Finite Group Databasebeta

← #34191491

Subgroups — #34191491

Order 20224.

396 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 77 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#23764384C211yesmaximal, characteristic
2#23761932C211yesmaximal, characteristic
3#23761919C211yesmaximal, characteristic
4#23761306C211yesmaximal, characteristic
5#23761286C211yesmaximal, characteristic
6#23761252C211yesmaximal, characteristic
7C79×#2667C211yesmaximal, characteristic
8C2×#14579140C2211yescharacteristic
9#19098527C2211yescharacteristic
10#19098527C2211yescharacteristic
11C4×C22⋊Dic79C2211yescharacteristic
12#19098329—21no—
13#19098329—21no—
14#19098299—21no—
15#19098299—21no—
16#19098299—21no—
17#19098299—21no—
18#399×C79—21no—
19#398×C79—21no—
20#398×C79—21no—
21#396×C79—21no—
22#389×C79—21no—
23#387×C79—21no—
24#381×C79C2211yescharacteristic
25#380×C79C2211yescharacteristic
26C4×C22⋊C4×C79C2211yescharacteristic
27C22×Dic158Q811yescharacteristic
28C2×#11929608—21no—
29C2×#11929608—21no—
30#14579140—21no—
31#14579140—21no—
32#14579140—21no—
33#14579140—21no—
34C2×C4⋊Dic79Q811yescharacteristic
35C2×Dic79⋊C4—21no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C4×Dic79—21no—
39C22⋊(C79⋊C8)—21no—
40C22⋊(C79⋊C8)—21no—
41C22⋊(C79⋊C8)—21no—
42C22⋊(C79⋊C8)—21no—
43C79⋊#95—21no—
44C23×C4×C79C2311yescharacteristic
45C2×C4⋊C4×C79—21no—
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×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C42×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79D411yescharacteristic
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64#95×C79D411yescharacteristic
65C2×Dic158—21no—
66C2×Dic158—21no—
67C2×Dic158—81no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—41no—
71#11929608—81no—
72#11929608—81no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C4⋊Dic79—21no—
76C4⋊Dic79—21no—
77Dic79⋊C4—41no—
78Dic79⋊C4—41no—
79C22⋊Dic79—22no—
80C22⋊Dic79—41no—
81C22⋊Dic79—41no—
82C22⋊Dic79—41no—
83C4×Dic79—81no—
84C24×C79C2×D411yescharacteristic
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—81no—
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×Q811yescharacteristic
95C22×C4×C79#4111yescharacteristic
96C22×C4×C79#4111yescharacteristic
97C22×C4×C79#4111yescharacteristic
98C4⋊C4×C79—42no—
99C4⋊C4×C79—42no—
100C4⋊C4×C79—81no—
101C4⋊C4×C79—81no—
102C4⋊C4×C79—81no—
103C4⋊C4×C79—81no—
104C4⋊C4×C79—81no—
105C22⋊C4×C79—42no—
106C22⋊C4×C79—41no—
107C22⋊C4×C79—41no—
108C22⋊C4×C79—81no—
109C42×C79—81no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79#12011yescharacteristic
115C23×C79C22⋊Q811yescharacteristic
116C23×C79#12411yescharacteristic
117C2×C4×C79—22no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
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×C79C22⋊Q811yescharacteristic
142C2×C4×C79C22⋊Q811yescharacteristic
143C2×C4×C79#12411yescharacteristic
144C2×C4×C79#12411yescharacteristic
145C2×Dic79—42no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—81no—
149C2×Dic79—81no—
150C2×Dic79—81no—
151C2×Dic79—81no—
152Dic158—81no—
153Dic158—81no—
154C79⋊C8—81no—
155C79⋊C8—81no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—81no—
165C22×C79—81no—
166C22×C79#39311yescharacteristic
167C22×C79#45511yescharacteristic
168C22×C79#45611yescharacteristic
169C4×C79—21no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—42no—
174C4×C79—81no—
175C4×C79—81no—
176C4×C79—81no—
177C4×C79—81no—
178C4×C79—81no—
179Dic79—82no—
180Dic79—81no—
181Dic79—81no—
182#10385—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79—41no—
186C2×C79—41no—
187C2×C79#202811yescharacteristic
