Finite Group Databasebeta

← #34185968

Subgroups — #34185968

Order 20224.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#23768353C211yesmaximal, characteristic
2#23762118C212yesmaximal, complemented
3#23761741C211yesmaximal, characteristic
4#23761257C212yesmaximal, complemented
5C79×#2973C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7C2×#14579140—21no—
8C2×#14579140—21no—
9#19098620C2212yes—
10#19098499—21no—
11#19098499—21no—
12#19098466C412yes—
13#19098432C412yes—
14C4×C4⋊Dic79C2211yescharacteristic
15#19098310—24no—
16#19098287—21no—
17#19098287—21no—
18C2×C4×D4×C79C2211yescharacteristic
19#435×C79—24no—
20#432×C79C2212yes—
21#407×C79—21no—
22C2×C22⋊C8×C79—21no—
23C22×Dic158—21no—
24#14579161—22no—
25#14579161—22no—
26#14579140—44no—
27#14579140—44no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79C2×C412yes—
31C2×Dic79⋊C4—22no—
32C2×Dic79⋊C4—22no—
33C2×C22⋊Dic79—22no—
34C2×C22⋊Dic79—22no—
35C2×C4×Dic79C2×C412yes—
36C4⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38#14579072—22no—
39C79⋊#95—21no—
40C22×D4×C79C2×C411yescharacteristic
41C23×C4×C79D411yescharacteristic
42C23×C4×C79D411yescharacteristic
43C2×M16×C79—22no—
44C22×C8×C79—22no—
45C4×D4×C79—22no—
46C4×D4×C79—22no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C42×C79C2311yescharacteristic
51C4⋊C8×C79D414yes—
52C22⋊C8×C79—22no—
53C22⋊C8×C79—22no—
54C22⋊C8×C79—22no—
55#97×C79—22no—
56C2×Dic158—42no—
57C2×Dic158—42no—
58C2×Dic158—41no—
59C2×Dic158—41no—
60C22×Dic79—22no—
61C22×Dic79—22no—
62C2×C79⋊C8—42no—
63C4⋊Dic79—24no—
64C4⋊Dic79—24no—
65C4⋊Dic79—42no—
66Dic79⋊C4—44no—
67Dic79⋊C4—44no—
68C22⋊Dic79—44no—
69C22⋊Dic79—44no—
70C4×Dic79—42no—
71C24×C79C22⋊C411yescharacteristic
72C24×C79C22⋊C411yescharacteristic
73C2×D4×C79—22no—
74C2×D4×C79—22no—
75C2×D4×C79—41no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—42no—
81C22×C4×C79—42no—
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87M16×C79—82no—
88C2×C8×C79—24no—
89C2×C8×C79—82no—
90C4⋊C4×C79—22no—
91C22⋊C4×C79—22no—
92C22⋊C4×C79—22no—
93C42×C79C2×D412yes—
94C42×C79#4112yes—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79#10112yes—
105C23×C79C2×C22⋊C411yescharacteristic
106D4×C79—44no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—22no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—81no—
118C2×C4×C79—81no—
119C2×C4×C79C4×D412yes—
120C2×C4×C79#12011yescharacteristic
121C2×C4×C79#12112yes—
122C2×C4×C79#12311yescharacteristic
123C2×Dic79—24no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129Dic158—82no—
130Dic158—82no—
131C8×C79—44no—
132C79⋊C8—82no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—22no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—81no—
144C22×C79—81no—
145C22×C79#38611yescharacteristic
146C22×C79C2×#10111yescharacteristic
147C22×C79#41311yescharacteristic
148C4×C79—21no—
149C4×C79—22no—
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79#45312yes—
154Dic79—44no—
155Dic79—82no—
156#9455—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—42no—
160C2×C79—81no—
161C2×C79#188711yescharacteristic
162C2×C79#190111yescharacteristic
163C2×C79#190911yescharacteristic
164#2973D7911yescharacteristic
165#2542—791no—
166#1556—792no—
167#1466—791no—
168#1285—792no—
169C79#945511yescharacteristic, complemented, Sylow
170C2×C4⋊Q8—791no—
171C2×C22⋊Q8—1581no—
172C2×C22⋊Q8—1581no—
173C2×C4×D4C2×D7911yescharacteristic
174#435—24no—
175#432—792no—
176#432C2×D7912yes—
177#407—21no—
178C2×C22⋊C8—21no—
179#399—1581no—
180#399—1581no—
181#390—792no—
182#384—792no—
183C4×C4⋊C4—791no—
184#332—1584no—
185#323—1581no—
186#323—1581no—
187C22×Q8—1581no—
188C22×D4C4×D7911yescharacteristic
189C23×C4C22⋊D7911yescharacteristic
190C23×C4C22⋊D7911yescharacteristic
191C2×M16—22no—
192C22×C8—22no—
193C4⋊Q8—1582no—
194C4⋊Q8—1582no—
195C22⋊Q8—3164no—
196C22⋊Q8—3164no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—792no—
200C2×C4⋊C4—1582no—
201C2×C4⋊C4—1582no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4C4×D7911yescharacteristic
205C2×C22⋊C4—1582no—
206C2×C22⋊C4—1582no—
207C2×C22⋊C4D31611yescharacteristic
208C2×C22⋊C4D31611yescharacteristic
209C2×C42—792no—
210C2×C42C22×D7911yescharacteristic
211C4⋊C8—1582no—
212C4⋊C8C22⋊D7914yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—1582no—
217#97—22no—
218#97—1582no—
219#95—1581no—
220C24(C2×D79)⋊C411yescharacteristic
221C24(C2×D79)⋊C411yescharacteristic
222C2×Q8—3162no—
223C2×Q8—3162no—
224C2×Q8—3161no—
225C2×Q8—3161no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1582no—
236C22×C4—1582no—
237C22×C4(C2×D79)⋊C411yescharacteristic
238C22×C4(C2×D79)⋊C411yescharacteristic
239C22×C4C2×C4×D7911yescharacteristic
240C22×C4C2×C22⋊D7911yescharacteristic
241C22×C4C2×D31611yescharacteristic
242M16—82no—
243C2×C8—24no—
244C2×C8—82no—
245C2×C8—3162no—
246C4⋊C4—22no—
247C4⋊C4—3164no—
248C4⋊C4—3164no—
249C4⋊C4—3162no—
250C4⋊C4—1584no—
251C4⋊C4—1584no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—3164no—
255C22⋊C4—3164no—
256C42—3162no—
257C42C2×C22⋊D7912yes—
258C42#1192962412yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1457907912yes—
269C23C2×(C2×D79)⋊C411yescharacteristic
270Q8—6322no—
271Q8—6322no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3162no—
289C2×C4—3162no—
290C2×C4—1584no—
291C2×C4C4×C22⋊D7912yes—
292C2×C4#1457913011yescharacteristic
293C2×C4#1457913512yes—
294C2×C4#1457914411yescharacteristic
295C8—44no—
296C8—6322no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1909844211yescharacteristic
310C22C2×#1457907911yescharacteristic
311C22#1909854511yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—3164no—
318C4—6322no—
319C4#1909873112yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#2376379811yescharacteristic
325C2#2376386811yescharacteristic
326C2#2376392211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2973C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#2973D79
hypocenterC79#9455
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #23762118C2semidirect
4. #23761257C2semidirect
169. C79#9455semidirect

Maximal subgroups

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

subgroupindex
#237683532
#237621182
#237617412
#237612572
C79×#29732
#945579