Finite Group Databasebeta

← #34184047

Subgroups — #34184047

Order 20224.

692 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 143 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#23771731C211yesmaximal, characteristic
2C2×#19098429C211yesmaximal, characteristic, complemented
3C2×#19098298C212yesmaximal
4#23761280C212yesmaximal, complemented
5C79×#1817C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C2211yescharacteristic
8C22×C4⋊Dic79—21no—
9#19098465C412yescomplemented
10#19098461C412yescomplemented
11#19098454—22no—
12#19098442—21no—
13#19098429—24no—
14#19098423C2211yescharacteristic
15C2×C79⋊#95—21no—
16C2×C79⋊#95C2212yes—
17#19098298—24no—
18C22×C42×C79C2211yescharacteristic
19C2×C22⋊C8×C79C2212yes—
20#336×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23C23×Dic79—22no—
24#14579144—42no—
25#14579144—42no—
26#14579135—22no—
27#14579135—22no—
28C2×C4⋊Dic79—24no—
29C2×C4⋊Dic79—42no—
30C2×C4⋊Dic79—42no—
31C2×C4⋊Dic79C2×C411yescharacteristic
32C2×C4⋊Dic79C2×C412yes—
33C2×Dic79⋊C4—22no—
34C2×C22⋊Dic79—21no—
35C2×(C2×D79)⋊C4—22no—
36C2×(C2×D79)⋊C4—22no—
37C2×(C2×D79)⋊C4—21no—
38C79⋊#95—24no—
39C79⋊#95—22no—
40C79⋊#95—22no—
41C79⋊#95—21no—
42C79⋊#95D414yes—
43C23×C4×C79D411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C23×C4×C79C2311yescharacteristic
46C22×C8×C79—22no—
47C2×C42×C79—22no—
48C2×C42×C79—22no—
49C2×C42×C79C2×C412yes—
50C2×C42×C79D412yes—
51C22⋊C8×C79—22no—
52C22⋊C8×C79—22no—
53C23×D79—21no—
54C2×D316—22no—
55C2×D316—42no—
56C2×C22⋊D79—41no—
57C2×C22⋊D79—41no—
58C2×C22⋊D79—41no—
59C2×C22⋊D79—41no—
60C2×C22⋊D79—41no—
61C2×C22⋊D79—41no—
62C22×Dic79—22no—
63C22×Dic79—21no—
64C22×Dic79—24no—
65C22×Dic79—44no—
66C22×Dic79—44no—
67C22×Dic79—42no—
68C22×Dic79—42no—
69C4⋊Dic79—22no—
70C4⋊Dic79—42no—
71C4⋊Dic79—48no—
72Dic79⋊C4—44no—
73C22⋊Dic79—41no—
74C22⋊Dic79—41no—
75(C2×D79)⋊C4—44no—
76(C2×D79)⋊C4—44no—
77(C2×D79)⋊C4—42no—
78C24×C79C2×D411yescharacteristic
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—42no—
86C22×C4×C79—42no—
87C22×C4×C79C22⋊C412yes—
88C22×C4×C79C22⋊C412yes—
89C22×C4×C79D812yes—
90C22×C4×C79SD1612yes—
91C22×C4×C79C22×C411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D412yes—
94C22×C4×C79#4112yes—
95C2×C8×C79—44no—
96C2×C8×C79—42no—
97C42×C79—24no—
98C42×C79—24no—
99C23×C79—22no—
100C23×C79—22no—
101C23×C79C2×C22⋊C411yescharacteristic
102C23×C79#12011yescharacteristic
103C23×C79#12311yescharacteristic
104C23×C79C2×D811yescharacteristic
105C23×C79C2×SD1611yescharacteristic
106C23×C79AGL(1,ℤ/8)11yescharacteristic
107C23×C79#13711yescharacteristic
108C22×D79—41no—
109C22×D79—42no—
110C22×D79—41no—
111C22×D79—41no—
112C22×D79—42no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—22no—
123C2×C4×C79—22no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79#10214yes—
129C2×C4×C79C4×D412yes—
130C2×C4×C79#12112yes—
131C22⋊D79—81no—
132C22⋊D79—81no—
133C22⋊D79—81no—
134C22⋊D79—81no—
135C2×Dic79—44no—
136C2×Dic79—44no—
137C2×Dic79—41no—
138C2×Dic79—42no—
139C2×Dic79—42no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—42no—
143C2×Dic79—44no—
144C2×Dic79—84no—
145C2×Dic79—84no—
146D316—82no—
147C8×C79—82no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—22no—
153C22×C79—22no—
154C22×C79—22no—
155C22×C79—22no—
156C22×C79#38611yescharacteristic
157C22×C79#41012yes—
158C22×C79C2×#10211yescharacteristic
159C22×C79#41711yescharacteristic
160C22×C79#42112yes—
161C22×C79#44711yescharacteristic
162C22×C79#44811yescharacteristic
163C22×C79#48011yescharacteristic
164C22×C79#48111yescharacteristic
165C2×D79—81no—
166C2×D79—81no—
167C2×D79—81no—
168C2×D79—82no—
169C2×D79—82no—
170C4×C79—24no—
171C4×C79—22no—
172C4×C79—22no—
173C4×C79—42no—
174Dic79—84no—
175Dic79—81no—
176Dic79—82no—
177#9156—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—21no—
182C2×C79#151212yes—
183C2×C79#151712yes—
