Finite Group Databasebeta

← #34188828

Subgroups — #34188828

Order 20224.

804 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#19098718C212yesmaximal
2C2×#19098300C212yesmaximal
3#23761670C211yesmaximal, characteristic
4#23761243C211yesmaximal, characteristic
5C79×#2659C211yesmaximal, characteristic
6C23×Dic158C2211yescharacteristic
7C22×#11929614—22no—
8#19098718—24no—
9#19098718—24no—
10C2×#14579084—22no—
11C2×C22⋊(C79⋊C8)C2212yes—
12C2×C79⋊#95—22no—
13C2×C79⋊#95C2211yescharacteristic
14#19098300—24no—
15C2×C22⋊Q8×C79C2212yes—
16#398×C79—22no—
17#395×C79—22no—
18#380×C79—22no—
19C2×#95×C79C2211yescharacteristic
20C22×Dic158—21no—
21C22×Dic158—22no—
22C22×Dic158—22no—
23C22×Dic158—44no—
24C22×Dic158D412yes—
25C23×Dic79—21no—
26C23×Dic79—21no—
27C2×#11929614—28no—
28C2×#11929614—48no—
29C22×C79⋊C8—22no—
30#14579084—28no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)D414yes—
33C79⋊#95—22no—
34C79⋊#95—44no—
35C22×Q8×C79D412yes—
36C23×C4×C79D412yes—
37C23×C4×C79C2311yescharacteristic
38C22⋊Q8×C79—22no—
39C22⋊Q8×C79—22no—
40C22⋊Q8×C79—22no—
41C22⋊Q8×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79D412yes—
46C2×C22⋊C4×C79—22no—
47#95×C79—24no—
48#95×C79—22no—
49C2×Dic158—28no—
50C2×Dic158—24no—
51C2×Dic158—44no—
52C2×Dic158—44no—
53C2×Dic158—48no—
54C2×Dic158—44no—
55C2×Dic158—42no—
56C22×Dic79—24no—
57C22×Dic79—22no—
58C22×Dic79—22no—
59C22×Dic79—42no—
60C22×Dic79—42no—
61C22×Dic79—41no—
62C22×Dic79—41no—
63C22×Dic79—41no—
64C22×Dic79—41no—
65C22×Dic79—42no—
66#11929614—416no—
67C2×C79⋊C8—24no—
68C2×C79⋊C8—44no—
69C24×C79C2×D411yescharacteristic
70C2×Q8×C79—22no—
71C2×Q8×C79—22no—
72C2×Q8×C79—44no—
73C22×C4×C79—22no—
74C22×C4×C79—22no—
75C22×C4×C79—22no—
76C22×C4×C79—22no—
77C22×C4×C79—42no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79Q1614yescomplemented
81C22×C4×C79C2×D412yes—
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D412yes—
84C22×C4×C79#4111yescharacteristic
85C4⋊C4×C79—22no—
86C4⋊C4×C79—22no—
87C4⋊C4×C79—42no—
88C4⋊C4×C79—42no—
89C4⋊C4×C79—82no—
90C4⋊C4×C79—82no—
91C22⋊C4×C79—42no—
92C22⋊C4×C79—42no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—22no—
96C23×C79#12011yescharacteristic
97C23×C79#12111yescharacteristic
98C23×C79#12711yescharacteristic
99C23×C79C2×Q1612yes—
100C23×C79#13712yes—
101Q8×C79—44no—
102Q8×C79—82no—
103C2×C4×C79—24no—
104C2×C4×C79—24no—
105C2×C4×C79—41no—
106C2×C4×C79—42no—
107C2×C4×C79—41no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—44no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—44no—
116C2×C4×C79—82no—
117C2×C4×C79#12012yes—
118C2×C4×C79#12112yes—
119C2×Dic79—24no—
120C2×Dic79—42no—
121C2×Dic79—44no—
122C2×Dic79—42no—
123C2×Dic79—44no—
124C2×Dic79—44no—
125C2×Dic79—42no—
126C2×Dic79—82no—
127C2×Dic79—81no—
128C2×Dic79—81no—
129Dic158—28no—
130Dic158—44no—
131Dic158—48no—
132C79⋊C8—44no—
133C22×C79—22no—
134C22×C79—22no—
135C22×C79—22no—
136C22×C79—22no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79#39211yescharacteristic
142C22×C79#45112yes—
143C22×C79#45511yescharacteristic
144C22×C79#45611yescharacteristic
145C22×C79B(4,𝔽2)12yes—
146C22×C79#46212yes—
147C22×C79#49411yescharacteristic
148C22×C79#49711yescharacteristic
149C4×C79—22no—
150C4×C79—22no—
151C4×C79—44no—
152C4×C79—82no—
153C4×C79—82no—
154Dic79—44no—
155Dic79—82no—
