Finite Group Databasebeta

← #34183819

Subgroups — #34183819

Order 20224.

560 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766065C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3C2×#19098288C212yesmaximal, complemented
4#23761246C212yesmaximal, complemented
5C79×#2973C211yesmaximal, characteristic
6C2×#14579142C411yescharacteristic
7C22×Dic79⋊C4C2211yescharacteristic
8C2×C22⋊(C79⋊C8)C2212yes—
9#19098458—21no—
10#19098457—21no—
11#19098447—21no—
12#19098443C411yescharacteristic
13#19098436—21no—
14#19098435C411yescharacteristic
15#19098424C411yescharacteristic
16C4×C22⋊Dic79C2211yescharacteristic
17#19098289—21no—
18#19098288—24no—
19#19098287—21no—
20C2×C4×D4×C79C2211yescharacteristic
21#435×C79—24no—
22#432×C79—21no—
23#407×C79—21no—
24C2×C22⋊C8×C79C2212yes—
25C23×Dic79C2×C411yescharacteristic
26C23×Dic79C2×C411yescharacteristic
27C22×C79⋊C8—22no—
28#14579142—22no—
29#14579142—22no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—22no—
34C2×Dic79⋊C4—22no—
35C2×Dic79⋊C4—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C22⋊Dic79C2×C411yescharacteristic
39C2×C22⋊Dic79C2×C411yescharacteristic
40C2×C4×Dic79—21no—
41C22⋊(C79⋊C8)—22no—
42C22⋊(C79⋊C8)—22no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C22×D4×C79C2×C411yescharacteristic
49C23×C4×C79D411yescharacteristic
50C23×C4×C79C2311yescharacteristic
51C2×M16×C79—22no—
52C22×C8×C79—22no—
53C4×D4×C79—22no—
54C4×D4×C79—22no—
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C22⋊C4×C79C2×C411yescharacteristic
57C2×C22⋊C4×C79D411yescharacteristic
58C2×C42×C79D411yescharacteristic
59C4⋊C8×C79—22no—
60C22⋊C8×C79—22no—
61C22⋊C8×C79—22no—
62C22⋊C8×C79D414yes—
63#97×C79—22no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—22no—
69C22×Dic79—22no—
70C22×Dic79—21no—
71C22×Dic79—22no—
72C22×Dic79—81no—
73C22×Dic79M1612yescomplemented
74C2×C79⋊C8—42no—
75C2×C79⋊C8—82no—
76Dic79⋊C4—24no—
77Dic79⋊C4—41no—
78Dic79⋊C4—42no—
79Dic79⋊C4—41no—
80Dic79⋊C4—81no—
81C22⋊Dic79—24no—
82C22⋊Dic79—21no—
83C22⋊Dic79—22no—
84C22⋊Dic79—81no—
85C22⋊Dic79M1612yescomplemented
86C4×Dic79—81no—
87C24×C79C22⋊C411yescharacteristic
88C24×C79C22×C411yescharacteristic
89C2×D4×C79—22no—
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—22no—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—81no—
96C22×C4×C79—81no—
97C22×C4×C79C22⋊C411yescharacteristic
98C22×C4×C79C22⋊C411yescharacteristic
99C22×C4×C79C22⋊C411yescharacteristic
100C22×C4×C79C2×D412yes—
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79#4112yes—
104M16×C79—82no—
105C2×C8×C79—24no—
106C2×C8×C79—82no—
107C4⋊C4×C79—22no—
108C22⋊C4×C79—22no—
109C22⋊C4×C79—22no—
110C42×C79—21no—
111C42×C79—21no—
112C23×C79—21no—
113C23×C79—21no—
114C23×C79—21no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79C2×C22⋊C411yescharacteristic
120C23×C79C4×D412yes—
121C23×C79C2×M1611yescharacteristic
122C23×C79C2×M1611yescharacteristic
123C23×C79#13112yes—
124D4×C79—42no—
125D4×C79—42no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—41no—
132C2×C4×C79—42no—
133C2×C4×C79—42no—
134C2×C4×C79—41no—
135C2×C4×C79—42no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—81no—
139C2×C4×C79#12011yescharacteristic
140C2×C4×C79#12112yes—
141C2×C4×C79#12311yescharacteristic
142C2×Dic79—22no—
143C2×Dic79—22no—
144C2×Dic79—24no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—42no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—82no—
157C8×C79—44no—
158C79⋊C8—82no—
159C22×C79—21no—
160C22×C79—22no—
161C22×C79—21no—
162C22×C79—22no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—42no—
166C22×C79—42no—
167C22×C79—41no—
168C22×C79—41no—
169C22×C79—41no—
170C22×C79—81no—
171C22×C79#38611yescharacteristic
172C22×C79#40711yescharacteristic
173C22×C79#40811yescharacteristic
174C22×C79#43512yes—
175C22×C79#43612yes—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—22no—
179C4×C79—41no—
180C4×C79—41no—
181C4×C79—41no—
182Dic79—42no—
183Dic79—42no—
184Dic79—81no—
185Dic79—81no—