184C2×C79#187411yescharacteristic
185C2×C79#189711yescharacteristic
186C2×C79#189911yescharacteristic
187D79—161no—
188#2854—791no—
189C2×#382—791no—
190#1817D7911yescharacteristic, complemented
191C2×#327—792no—
192#1294—792no—
193C79#915611yescharacteristic, complemented, Sylow
194C2×#123—1581no—
195C2×#121—791no—
196C22×C4⋊C4—1581no—
197C22×C42C2×D7911yescharacteristic
198C2×C22⋊C8C2×D7912yes—
199#390—792no—
200#389—792no—
201#388—1582no—
202#386—1581no—
203#382—1584no—
204#380—791no—
205C2×#95—792no—
206C2×#95—1581no—
207#336—22no—
208#327—1584no—
209C22×D4—791no—
210C22×D4—1581no—
211C23×C4—1582no—
212C23×C4D31611yescharacteristic
213C23×C4D31611yescharacteristic
214C23×C4C22×D7911yescharacteristic
215C22×C8—22no—
216#123—3162no—
217#123—3162no—
218#121—1582no—
219#121—1582no—
220C2×C4⋊C4—3162no—
221C2×C4⋊C4—3162no—
222C2×C4⋊C4—792no—
223C2×C4⋊C4—791no—
224C2×C4⋊C4—1584no—
225C2×C4⋊C4—1582no—
226C2×C22⋊C4—1581no—
227C2×C22⋊C4—1582no—
228C2×C22⋊C4—1582no—
229C2×C22⋊C4—1581no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D7912yes—
233C2×C42C22⋊D7912yes—
234C22⋊C8—22no—
235C22⋊C8—22no—
236#95—794no—
237#95—1584no—
238#95—1582no—
239#95—1582no—
240#95—1581no—
241C24—1581no—
242C24C2×D31611yescharacteristic
243C2×D4—3162no—
244C2×D4—3161no—
245C2×D4—3161no—
246C2×D4—3161no—
247C2×D4—3161no—
248C2×D4—3161no—
249C2×D4—3161no—
250C2×D4—1582no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—3164no—
260C22×C4—3164no—
261C22×C4—3162no—
262C22×C4—3162no—
263C22×C4—1582no—
264C22×C4—1581no—
265C22×C4—1584no—
266C22×C4(C2×D79)⋊C412yes—
267C22×C4(C2×D79)⋊C412yes—
268C22×C4D63212yes—
269C22×C4#1192961312yes—
270C22×C4C2×C4×D7911yescharacteristic
271C22×C4D4×D7912yes—
272C22×C4C2×C22⋊D7911yescharacteristic
273C22×C4#1192962312yes—
274C2×C8—44no—
275C2×C8—42no—
276C4⋊C4—3162no—
277C4⋊C4—3168no—
278C4⋊C4—3164no—
279C4⋊C4—1582no—
280C22⋊C4—3164no—
281C22⋊C4—3161no—
282C22⋊C4—3161no—
283C22⋊C4—3164no—
284C22⋊C4—3162no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—3161no—
290C23—3162no—
291C23—3161no—
292C23—3161no—
293C23—3162no—
294C23C2×(C2×D79)⋊C411yescharacteristic
295C23#1457913111yescharacteristic
296C23#1457914511yescharacteristic
297C23C2×D63211yescharacteristic
298C23C2×#1192961311yescharacteristic
299C23#1457918911yescharacteristic
300C23#1457919411yescharacteristic
301D4—6322no—
302D4—6321no—
303D4—6321no—
304D4—6321no—
305D4—6321no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—3164no—
322C2×C4—3164no—
323C2×C4—3161no—
324C2×C4—3164no—
325C2×C4—3162no—
326C2×C4—3162no—
327C2×C4—3161no—
328C2×C4—3161no—
329C2×C4—3162no—
330C2×C4—6324no—
331C2×C4—6324no—
332C2×C4#1457908314yes—
333C2×C4(C22⋊D79)⋊C412yes—
334C2×C4#1457913312yes—
335C8—82no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—6321no—
345C22—6322no—
346C22—6322no—
347C22—6321no—
348C22—6321no—
349C22#1909844011yescharacteristic
350C22#1909853612yes—
351C22C2×#1457908311yescharacteristic
352C22#1909856511yescharacteristic
353C22#1909857912yes—
354C22#1909869911yescharacteristic
355C22#1909870411yescharacteristic
356C22#1909887711yescharacteristic
357C22#1909888211yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—6324no—
363C4—6321no—
364C4—6322no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—12641no—
370C2#2376193912yes—
371C2#2376196112yes—
372C2#2376371511yescharacteristic
373C2#2376384611yescharacteristic
374C2#2376385611yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#1817D79
hypocenterC79#9156
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#19098429C2semidirect
4. #23761280C2semidirect
5. C79×#1817C2semidirect
9. #19098465C4semidirect
10. #19098461C4semidirect
190. #1817D79semidirect
193. C79#9156semidirect

Maximal subgroups

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

subgroupindex
#237717312
C2×#190984292
C2×#190982982
#237612802
C79×#18172
#915679