Finite Group Databasebeta

← #34186038

Subgroups — #34186038

Order 20224.

578 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2972C211yesmaximal, 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
19C8×D4×C79—24no—
20#431×C79—21no—
21C2×C22⋊C8×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
42C22×C8×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—
54C4×C8×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—42no—
67C22⋊Dic79—42no—
68C22⋊Dic79—42no—
69C22⋊Dic79—82no—
70C24×C79C22⋊C411yescharacteristic
71C24×C79C2×D411yescharacteristic
72C2×D4×C79—22no—
73C2×D4×C79—22no—
74C2×D4×C79SD1612yes—
75C2×D4×C79Q1612yes—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D412yes—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79#4112yes—
87C2×C8×C79—24no—
88C2×C8×C79—44no—
89C2×C8×C79—82no—
90C4⋊C4×C79—22no—
91C22⋊C4×C79—22no—
92C22⋊C4×C79—22no—
93C42×C79—21no—
94C42×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79#10312yes—
102C23×C79C2×C22⋊C411yescharacteristic
103C23×C79#12112yes—
104C23×C79C2×SD1611yescharacteristic
105C23×C79C2×Q1611yescharacteristic
106C23×C79#13512yes—
107D4×C79—24no—
108D4×C79—24no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—24no—
112C2×C4×C79—22no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—41no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—81no—
121C2×C4×C79#10312yes—
122C2×C4×C79C4×D412yes—
123C2×C4×C79#12011yescharacteristic
124C2×C4×C79#12311yescharacteristic
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—42no—
131C2×Dic79—84no—
132C2×Dic79—82no—
133C2×Dic79—82no—
134C2×Dic79—82no—
135C8×C79—44no—
136C22×C79—21no—
137C22×C79—22no—
138C22×C79—21no—
139C22×C79—22no—
140C22×C79—22no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—42no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—81no—
148C22×C79#38611yescharacteristic
149C22×C79C2×#10311yescharacteristic
150C22×C79#41611yescharacteristic
151C22×C79#46512yes—
152C22×C79#46712yes—
153C4×C79—21no—
154C4×C79—22no—
155C4×C79—22no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79#45812yes—
159Dic79—82no—
160Dic79—82no—
161Dic79—162no—
162#9469—791nomaximal, Sylow
163C2×C79—21no—
164C2×C79—22no—
165C2×C79—21no—
166C2×C79—22no—
167C2×C79—42no—
168C2×C79#189011yescharacteristic
169C2×C79#189211yescharacteristic
170C2×C79#191011yescharacteristic
171#2972D7911yescharacteristic
172#2487—791no—
173#1824—792no—
174#1799—791no—
175#1289—792no—
176C79#946911yescharacteristic, complemented, Sylow
177C2×C4×D4C2×D7911yescharacteristic
178C22×C4⋊C4—791no—
179C8×D4—24no—
180#431—21no—
181C2×#99—1581no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8C2×D7912yes—
184#400—792no—
185#399—792no—
186#389—1581no—
187#386—1581no—
188#385—1581no—
189#380—791no—
190#380—792no—
191#380—1581no—
192#342—1581no—
193#340—1584no—
194C22×D4D31611yescharacteristic
195C23×C4—1581no—
196C23×C4C22⋊D7911yescharacteristic
197C23×C4C22×D7911yescharacteristic
198C22×C8—22no—
199C22×C8C22⋊D7914yes—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—792no—
203C2×C4⋊C4—1582no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1582no—
207C2×C4⋊C4—1582no—
208C2×C4⋊C4—1582no—
209C2×C4⋊C4C4×D7911yescharacteristic
210C2×C22⋊C4—1582no—
211C2×C22⋊C4—1581no—
212C2×C22⋊C4—1582no—
213C2×C22⋊C4C4×D7911yescharacteristic
214C2×C22⋊C4D31611yescharacteristic
215C2×C42C22⋊D7911yescharacteristic
216C4⋊C8—22no—
217#99—3162no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C4×C8—22no—
222#95—792no—
223#95—1582no—
224#95—1582no—
225#95—1582no—
226C24(C2×D79)⋊C411yescharacteristic
227C24C2×D31611yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#1192961312yes—
231C2×D4Dic31612yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—3162no—
237C22×C4—3162no—
238C22×C4—6322no—
239C22×C4—1582no—
240C22×C4—1582no—
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4C2×C4×D7911yescharacteristic
245C22×C4C2×C22⋊D7912yes—
246C22×C4C2×C22⋊D7911yescharacteristic
247C22×C4#1192962412yes—
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—82no—
251C4⋊C4—22no—
252C4⋊C4—3162no—
253C4⋊C4—3164no—
254C4⋊C4—3161no—
255C4⋊C4—3161no—
256C4⋊C4—6322no—
257C4⋊C4—1584no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—3162no—
261C22⋊C4—3162no—
262C22⋊C4—3162no—
263C22⋊C4—6322no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1457908612yes—
273C23C2×(C2×D79)⋊C411yescharacteristic
274C23#1457913512yes—
275C23C2×#1192961311yescharacteristic
276C23C2×Dic31611yescharacteristic
277C23#1457918312yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—3162no—
293C2×C4—3162no—
294C2×C4—3162no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—6324no—
299C2×C4—6322no—
300C2×C4—6322no—
301C2×C4—6322no—
302C2×C4#1457908612yes—
303C2×C4C4×C22⋊D7912yes—
304C2×C4#1457913011yescharacteristic
305C2×C4#1457914411yescharacteristic
306C8—44no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1909844211yescharacteristic
320C22C2×#1457908611yescharacteristic
321C22#1909856011yescharacteristic
322C22#1909879212yes—
323C22#1909880412yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—6322no—
330C4—6322no—
331C4—12642no—
332C4#1909875312yes—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#2376381311yescharacteristic
339C2#2376382311yescharacteristic
340C2#2376392711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2972C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC23×C4×C79C23
hypercenter#2972D79
hypocenterC79#9469
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #23763491C2semidirect
4. #23761268C2semidirect
8. #19098502C4semidirect
9. #19098499C4semidirect
176. C79#9469semidirect

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×#29722
#946979