188C2×C79#203111yescharacteristic
189C2×C79#203211yescharacteristic
190#2667D7911yescharacteristic
191#1996—791no—
192#1511—791no—
193#1508—791no—
194#1305—791no—
195#1297—791no—
196#1284—791no—
197C79#1038511yescharacteristic, complemented, Sylow
198C2×C22⋊Q8—791no—
199#407—791no—
200#407—791no—
201#399—21no—
202#398—21no—
203#398—21no—
204#396—21no—
205#389—21no—
206#387—21no—
207#381C2×D7911yescharacteristic
208#380C2×D7911yescharacteristic
209C4×C22⋊C4—791no—
210C4×C22⋊C4C2×D7911yescharacteristic
211#342—1581no—
212#342—1581no—
213#328—1581no—
214#328—1581no—
215#328—1581no—
216#328—1581no—
217C22×Q8—791no—
218C23×C4C22×D7911yescharacteristic
219C2×M16—1581no—
220C2×M16—1581no—
221C22⋊Q8—1581no—
222C22⋊Q8—1581no—
223C22⋊Q8—1581no—
224C22⋊Q8—1581no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—791no—
233C2×C4⋊C4—1581no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—1581no—
237C2×C22⋊C4—1581no—
238C2×C22⋊C4C22⋊D7911yescharacteristic
239C2×C22⋊C4C22⋊D7911yescharacteristic
240C2×C42—21no—
241C2×C42—1581no—
242C22⋊C8—1581no—
243C22⋊C8—1581no—
244C22⋊C8—1581no—
245C22⋊C8—1581no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—1581no—
251#95C22⋊D7911yescharacteristic
252#95C22⋊D7911yescharacteristic
253#95C22⋊D7911yescharacteristic
254#95C22⋊D7911yescharacteristic
255C24C2×C22⋊D7911yescharacteristic
256C2×Q8—6321no—
257C2×Q8—1581no—
258C2×Q8—1581no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—81no—
266C22×C4—3161no—
267C22×C4—1581no—
268C22×C4—1581no—
269C22×C4C2×C22⋊D7911yescharacteristic
270C22×C4C2×C22⋊D7911yescharacteristic
271C22×C4Q8×D7911yescharacteristic
272C22×C4#1192962511yescharacteristic
273C22×C4#1192962511yescharacteristic
274C22×C4#1192962511yescharacteristic
275M16—6321no—
276M16—6321no—
277C2×C8—3161no—
278C2×C8—3161no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—3161no—
287C4⋊C4—3161no—
288C4⋊C4—1581no—
289C4⋊C4—1581no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—81no—
294C22⋊C4—3161no—
295C22⋊C4—3161no—
296C22⋊C4—3161no—
297C22⋊C4—1582no—
298C42—81no—
299C42—6321no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23#1457913011yescharacteristic
305C23#1457913711yescharacteristic
306C23#1457915011yescharacteristic
307Q8—6321no—
308Q8—6321no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
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—41no—
333C2×C4—3162no—
334C2×C4—3161no—
335C2×C4—3161no—
336C2×C4—6321no—
337C2×C4—6321no—
338C2×C4—6321no—
339C2×C4—6321no—
340C2×C4#1457913711yescharacteristic
341C2×C4#1457913711yescharacteristic
342C2×C4#1457915011yescharacteristic
343C2×C4#1457915011yescharacteristic
344C8—6321no—
345C8—6321no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22#1909847411yescharacteristic
357C22#1909874111yescharacteristic
358C22#1909874611yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—42no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—6322no—
370C4—6321no—
371C4—6321no—
372C2—21no—
373C2—21no—
374C2—41no—
375C2—41no—
376C2#2376455811yescharacteristic
377C2#2376456911yescharacteristic
378C2#2376457611yescharacteristic
subgroupG/H
centerC22#19098474
commutator, cosocleC23×C4×C79C23
FrattiniC23×C4C22×D79
FittingC79×#2667C2
socleC22×C79#393
prime radicalC24×C79C2×D4
hypercenter#2667D79
hypocenterC79#10385
LCSG ▷ C23×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2667

Complements

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

NH ≅ G/Ntype
197. C79#10385semidirect

Maximal subgroups

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

subgroupindex
#237643842
#237619322
#237619192
#237613062
#237612862
#237612522
C79×#26672
#1038579