Finite Group Databasebeta

← #34198477

Subgroups — #34198477

Order 20224.

514 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 107 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#23769142C211yesmaximal, characteristic
2C2×#19098332C211yesmaximal, characteristic, complemented
3#23761407C212yesmaximal, complemented
4#23761404C212yesmaximal, complemented
5C2×#324×C79C211yesmaximal, characteristic, complemented
6C22×#11929608—21no—
7C2×#14579144—21no—
8C2×#14579135C411yescharacteristic
9#19098493C411yescharacteristic
10#19098493C2211yescharacteristic
11#19098485—21no—
12#19098485—21no—
13#19098480C411yescharacteristic
14#19098480C2211yescharacteristic
15#19098442—21no—
16#19098423C411yescharacteristic
17#19098332—21no—
18#19098332C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C2×#11929608—24no—
26C2×#11929608—22no—
27C2×#11929608—41no—
28C2×#11929608—42no—
29C22×C79⋊C8—41no—
30#14579144—42no—
31#14579144—42no—
32#14579135—41no—
33#14579135—41no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79C2×C411yescharacteristic
36C2×Dic79⋊C4—21no—
37C2×Dic79⋊C4—21no—
38C2×C22⋊Dic79—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—21no—
41C2×(C2×D79)⋊C4—21no—
42C2×(C2×D79)⋊C4—21no—
43C2×(C2×D79)⋊C4—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95C2×C411yescharacteristic
47C79⋊#95C2×C411yescharacteristic
48C23×C4×C79C2×C411yescharacteristic
49C23×C4×C79D411yescharacteristic
50C23×C4×C79D411yescharacteristic
51C22×C8×C79—41no—
52C2×C4⋊C4×C79—41no—
53C2×C4⋊C4×C79—41no—
54C2×C4⋊C4×C79C2×C411yescharacteristic
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C4⋊C4×C79D411yescharacteristic
57C2×C4⋊C4×C79C2311yescharacteristic
58C22⋊C8×C79—22no—
59C22⋊C8×C79—41no—
60C23×D79—21no—
61C2×D316—41no—
62C2×D316—81no—
63C2×C22⋊D79—41no—
64C2×C22⋊D79—41no—
65C2×C22⋊D79—42no—
66C2×C22⋊D79—42no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70#11929608—24no—
71#11929608—44no—
72#11929608—42no—
73C2×C79⋊C8—44no—
74C2×C79⋊C8—42no—
75C4⋊Dic79—41no—
76C4⋊Dic79—81no—
77Dic79⋊C4—42no—
78Dic79⋊C4—41no—
79Dic79⋊C4—41no—
80C22⋊Dic79—42no—
81(C2×D79)⋊C4—42no—
82(C2×D79)⋊C4—42no—
83(C2×D79)⋊C4—41no—
84(C2×D79)⋊C4—41no—
85(C2×D79)⋊C4—81no—
86(C2×D79)⋊C4—81no—
87C24×C79C22⋊C411yescharacteristic
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—22no—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
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×C79D812yes—
101C22×C4×C79SD1612yes—
102C22×C4×C79C22×C411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C2×C8×C79—42no—
106C2×C8×C79—82no—
107C4⋊C4×C79—41no—
108C4⋊C4×C79—41no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79#9911yescharacteristic
114C23×C79#9911yescharacteristic
115C23×C79#10212yes—
116C23×C79#10412yes—
117C23×C79C2×C22⋊C411yescharacteristic
118C22×D79—41no—
119C22×D79—41no—
120C22×D79—41no—
121C22×D79—41no—
122C22×D79—41no—
123C22×D79—41no—
124C22×D79—41no—
125C2×C4×C79—22no—
126C2×C4×C79—22no—
127C2×C4×C79—22no—
128C2×C4×C79—22no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—42no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—81no—
143C2×C4×C79—81no—
144C2×C4×C79—81no—
145C2×C4×C79#10212yes—
146C2×C4×C79C2×D811yescharacteristic
147C2×C4×C79C2×SD1611yescharacteristic
148C22⋊D79—82no—
149C22⋊D79—82no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159D316—82no—
160C8×C79—82no—
161C79⋊C8—44no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—21no—
165C22×C79—21no—
166C22×C79—21no—
167C22×C79—21no—
168C22×C79—42no—
169C22×C79—42no—
170C22×C79#32712yes—
171C22×C79#32712yes—
172C22×C79C2×#9911yescharacteristic
173C22×C79C2×#10211yescharacteristic
174C22×C79C2×#10411yescharacteristic
175C2×D79—81no—
176C2×D79—81no—
177C2×D79—81no—
178C2×D79—81no—
179C2×D79—81no—
180C2×D79—81no—
181C2×D79—81no—
182C4×C79—22no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—41no—
186C4×C79—41no—
187C4×C79—41no—
188C4×C79—41no—
189Dic79—81no—
190Dic79—81no—
191Dic79—81no—
192#11698—791nomaximal, Sylow
193C2×C79—21no—
