Finite Group Databasebeta

← #34263067

Subgroups — #34263067

Order 20224.

974 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 207 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#23774541C212yesmaximal, complemented
2#23768392C211yesmaximal, characteristic
3#23762391C218yesmaximal, complemented
4#23761899C212yesmaximal, complemented
5#23761721C211yesmaximal, characteristic
6C79×#3501C211yesmaximal, characteristic
7C2×#14579161C2211yescharacteristic
8C2×#14579148—21no—
9#19098959—22no—
10#19098913—22no—
11#19098891C2218yes—
12#19098848—24no—
13#19098616C2212yes—
14#19098488C2214yes—
15#19098468—21no—
16#19098423—22no—
17C4×C4⋊Dic79C2211yescharacteristic
18#19098300C2218yes—
19#19098287C2214yes—
20#554×C79—22no—
21#554×C79—24no—
22D4×Q8×C79—22no—
23#544×C79—21no—
24#543×C79C2214yes—
25#533×C79C2212yes—
26C2×C22⋊Q8×C79—22no—
27#518×C79—21no—
28C2×C4×D4×C79C2211yescharacteristic
29C22×Dic158—21no—
30#14579161—22no—
31#14579161—22no—
32#14579148—44no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79C2312yes—
36C2×C22⋊Dic79—24no—
37C2×C4×Dic79—21no—
38#14579092—28no—
39C4⋊(C79⋊C8)—22no—
40#14579085—28no—
41C22⋊(C79⋊C8)C2314yes—
42C4×C79⋊C8—22no—
43C79⋊#95—24no—
44C79⋊#95C2312yes—
45C22×Q8×C79—22no—
46C22×D4×C79D411yescharacteristic
47C23×C4×C79D412yes—
48C4⋊Q8×C79—24no—
49C4⋊Q8×C79D412yes—
50C4⋊Q8×C79D412yes—
51#125×C79—24no—
52#125×C79—22no—
53C22⋊Q8×C79—24no—
54C22⋊Q8×C79—24no—
55C22⋊Q8×C79—28no—
56C22⋊Q8×C79—22no—
57C22⋊Q8×C79—24no—
58C22⋊Q8×C79—28no—
59C22⋊Q8×C79C2314yes—
60C4×Q8×C79—22no—
61C4×Q8×C79D414yes—
62C4×D4×C79—24no—
63C4×D4×C79—24no—
64C4×D4×C79—22no—
65C4×D4×C79—22no—
66#117×C79—22no—
67C2×C4⋊C4×C79—22no—
68C2×C4⋊C4×C79—24no—
69C2×C4⋊C4×C79D411yescharacteristic
70C2×C22⋊C4×C79—24no—
71C2×C22⋊C4×C79C2312yes—
72C2×C42×C79C2311yescharacteristic
73C2×Dic158—42no—
74C2×Dic158—42no—
75C2×Dic158—42no—
76C22×Dic79—22no—
77C22×Dic79—22no—
78C2×C79⋊C8—24no—
79C4⋊Dic79—24no—
80C4⋊Dic79—42no—
81C4⋊Dic79—41no—
82C4⋊Dic79—41no—
83C22⋊Dic79—48no—
84C4×Dic79—42no—
85C24×C79C2×D412yes—
86C2×Q8×C79—24no—
87C2×Q8×C79—22no—
88C2×Q8×C79—84no—
89C2×Q8×C79C2×D414yes—
90C2×D4×C79—22no—
91C2×D4×C79—22no—
92C2×D4×C79SD1614yes—
93C22×C4×C79—24no—
94C22×C4×C79—24no—
95C22×C4×C79—22no—
96C22×C4×C79—22no—
97C22×C4×C79—44no—
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79C2×D412yes—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2411yescharacteristic
102C4⋊C4×C79—24no—
103C4⋊C4×C79—28no—
104C4⋊C4×C79—22no—
105C4⋊C4×C79—24no—
106C4⋊C4×C79—22no—
107C4⋊C4×C79—22no—
108C4⋊C4×C79—22no—
109C4⋊C4×C79—24no—
110C4⋊C4×C79C2×D414yes—
111C22⋊C4×C79—28no—
112C22⋊C4×C79—24no—
113C22⋊C4×C79—24no—
114C42×C79—24no—
115C42×C79C2×D412yes—
116C42×C79C2×D412yes—
117C23×C79—22no—
118C23×C79—22no—
119C23×C79—42no—
120C23×C79—42no—
121C23×C79C2×SD1614yes—
122C23×C79C22×D411yescharacteristic
123Q8×C79—44no—
124Q8×C79—44no—
125Q8×C79—84no—
126D4×C79—28no—
127C2×C4×C79—22no—
128C2×C4×C79—24no—
129C2×C4×C79—24no—
130C2×C4×C79—22no—
131C2×C4×C79—41no—
132C2×C4×C79—44no—
133C2×C4×C79—42no—
134C2×C4×C79—82no—
135C2×C4×C79—84no—
136C2×C4×C79—84no—
137C2×C4×C79#12011yescharacteristic
138C2×C4×C79#12012yes—
139C2×C4×C79#12011yescharacteristic
140C2×C4×C79C2×SD1612yes—
141C2×C4×C79#13712yes—
142C2×C4×C79C22×D412yes—
143C2×Dic79—42no—
144C2×Dic79—42no—
145C2×Dic79—42no—
146C2×Dic79—42no—
147C2×Dic79—42no—
148C2×Dic79—42no—
149Dic158—84no—
150C79⋊C8—44no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—24no—
154C22×C79—44no—
155C22×C79—42no—
156C22×C79—42no—
157C22×C79—82no—
158C22×C79C2×#12011yescharacteristic
159C22×C79C22×SD1611yescharacteristic
160C22×C79C2×#13711yescharacteristic
161C4×C79—21no—
162C4×C79—22no—
163C4×C79—42no—
164C4×C79—44no—
165C4×C79—44no—
166C4×C79#44814yes—
167C4×C79#45512yes—
168Dic79—82no—
169Dic79—82no—
170#18396—791nomaximal, Sylow
