Finite Group Databasebeta

← #34186118

Subgroups — #34186118

Order 20224.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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#23768297C211yesmaximal, characteristic
2#23763392C211yesmaximal, characteristic, complemented
3#23761826C212yesmaximal, complemented
4#23761259C212yesmaximal, complemented
5C79×#2972C211yesmaximal, characteristic, complemented
6C2×#14579132—21no—
7C2×C4×D316C2211yescharacteristic
8#19098587C2212yes—
9C2×#14579078—21no—
10#19098488C412yes—
11#19098472C412yes—
12#19098442—22no—
13#19098429C2211yescharacteristic
14#19098420—21no—
15#19098332—21no—
16#19098292—24no—
17C2×C4×D4×C79C2211yescharacteristic
18C8×D4×C79—24no—
19#431×C79C2212yes—
20C2×C22⋊C8×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—22no—
23C22×C4×D79—21no—
24C2×#11929608—22no—
25#14579132—42no—
26#14579132—42no—
27C4×D316—22no—
28C4×D316—22no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×C22⋊Dic79—22no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—22no—
34C4⋊(C79⋊C8)—22no—
35C4⋊(C79⋊C8)—22no—
36#14579078—42no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95C2×C412yes—
40C22×D4×C79D411yescharacteristic
41C23×C4×C79D412yes—
42C22×C8×C79—24no—
43C4×D4×C79—22no—
44C4×D4×C79—22no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C22⋊C4×C79C2×C412yes—
47C2×C42×C79C2311yescharacteristic
48C4⋊C8×C79—22no—
49C22⋊C8×C79—24no—
50C22⋊C8×C79—22no—
51C4×C8×C79D414yescomplemented
52C23×D79—21no—
53C2×D316—22no—
54C2×D316—42no—
55C2×C22⋊D79—44no—
56C2×C22⋊D79—44no—
57C2×C22⋊D79—82no—
58C22×Dic79—22no—
59C22×Dic79—21no—
60C2×C4×D79—42no—
61C2×C4×D79—41no—
62C2×C4×D79—41no—
63C2×C4×D79—42no—
64#11929608—42no—
65#11929608—42no—
66C2×C79⋊C8—42no—
67C4⋊Dic79—22no—
68C4⋊Dic79—41no—
69C4⋊Dic79—41no—
70C22⋊Dic79—44no—
71(C2×D79)⋊C4—41no—
72(C2×D79)⋊C4—41no—
73(C2×D79)⋊C4—82no—
74C24×C79C22⋊C412yes—
75C2×D4×C79—22no—
76C2×D4×C79—22no—
77C2×D4×C79D812yes—
78C2×D4×C79SD1612yes—
79C22×C4×C79—22no—
80C22×C4×C79—22no—
81C22×C4×C79—82no—
82C22×C4×C79C22⋊C412yes—
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C2×C8×C79—24no—
87C2×C8×C79—84no—
88C4⋊C4×C79—22no—
89C22⋊C4×C79—24no—
90C42×C79C2×D412yes—
91C42×C79#4112yes—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—42no—
95C23×C79—42no—
96C23×C79#10214yes—
97C23×C79C2×C22⋊C411yescharacteristic
98C22×D79—22no—
99C22×D79—41no—
100C22×D79—41no—
101C22×D79—41no—
102C22×D79—41no—
103C22×D79—41no—
104C22×D79—41no—
105D4×C79—24no—
106D4×C79—24no—
107C2×C4×C79—22no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—44no—
111C2×C4×C79—42no—
112C2×C4×C79—82no—
113C2×C4×C79C4×D412yes—
114C2×C4×C79#12011yescharacteristic
115C2×C4×C79#12112yes—
116C2×C4×C79#12311yescharacteristic
117C2×C4×C79C2×D811yescharacteristic
118C2×C4×C79C2×SD1611yescharacteristic
119C2×C4×C79#13512yes—
120C22⋊D79—88no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126C2×Dic79—41no—
127D316—24no—
128D316—42no—
129C4×D79—82no—
130C4×D79—82no—
131C8×C79—44no—
132C79⋊C8—82no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—24no—
136C22×C79—44no—
137C22×C79—42no—
138C22×C79—42no—
139C22×C79—82no—
140C22×C79#38611yescharacteristic
141C22×C79C2×#10211yescharacteristic
142C22×C79#41611yescharacteristic
143C2×D79—42no—
144C2×D79—42no—
145C2×D79—42no—
146C2×D79—81no—
147C2×D79—81no—
148C2×D79—81no—
149C2×D79—81no—
150C4×C79—22no—
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—42no—
154C4×C79#46512yes—
155C4×C79#46612yes—
156Dic79—82no—
157Dic79—81no—
158#9485—791nomaximal, Sylow
159C2×C79—21no—
160C2×C79—21no—
161C2×C79—44no—
162C2×C79#189111yescharacteristic