194C2×C79—21no—
195C2×C79—22no—
196C2×C79—22no—
197C2×C79C2×#32711yescharacteristic
198C2×C79#212111yescharacteristic
199C2×C79#213311yescharacteristic
200D79—161no—
201#2618—791no—
202C2×#343—791no—
203C2×#324D7911yescharacteristic, complemented
204#1346—792no—
205#1345—792no—
206C79#1169811yescharacteristic, complemented, Sylow
207C22×M16—1581no—
208C2×#123—1581no—
209C2×#121—791no—
210C22×C4⋊C4C2×D7911yescharacteristic
211C2×C22⋊C8—21no—
212#398—791no—
213#398—791no—
214#396—1581no—
215#396—1581no—
216#394—791no—
217#394—791no—
218#386—1581no—
219#380—791no—
220#343—792no—
221#343—1581no—
222#324—21no—
223#324C2×D7912yes—
224C22×D4—791no—
225C22×D4—1581no—
226C23×C4C4×D7911yescharacteristic
227C23×C4D31611yescharacteristic
228C23×C4C22⋊D7911yescharacteristic
229C2×M16—3161no—
230C2×M16—3162no—
231C2×M16—1584no—
232C2×M16—1582no—
233C22×C8—41no—
234C22×C8—3161no—
235#123—3162no—
236#123—3162no—
237#121—3161no—
238#121—3161no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—791no—
242C2×C4⋊C4—1581no—
243C2×C4⋊C4—1581no—
244C2×C4⋊C4—1581no—
245C2×C4⋊C4C4×D7911yescharacteristic
246C2×C4⋊C4D31611yescharacteristic
247C2×C4⋊C4C22⋊D7911yescharacteristic
248C2×C4⋊C4C22×D7911yescharacteristic
249C2×C22⋊C4—1581no—
250C2×C22⋊C4—1581no—
251C2×C22⋊C4—1581no—
252C2×C22⋊C4—1581no—
253C2×C22⋊C4—1581no—
254C2×C22⋊C4—1581no—
255C22⋊C8—22no—
256C22⋊C8—41no—
257#95—791no—
258#95—791no—
259#95—1581no—
260#95—1581no—
261C24—1581no—
262C24(C2×D79)⋊C411yescharacteristic
263C2×D4—3161no—
264C2×D4—3161no—
265C2×D4—3161no—
266C2×D4—3162no—
267C2×D4—3162no—
268C2×D4—6321no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—1581no—
279C22×C4—1581no—
280C22×C4—1581no—
281C22×C4(C2×D79)⋊C411yescharacteristic
282C22×C4(C2×D79)⋊C411yescharacteristic
283C22×C4(C2×D79)⋊C411yescharacteristic
284C22×C4D63212yes—
285C22×C4#1192961312yes—
286C22×C4C2×C4×D7911yescharacteristic
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4C2×D31611yescharacteristic
289M16—3164no—
290M16—3162no—
291M16—1584no—
292C2×C8—42no—
293C2×C8—82no—
294C2×C8—3162no—
295C2×C8—3164no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—3161no—
301C4⋊C4—3162no—
302C4⋊C4—3161no—
303C4⋊C4—3161no—
304C4⋊C4—6321no—
305C22⋊C4—3162no—
306C22⋊C4—3162no—
307C22⋊C4—3162no—
308C22⋊C4—3161no—
309C22⋊C4—3161no—
310C22⋊C4—6321no—
311C22⋊C4—6321no—
312C23—41no—
313C23—41no—
314C23—3161no—
315C23—3161no—
316C23—3161no—
317C23—3161no—
318C23—3161no—
319C23—3161no—
320C23—3161no—
321C23#1457907511yescharacteristic
322C23#1457907511yescharacteristic
323C23#1457908312yes—
324C23#1457908712yes—
325C23C2×(C2×D79)⋊C411yescharacteristic
326D4—6322no—
327D4—6322no—
328D4—6322no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—3161no—
355C2×C4—3161no—
356C2×C4—3161no—
357C2×C4—3161no—
358C2×C4#1457908312yes—
359C2×C4C2×D63211yescharacteristic
360C2×C4C2×#1192961311yescharacteristic
361C8—82no—
362C8—3164no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—6321no—
372C22—6321no—
373C22—6321no—
374C22—6321no—
375C22—6321no—
376C22—6321no—
377C22—6321no—
378C22#1909829812yes—
379C22#1909829812yes—
380C22C2×#1457907511yescharacteristic
381C22C2×#1457908311yescharacteristic
382C22C2×#1457908711yescharacteristic
383C4—21no—
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—6321no—
391C4—6321no—
392C4—6321no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—12641no—
398C2C2×#1909829811yescharacteristic
399C2#2376504411yescharacteristic
400C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC2×#324×C79C2
socleC22×C79C2×#99
prime radicalC2×#14579135C4
cosocleC2×C4⋊C4×C79C23
hypercenterC2×#324D79
hypocenterC79#11698
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×#19098332C2semidirect
3. #23761407C2semidirect
4. #23761404C2semidirect
5. C2×#324×C79C2semidirect
203. C2×#324D79semidirect
206. C79#11698semidirect

Maximal subgroups

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

subgroupindex
#237691422
C2×#190983322
#237614072
#237614042
C2×#324×C792
#1169879