156#9967—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—22no—
159C2×C79—21no—
160C2×C79#160212yes—
161C2×C79#160512yes—
162C2×C79#200711yescharacteristic
163C2×C79#201611yescharacteristic
164C2×C79#202111yescharacteristic
165C2×#451—792no—
166#2659D7911yescharacteristic
167C2×#328—792no—
168#1446—791no—
169#1280—791no—
170C79#996711yescharacteristic, complemented, Sylow
171C23×Q8—791no—
172C22×Q16—1582no—
173C2×C22⋊Q8C2×D7912yes—
174#451—1584no—
175#451—1584no—
176C2×#103—1582no—
177C2×C22⋊C8—792no—
178#398—22no—
179#395—22no—
180#380—22no—
181C2×#95—791no—
182C2×#95—1582no—
183C2×#95C2×D7911yescharacteristic
184#328—1584no—
185C22×Q8—3164no—
186C22×Q8—792no—
187C22×Q8—1582no—
188C22×Q8—1582no—
189C22×Q8—1581no—
190C22×Q8C22⋊D7912yes—
191C23×C4—1581no—
192C23×C4—1581no—
193C23×C4C22⋊D7912yes—
194C23×C4C22×D7911yescharacteristic
195C2×Q16—3168no—
196C2×Q16—1588no—
197C22×C8—1582no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C22⋊D7912yes—
206C2×C22⋊C4—22no—
207#103—1588no—
208C22⋊C8—794no—
209C22⋊C8—1582no—
210#95—24no—
211#95—22no—
212#95—3164no—
213#95—1582no—
214C24C2×C22⋊D7911yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—3164no—
219C2×Q8—3168no—
220C2×Q8—3164no—
221C2×Q8—3164no—
222C2×Q8—3162no—
223C2×Q8—1588no—
224C2×Q8—1584no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—3162no—
233C22×C4—3161no—
234C22×C4—3161no—
235C22×C4—3162no—
236C22×C4—3161no—
237C22×C4—3161no—
238C22×C4—3162no—
239C22×C4—1582no—
240C22×C4—1584no—
241C22×C4—1582no—
242C22×C4#1192961414yescomplemented
243C22×C4D4×D7911yescharacteristic
244C22×C4D4×D7912yes—
245C22×C4C2×C22⋊D7912yes—
246C22×C4#1192962311yescharacteristic
247Q16—31616no—
248C2×C8—3164no—
249C2×C8—1584no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#1457913011yescharacteristic
262C23#1457913211yescharacteristic
263C23#1457915711yescharacteristic
264C23C2×#1192961412yes—
265C23#1457919212yes—
266Q8—44no—
267Q8—82no—
268Q8—3164no—
269Q8—3168no—
270Q8—1588no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—3164no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3164no—
289C2×C4—3164no—
290C2×C4—3162no—
291C2×C4—6321no—
292C2×C4—6322no—
293C2×C4—6321no—
294C2×C4—1584no—
295C2×C4#1457912912yes—
296C2×C4#1457913312yes—
297C8—3164no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22#1909847111yescharacteristic
307C22#1909871912yes—
308C22#1909874111yescharacteristic
309C22#1909874611yescharacteristic
310C22#1909875012yes—
311C22#1909877012yes—
312C22#1909894211yescharacteristic
313C22#1909895511yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—82no—
318C4—82no—
319C4—3164no—
320C4—6322no—
321C2—21no—
322C2—21no—
323C2—22no—
324C2#2376235712yes—
325C2#2376237412yes—
326C2#2376444711yescharacteristic
327C2#2376450011yescharacteristic
328C2#2376452411yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC23×C4×C79C23
FrattiniC23×C4C22×D79
FittingC79×#2659C2
socleC23×C79#120
prime radicalC24×C79C2×D4
hypercenter#2659D79
hypocenterC79#9967
LCSG ▷ C23×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
80. C22×C4×C79Q16semidirect
170. C79#9967semidirect
242. C22×C4#11929614semidirect

Maximal subgroups

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

subgroupindex
C2×#190987182
C2×#190983002
#237616702
#237612432
C79×#26592
#996779