Finite Group Databasebeta

← #34201179

Subgroups — #34201179

Order 20224.

448 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 59 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#23764745C211yesmaximal, characteristic, complemented
2#23762373C211yesmaximal, characteristic, complemented
3#23762339C211yesmaximal, characteristic
4#23761430C211yesmaximal, characteristic, complemented
5#23761403C211yesmaximal, characteristic
6#23761346C211yesmaximal, characteristic, complemented
7C79×#2680C211yesmaximal, characteristic
8#19099194C2211yescharacteristic
9#19098719—21no—
10#19098714—21no—
11#19098603—21no—
12#19098332C2211yescharacteristic
13#19098323—21no—
14#19098299—21no—
15#19098299—21no—
16#19098289C2211yescharacteristic
17#19098289C2211yescharacteristic
18C2×#123×C79—21no—
19C2×C22⋊Q8×C79—21no—
20#398×C79—21no—
21#397×C79—21no—
22#395×C79C2211yescharacteristic
23#394×C79C2211yescharacteristic
24#388×C79—21no—
25#385×C79—21no—
26#384×C79C2211yescharacteristic
27C2×#11929614—41no—
28C2×#11929612—41no—
29C2×#11929608—21no—
30#14579160—41no—
31#14579152—41no—
32#14579140—21no—
33#14579140—21no—
34#14579120—21no—
35#14579094—41no—
36#14579092—41no—
37#14579085—41no—
38#14579082—41no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22×Q8×C79D411yescharacteristic
42C22×D4×C79D411yescharacteristic
43C23×C4×C79—21no—
44#123×C79—44no—
45C22⋊Q8×C79—44no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79C2311yescharacteristic
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79—21no—
57C2×C22⋊C4×C79—21no—
58C2×C42×C79D411yescharacteristic
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C2×Dic158—21no—
65C2×Dic158—21no—
66#11929614—82no—
67#11929612—82no—
68#11929608—22no—
69#11929608—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C2×C79⋊C8—41no—
73C4⋊Dic79—21no—
74C4⋊Dic79—21no—
75C4⋊Dic79—21no—
76C4⋊Dic79—21no—
77Dic79⋊C4—41no—
78Dic79⋊C4—41no—
79C22⋊Dic79—41no—
80C4×Dic79—41no—
81C24×C79—21no—
82C2×Q8×C79—42no—
83C2×Q8×C79—41no—
84C2×D4×C79—41no—
85C2×D4×C79—42no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—42no—
92C22×C4×C79—42no—
93C22×C4×C79—81no—
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C4⋊C4×C79—41no—
98C4⋊C4×C79—42no—
99C4⋊C4×C79—42no—
100C4⋊C4×C79—42no—
101C4⋊C4×C79—42no—
102C4⋊C4×C79—42no—
103C4⋊C4×C79—41no—
104C22⋊C4×C79—24no—
105C22⋊C4×C79—42no—
106C22⋊C4×C79—42no—
107C22⋊C4×C79—42no—
108C22⋊C4×C79—41no—
109C22⋊C4×C79—41no—
110C42×C79—21no—
111C42×C79—21no—
112C23×C79—22no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79#12011yescharacteristic
120Q8×C79—42no—
121Q8×C79—81no—
122D4×C79—42no—
123D4×C79—81no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—22no—
127C2×C4×C79—22no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
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—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—84no—
146C2×C4×C79#13511yescharacteristic
147C2×C4×C79#13711yescharacteristic
148C2×Dic79—41no—
149C2×Dic79—41no—
150Dic158—81no—
151Dic158—81no—
152C79⋊C8—42no—
153C79⋊C8—81no—
154C79⋊C8—81no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—42no—
158C22×C79—42no—
159C22×C79—42no—
160C22×C79—81no—
161C22×C79—81no—
162C22×C79—81no—
163C22×C79—81no—
164C22×C79#45211yescharacteristic
165C22×C79#45611yescharacteristic
166C22×C79B(4,𝔽2)11yescharacteristic
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—42no—
170C4×C79—42no—
171C4×C79—41no—
172C4×C79—41no—
173C4×C79—81no—
174C4×C79—81no—
175C4×C79—81no—
176Dic79—81no—
177Dic79—81no—
178#12234—791nomaximal, Sylow
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—82no—
182C2×C79#160311yescharacteristic
183C2×C79#219611yescharacteristic
184C2×C79#220311yescharacteristic
185#2680D7911yescharacteristic
186#2060—791no—
187#1605—791no—
188#1599—791no—
189#1354—791no—
190#1345—791no—
191#1325—791no—
192C79#1223411yescharacteristic, complemented, Sylow
193#544—791no—
194C2×#123—21no—
195C2×C22⋊Q8—21no—
196#451—1581no—
197#450—1581no—
198#428—1581no—
199#398—21no—
200#397—21no—
201#395C2×D7911yescharacteristic
202#394C2×D7911yescharacteristic
203#388—21no—
204#385—21no—
205#384C2×D7911yescharacteristic
206#343—791no—
207#339—1581no—
208#328—1581no—
209#328—1581no—
210#324—791no—
211#324—791no—
212C22×Q8C22⋊D7911yescharacteristic
213C22×D4C22⋊D7911yescharacteristic
214C23×C4—21no—
215C2×Q16—3161no—
216C2×SD16—3161no—
217C2×M16—1581no—
218C4⋊Q8—3161no—
219#125—3161no—
220#123—44no—
221C22⋊Q8—44no—
222C22⋊Q8—1581no—
223C22⋊Q8—1581no—
224#117—1581no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊D7911yescharacteristic
231C2×C4⋊C4C22×D7911yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42C22⋊D7911yescharacteristic
238#107—3161no—
239#106—3161no—
240#103—3161no—
241#102—3161no—
242C22⋊C8—1581no—
243C22⋊C8—1581no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95C22⋊D7911yescharacteristic
248#95C22⋊D7911yescharacteristic
249C24—21no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—1581no—
253C2×Q8—1581no—
254C2×D4—41no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4C2×C22⋊D7911yescharacteristic
265C22×C4C2×C22⋊D7911yescharacteristic
266C22×C4C2×C22⋊D7911yescharacteristic
267Q16—6322no—
268SD16—6322no—
269M16—3161no—
270M16—1582no—
271C2×C8—3161no—
272C2×C8—3161no—
273C2×C8—3161no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—3161no—
282C4⋊C4—3161no—
283C4⋊C4—1581no—
284C4⋊C4—1581no—
285C4⋊C4—1581no—
286C4⋊C4—1581no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—3161no—
294C42—21no—
295C42—21no—
296C42—3161no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#1457913011yescharacteristic
305Q8—42no—
306Q8—81no—
307Q8—6321no—
308Q8—6321no—
309D4—42no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4—3161no—
334C2×C4—3161no—
335C2×C4#1457918111yescharacteristic
336C2×C4#1457919111yescharacteristic
337C8—3162no—
338C8—6321no—
339C8—6321no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#1909872511yescharacteristic
350C22#1909874611yescharacteristic
351C22#1909875011yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—6321no—
362C4—6321no—
363C2—21no—
364C2—21no—
365C2—82no—
366C2#2376236311yescharacteristic
367C2#2376538411yescharacteristic
368C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2680C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2680D79
hypocenterC79#12234
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2680

Complements

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

NH ≅ G/Ntype
1. #23764745C2semidirect
2. #23762373C2semidirect
4. #23761430C2semidirect
6. #23761346C2semidirect
192. C79#12234semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237623732
#237623392
#237614302
#237614032
#237613462
C79×#26802
#1223479