Finite Group Databasebeta

← #34186058

Subgroups — #34186058

Order 20224.

556 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 119 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#23767717C211yesmaximal, characteristic
2#23763491C212yesmaximal, complemented
3#23763388C211yesmaximal, characteristic
4#23761268C212yesmaximal, complemented
5C79×#2973C211yesmaximal, characteristic
6C22×C4⋊Dic79C2211yescharacteristic
7C2×#14579076—21no—
8#19098502C412yescomplemented
9#19098499C412yescomplemented
10#19098461—21no—
11#19098443—21no—
12#19098435—21no—
13#19098423—21no—
14#19098423C2212yes—
15#19098423C2211yescharacteristic
16#19098330—21no—
17#19098326—24no—
18C2×C4×D4×C79C2211yescharacteristic
19#435×C79—24no—
20#432×C79—21no—
21#407×C79—21no—
22C2×C22⋊C8×C79C2212yes—
23C23×Dic79—21no—
24C2×C4⋊Dic79—22no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—22no—
29C2×C4⋊Dic79C2×C412yes—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—22no—
33C2×C22⋊Dic79—22no—
34#14579076—42no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C79⋊#95C2×C412yes—
39C22×D4×C79D411yescharacteristic
40C23×C4×C79D411yescharacteristic
41C23×C4×C79C2311yescharacteristic
42C2×M16×C79—22no—
43C22×C8×C79D414yes—
44C4×D4×C79—22no—
45C4×D4×C79—22no—
46C2×C4⋊C4×C79C2×C411yescharacteristic
47C2×C22⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79D411yescharacteristic
49C2×C42×C79D411yescharacteristic
50C4⋊C8×C79—22no—
51C22⋊C8×C79—22no—
52C22⋊C8×C79—22no—
53C22⋊C8×C79—22no—
54#97×C79—22no—
55C22×Dic79—22no—
56C22×Dic79—22no—
57C22×Dic79—42no—
58C22×Dic79—42no—
59C22×Dic79—82no—
60C4⋊Dic79—24no—
61C4⋊Dic79—42no—
62C4⋊Dic79—44no—
63C4⋊Dic79—41no—
64C4⋊Dic79—41no—
65Dic79⋊C4—82no—
66C22⋊Dic79—41no—
67C22⋊Dic79—42no—
68C22⋊Dic79—41no—
69C22⋊Dic79—42no—
70C22⋊Dic79—82no—
71C24×C79C22⋊C411yescharacteristic
72C24×C79C2×D411yescharacteristic
73C2×D4×C79—22no—
74C2×D4×C79—21no—
75C2×D4×C79—21no—
76C2×D4×C79—22no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D412yes—
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79#4112yes—
88M16×C79—82no—
89C2×C8×C79—24no—
90C2×C8×C79—44no—
91C4⋊C4×C79—22no—
92C22⋊C4×C79—22no—
93C22⋊C4×C79—22no—
94C42×C79—21no—
95C42×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79C2×C22⋊C411yescharacteristic
104C23×C79#12112yes—
105C23×C79AGL(1,ℤ/8)11yescharacteristic
106C23×C79#13711yescharacteristic
107C23×C79#13712yes—
108D4×C79—42no—
109D4×C79—42no—
110C2×C4×C79—24no—
111C2×C4×C79—22no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—41no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—81no—
123C2×C4×C79C4×D412yes—
124C2×C4×C79#12011yescharacteristic
125C2×C4×C79#12311yescharacteristic
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—42no—
131C2×Dic79—42no—
132C2×Dic79—84no—
133C2×Dic79—82no—
134C2×Dic79—82no—
135C2×Dic79—82no—
136C8×C79—44no—
137C22×C79—21no—
138C22×C79—22no—
139C22×C79—21no—
140C22×C79—22no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—42no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—81no—
149C22×C79#38611yescharacteristic
150C22×C79#41711yescharacteristic
151C22×C79#41911yescharacteristic
152C22×C79#46812yes—
153C22×C79#47012yes—
154C4×C79—21no—
155C4×C79—22no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79#45812yes—
160Dic79—82no—
161Dic79—82no—
162Dic79—162no—
163#9473—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—22no—
166C2×C79—21no—
