Finite Group Databasebeta

← #34177339

Subgroups — #34177339

Order 20224.

420 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 81 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#23770359C211yesmaximal, characteristic
2#23764877C211yesmaximal, characteristic, complemented
3#23764872C211yesmaximal, characteristic, complemented
4#23761224C211yesmaximal, characteristic, complemented
5#23761222C211yesmaximal, characteristic, complemented
6#23761221C211yesmaximal, characteristic
7C79×#2093C211yesmaximal, characteristic
8C2×#14579154—21no—
9C2×#14579094—21no—
10C2×#14579092—21no—
11#19098499—21no—
12#19098486—21no—
13#19098477—21no—
14#19098461C2211yescharacteristic
15#19098461C2211yescharacteristic
16#19098458—21no—
17#19098456—21no—
18#19098325—21no—
19#19098325—21no—
20#19098318C2211yescharacteristic
21#19098318C2211yescharacteristic
22C2×#107×C79—21no—
23C2×#102×C79—21no—
24#390×C79C2211yescharacteristic
25#384×C79C2211yescharacteristic
26#336×C79C2211yescharacteristic
27C22×C79⋊C8—21no—
28C22×C79⋊C8—21no—
29#14579154—42no—
30#14579154—42no—
31C2×C4⋊Dic79D411yescharacteristic
32C2×C4⋊Dic79D411yescharacteristic
33C2×C4⋊Dic79D411yescharacteristic
34C2×C4⋊Dic79D411yescharacteristic
35C2×Dic79⋊C4—21no—
36C2×Dic79⋊C4—21no—
37C2×Dic79⋊C4—21no—
38C2×Dic79⋊C4—21no—
39C2×C22⋊Dic79—21no—
40C2×C22⋊Dic79—21no—
41C2×C22⋊Dic79—21no—
42C2×C22⋊Dic79—21no—
43C2×C4×Dic79—21no—
44#14579094—42no—
45#14579092—42no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C22×D4×C79D411yescharacteristic
50C22×C8×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C4⋊C4×C79—21no—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79—21no—
57C2×C42×C79C2311yescharacteristic
58#107×C79—42no—
59#102×C79—42no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C2×C79⋊C8—22no—
65C2×C79⋊C8—22no—
66C2×C79⋊C8—41no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—41no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73C4⋊Dic79—41no—
74C4⋊Dic79—41no—
75C4⋊Dic79D812yes—
76Dic79⋊C4—42no—
77Dic79⋊C4—41no—
78Dic79⋊C4—41no—
79Dic79⋊C4—41no—
80Dic79⋊C4—41no—
81Dic79⋊C4—81no—
82C22⋊Dic79—42no—
83C22⋊Dic79—42no—
84C22⋊Dic79—81no—
85C22⋊Dic79—81no—
86C4×Dic79—42no—
87C24×C79—21no—
88C2×D4×C79—21no—
89C2×D4×C79—21no—
90C2×D4×C79—41no—
91C2×D4×C79—41no—
92C22×C4×C79—21no—
93C22×C4×C79—41no—
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C2×C8×C79—22no—
98C2×C8×C79—41no—
99C2×C8×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—41no—
105C4⋊C4×C79—41no—
106C4⋊C4×C79—41no—
107C22⋊C4×C79—41no—
108C22⋊C4×C79—41no—
109C22⋊C4×C79—41no—
110C22⋊C4×C79—41no—
111C42×C79—41no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79#12011yescharacteristic
120D4×C79—42no—
121D4×C79—81no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—81no—
132C2×C4×C79—81no—
133C2×C4×C79—81no—
134C2×C4×C79C2×D811yescharacteristic
135C2×C4×C79#13511yescharacteristic
136C2×C4×C79#13511yescharacteristic
137C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C79#13711yescharacteristic
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—
152C8×C79—42no—
153C79⋊C8—42no—
154C79⋊C8—42no—
155C22×C79—81no—
156C22×C79—81no—
157C22×C79—81no—
158C22×C79—81no—
159C22×C79—81no—
160C22×C79—81no—
161C22×C79—81no—
162C22×C79#44711yescharacteristic
163C22×C79#44911yescharacteristic
164C22×C79#45211yescharacteristic
165C22×C79#45311yescharacteristic
166C22×C79#45611yescharacteristic
167C22×C79#45611yescharacteristic
168C22×C79#45811yescharacteristic
169C4×C79—21no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—81no—
176C4×C79—161no—
177Dic79—42no—
178Dic79—81no—
179Dic79—81no—
180Dic79—81no—
181#7793—791nomaximal, Sylow
182C2×C79—161no—
183C2×C79#160911yescharacteristic
184C2×C79#161011yescharacteristic