171C2×C79—21no—
172C2×C79—21no—
173C2×C79—44no—
174C2×C79C2×#44811yescharacteristic
175C2×C79C2×#45511yescharacteristic
176C2×C79#302811yescharacteristic
177#3501D7911yescharacteristic
178#3229—792no—
179#2547—791no—
180#1608—798no—
181#1503—792no—
182#1460—791no—
183C79#1839611yescharacteristic, complemented, Sylow
184#554—22no—
185#554—24no—
186D4×Q8—22no—
187#544—21no—
188#543C2×D7914yes—
189#533C2×D7912yes—
190C2×C4⋊Q8—791no—
191C2×#124—1581no—
192C2×C22⋊Q8—22no—
193#518—21no—
194C2×C4×D4C2×D7911yescharacteristic
195#498—1582no—
196#487—1582no—
197#483—798no—
198#475—1584no—
199#431—792no—
200#397—794no—
201#391—1581no—
202#380—1582no—
203C4×C4⋊C4—791no—
204#328—798no—
205#323—794no—
206C22×Q8—22no—
207C22×Q8—1581no—
208C22×D4C22⋊D7911yescharacteristic
209C23×C4C22⋊D7912yes—
210C4⋊Q8—24no—
211C4⋊Q8—1582no—
212C4⋊Q8—1582no—
213C4⋊Q8C22⋊D7912yes—
214C4⋊Q8C22⋊D7912yes—
215#125—24no—
216#125—22no—
217#124—3164no—
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—28no—
221C22⋊Q8—22no—
222C22⋊Q8—24no—
223C22⋊Q8—28no—
224C22⋊Q8C22×D7914yes—
225C4×Q8—22no—
226C4×Q8C22⋊D7914yes—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—22no—
230C4×D4—22no—
231#117—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—792no—
235C2×C4⋊C4—1581no—
236C2×C4⋊C4—1581no—
237C2×C4⋊C4C22⋊D7911yescharacteristic
238C2×C22⋊C4—24no—
239C2×C22⋊C4—1584no—
240C2×C22⋊C4C22×D7912yes—
241C2×C42—1581no—
242C2×C42C22×D7911yescharacteristic
243#106—1588no—
244C4⋊C8—1582no—
245#103—1588no—
246C22⋊C8—794no—
247C4×C8—1582no—
248#95—792no—
249#95—1584no—
250C24C2×C22⋊D7912yes—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—84no—
254C2×Q8—3162no—
255C2×Q8—3162no—
256C2×Q8—3162no—
257C2×Q8C2×C22⋊D7914yes—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4#1192961114yes—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—44no—
266C22×C4—1582no—
267C22×C4—1582no—
268C22×C4C2×C22⋊D7911yescharacteristic
269C22×C4C2×C22⋊D7912yes—
270C22×C4C2×C22⋊D7911yescharacteristic
271C22×C4C23×D7911yescharacteristic
272C2×C8—1584no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—3162no—
282C4⋊C4—3161no—
283C4⋊C4—3161no—
284C4⋊C4—1584no—
285C4⋊C4C2×C22⋊D7914yes—
286C22⋊C4—28no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—3168no—
290C42—24no—
291C42—3162no—
292C42C2×C22⋊D7912yes—
293C42C2×C22⋊D7912yes—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23C2×#1192961114yes—
299C23C22×C22⋊D7911yescharacteristic
300Q8—44no—
301Q8—44no—
302Q8—84no—
303Q8—6324no—
304D4—28no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—3162no—
316C2×C4—3162no—
317C2×C4—3162no—
318C2×C4—3162no—
319C2×C4—3162no—
320C2×C4—3162no—
321C2×C4#1457913011yescharacteristic
322C2×C4#1457913012yes—
323C2×C4#1457913011yescharacteristic
324C2×C4C2×#1192961112yes—
325C2×C4#1457919212yes—
326C2×C4C22×C22⋊D7912yes—
327C8—3164no—
328C22—21no—
329C22—21no—
330C22—24no—
331C22—44no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22C2×#1457913011yescharacteristic
336C22C22×#1192961111yescharacteristic
337C22C2×#1457919211yescharacteristic
338C4—21no—
339C4—22no—
340C4—42no—
341C4—44no—
342C4—44no—
343C4—6322no—
344C4—6322no—
345C4#1909870214yes—
346C4#1909874112yes—
347C2—21no—
348C2—21no—
349C2—44no—
350C2C2×#1909870211yescharacteristic
351C2C2×#1909874111yescharacteristic
352C2#2377280611yescharacteristic
subgroupG/H
centerC22C2×#14579130
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#3501C2
socleC22×C79C2×#120
prime radicalC22×D4×C79D4
hypercenter#3501D79
hypocenterC79#18396
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #23774541C2semidirect
3. #23762391C2semidirect
4. #23761899C2semidirect
183. C79#18396semidirect

Maximal subgroups

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

subgroupindex
#237745412
#237683922
#237623912
#237618992
#237617212
C79×#35012
#1839679