167C2×C79—22no—
168C2×C79—81no—
169C2×C79#189011yescharacteristic
170C2×C79#189411yescharacteristic
171C2×C79#191011yescharacteristic
172#2973D7911yescharacteristic
173#2487—791no—
174#1824—792no—
175#1799—791no—
176#1289—792no—
177C79#947311yescharacteristic, complemented, Sylow
178C2×C4×D4C2×D7911yescharacteristic
179C22×C4⋊C4—791no—
180#435—24no—
181#432—21no—
182C2×#99—1581no—
183#407—21no—
184C2×C22⋊C8C2×D7912yes—
185#400—792no—
186#399—792no—
187#389—1581no—
188#386—1581no—
189#385—1581no—
190#380—791no—
191#380—792no—
192#380—1581no—
193#342—1581no—
194#340—1584no—
195C22×D4D31611yescharacteristic
196C23×C4—1581no—
197C23×C4C22⋊D7911yescharacteristic
198C23×C4C22×D7911yescharacteristic
199C2×M16—22no—
200C22×C8C22⋊D7914yes—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—792no—
204C2×C4⋊C4—1582no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1582no—
208C2×C4⋊C4—1582no—
209C2×C4⋊C4—1582no—
210C2×C4⋊C4C4×D7911yescharacteristic
211C2×C22⋊C4—1582no—
212C2×C22⋊C4—1581no—
213C2×C22⋊C4—1582no—
214C2×C22⋊C4C4×D7911yescharacteristic
215C2×C22⋊C4D31611yescharacteristic
216C2×C42C22⋊D7911yescharacteristic
217C4⋊C8—22no—
218#99—3162no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222#97—22no—
223#95—792no—
224#95—1582no—
225#95—1582no—
226#95—1582no—
227C24(C2×D79)⋊C411yescharacteristic
228C24C2×D31611yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—3162no—
238C22×C4—3162no—
239C22×C4—6322no—
240C22×C4—1582no—
241C22×C4—1582no—
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4(C2×D79)⋊C411yescharacteristic
245C22×C4C2×C4×D7911yescharacteristic
246C22×C4C2×C22⋊D7912yes—
247C22×C4C2×C22⋊D7911yescharacteristic
248C22×C4#1192962412yes—
249M16—82no—
250C2×C8—24no—
251C2×C8—44no—
252C4⋊C4—22no—
253C4⋊C4—3162no—
254C4⋊C4—3164no—
255C4⋊C4—3161no—
256C4⋊C4—3161no—
257C4⋊C4—6322no—
258C4⋊C4—1584no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—3161no—
262C22⋊C4—3161no—
263C22⋊C4—3162no—
264C22⋊C4—3162no—
265C22⋊C4—6322no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×(C2×D79)⋊C411yescharacteristic
276C23#1457913512yes—
277C23#1457918911yescharacteristic
278C23#1457919412yes—
279C23#1457919411yescharacteristic
280D4—42no—
281D4—42no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—3162no—
299C2×C4—3162no—
300C2×C4—3162no—
301C2×C4—6324no—
302C2×C4—6322no—
303C2×C4—6322no—
304C2×C4—6322no—
305C2×C4C4×C22⋊D7912yes—
306C2×C4#1457913011yescharacteristic
307C2×C4#1457914411yescharacteristic
308C8—44no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1909844211yescharacteristic
322C22#1909856511yescharacteristic
323C22#1909857311yescharacteristic
324C22#1909880912yes—
325C22#1909882112yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—6322no—
332C4—6322no—
333C4—12642no—
334C4#1909875312yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2376381311yescharacteristic
341C2#2376383311yescharacteristic
342C2#2376392711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2973C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC23×C4×C79C23
hypercenter#2973D79
hypocenterC79#9473
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #23763491C2semidirect
4. #23761268C2semidirect
8. #19098502C4semidirect
9. #19098499C4semidirect
177. C79#9473semidirect

Maximal subgroups

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

subgroupindex
#237677172
#237634912
#237633882
#237612682
C79×#29732
#947379