Finite Group Databasebeta

← #34203441

Subgroups — #34203441

Order 20224.

564 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 125 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#23766113C211yesmaximal, characteristic
2C2×#19098358C211yesmaximal, characteristic
3#99×Dic79C212yesmaximal
4#23761524C212yesmaximal
5C2×#354×C79C211yesmaximal, characteristic
6C2×Q8×Dic79C411yescharacteristic, complemented
7C2×D4×Dic79C2211yescharacteristic
8C2×#14579076C2212yes—
9#19098470C411yescharacteristic, complemented
10#19098466C2211yescharacteristic
11#19098456—22no—
12#19098438—22no—
13#19098433C411yescharacteristic, complemented
14C22⋊C4×Dic79—22no—
15#19098409C411yescharacteristic, complemented
16#19098358C414yescomplemented
17#19098329—22no—
18C2×#124×C79C2211yescharacteristic
19C2×#99×C79C2212yes—
20#354×C79C414yescomplemented
21C23×Dic79D412yes—
22Q8×Dic79—42no—
23D4×Dic79—42no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79D412yes—
31C2×C4×Dic79—21no—
32C2×C4×Dic79—42no—
33C2×C4×Dic79C2×C411yescharacteristic
34C2×C4×Dic79C2×C411yescharacteristic
35#14579076—22no—
36#14579076C2×C414yes—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—22no—
40C79⋊#95—42no—
41C22×Q8×C79C2×C411yescharacteristic
42C22×D4×C79C2311yescharacteristic
43#124×C79—22no—
44#124×C79C2×C412yes—
45#124×C79C2×C412yes—
46C2×C22⋊C4×C79—22no—
47C2×C22⋊C4×C79—22no—
48C2×C42×C79C2×C411yescharacteristic
49#99×C79—22no—
50#99×C79C2×C414yes—
51C22×Dic79—22no—
52C22×Dic79—22no—
53C22×Dic79—21no—
54C22×Dic79—21no—
55C22×Dic79—42no—
56C22×Dic79—82no—
57C22×Dic79C22⋊C411yescharacteristic
58C22×Dic79C22⋊C412yes—
59C22×Dic79C22⋊C411yescharacteristic
60C4⋊Dic79—41no—
61C4⋊Dic79—41no—
62C4⋊Dic79—42no—
63C4⋊Dic79—82no—
64C22⋊Dic79—24no—
65C22⋊Dic79—42no—
66C22⋊Dic79—42no—
67C22⋊Dic79—82no—
68C4×Dic79—22no—
69C4×Dic79—41no—
70C4×Dic79—42no—
71C4×Dic79—84no—
72C24×C79C2×D412yes—
73C2×Q8×C79—21no—
74C2×Q8×C79—41no—
75C2×Q8×C79—41no—
76C2×Q8×C79C4212yescomplemented
77C2×D4×C79—41no—
78C2×D4×C79—41no—
79C2×D4×C79C22×C411yescharacteristic
80C2×D4×C79C22×C411yescharacteristic
81C2×D4×C79#4112yes—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—42no—
85C22×C4×C79C22×C411yescharacteristic
86C22⋊C4×C79—24no—
87C22⋊C4×C79—24no—
88C22⋊C4×C79—24no—
89C22⋊C4×C79—42no—
90C42×C79—21no—
91C42×C79C4212yescomplemented
92C23×C79—22no—
93C23×C79—22no—
94C23×C79—42no—
95C23×C79—42no—
96C23×C79C2×C22⋊C411yescharacteristic
97C23×C79C4×D412yes—
98C23×C79C4×D412yes—
99Q8×C79—42no—
100Q8×C79—42no—
101D4×C79—41no—
102D4×C79—41no—
103D4×C79—42no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—44no—
113C2×C4×C79—82no—
114C2×C4×C79—82no—
115C2×C4×C79C2×C4211yescharacteristic
116C2×C4×C79#11711yescharacteristic
117C2×Dic79—21no—
118C2×Dic79—22no—
119C2×Dic79—41no—
120C2×Dic79—44no—
121C2×Dic79—42no—
122C2×Dic79—41no—
123C2×Dic79—44no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—82no—
130C2×Dic79—82no—
131C2×Dic79—82no—
132C2×Dic79#9912yes—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—22no—
136C22×C79—22no—
137C22×C79—42no—
138C22×C79—42no—
139C22×C79—42no—
140C22×C79—42no—
141C22×C79—82no—
142C22×C79C4×C22⋊C411yescharacteristic
143C22×C79C2×#9911yescharacteristic
144C22×C79#41011yescharacteristic
145C4×C79—21no—
146C4×C79—21no—
147C4×C79—42no—
148C4×C79—42no—
149C4×C79—84no—
150Dic79—22no—
151Dic79—42no—
152Dic79—41no—
153Dic79—81no—
154Dic79—84no—
155Dic79—81no—
156#12627—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—42no—
160C2×C79—42no—
161C2×C79C4×#9911yescharacteristic
