Finite Group Databasebeta

← #34181820

Subgroups — #34181820

Order 20224.

530 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23763388C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761279C212yesmaximal, complemented
4#23761267C212yesmaximal, complemented
5C79×#2349C211yesmaximal, characteristic
6C2×#14579076—21no—
7C2×C22⋊(C79⋊C8)C2212yes—
8#19098423C2212yes—
9#19098330—21no—
10#19098289—21no—
11#19098287—21no—
12C2×C4×D4×C79—21no—
13C2×C4×D4×C79C2211yescharacteristic
14#389×C79—21no—
15#385×C79—21no—
16#385×C79—21no—
17#382×C79C411yescharacteristic, complemented
18#380×C79C411yescharacteristic, complemented
19#380×C79C2211yescharacteristic
20C4×C4⋊C4×C79C411yescharacteristic, complemented
21C2×#95×C79C411yescharacteristic, complemented
22C2×#95×C79C2211yescharacteristic
23C22×C79⋊C8—22no—
24C2×C4⋊Dic79—22no—
25C2×C22⋊Dic79—22no—
26#14579076—42no—
27C22⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—22no—
29C79⋊#95—22no—
30C22×D4×C79D411yescharacteristic
31C23×C4×C79—21no—
32C23×C4×C79—21no—
33C23×C4×C79C2×C411yescharacteristic
34C23×C4×C79C2311yescharacteristic
35C4×D4×C79—22no—
36C4×D4×C79—22no—
37C4×D4×C79—44no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79D411yescharacteristic
46C2×C22⋊C4×C79D411yescharacteristic
47C2×C42×C79—21no—
48C2×C42×C79C2×C411yescharacteristic
49#95×C79—21no—
50#95×C79—21no—
51#95×C79—21no—
52#95×C79C2×C411yescharacteristic
53#95×C79C2×C411yescharacteristic
54#95×C79C2×C411yescharacteristic
55#95×C79C2×C411yescharacteristic
56C22×Dic79—42no—
57C2×C79⋊C8—42no—
58C2×C79⋊C8—82no—
59C4⋊Dic79—82no—
60C22⋊Dic79—42no—
61C22⋊Dic79—42no—
62C24×C79C22⋊C411yescharacteristic
63C24×C79C2×D411yescharacteristic
64C2×D4×C79—22no—
65C2×D4×C79—22no—
66C2×D4×C79SD1612yes—
67C2×D4×C79Q1612yes—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79—42no—
78C22×C4×C79—81no—
79C22×C4×C79—81no—
80C22×C4×C79—81no—
81C22×C4×C79—81no—
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79#4112yes—
88C22×C4×C79#4112yes—
89C4⋊C4×C79—22no—
90C4⋊C4×C79—42no—
91C4⋊C4×C79—42no—
92C4⋊C4×C79—81no—
93C4⋊C4×C79—81no—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—42no—
97C22⋊C4×C79—42no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—42no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79#10312yes—
108C23×C79C2×C22⋊C411yescharacteristic
109C23×C79#12312yes—
110C23×C79C2×SD1611yescharacteristic
111C23×C79C2×Q1611yescharacteristic
112C23×C79AGL(1,ℤ/8)12yes—
113D4×C79—24no—
114D4×C79—24no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—22no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79—84no—
140C2×C4×C79—84no—
141C2×C4×C79#10312yes—
142C2×C4×C79#11711yescharacteristic
143C2×C4×C79#11711yescharacteristic
144C2×C4×C79#12312yes—
145C2×Dic79—82no—
146C2×Dic79—82no—
147C2×Dic79—82no—
148C79⋊C8—82no—
149C22×C79—21no—
150C22×C79—22no—
151C22×C79—21no—
152C22×C79—22no—
153C22×C79—22no—
154C22×C79—42no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—42no—
158C22×C79—42no—
159C22×C79—41no—
160C22×C79—41no—
161C22×C79#38111yescharacteristic
162C22×C79C2×#10311yescharacteristic
163C22×C79#41811yescharacteristic
164C22×C79#48112yes—
165C22×C79#48412yes—
166C4×C79—22no—
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—22no—
170C4×C79—42no—
171C4×C79—41no—
172C4×C79—41no—
173C4×C79—81no—
174C4×C79—81no—
175C4×C79—81no—
176Dic79—162no—
177#8761—791nomaximal, Sylow
178C2×C79—22no—
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—22no—
182C2×C79—42no—
183C2×C79#182411yescharacteristic
184C2×C79#182711yescharacteristic
