Finite Group Databasebeta

← #34206948

Subgroups — #34206948

Order 20224.

492 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 103 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#23768596C211yesmaximal, characteristic
2#23763388C211yesmaximal, characteristic
3#23761524C212yesmaximal
4#23761524C212yesmaximal
5C4×#99×C79C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic, complemented
7C2×#14579076C2212yes—
8#19098486—21no—
9#19098486—21no—
10#19098466C2211yescharacteristic
11#19098466C2211yescharacteristic
12#19098436—21no—
13#19098436—21no—
14#19098432C411yescharacteristic, complemented
15#19098428C411yescharacteristic, complemented
16#19098423—21no—
17C4×C4⋊Dic79C411yescharacteristic, complemented
18#19098330—21no—
19C2×C4×D4×C79C2211yescharacteristic
20C2×#99×C79C2212yes—
21C4×C22⋊C4×C79—21no—
22#342×C79—21no—
23#14579153—41no—
24#14579153—41no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—22no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C4×Dic79C2×C411yescharacteristic
39C2×C4×Dic79C2×C411yescharacteristic
40#14579076—24no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—41no—
45C22×D4×C79C2311yescharacteristic
46C23×C4×C79D411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C4×D4×C79—41no—
49C4×D4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79—22no—
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C42×C79—41no—
55C2×C42×C79C2×C411yescharacteristic
56#99×C79—22no—
57#99×C79Q814yes—
58#95×C79—41no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—41no—
66C4⋊Dic79—42no—
67C4⋊Dic79—82no—
68Dic79⋊C4—42no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71Dic79⋊C4—81no—
72Dic79⋊C4—81no—
73C22⋊Dic79—42no—
74C22⋊Dic79—42no—
75C22⋊Dic79—42no—
76C22⋊Dic79—42no—
77C22⋊Dic79—81no—
78C22⋊Dic79—81no—
79C4×Dic79—41no—
80C4×Dic79—41no—
81C24×C79C2×D411yescharacteristic
82C24×C79C2×D411yescharacteristic
83C2×D4×C79—41no—
84C2×D4×C79—41no—
85C2×D4×C79C2×Q812yes—
86C2×D4×C79#4112yes—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—42no—
92C22×C4×C79—81no—
93C22×C4×C79—81no—
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C22⋊C411yescharacteristic
96C22×C4×C79C22⋊C411yescharacteristic
97C22×C4×C79C22⋊C411yescharacteristic
98C22×C4×C79C22×C411yescharacteristic
99C4⋊C4×C79—41no—
100C22⋊C4×C79—24no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C22⋊C4×C79—42no—
104C42×C79—21no—
105C42×C79—21no—
106C42×C79—82no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79C2×C22⋊C411yescharacteristic
116C23×C79C22⋊Q812yes—
117C23×C79#12411yescharacteristic
118C23×C79C4⋊Q811yescharacteristic
119D4×C79—42no—
120D4×C79—42no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—41no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—44no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79—82no—
136C2×C4×C79—82no—
137C2×C4×C79#9911yescharacteristic
138C2×C4×C79#9911yescharacteristic
139C2×C4×C79C4×Q812yes—
140C2×Dic79—22no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—82no—
153C2×Dic79—82no—
154C2×Dic79—82no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—22no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—42no—
161C22×C79—41no—
162C22×C79—41no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—41no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79#39111yescharacteristic
170C22×C79C2×#9911yescharacteristic
171C22×C79#41011yescharacteristic
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—22no—
175C4×C79—41no—
176C4×C79—41no—
177C4×C79—41no—
178C4×C79—84no—
179Dic79—42no—
180Dic79—81no—
181Dic79—81no—
182Dic79—81no—
183Dic79—162no—
184#13300—791nomaximal, Sylow
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—41no—
188C2×C79—42no—
189C2×C79—41no—