162C2×C79#212711yescharacteristic
163C2×C79#212911yescharacteristic
164#2326—791no—
165C2×#354—791no—
166C2×#354D7911yescharacteristic
167C4×#99—792no—
168#1393—792no—
169C79#1262711yescharacteristic, complemented, Sylow
170C2×#124C2×D7911yescharacteristic
171C2×C4×Q8—791no—
172C2×C4×D4—791no—
173C2×#99—792no—
174C2×#99C2×D7912yes—
175#391—791no—
176#390—791no—
177#388—1582no—
178#385—1582no—
179#384—791no—
180C4×C22⋊C4—1582no—
181#376—791no—
182#354—794no—
183#354Dic7914yescomplemented
184#342—1582no—
185C22×Q8C4×D7911yescharacteristic
186C22×D4C22×D7911yescharacteristic
187C23×C4—792no—
188#124—22no—
189#124C4×D7912yes—
190#124C2×Dic7912yes—
191C4×Q8—3162no—
192C4×D4—3162no—
193C2×C4⋊C4—791no—
194C2×C4⋊C4—791no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1582no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—792no—
200C2×C22⋊C4—1582no—
201C2×C22⋊C4—1582no—
202C2×C42—3162no—
203C2×C42—791no—
204C2×C42—791no—
205C2×C42—1581no—
206C2×C42C4×D7911yescharacteristic
207#99—22no—
208#99—794no—
209#99—1582no—
210#99C2×Dic7914yes—
211#95—3162no—
212#95—1581no—
213#95—1581no—
214#95—1582no—
215C24D4×D7912yes—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8C4×Dic7912yescomplemented
220C2×D4—41no—
221C2×D4—41no—
222C2×D4C2×C4×D7911yescharacteristic
223C2×D4C22×Dic7911yescharacteristic
224C2×D4#1192962312yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—3162no—
229C22×C4—791no—
230C22×C4—791no—
231C22×C4—792no—
232C22×C4—6322no—
233C22×C4—1581no—
234C22×C4—1581no—
235C22×C4—1582no—
236C22×C4—1582no—
237C22×C4C2×C4×D7911yescharacteristic
238C4⋊C4—3161no—
239C4⋊C4—3161no—
240C4⋊C4—3162no—
241C4⋊C4—6322no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—42no—
246C22⋊C4—3162no—
247C22⋊C4—3162no—
248C22⋊C4—6322no—
249C22⋊C4—1584no—
250C42—21no—
251C42—3161no—
252C42—3162no—
253C42—6324no—
254C42—1582no—
255C42C4×Dic7912yescomplemented
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23C22⋊C4×D7911yescharacteristic
261C23(C22⋊D79)⋊C412yes—
262C23D4×Dic7912yes—
263Q8—42no—
264Q8—42no—
265D4—41no—
266D4—41no—
267D4—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—3164no—
280C2×C4—3162no—
281C2×C4—3161no—
282C2×C4—3164no—
283C2×C4—3161no—
284C2×C4—3162no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—792no—
290C2×C4—6322no—
291C2×C4—6322no—
292C2×C4—6322no—
293C2×C4—1581no—
294C2×C4—1582no—
295C2×C4C2×C4×Dic7911yescharacteristic
296C2×C4#1457911711yescharacteristic
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22C22⋊C4×Dic7911yescharacteristic
307C22#99×D7911yescharacteristic
308C22#1909853511yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—84no—
314C4—3162no—
315C4—3161no—
316C4—6321no—
317C4—6324no—
318C4—6321no—
319C4—1582no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#99×Dic7911yescharacteristic
325C2#2376506811yescharacteristic
326C2#2376507311yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC2×C4×C79C2×C42
FrattiniC22×D4C22×D79
FittingC2×#354×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC2×#354D79
hypocenterC79#12627
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic79C4semidirect
9. #19098470C4semidirect
13. #19098433C4semidirect
15. #19098409C4semidirect
16. #19098358C4semidirect
20. #354×C79C4semidirect
76. C2×Q8×C79C42semidirect
91. C42×C79C42semidirect
169. C79#12627semidirect
183. #354Dic79semidirect
219. C2×Q8C4×Dic79semidirect
255. C42C4×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237661132
C2×#190983582
#99×Dic792
#237615242
C2×#354×C792
#1262779