Finite Group Databasebeta

← #34206933

Subgroups — #34206933

Order 20224.

488 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 99 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#23768743C211yesmaximal, characteristic
2C4×#14579076C211yesmaximal, characteristic
3#23761519C212yesmaximal
4#23761519C212yesmaximal
5C79×#1800C211yesmaximal, characteristic
6C22×Dic79⋊C4C411yescharacteristic, complemented
7C2×#14579076C2212yes—
8#19098496—21no—
9#19098486—21no—
10#19098447—21no—
11#19098443C2211yescharacteristic
12#19098443C2211yescharacteristic
13#19098436—21no—
14#19098435C411yescharacteristic, complemented
15#19098435C411yescharacteristic, complemented
16#19098423C411yescharacteristic, complemented
17C4×C22⋊Dic79—21no—
18#19098330—21no—
19C2×C4×D4×C79C2211yescharacteristic
20#423×C79—21no—
21C2×#100×C79C2212yes—
22#343×C79—21no—
23C23×Dic79C2×C411yescharacteristic
24C23×Dic79C2×C411yescharacteristic
25C2×C4⋊Dic79—21no—
26C2×Dic79⋊C4—21no—
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—41no—
31C2×Dic79⋊C4—41no—
32C2×C22⋊Dic79—22no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79C2×C411yescharacteristic
36C2×C22⋊Dic79C2×C411yescharacteristic
37C2×C4×Dic79—41no—
38#14579076—22no—
39#14579076Q814yes—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—41no—
46C22×D4×C79C2311yescharacteristic
47C23×C4×C79C2×C411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C2×M16×C79—22no—
50C4×D4×C79—41no—
51C4×D4×C79—41no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79C2×C411yescharacteristic
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C42×C79D411yescharacteristic
56C4⋊C8×C79—41no—
57C4⋊C8×C79—41no—
58#100×C79—24no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—41no—
64C22×Dic79—42no—
65C22×Dic79—41no—
66C22×Dic79—42no—
67C22×Dic79—81no—
68C4⋊Dic79—81no—
69Dic79⋊C4—42no—
70Dic79⋊C4—42no—
71Dic79⋊C4—41no—
72Dic79⋊C4—41no—
73Dic79⋊C4—81no—
74C22⋊Dic79—24no—
75C22⋊Dic79—41no—
76C22⋊Dic79—41no—
77C22⋊Dic79—42no—
78C22⋊Dic79—42no—
79C22⋊Dic79—81no—
80C4×Dic79—82no—
81C24×C79C22×C411yescharacteristic
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—81no—
92C22×C4×C79—81no—
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C22⋊C411yescharacteristic
96C22×C4×C79C22⋊C411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98M16×C79—42no—
99M16×C79—42no—
100C2×C8×C79—42no—
101C4⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C22⋊C4×C79—41no—
104C42×C79—21no—
105C42×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—41no—
111C23×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79C2×C22⋊C411yescharacteristic
115C23×C79C4×Q812yes—
116C23×C79C22⋊Q812yes—
117D4×C79—42no—
118D4×C79—42no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—41no—
125C2×C4×C79—42no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79#9911yescharacteristic
133C2×C4×C79#9911yescharacteristic
134C2×C4×C79#12411yescharacteristic
135C2×C4×C79C4⋊Q811yescharacteristic
136C2×Dic79—22no—
137C2×Dic79—41no—
138C2×Dic79—42no—
139C2×Dic79—41no—
140C2×Dic79—42no—
141C2×Dic79—44no—
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—82no—
152C2×Dic79—82no—
153C2×Dic79—82no—
154C8×C79—82no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—22no—
158C22×C79—22no—
159C22×C79—42no—
160C22×C79—42no—
161C22×C79—42no—
162C22×C79—41no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—81no—
167C22×C79#39111yescharacteristic
168C22×C79C2×#9911yescharacteristic
169C22×C79#41011yescharacteristic
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—41no—
175C4×C79—41no—
176C4×C79—41no—
177Dic79—42no—
178Dic79—81no—
179Dic79—81no—
180Dic79—84no—
