Finite Group Databasebeta

← #34203416

Subgroups — #34203416

Order 20224.

636 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098350C212yesmaximal
2C4×#14579078C211yesmaximal, characteristic, complemented
3C4×#14579076C211yesmaximal, characteristic, complemented
4#23761518C212yesmaximal, complemented
5C79×#2321C211yesmaximal, characteristic
6C2×#14579078C2212yes—
7C2×#14579076C2212yes—
8C4×#11929608—21no—
9C4×C22⋊Dic79—21no—
10#19098350C418yescomplemented
11#19098332—21no—
12#19098330—21no—
13C2×#120×C79C2211yescharacteristic
14C2×C4×D4×C79C2211yescharacteristic
15C22×C22⋊C4×C79C411yescharacteristic, complemented
16#390×C79—21no—
17#388×C79—21no—
18#388×C79—21no—
19#386×C79—21no—
20#386×C79C2211yescharacteristic
21#385×C79C411yescharacteristic, complemented
22#379×C79C411yescharacteristic, complemented
23C4×C22⋊C4×C79C411yescharacteristic, complemented
24C2×#11929608—22no—
25C2×C22⋊Dic79—22no—
26C2×C4×Dic79—41no—
27#14579078—22no—
28#14579078C2×C414yes—
29#14579076—22no—
30#14579076C2×C414yes—
31#14579072—41no—
32C4×C79⋊C8—41no—
33C79⋊#95—41no—
34C25×C79C2×C411yescharacteristic
35C22×D4×C79—21no—
36C22×D4×C79C2311yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79C2×C411yescharacteristic
39C23×C4×C79D411yescharacteristic
40#120×C79—22no—
41#120×C79C2×C414yes—
42C4×D4×C79—41no—
43C4×D4×C79—41no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79—21no—
48C2×C22⋊C4×C79—21no—
49C2×C22⋊C4×C79—21no—
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—41no—
53C2×C22⋊C4×C79—41no—
54C2×C22⋊C4×C79C2×C411yescharacteristic
55C2×C22⋊C4×C79C2×C411yescharacteristic
56C2×C22⋊C4×C79C2×C411yescharacteristic
57C2×C22⋊C4×C79D411yescharacteristic
58C2×C42×C79—21no—
59C2×C42×C79D411yescharacteristic
60#95×C79—21no—
61#95×C79—21no—
62#95×C79—21no—
63C22×Dic79—42no—
64#11929608—24no—
65#11929608—42no—
66C2×C79⋊C8—42no—
67C22⋊Dic79—24no—
68C22⋊Dic79—42no—
69C4×Dic79—82no—
70C24×C79—21no—
71C24×C79—21no—
72C24×C79—41no—
73C24×C79—41no—
74C24×C79—81no—
75C24×C79—81no—
76C24×C79C4212yescomplemented
77C24×C79C22×C411yescharacteristic
78C24×C79C2×D411yescharacteristic
79C2×D4×C79—24no—
80C2×D4×C79—41no—
81C2×D4×C79—41no—
82C2×D4×C79—81no—
83C2×D4×C79—81no—
84C2×D4×C79C22×C412yes—
85C2×D4×C79#4112yes—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—41no—
94C22×C4×C79—81no—
95C22×C4×C79—81no—
96C22×C4×C79—81no—
97C22×C4×C79C22⋊C411yescharacteristic
98C22×C4×C79C22⋊C411yescharacteristic
99C22×C4×C79C22⋊C411yescharacteristic
100C22×C4×C79C22⋊C411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C4⋊C4×C79—41no—
103C4⋊C4×C79—81no—
104C22⋊C4×C79—21no—
105C22⋊C4×C79—24no—
106C22⋊C4×C79—41no—
107C22⋊C4×C79—41no—
108C22⋊C4×C79—42no—
109C22⋊C4×C79—41no—
110C22⋊C4×C79—41no—
111C22⋊C4×C79—41no—
112C22⋊C4×C79—81no—
113C22⋊C4×C79—81no—
114C22⋊C4×C79—81no—
115C22⋊C4×C79C4212yescomplemented
116C42×C79—21no—
117C42×C79—21no—
118C42×C79—81no—
