C2#2376558211yescharacteristic
410C2#2376558711yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#1380C2
socleC23×C79C2×D8
prime radicalC2×#19098800C2
cosocleC22×C8×C79C23
hypercenter#1380D79
hypocenterC79#12991
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#107×D79C2semidirect
3. C2×#19098365C2semidirect
4. C2×#19098363C2semidirect
5. C2×(C2×D79)⋊C16C2semidirect
6. #23761491C2semidirect
7. C79×#1380C2semidirect
14. #19098800C4semidirect
35. C2×D632C2×C4semidirect
54. #14579095D4semidirect
65. #107×C79C2×C4semidirect
66. #107×C79D4semidirect
187. C2×D79#357semidirect
212. #1380D79semidirect
213. C79#12991semidirect
270. #107C4×D79semidirect
271. #107D316semidirect

Maximal subgroups

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

subgroupindex
C2×#190988002
C2×#107×D792
C2×#190983652
C2×#190983632
C2×(C2×D79)⋊C162
#237614912
C79×#13802
#1299179