185C2×C79#182811yescharacteristic
186#2349D7911yescharacteristic
187#1799—791no—
188#1459—791no—
189#1294—792no—
190#1289—792no—
191C79#876111yescharacteristic, complemented, Sylow
192C2×C4×D4—21no—
193C2×C4×D4C2×D7911yescharacteristic
194C2×#99—1581no—
195C2×C22⋊C8—792no—
196#389—21no—
197#385—21no—
198#385—21no—
199#382Dic7911yescharacteristic, complemented
200#380—792no—
201#380Dic7911yescharacteristic, complemented
202#380C2×D7911yescharacteristic
203C4×C4⋊C4Dic7911yescharacteristic, complemented
204C2×#95Dic7911yescharacteristic, complemented
205C2×#95C2×D7911yescharacteristic
206#342—1581no—
207#324—1581no—
208#323—1581no—
209C22×D4C22⋊D7911yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4C2×Dic7911yescharacteristic
213C23×C4C22×D7911yescharacteristic
214C22×C8—1582no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—1582no—
223C2×C4⋊C4C22⋊D7911yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—1582no—
227C2×C22⋊C4C22⋊D7911yescharacteristic
228C2×C22⋊C4C22⋊D7911yescharacteristic
229C2×C42—21no—
230C2×C42C2×Dic7911yescharacteristic
231#99—3162no—
232C22⋊C8—1582no—
233C22⋊C8—1582no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—1582no—
238#95C2×Dic7911yescharacteristic
239#95C2×Dic7911yescharacteristic
240#95C2×Dic7911yescharacteristic
241#95C2×Dic7911yescharacteristic
242C24C22⋊Dic7911yescharacteristic
243C24C2×C22⋊D7911yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4#1192961112yes—
247C2×D4#1192961412yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—3162no—
263C22×C4C22⋊Dic7911yescharacteristic
264C22×C4C22⋊Dic7911yescharacteristic
265C22×C4C22⋊Dic7911yescharacteristic
266C22×C4C22×Dic7911yescharacteristic
267C22×C4C2×C22⋊D7911yescharacteristic
268C22×C4#1192962412yes—
269C22×C4#1192962412yes—
270C2×C8—3162no—
271C2×C8—6322no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—6322no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—3162no—
283C22⋊C4—3162no—
284C42—21no—
285C42—21no—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#1457908512yes—
294C23C2×C22⋊Dic7911yescharacteristic
295C23#1457914412yes—
296C23C2×#1192961111yescharacteristic
297C23C2×#1192961411yescharacteristic
298C23#1457918712yes—
299D4—24no—
300D4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—6322no—
328C2×C4—6322no—
329C2×C4—6322no—
330C2×C4#1457908512yes—
331C2×C4#1457912011yescharacteristic
332C2×C4#1457912011yescharacteristic
333C2×C4#1457914412yes—
334C8—6322no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22#1909842611yescharacteristic
348C22C2×#1457908511yescharacteristic
349C22#1909856811yescharacteristic
350C22#1909888112yes—
351C22#1909889812yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—12642no—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2#2376349011yescharacteristic
369C2#2376350411yescharacteristic
370C2#2376350811yescharacteristic
subgroupG/H
centerC22#19098426
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2349C2
socleC22×C79#381
prime radicalC22×D4×C79D4
cosocleC23×C4×C79C23
hypercenter#2349D79
hypocenterC79#8761
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #23761279C2semidirect
4. #23761267C2semidirect
17. #382×C79C4semidirect
18. #380×C79C4semidirect
20. C4×C4⋊C4×C79C4semidirect
21. C2×#95×C79C4semidirect
191. C79#8761semidirect
199. #382Dic79semidirect
201. #380Dic79semidirect
203. C4×C4⋊C4Dic79semidirect
204. C2×#95Dic79semidirect

Maximal subgroups

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

subgroupindex
#237633882
#237617182
#237612792
#237612672
C79×#23492
#876179