119C23×C79—21no—
120C23×C79—21no—
121C23×C79—21no—
122C23×C79—21no—
123C23×C79—22no—
124C23×C79—41no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79—41no—
128C23×C79—41no—
129C23×C79—41no—
130C23×C79—41no—
131C23×C79—41no—
132C23×C79—42no—
133C23×C79—42no—
134C23×C79—41no—
135C23×C79—41no—
136C23×C79—82no—
137C23×C79—82no—
138C23×C79—82no—
139C23×C79—82no—
140C23×C79—84no—
141C23×C79C2×C4211yescharacteristic
142C23×C79C2×C22⋊C411yescharacteristic
143C23×C79#11711yescharacteristic
144C23×C79C4×D412yes—
145D4×C79—42no—
146D4×C79—42no—
147D4×C79—84no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—22no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—41no—
155C2×C4×C79—42no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—81no—
168C2×C4×C79—81no—
169C2×C4×C79—81no—
170C2×C4×C79—82no—
171C2×C4×C79#9911yescharacteristic
172C2×C4×C79#9911yescharacteristic
173C2×C4×C79C4×D412yes—
174C2×Dic79—44no—
175C2×Dic79—82no—
176C2×Dic79—82no—
177C79⋊C8—44no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—22no—
183C22×C79—42no—
184C22×C79—41no—
185C22×C79—41no—
186C22×C79—42no—
187C22×C79—42no—
188C22×C79—42no—
189C22×C79—41no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79—41no—
193C22×C79—42no—
194C22×C79—81no—
195C22×C79—81no—
196C22×C79—81no—
197C22×C79—84no—
198C22×C79—84no—
199C22×C79—81no—
200C22×C79—81no—
201C22×C79C4×C22⋊C411yescharacteristic
202C22×C79C2×#9911yescharacteristic
203C22×C79#41011yescharacteristic
204C4×C79—21no—
205C4×C79—21no—
206C4×C79—22no—
207C4×C79—42no—
208C4×C79—41no—
209C4×C79—41no—
210C4×C79—41no—
211C4×C79—81no—
212C4×C79—81no—
213Dic79—84no—
214#12622—791nomaximal, Sylow
215C2×C79—21no—
216C2×C79—21no—
217C2×C79—42no—
218C2×C79—41no—
219C2×C79—41no—
220C2×C79—82no—
221C2×C79C4×#9911yescharacteristic
222C2×C79#212111yescharacteristic
223C2×C79#212211yescharacteristic
224#2321D7911yescharacteristic
225C2×AΣL(1,𝔽16)—792no—
226C4×#100—791no—
227C4×#99—791no—
228#1390—792no—
229C79#1262211yescharacteristic, complemented, Sylow
230C2×#120C2×D7911yescharacteristic
231C2×C4×D4C2×D7911yescharacteristic
232C22×C22⋊C4Dic7911yescharacteristic, complemented
233C2×#100—792no—
234C2×#99—792no—
235C4×M16—1581no—
236#390—21no—
237#388—21no—
238#388—21no—
239#386—21no—
240#386C2×D7911yescharacteristic
241#385Dic7911yescharacteristic, complemented
242#379Dic7911yescharacteristic, complemented
243C4×C22⋊C4—1581no—
244C4×C22⋊C4Dic7911yescharacteristic, complemented
245AΣL(1,𝔽16)—798no—
246#343—1581no—
247#342—1581no—
248C25C2×Dic7911yescharacteristic
249C22×D4—21no—
250C22×D4C22×D7911yescharacteristic
251C23×C4C2×Dic7911yescharacteristic
252C23×C4C2×Dic7911yescharacteristic
253C23×C4C22⋊D7911yescharacteristic
254C2×M16—1582no—
255#120—22no—
256#120C4×D7914yes—
257C4×D4—41no—
258C4×D4—41no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D7911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—1582no—