163C2×C79#189311yescharacteristic
164C2×C79#191111yescharacteristic
165D79—82no—
166#2972D7911yescharacteristic, complemented
167#2535—791no—
168#1800—791no—
169#1486—792no—
170#1286—792no—
171C79#948511yescharacteristic, complemented, Sylow
172C2×#121—1581no—
173C2×C4×D4—791no—
174C2×C4×D4C2×D7911yescharacteristic
175C8×D4—24no—
176#431C2×D7912yes—
177#423—792no—
178C2×#100—1581no—
179C2×C22⋊C8—22no—
180#397—792no—
181#392—792no—
182#386—1582no—
183#382—791no—
184#380—1581no—
185#343—1581no—
186D4⋊C8—1584no—
187C22×D4—791no—
188C22×D4—1582no—
189C22×D4D31611yescharacteristic
190C23×C4—1581no—
191C23×C4C22⋊D7912yes—
192C2×M16—1582no—
193C22×C8—24no—
194#121—3162no—
195#121—3162no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—1582no—
199C4×D4—1582no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4D31611yescharacteristic
203C2×C22⋊C4—1582no—
204C2×C22⋊C4—1581no—
205C2×C22⋊C4—1582no—
206C2×C22⋊C4C4×D7912yes—
207C2×C42C22×D7911yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—1582no—
210C4⋊C8—1582no—
211#100—3162no—
212C22⋊C8—24no—
213C22⋊C8—22no—
214C4×C8C22⋊D7914yescomplemented
215#95—792no—
216#95—1582no—
217#95—1581no—
218C24—1581no—
219C24(C2×D79)⋊C412yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—3162no—
223C2×D4—3164no—
224C2×D4—3164no—
225C2×D4—6322no—
226C2×D4—1582no—
227C2×D4D63212yes—
228C2×D4#1192961312yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—3162no—
233C22×C4—3161no—
234C22×C4—3161no—
235C22×C4—3162no—
236C22×C4—1582no—
237C22×C4—1581no—
238C22×C4(C2×D79)⋊C412yes—
239C22×C4C2×C4×D7911yescharacteristic
240C22×C4C2×C22⋊D7911yescharacteristic
241C22×C4C2×D31611yescharacteristic
242M16—3162no—
243M16—3162no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—3162no—
247C4⋊C4—22no—
248C4⋊C4—3161no—
249C4⋊C4—3161no—
250C4⋊C4—1582no—
251C22⋊C4—24no—
252C22⋊C4—3164no—
253C22⋊C4—3161no—
254C22⋊C4—3161no—
255C22⋊C4—6322no—
256C42C2×C22⋊D7912yes—
257C42#1192962412yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—3161no—
263C23—3161no—
264C23—3161no—
265C23—3161no—
266C23—3161no—
267C23—3161no—
268C23—1582no—
269C23#1457908314yes—
270C23C2×(C2×D79)⋊C411yescharacteristic
271D4—24no—
272D4—24no—
273D4—3162no—
274D4—6328no—
275D4—1584no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3162no—
287C2×C4—3161no—
288C2×C4—6322no—
289C2×C4—6322no—
290C2×C4C4×C22⋊D7912yes—
291C2×C4#1457913011yescharacteristic
292C2×C4#1457913512yes—
293C2×C4#1457914411yescharacteristic
294C2×C4C2×D63211yescharacteristic
295C2×C4C2×#1192961311yescharacteristic
296C2×C4#1457918312yes—
297C8—44no—
298C8—6322no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—3162no—
307C22—3162no—
308C22—3162no—
309C22—6321no—
310C22—6321no—
311C22—6321no—
312C22—6321no—
313C22#1909844211yescharacteristic
314C22C2×#1457908311yescharacteristic
315C22#1909856011yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—6322no—
321C4—6321no—
322C4#1909879212yes—
323C4#1909879912yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—6322no—
328C2#2376381811yescharacteristic
329C2#2376382811yescharacteristic
330C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2972C2
socleC22×C79#386
prime radical#23768297C2
cosocleC2×C42×C79C23
hypercenter#2972D79
hypocenterC79#9485
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #23763392C2semidirect
3. #23761826C2semidirect
4. #23761259C2semidirect
5. C79×#2972C2semidirect
51. C4×C8×C79D4semidirect
166. #2972D79semidirect
171. C79#9485semidirect
214. C4×C8C22⋊D79semidirect

Maximal subgroups

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

subgroupindex
#237682972
#237633922
#237618262
#237612592
C79×#29722
#948579