Finite Group Databasebeta

← #34181726

Subgroups — #34181726

Order 20224.

622 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765753C211yesmaximal, characteristic
2C2×#19098287C211yesmaximal, characteristic
3#23761247C212yesmaximal, complemented
4#23761246C212yesmaximal, complemented
5C2×#323×C79C211yesmaximal, characteristic
6C22×C22⋊Dic79C411yescharacteristic
7C22×C22⋊Dic79C411yescharacteristic
8C2×C22⋊(C79⋊C8)C2212yes—
9C22⋊C4×Dic79—21no—
10C22⋊C4×Dic79—21no—
11C22⋊C4×Dic79—21no—
12C22⋊C4×Dic79—21no—
13C4×C22⋊Dic79C411yescharacteristic
14C4×C22⋊Dic79C411yescharacteristic
15C4×C22⋊Dic79C2211yescharacteristic
16C4×C22⋊Dic79C2211yescharacteristic
17#19098287—21no—
18#19098287—21no—
19C22×C22⋊C4×C79C2211yescharacteristic
20C2×C22⋊C8×C79C2212yes—
21#323×C79—21no—
22#323×C79—21no—
23C23×Dic79—21no—
24C23×Dic79—21no—
25C22×C79⋊C8—22no—
26C2×C22⋊Dic79—21no—
27C2×C22⋊Dic79—21no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79C814yescomplemented
32C2×C22⋊Dic79C2×C411yescharacteristic
33C2×C22⋊Dic79C2×C411yescharacteristic
34C2×C22⋊Dic79C2×C411yescharacteristic
35C2×C22⋊Dic79C2×C411yescharacteristic
36C2×C4×Dic79—21no—
37C2×C4×Dic79—21no—
38C2×C4×Dic79—21no—
39C2×C4×Dic79—21no—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—22no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C25×C79C2×C411yescharacteristic
47C23×C4×C79C2×C411yescharacteristic
48C23×C4×C79C2311yescharacteristic
49C22×C8×C79—22no—
50C2×C22⋊C4×C79—22no—
51C2×C22⋊C4×C79—22no—
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—22no—
58C22×Dic79—24no—
59C22×Dic79—24no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—81no—
65C22×Dic79—81no—
66C22×Dic79—81no—
67C22×Dic79—81no—
68C2×C79⋊C8—42no—
69C2×C79⋊C8—82no—
70C22⋊Dic79—24no—
71C22⋊Dic79—24no—
72C22⋊Dic79—21no—
73C22⋊Dic79—21no—
74C22⋊Dic79—21no—
75C22⋊Dic79—21no—
76C22⋊Dic79C2×C812yes—
77C22⋊Dic79C2×C812yes—
78C22⋊Dic79M1612yes—
79C22⋊Dic79M1612yes—
80C4×Dic79—81no—
81C4×Dic79—81no—
82C4×Dic79—81no—
83C4×Dic79—81no—
84C24×C79—21no—
85C24×C79—21no—
86C24×C79—21no—
87C24×C79—21no—
88C24×C79—81no—
89C24×C79—81no—
90C24×C79C22⋊C411yescharacteristic
91C24×C79C22⋊C411yescharacteristic
92C24×C79C2×C812yes—
93C24×C79M1612yes—
94C24×C79C22×C411yescharacteristic
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—81no—
98C22×C4×C79—81no—
99C22×C4×C79C22⋊C411yescharacteristic
100C22×C4×C79C22⋊C411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79#4112yes—
104C22×C4×C79#4112yes—
105C2×C8×C79—42no—
106C2×C8×C79—82no—
107C22⋊C4×C79—22no—
108C22⋊C4×C79—22no—
109C22⋊C4×C79—22no—
110C22⋊C4×C79—22no—
111C23×C79—21no—
112C23×C79—21no—
113C23×C79—21no—
114C23×C79—21no—
115C23×C79—22no—
116C23×C79—22no—
117C23×C79—22no—
118C23×C79—22no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79—41no—
123C23×C79—41no—
124C23×C79—41no—
125C23×C79—81no—
126C23×C79—81no—
127C23×C79—81no—
128C23×C79—81no—
129C23×C79—82no—
130C23×C79—82no—
131C23×C79—82no—