185C2×C79#161811yescharacteristic
186C2×C79#162111yescharacteristic
187C2×C79#164711yescharacteristic
188C2×C79#165211yescharacteristic
189C2×C79#166011yescharacteristic
190#2727—791no—
191#2093D7911yescharacteristic
192#2084—791no—
193#2083—791no—
194#1271—791no—
195#1271—791no—
196#1270—791no—
197C79#779311yescharacteristic, complemented, Sylow
198C2×#126—1581no—
199C2×#107—21no—
200C2×#107—1581no—
201C2×#106—1581no—
202C2×#102—21no—
203#399—1581no—
204#396—1581no—
205#394—1581no—
206#390C2×D7911yescharacteristic
207#389—791no—
208#389—791no—
209#389—1581no—
210#388—1581no—
211#384C2×D7911yescharacteristic
212#340—1581no—
213#340—1581no—
214#336—791no—
215#336—791no—
216#336C2×D7911yescharacteristic
217C22×D4C22⋊D7911yescharacteristic
218C22×C8—21no—
219C22×C8—1581no—
220C22×C8—1581no—
221#126—3162no—
222#126—3162no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—791no—
227C2×C4⋊C4—791no—
228C2×C4⋊C4—791no—
229C2×C4⋊C4—791no—
230C2×C4⋊C4—1581no—
231C2×C4⋊C4—1581no—
232C2×C4⋊C4—1581no—
233C2×C4⋊C4—1581no—
234C2×C4⋊C4C22⋊D7911yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—1581no—
238C2×C22⋊C4—1581no—
239C2×C22⋊C4—1581no—
240C2×C22⋊C4—1581no—
241C2×C42—1581no—
242C2×C42C22×D7911yescharacteristic
243#107—42no—
244#107—3162no—
245#106—3162no—
246#102—42no—
247#95—1581no—
248#95—1581no—
249#95—1581no—
250C24—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—1581no—
258C22×C4—1581no—
259C22×C4—1581no—
260C22×C4—1581no—
261C22×C4D4×D7911yescharacteristic
262C22×C4D4×D7911yescharacteristic
263C22×C4C2×C22⋊D7911yescharacteristic
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—3161no—
268C2×C8—3161no—
269C2×C8—3161no—
270C2×C8—3161no—
271C2×C8—1582no—
272C2×C8—1582no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—3161no—
281C4⋊C4—3161no—
282C4⋊C4—3162no—
283C4⋊C4—3161no—
284C4⋊C4—3161no—
285C4⋊C4—3161no—
286C4⋊C4—3161no—
287C4⋊C4—792no—
288C4⋊C4—6321no—
289C4⋊C4—1581no—
290C4⋊C4—1581no—
291C4⋊C4—1581no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—3162no—
297C22⋊C4—3162no—
298C22⋊C4—6321no—
299C22⋊C4—6321no—
300C42—41no—
301C42—3162no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1457912911yescharacteristic
310D4—42no—
311D4—81no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—3161no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4—3161no—
334C2×C4—3161no—
335C2×C4—1582no—
336C2×C4D8×D7911yescharacteristic
337C2×C4#1457918011yescharacteristic
338C2×C4#1457918111yescharacteristic
339C2×C4#1457918511yescharacteristic
340C2×C4#1457918511yescharacteristic
341C2×C4#1457919211yescharacteristic
342C8—42no—
343C8—3162no—
344C8—3162no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#1909869511yescharacteristic
353C22#1909870511yescharacteristic
354C22#1909872311yescharacteristic
355C22#1909873011yescharacteristic
356C22#1909874411yescharacteristic
357C22#1909874411yescharacteristic
358C22#1909875211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—161no—
367C4—3162no—
368C4—6321no—
369C4—6321no—
370C4—6321no—
371C2—161no—
372C2#2376239411yescharacteristic
373C2#2376240111yescharacteristic
374C2#2376244611yescharacteristic
375C2#2376246211yescharacteristic
376C2#2376263411yescharacteristic
377C2#2376266911yescharacteristic
378C2#2376270911yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2093C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2093D79
hypocenterC79#7793
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #23764877C2semidirect
3. #23764872C2semidirect
4. #23761224C2semidirect
5. #23761222C2semidirect
197. C79#7793semidirect

Maximal subgroups

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

subgroupindex
#237703592
#237648772
#237648722
#237612242
#237612222
#237612212
C79×#20932
#779379