Finite Group Databasebeta

← #34186219

Subgroups — #34186219

Order 20224.

684 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 111 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#23772429C211yesmaximal, characteristic
2#23763324C212yesmaximal, complemented
3#23761738C211yesmaximal, characteristic
4#23761306C212yesmaximal, complemented
5C79×#2409C211yesmaximal, characteristic
6#19098635—24no—
7#19098620—21no—
8C2×#14579080—24no—
9#19098527C2212yes—
10#19098527C2212yes—
11C2×C22⋊(C79⋊C8)—21no—
12#19098289—21no—
13#19098287—21no—
14C23×Q8×C79C2211yescharacteristic
15C2×C22⋊Q8×C79—21no—
16C2×C4×D4×C79C2211yescharacteristic
17#395×C79—21no—
18#393×C79—21no—
19#391×C79—21no—
20#386×C79C412yes—
21#385×C79C412yes—
22#380×C79C2211yescharacteristic
23C2×#11929608—22no—
24C2×#11929608D414yes—
25C22×C79⋊C8—22no—
26C4⋊(C79⋊C8)—22no—
27#14579080—48no—
28C22⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33#14579072—22no—
34C22×Q8×C79—21no—
35C22×Q8×C79—22no—
36C22×Q8×C79—21no—
37C22×Q8×C79—22no—
38C22×Q8×C79—44no—
39C22×D4×C79C2×C411yescharacteristic
40C23×C4×C79C2×C412yes—
41C23×C4×C79D411yescharacteristic
42C23×C4×C79C2311yescharacteristic
43C22⋊Q8×C79—44no—
44C4×D4×C79—22no—
45C4×D4×C79—22no—
46C2×C4⋊C4×C79—22no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79—22no—
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79C2×C412yes—
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54#95×C79—22no—
55#95×C79—22no—
56#95×C79—21no—
57#11929608—44no—
58#11929608—44no—
59C2×C79⋊C8—24no—
60C2×C79⋊C8—42no—
61C2×C79⋊C8—82no—
62C24×C79C22⋊C411yescharacteristic
63C24×C79C22×C411yescharacteristic
64C2×Q8×C79—24no—
65C2×Q8×C79—28no—
66C2×Q8×C79—44no—
67C2×Q8×C79—42no—
68C2×Q8×C79—44no—
69C2×Q8×C79—48no—
70C2×Q8×C79—41no—
71C2×Q8×C79—42no—
72C2×Q8×C79—41no—
73C2×D4×C79—22no—
74C2×D4×C79—22no—
75C2×D4×C79—41no—
76C22×C4×C79—22no—
77C22×C4×C79—24no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—22no—
81C22×C4×C79—44no—
82C22×C4×C79—42no—
83C22×C4×C79—42no—
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C2×D412yes—
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79#4112yes—
91C4⋊C4×C79—22no—
92C4⋊C4×C79—42no—
93C4⋊C4×C79—44no—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—44no—
97C22⋊C4×C79—42no—
98C42×C79—21no—
99C42×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79—41no—
108C23×C79#10112yescomplemented
109C23×C79C2×C22⋊C411yescharacteristic
110C23×C79C4×D412yes—
111Q8×C79—44no—
112Q8×C79—48no—
113Q8×C79—44no—
114D4×C79—44no—
115C2×C4×C79—24no—
116C2×C4×C79—22no—
117C2×C4×C79—24no—
118C2×C4×C79—28no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—41no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79—44no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—42no—
133C2×C4×C79—42no—
134C2×C4×C79—42no—
135C2×C4×C79—81no—
136C2×C4×C79#12011yescharacteristic
137C2×C4×C79#12112yes—
138C2×C4×C79#12311yescharacteristic
139C79⋊C8—44no—
140C79⋊C8—82no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—22no—
145C22×C79—22no—
146C22×C79—22no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79—42no—
150C22×C79—41no—
151C22×C79—41no—
152C22×C79—81no—
153C22×C79#38611yescharacteristic
154C22×C79C2×#10111yescharacteristic
155C22×C79#41311yescharacteristic
156C4×C79—21no—
157C4×C79—22no—
158C4×C79—44no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—41no—
162C4×C79—82no—
163C4×C79B(4,𝔽2)12yes—
164#9505—791nomaximal, Sylow
165C2×C79—21no—
166C2×C79—21no—
167C2×C79—42no—
168C2×C79—42no—
169C2×C79#188911yescharacteristic
170C2×C79#190111yescharacteristic
171C2×C79#191511yescharacteristic
172#2973—791no—
173#2409D7911yescharacteristic
174#1784—792no—
175#1465—791no—
176#1305—792no—
177C79#950511yescharacteristic, complemented, Sylow
178C23×Q8C2×D7911yescharacteristic
179C2×C22⋊Q8—21no—
180C2×C4×D4C2×D7911yescharacteristic
181#435—1584no—
182#432—1581no—
183C2×#101—1584no—
184#407—792no—
185#407—792no—
186C2×C22⋊C8—1581no—
187#395—21no—
188#393—21no—
189#391—21no—
190#386Dic7912yes—
191#385Dic7912yes—
192#380C2×D7911yescharacteristic
193#324—1581no—
194#323—1581no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×D4C2×Dic7911yescharacteristic
201C23×C4C2×Dic7912yes—
202C23×C4C22⋊D7911yescharacteristic
203C23×C4C22×D7911yescharacteristic
204C2×M16—794no—
205C2×M16—1582no—
206C22×C8—1582no—
207C22⋊Q8—44no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22⋊D7911yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic7911yescharacteristic
215C2×C22⋊C4C2×Dic7912yes—
216C2×C22⋊C4C22⋊D7911yescharacteristic
217C2×C42C22⋊D7911yescharacteristic
218C4⋊C8—1582no—
219#101—3168no—
220C22⋊C8—1582no—
221C22⋊C8—1582no—
222C22⋊C8—1582no—
223C22⋊C8—1582no—
224C22⋊C8—1582no—
225#97—1582no—
226#95—22no—
227#95—22no—
228#95—21no—
229C24C22⋊Dic7911yescharacteristic
230C24C22×Dic7911yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4C22⋊Dic7911yescharacteristic
252C22×C4C22⋊Dic7911yescharacteristic
253C22×C4C22⋊Dic7911yescharacteristic
254C22×C4D4×D7912yes—
255C22×C4C2×C22⋊D7911yescharacteristic
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4#1192962312yes—
258M16—3164no—
259M16—3164no—
260C2×C8—3162no—
261C2×C8—6322no—
262C2×C8—1584no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1457908012yescomplemented
281C23C2×C22⋊Dic7911yescharacteristic
282C23D4×Dic7912yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4#1457912911yescharacteristic
309C2×C4#1457913312yes—
310C2×C4#1457914211yescharacteristic
311C8—3164no—
312C8—6322no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1909844311yescharacteristic
326C22C2×#1457908011yescharacteristic
327C22#1909854711yescharacteristic
328C4—22no—
329C4—21no—
330C4—44no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4#1909874912yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#2376380911yescharacteristic
341C2#2376386911yescharacteristic
342C2#2376395311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2409C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2409D79
hypocenterC79#9505
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #23763324C2semidirect
4. #23761306C2semidirect
108. C23×C79#101semidirect
177. C79#9505semidirect
280. C23#14579080semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237633242
#237617382
#237613062
C79×#24092
#950579