132C23×C79—82no—
133C23×C79C22⋊C812yes—
134C23×C79C22⋊C812yes—
135C23×C79C2×C22⋊C411yescharacteristic
136C23×C79#11711yescharacteristic
137C23×C79#11711yescharacteristic
138C23×C79C22×C811yescharacteristic
139C23×C79C2×M1611yescharacteristic
140C23×C79C2×M1611yescharacteristic
141C23×C79C2×M1611yescharacteristic
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—42no—
147C2×C4×C79—42no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—81no—
151C2×C4×C79—81no—
152C2×C4×C79#12312yes—
153C2×C4×C79#12312yes—
154C2×Dic79—22no—
155C2×Dic79—22no—
156C2×Dic79—22no—
157C2×Dic79—22no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161C2×Dic79—41no—
162C2×Dic79—41no—
163C2×Dic79—41no—
164C2×Dic79—41no—
165C2×Dic79—41no—
166C2×Dic79—84no—
167C2×Dic79—84no—
168C8×C79—82no—
169C79⋊C8—82no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—21no—
176C22×C79—22no—
177C22×C79—22no—
178C22×C79—41no—
179C22×C79—41no—
180C22×C79—41no—
181C22×C79—41no—
182C22×C79—42no—
183C22×C79—42no—
184C22×C79—42no—
185C22×C79—42no—
186C22×C79—41no—
187C22×C79—41no—
188C22×C79—41no—
189C22×C79—41no—
190C22×C79—82no—
191C22×C79—82no—
192C22×C79—82no—
193C22×C79—82no—
194C22×C79#38111yescharacteristic
195C22×C79C2×C22⋊C811yescharacteristic
196C22×C79#40711yescharacteristic
197C22×C79#43111yescharacteristic
198C22×C79#43111yescharacteristic
199C22×C79#43211yescharacteristic
200C22×C79#43211yescharacteristic
201C4×C79—22no—
202C4×C79—22no—
203C4×C79—41no—
204C4×C79—41no—
205Dic79—42no—
206Dic79—42no—
207Dic79—42no—
208Dic79—42no—
209#8734—791nomaximal, Sylow
210C2×C79—21no—
211C2×C79—21no—
212C2×C79—21no—
213C2×C79—21no—
214C2×C79—21no—
215C2×C79—21no—
216C2×C79—42no—
217C2×C79—42no—
218C2×C79#181711yescharacteristic
219C2×C79#182811yescharacteristic
220C2×C79#182911yescharacteristic
221#2273—791no—
222C2×#323—791no—
223C2×#323D7911yescharacteristic
224#1282—792no—
225#1282—792no—
226C79#873411yescharacteristic, complemented, Sylow
227C22×C22⋊C4—791no—
228C22×C22⋊C4—791no—
229C22×C22⋊C4C2×D7911yescharacteristic
230C2×C22⋊C8—792no—
231C2×C22⋊C8C2×D7912yes—
232C4×C22⋊C4—791no—
233C4×C22⋊C4—791no—
234C4×C22⋊C4—791no—
235C4×C22⋊C4—791no—
236C4×C22⋊C4—1581no—
237C4×C22⋊C4—1581no—
238C4×C22⋊C4—1581no—
239C4×C22⋊C4—1581no—
240#323—21no—
241#323—21no—
242#323—1581no—
243#323—1581no—
244C25C4×D7911yescharacteristic
245C23×C4—1581no—
246C23×C4—1581no—
247C23×C4C4×D7911yescharacteristic
248C23×C4C22×D7911yescharacteristic
249C22×C8—22no—
250C22×C8—1582no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—794no—
254C2×C22⋊C4—791no—
255C2×C22⋊C4—791no—
256C2×C22⋊C4—791no—
257C2×C22⋊C4—791no—
258C2×C22⋊C4—1582no—
259C2×C22⋊C4—1582no—
260C2×C22⋊C4—1581no—
261C2×C22⋊C4—1581no—
262C2×C22⋊C4—1582no—
263C2×C22⋊C4D31611yescharacteristic
264C2×C22⋊C4D31611yescharacteristic
265C2×C22⋊C4C22⋊D7911yescharacteristic
266C2×C22⋊C4C22⋊D7911yescharacteristic
267C2×C42—1581no—
268C2×C42—1581no—
269C2×C42—1581no—