270C2×C22⋊C4C2×Dic7911yescharacteristic
271C2×C22⋊C4C2×Dic7911yescharacteristic
272C2×C22⋊C4C2×Dic7911yescharacteristic
273C2×C22⋊C4C22⋊D7911yescharacteristic
274C2×C42—21no—
275C2×C42—3161no—
276C2×C42C22⋊D7911yescharacteristic
277#100—794no—
278#100—1582no—
279#99—794no—
280#99—1582no—
281#97—3161no—
282C4×C8—3161no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—3161no—
287C24—21no—
288C24—21no—
289C24—41no—
290C24—41no—
291C24—81no—
292C24—81no—
293C24C4×Dic7912yescomplemented
294C24C22×Dic7911yescharacteristic
295C24C2×C22⋊D7911yescharacteristic
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4C2×C4×D7912yes—
302C2×D4#1192962412yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—3162no—
315C22×C4C22⋊Dic7911yescharacteristic
316C22×C4C22⋊Dic7911yescharacteristic
317C22×C4C22⋊Dic7911yescharacteristic
318C22×C4C22⋊Dic7911yescharacteristic
319C22×C4C2×C22⋊D7911yescharacteristic
320M16—3162no—
321M16—1584no—
322C2×C8—3162no—
323C4⋊C4—41no—
324C4⋊C4—81no—
325C22⋊C4—21no—
326C22⋊C4—24no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C22⋊C4—3162no—
337C22⋊C4—1584no—
338C22⋊C4C4×Dic7912yescomplemented
339C42—21no—
340C42—21no—
341C42—81no—
342C42—6322no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—84no—
365C23C2×C4×Dic7911yescharacteristic
366C23C2×C22⋊Dic7911yescharacteristic
367C23#1457912011yescharacteristic
368C23C4×C22⋊D7912yes—
369D4—42no—
370D4—42no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—3164no—
396C2×C4—6322no—
397C2×C4—6322no—
398C2×C4#1457907611yescharacteristic
399C2×C4#1457907611yescharacteristic
400C2×C4C4×C22⋊D7912yes—
401C8—3164no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22C4×C22⋊Dic7911yescharacteristic
426C22C2×#1457907611yescharacteristic
427C22#1909853711yescharacteristic
428C4—21no—
429C4—21no—
430C4—22no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—81no—
437C4—6324no—
438C2—21no—
439C2—21no—
440C2—42no—
441C2—41no—
442C2—41no—
443C2—82no—
444C2C4×#1457907611yescharacteristic
445C2#2376504611yescharacteristic
446C2#2376505011yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC23×C79C2×C42
FrattiniC22×D4C22×D79
FittingC79×#2321C2
socleC22×C79C2×#99
prime radicalC2×#120×C79C22
cosocleC22×D4×C79C23
hypercenter#2321D79
hypocenterC79#12622
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
2. C4×#14579078C2semidirect
3. C4×#14579076C2semidirect
4. #23761518C2semidirect
10. #19098350C4semidirect
15. C22×C22⋊C4×C79C4semidirect
21. #385×C79C4semidirect
22. #379×C79C4semidirect
23. C4×C22⋊C4×C79C4semidirect
76. C24×C79C42semidirect
115. C22⋊C4×C79C42semidirect
229. C79#12622semidirect
232. C22×C22⋊C4Dic79semidirect
241. #385Dic79semidirect
242. #379Dic79semidirect
244. C4×C22⋊C4Dic79semidirect
293. C24C4×Dic79semidirect
338. C22⋊C4C4×Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190983502
C4×#145790782
C4×#145790762
#237615182
C79×#23212
#1262279