190C2×C79#199611yescharacteristic
191C2×C79#212611yescharacteristic
192C2×C79#212711yescharacteristic
193#2566—791no—
194#1799—791no—
195C4×#99D7911yescharacteristic
196#1393—792no—
197#1393—792no—
198C79#1330011yescharacteristic, complemented, Sylow
199C2×#125—791no—
200C2×C4×D4C2×D7911yescharacteristic
201C2×#99—792no—
202C2×#99C2×D7912yes—
203#396—1581no—
204#396—1581no—
205#390—791no—
206#390—791no—
207#385—1581no—
208#385—1581no—
209#384—791no—
210#382—791no—
211#380—1581no—
212C4×C4⋊C4—791no—
213C4×C22⋊C4—21no—
214#342—21no—
215#342—1581no—
216C22×D4C22×D7911yescharacteristic
217C23×C4D31611yescharacteristic
218C23×C4C22⋊D7911yescharacteristic
219#125—3161no—
220#125—3161no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—3161no—
224C2×C4⋊C4—791no—
225C2×C4⋊C4—791no—
226C2×C4⋊C4—1581no—
227C2×C4⋊C4—1581no—
228C2×C4⋊C4—1581no—
229C2×C4⋊C4—1581no—
230C2×C4⋊C4—1581no—
231C2×C4⋊C4C4×D7911yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1582no—
234C2×C22⋊C4—1581no—
235C2×C22⋊C4—1581no—
236C2×C22⋊C4—1581no—
237C2×C22⋊C4—1581no—
238C2×C22⋊C4D31611yescharacteristic
239C2×C22⋊C4C22⋊D7911yescharacteristic
240C2×C42—41no—
241C2×C42—791no—
242C2×C42—791no—
243C2×C42C4×D7911yescharacteristic
244#99—22no—
245#99—1584no—
246#99Dic15814yes—
247#95—41no—
248#95—3161no—
249#95—1581no—
250#95—1581no—
251#95—1581no—
252C24C2×C22⋊D7911yescharacteristic
253C24C2×D31611yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4C2×Dic15812yes—
257C2×D4#1192962412yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—3162no—
266C22×C4—1581no—
267C22×C4—1581no—
268C22×C4—1581no—
269C22×C4—1581no—
270C22×C4(C2×D79)⋊C411yescharacteristic
271C22×C4(C2×D79)⋊C411yescharacteristic
272C22×C4(C2×D79)⋊C411yescharacteristic
273C22×C4(C2×D79)⋊C411yescharacteristic
274C22×C4C2×C4×D7911yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—3161no—
277C4⋊C4—3161no—
278C4⋊C4—3162no—
279C4⋊C4—3162no—
280C4⋊C4—3161no—
281C4⋊C4—3161no—
282C4⋊C4—6321no—
283C4⋊C4—6321no—
284C4⋊C4—6322no—
285C22⋊C4—24no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—3162no—
290C22⋊C4—3162no—
291C22⋊C4—3162no—
292C22⋊C4—3162no—
293C22⋊C4—6321no—
294C22⋊C4—6321no—
295C42—21no—
296C42—21no—
297C42—82no—
298C42—3161no—
299C42—3161no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×(C2×D79)⋊C411yescharacteristic
309C23#1457914012yes—
310C23#1457914911yescharacteristic
311C23#1457916111yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—44no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4—3161no—
334C2×C4—3161no—
335C2×C4—3161no—
336C2×C4—3161no—
337C2×C4—3161no—
338C2×C4—3161no—
339C2×C4—3161no—
340C2×C4—3161no—
341C2×C4—6322no—
342C2×C4—6322no—
343C2×C4—6322no—
344C2×C4—1582no—
345C2×C4#1457907511yescharacteristic
346C2×C4#1457907511yescharacteristic
347C2×C4C4×Dic15812yes—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22#1909846911yescharacteristic
363C22C2×#1457907511yescharacteristic
364C22#1909853611yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—84no—
372C4—3162no—
373C4—6321no—
374C4—6321no—
375C4—6321no—
376C4—12642no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2#2376438511yescharacteristic
383C2#2376506311yescharacteristic
384C2#2376506711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC22×D4C22×D79
FittingC4×#99×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#99D79
hypocenterC79#13300
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
6. C2×#14579153C4semidirect
14. #19098432C4semidirect
15. #19098428C4semidirect
17. C4×C4⋊Dic79C4semidirect
198. C79#13300semidirect

Maximal subgroups

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

subgroupindex
#237685962
#237633882
#237615242
C4×#99×C792
#1330079