270C2×C42—1581no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—1582no—
274C22⋊C8—1582no—
275#95—1581no—
276#95—1581no—
277#95—1581no—
278#95—1581no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—81no—
284C24—81no—
285C24(C2×D79)⋊C411yescharacteristic
286C24(C2×D79)⋊C411yescharacteristic
287C24C8×D7912yes—
288C24#1192960712yes—
289C24C2×C4×D7911yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—6321no—
295C22×C4—6321no—
296C22×C4—6321no—
297C22×C4—6321no—
298C22×C4—1581no—
299C22×C4—1581no—
300C22×C4—1581no—
301C22×C4—1581no—
302C22×C4—1584no—
303C22×C4—1584no—
304C22×C4(C2×D79)⋊C411yescharacteristic
305C22×C4(C2×D79)⋊C411yescharacteristic
306C22×C4C2×C22⋊D7911yescharacteristic
307C22×C4C2×D31611yescharacteristic
308C22×C4#1192962312yes—
309C22×C4#1192962312yes—
310C2×C8—42no—
311C2×C8—82no—
312C2×C8—3162no—
313C2×C8—6322no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—792no—
319C22⋊C4—792no—
320C22⋊C4—792no—
321C22⋊C4—792no—
322C22⋊C4—1584no—
323C22⋊C4—1584no—
324C22⋊C4—1581no—
325C22⋊C4—1581no—
326C22⋊C4—1581no—
327C22⋊C4—1581no—
328C42—6321no—
329C42—6321no—
330C42—6321no—
331C42—6321no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23(C2×D79)⋊C812yes—
355C23(C2×D79)⋊C812yes—
356C23C2×(C2×D79)⋊C411yescharacteristic
357C23#1457911711yescharacteristic
358C23#1457911711yescharacteristic
359C23C2×C8×D7911yescharacteristic
360C23M16×D7911yescharacteristic
361C23M16×D7911yescharacteristic
362C23C2×#1192960711yescharacteristic
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—3161no—
374C2×C4—3161no—
375C2×C4—3161no—
376C2×C4—3161no—
377C2×C4—3161no—
378C2×C4—3161no—
379C2×C4—3161no—
380C2×C4—3161no—
381C2×C4—6324no—
382C2×C4—6324no—
383C2×C4—1582no—
384C2×C4—1582no—
385C2×C4—1582no—
386C2×C4—1582no—
387C2×C4#1457914212yes—
388C2×C4#1457914512yes—
389C8—82no—
390C8—6322no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22#1909842411yescharacteristic
416C22C2×(C2×D79)⋊C811yescharacteristic
417C22#1909852611yescharacteristic
418C22#1909861311yescharacteristic
419C22#1909861311yescharacteristic
420C22#1909861711yescharacteristic
421C22#1909861711yescharacteristic
422C4—22no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—3162no—
427C4—3162no—
428C4—3162no—
429C4—3162no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—42no—
437C2—42no—
438C2#2376346611yescharacteristic
439C2#2376350611yescharacteristic
440C2#2376350911yescharacteristic
subgroupG/H
centerC22#19098424
commutatorC23×C79C22×C8
FrattiniC23×C4C22×D79
FittingC2×#323×C79C2
socleC22×C79#381
prime radicalC25×C79C2×C4
cosocleC23×C4×C79C23
hypercenterC2×#323D79
hypocenterC79#8734
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #23761247C2semidirect
4. #23761246C2semidirect
31. C2×C22⋊Dic79C8semidirect
226. C79#8734semidirect

Maximal subgroups

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

subgroupindex
#237657532
C2×#190982872
#237612472
#237612462
C2×#323×C792
#873479