Finite Group Databasebeta

← #34184307

Subgroups — #34184307

Order 20224.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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#23769150C211yesmaximal, characteristic
2#23762069C212yesmaximal, complemented
3#23761714C211yesmaximal, characteristic, complemented
4#23761260C212yesmaximal, complemented
5C79×#2921C211yesmaximal, characteristic, complemented
6C2×#14579158C2211yescharacteristic
7C2×#14579135—21no—
8#19098616C2212yes—
9#19098480—21no—
10#19098480C412yes—
11#19098472—21no—
12#19098472C412yes—
13#19098465—21no—
14#19098429C2211yescharacteristic
15#19098306—24no—
16#19098289—21no—
17#19098287—21no—
18C2×#117×C79C2211yescharacteristic
19#433×C79—24no—
20#423×C79C2212yes—
21#407×C79—21no—
22C4×M16×C79—21no—
23C22×D316—21no—
24C22×D316—21no—
25C22×D316C2×C412yes—
26C22×C22⋊D79—22no—
27#14579158—22no—
28#14579158—22no—
29#14579135—44no—
30C2×C4⋊Dic79—21no—
31C2×(C2×D79)⋊C4—22no—
32C2×(C2×D79)⋊C4—22no—
33C2×(C2×D79)⋊C4—22no—
34C4⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C4×C79⋊C8—22no—
37C79⋊#95—21no—
38C79⋊#95C2×C412yes—
39C23×C4×C79D411yescharacteristic
40C2×M16×C79—22no—
41#117×C79—22no—
42#117×C79—22no—
43C2×C4⋊C4×C79C2×C411yescharacteristic
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79D411yescharacteristic
47C2×C42×C79D411yescharacteristic
48C2×C42×C79C2311yescharacteristic
49C4⋊C8×C79—22no—
50C4⋊C8×C79D414yescomplemented
51C22⋊C8×C79—24no—
52#97×C79—22no—
53C4×C8×C79—22no—
54C23×D79—22no—
55C2×D316—24no—
56C2×D316—41no—
57C2×D316—42no—
58C2×D316—41no—
59C2×D316—42no—
60C2×D316—42no—
61C2×D316—82no—
62C2×D316—82no—
63C2×C22⋊D79—44no—
64C2×C22⋊D79—82no—
65C2×C22⋊D79—82no—
66C22×Dic79—22no—
67C2×C79⋊C8—42no—
68C4⋊Dic79—42no—
69(C2×D79)⋊C4—44no—
70(C2×D79)⋊C4—82no—
71(C2×D79)⋊C4—82no—
72C24×C79C22⋊C411yescharacteristic
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—22no—
76C22×C4×C79—22no—
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22×C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83M16×C79—42no—
84M16×C79—42no—
85C2×C8×C79—24no—
86C4⋊C4×C79—22no—
87C4⋊C4×C79—22no—
88C22⋊C4×C79—22no—
89C22⋊C4×C79—22no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79C2×D412yes—
93C42×C79#4112yes—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79C2×C22⋊C411yescharacteristic
100C22×D79—42no—
101C22×D79—42no—
102C22×D79—42no—
103C22×D79—42no—
104C22×D79—42no—
105C22×D79—42no—
106C22×D79—42no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79C4×D412yes—
119C2×C4×C79#12011yescharacteristic
120C2×C4×C79#12112yes—
121C2×C4×C79#12311yescharacteristic
122C2×C4×C79#13512yes—
123C2×C4×C79AGL(1,ℤ/8)12yes—
124C22⋊D79—88no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128D316—84no—
129D316—84no—
130D316—82no—
131D316—82no—
132C8×C79—44no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—42no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—81no—
141C22×C79#38611yescharacteristic
142C22×C79#41611yescharacteristic
143C22×C79#41811yescharacteristic
144C2×D79—82no—
145C2×D79—82no—
146C2×D79—82no—
147C2×D79—82no—
148C2×D79—82no—
149C2×D79—82no—
150C2×D79—82no—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79#44912yes—
157C4×C79#48212yes—
158Dic79—82no—
159#9198—791nomaximal, Sylow
160C2×C79—21no—
161C2×C79—21no—
162C2×C79—81no—
163C2×C79#187511yescharacteristic
164C2×C79#191011yescharacteristic
165C2×C79#191411yescharacteristic
166D79—162no—
167#2921D7911yescharacteristic, complemented