186#9110—791nomaximal, Sylow
187C2×C79—21no—
188C2×C79—21no—
189C2×C79—22no—
190C2×C79—22no—
191C2×C79—81no—
192C2×C79#187011yescharacteristic
193C2×C79#191311yescharacteristic
194C2×C79#191511yescharacteristic
195#2973D7911yescharacteristic
196#2322—791no—
197#1459—791no—
198C2×#324—792no—
199#1282—792no—
200C79#911011yescharacteristic, complemented, Sylow
201C2×#123—791no—
202C2×C4×D4C2×D7911yescharacteristic
203C22×C4⋊C4—791no—
204#435—24no—
205#432—21no—
206#407—21no—
207C2×C22⋊C8—792no—
208C2×C22⋊C8C2×D7912yes—
209#389—1581no—
210#388—1581no—
211#387—1581no—
212#386—791no—
213#385—791no—
214#385—1581no—
215#381—791no—
216C4×C22⋊C4—791no—
217#324—1581no—
218#324—1584no—
219#323—1581no—
220C22×D4C4×D7911yescharacteristic
221C23×C4—791no—
222C23×C4—791no—
223C23×C4C22⋊D7911yescharacteristic
224C23×C4C22×D7911yescharacteristic
225C2×M16—22no—
226C22×C8—22no—
227C22×C8—1582no—
228#123—1582no—
229#123—1582no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—1582no—
233C2×C4⋊C4—1581no—
234C2×C4⋊C4—1581no—
235C2×C4⋊C4—1582no—
236C2×C4⋊C4—1581no—
237C2×C4⋊C4—1581no—
238C2×C4⋊C4D31611yescharacteristic
239C2×C22⋊C4—791no—
240C2×C22⋊C4—791no—
241C2×C22⋊C4—1581no—
242C2×C22⋊C4—1581no—
243C2×C22⋊C4C4×D7911yescharacteristic
244C2×C22⋊C4D31611yescharacteristic
245C2×C42—1581no—
246C2×C42C22⋊D7911yescharacteristic
247C4⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—1582no—
251C22⋊C8—1582no—
252C22⋊C8C22⋊D7914yes—
253#97—22no—
254#95—1581no—
255#95—1581no—
256#95—1581no—
257#95—1581no—
258#95—1581no—
259C24(C2×D79)⋊C411yescharacteristic
260C24C2×C4×D7911yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—792no—
270C22×C4—6321no—
271C22×C4—1581no—
272C22×C4—1581no—
273C22×C4—1581no—
274C22×C4—1581no—
275C22×C4—1582no—
276C22×C4—1581no—
277C22×C4—1582no—
278C22×C4—1582no—
279C22×C4(C2×D79)⋊C411yescharacteristic
280C22×C4(C2×D79)⋊C411yescharacteristic
281C22×C4(C2×D79)⋊C411yescharacteristic
282C22×C4C2×C22⋊D7912yes—
283C22×C4C2×C22⋊D7911yescharacteristic
284C22×C4C2×D31611yescharacteristic
285C22×C4#1192962412yes—
286M16—82no—
287C2×C8—24no—
288C2×C8—82no—
289C2×C8—3162no—
290C2×C8—6322no—
291C4⋊C4—22no—
292C4⋊C4—3161no—
293C4⋊C4—3162no—
294C4⋊C4—3161no—
295C4⋊C4—6321no—
296C4⋊C4—1584no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—792no—
300C22⋊C4—6321no—
301C22⋊C4—1582no—
302C22⋊C4—1584no—
303C22⋊C4—1581no—
304C42—21no—
305C42—21no—
306C42—6321no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×(C2×D79)⋊C411yescharacteristic
315C23C4×C22⋊D7912yes—
316C23M16×D7911yescharacteristic
317C23M16×D7911yescharacteristic
318C23#1457916912yes—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—3161no—
335C2×C4—3161no—
336C2×C4—3162no—
337C2×C4—3161no—
338C2×C4—3161no—
339C2×C4—3161no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—3161no—
343C2×C4—3161no—
344C2×C4—3161no—
345C2×C4—6322no—
346C2×C4—1584no—
347C2×C4—1582no—
348C2×C4—1582no—
349C2×C4#1457913011yescharacteristic
350C2×C4#1457913512yes—
351C2×C4#1457914411yescharacteristic
352C8—44no—
353C8—6322no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1909844211yescharacteristic
367C22#1909852611yescharacteristic
368C22#1909853111yescharacteristic
369C22#1909863212yes—
370C22#1909863912yes—
371C4—21no—
372C4—22no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—3162no—
378C4—3162no—
379C4—6321no—
380C4—6321no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2376369711yescharacteristic
387C2#2376394211yescharacteristic
388C2#2376395211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2973C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2973D79
hypocenterC79#9110
LCSG ▷ C24×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. C2×#19098288C2semidirect
4. #23761246C2semidirect
73. C22×Dic79M16semidirect
85. C22⋊Dic79M16semidirect
200. C79#9110semidirect

Maximal subgroups

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

subgroupindex
#237660652
#237617182
C2×#190982882
#237612462
C79×#29732
#911079