181Dic79—81no—
182#13297—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79—42no—
186C2×C79—42no—
187C2×C79#199611yescharacteristic
188C2×C79#212011yescharacteristic
189C2×C79#212111yescharacteristic
190#2579—791no—
191#1800D7911yescharacteristic
192C4×#99—791no—
193#1390—792no—
194#1390—792no—
195C79#1329711yescharacteristic, complemented, Sylow
196C2×C4×D4C2×D7911yescharacteristic
197C22×C4⋊C4—791no—
198#423—21no—
199C2×#100C2×D7912yes—
200C2×#99—792no—
201#398—1581no—
202#396—1581no—
203#387—1581no—
204#386—791no—
205#386—791no—
206#385—791no—
207#385—791no—
208#385—1581no—
209#380—791no—
210C4×C22⋊C4—1581no—
211#343—21no—
212#342—1581no—
213C22×D4C22×D7911yescharacteristic
214C23×C4—791no—
215C23×C4—791no—
216C23×C4C4×D7911yescharacteristic
217C23×C4D31611yescharacteristic
218C2×M16—22no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—3161no—
222C2×C4⋊C4—3161no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4—1581no—
225C2×C4⋊C4—1581no—
226C2×C4⋊C4—1581no—
227C2×C4⋊C4—1581no—
228C2×C4⋊C4C22⋊D7911yescharacteristic
229C2×C22⋊C4—791no—
230C2×C22⋊C4—791no—
231C2×C22⋊C4—1581no—
232C2×C22⋊C4—1582no—
233C2×C22⋊C4—1581no—
234C2×C22⋊C4C4×D7911yescharacteristic
235C2×C22⋊C4D31611yescharacteristic
236C2×C42—3161no—
237C2×C42C22⋊D7911yescharacteristic
238C4⋊C8—41no—
239C4⋊C8—41no—
240#100—24no—
241#99—794no—
242#99—1582no—
243#95—3161no—
244#95—1581no—
245#95—1581no—
246#95—1581no—
247#95—1581no—
248#95—1581no—
249C24C2×C4×D7911yescharacteristic
250C24C2×D31611yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Q8×D7912yes—
254C2×D4#1192962312yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—3161no—
262C22×C4—3162no—
263C22×C4—3162no—
264C22×C4—3161no—
265C22×C4—6321no—
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×C22⋊D7911yescharacteristic
275M16—42no—
276M16—42no—
277C2×C8—42no—
278C4⋊C4—41no—
279C4⋊C4—3162no—
280C4⋊C4—3162no—
281C4⋊C4—3161no—
282C4⋊C4—3161no—
283C4⋊C4—6321no—
284C4⋊C4—6321no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—3162no—
288C22⋊C4—3162no—
289C22⋊C4—3161no—
290C22⋊C4—3161no—
291C22⋊C4—6321no—
292C22⋊C4—1584no—
293C42—21no—
294C42—21no—
295C42—6322no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×(C2×D79)⋊C411yescharacteristic
305C23#1457912612yes—
306C23#1457914112yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3161no—
323C2×C4—3162no—
324C2×C4—3161no—
325C2×C4—3162no—
326C2×C4—3161no—
327C2×C4—3164no—
328C2×C4—3161no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4—3161no—
334C2×C4—3161no—
335C2×C4—3161no—
336C2×C4—6322no—
337C2×C4—6322no—
338C2×C4—6322no—
339C2×C4—1582no—
340C2×C4#1457907511yescharacteristic
341C2×C4#1457907511yescharacteristic
342C2×C4#1457914811yescharacteristic
343C2×C4#1457915911yescharacteristic
344C8—82no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1909846811yescharacteristic
358C22C2×#1457907511yescharacteristic
359C22#1909853611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—3162no—
368C4—6321no—
369C4—6321no—
370C4—6324no—
371C4—6321no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2376438411yescharacteristic
377C2#2376504211yescharacteristic
378C2#2376504711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC79×#1800C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenter#1800D79
hypocenterC79#13297
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
6. C22×Dic79⋊C4C4semidirect
14. #19098435C4semidirect
15. #19098435C4semidirect
16. #19098423C4semidirect
195. C79#13297semidirect

Maximal subgroups

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

subgroupindex
#237687432
C4×#145790762
#237615192
C79×#18002
#1329779