168#2619—791no—
169#1545—792no—
170#1458—791no—
171#1286—792no—
172C79#919811yescharacteristic, complemented, Sylow
173C2×#127—791no—
174C2×#121—1581no—
175C2×#117C2×D7911yescharacteristic
176#433—24no—
177#431—792no—
178#423C2×D7912yes—
179#407—21no—
180C4×M16—21no—
181#394—792no—
182#394—1581no—
183#392—792no—
184#392—1581no—
185#390—1581no—
186#382—791no—
187#331—1584no—
188#324—1581no—
189#323—1581no—
190C22×D4—792no—
191C22×D4—1582no—
192C22×D4—1581no—
193C22×D4—1581no—
194C23×C4C22⋊D7911yescharacteristic
195C2×M16—22no—
196#127—1582no—
197#127—1582no—
198#121—3164no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4C4×D7911yescharacteristic
203C2×C4⋊C4D31611yescharacteristic
204C2×C22⋊C4—1582no—
205C2×C22⋊C4—1582no—
206C2×C22⋊C4—1582no—
207C2×C22⋊C4C4×D7911yescharacteristic
208C2×C22⋊C4D31611yescharacteristic
209C2×C42C22⋊D7911yescharacteristic
210C2×C42C22×D7911yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—1582no—
213C4⋊C8C22⋊D7914yescomplemented
214C22⋊C8—24no—
215C22⋊C8—1582no—
216#97—22no—
217C4×C8—22no—
218C4×C8—1582no—
219#95—792no—
220#95—1581no—
221C24—1582no—
222C24(C2×D79)⋊C411yescharacteristic
223C2×D4—3164no—
224C2×D4—3161no—
225C2×D4—3162no—
226C2×D4—3161no—
227C2×D4—3162no—
228C2×D4—3162no—
229C2×D4—6322no—
230C2×D4—6322no—
231C2×D4—6322no—
232C2×D4—6322no—
233C2×D4—1584no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—1582no—
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4C2×C4×D7911yescharacteristic
243C22×C4C2×C22⋊D7911yescharacteristic
244C22×C4C2×D31611yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—24no—
248C2×C8—3162no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—3162no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—3164no—
255C22⋊C4—6322no—
256C22⋊C4—6322no—
257C42—21no—
258C42—21no—
259C42C2×C22⋊D7912yes—
260C42#1192962412yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—3162no—
267C23—3162no—
268C23—3162no—
269C23—3162no—
270C23—3162no—
271C23—3162no—
272C23—3162no—
273C23C2×(C2×D79)⋊C411yescharacteristic
274D4—6324no—
275D4—6324no—
276D4—6328no—
277D4—6322no—
278D4—6322no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—3162no—
293C2×C4C4×C22⋊D7912yes—
294C2×C4#1457913011yescharacteristic
295C2×C4#1457913512yes—
296C2×C4#1457914411yescharacteristic
297C2×C4#1457918112yes—
298C2×C4#1457918612yes—
299C8—44no—
300C8—6322no—
301C22—21no—
302C22—21no—
303C22—42no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—6322no—
309C22—6322no—
310C22—6322no—
311C22—6322no—
312C22—6322no—
313C22—6322no—
314C22—6322no—
315C22#1909844211yescharacteristic
316C22#1909855711yescharacteristic
317C22#1909856711yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—6322no—
324C4#1909870812yes—
325C4#1909888712yes—
326C2—21no—
327C2—21no—
328C2—81no—
329C2—12642no—
330C2#2376372211yescharacteristic
331C2#2376392711yescharacteristic
332C2#2376394711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2921C2
socleC22×C79#386
prime radical#23769150C2
cosocleC2×C42×C79C23
hypercenter#2921D79
hypocenterC79#9198
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #23762069C2semidirect
3. #23761714C2semidirect
4. #23761260C2semidirect
5. C79×#2921C2semidirect
50. C4⋊C8×C79D4semidirect
167. #2921D79semidirect
172. C79#9198semidirect
213. C4⋊C8C22⋊D79semidirect

Maximal subgroups

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

subgroupindex
#237691502
#237620692
#237617142
#237612602
C79×#29212
#919879