Finite Group Databasebeta

← #34206926

Subgroups — #34206926

Order 20224.

488 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 107 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#23767717C211yesmaximal, characteristic
2#23763392C211yesmaximal, characteristic
3#23761517C214yesmaximal
4C4×#99×C79C211yesmaximal, characteristic
5C22×C4⋊Dic79C411yescharacteristic, complemented
6#19098587—21no—
7C2×#14579078C2212yes—
8#19098502—21no—
9#19098499—21no—
10#19098461—21no—
11#19098443C2212yes—
12#19098435C412yescomplemented
13#19098423—21no—
14#19098423C411yescharacteristic, complemented
15#19098332—21no—
16C2×C4×D4×C79C2211yescharacteristic
17C2×#99×C79C2212yes—
18C4×C22⋊C4×C79—21no—
19#342×C79—21no—
20C23×Dic79C2×C412yes—
21C2×#11929608—22no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79—41no—
27C2×Dic79⋊C4—22no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79C2×C412yes—
30C4⋊(C79⋊C8)—41no—
31C4⋊(C79⋊C8)—41no—
32#14579078—24no—
33C79⋊#95—22no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C22×D4×C79C2311yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C4×D4×C79—41no—
40C4×D4×C79—41no—
41C2×C4⋊C4×C79D411yescharacteristic
42C2×C22⋊C4×C79—22no—
43C2×C22⋊C4×C79C2×C411yescharacteristic
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C42×C79—41no—
46C2×C42×C79D411yescharacteristic
47#99×C79—22no—
48#99×C79Q814yes—
49#95×C79—41no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—42no—
53C22×Dic79—82no—
54#11929608—42no—
55#11929608—42no—
56C2×C79⋊C8—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—42no—
59C4⋊Dic79—41no—
60C4⋊Dic79—41no—
61Dic79⋊C4—82no—
62C22⋊Dic79—42no—
63C22⋊Dic79—82no—
64C24×C79C22×C411yescharacteristic
65C24×C79C2×D411yescharacteristic
66C2×D4×C79—41no—
67C2×D4×C79—41no—
68C2×D4×C79C2×Q812yes—
69C2×D4×C79#4112yes—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C4⋊C4×C79—41no—
83C22⋊C4×C79—24no—
84C22⋊C4×C79—41no—
85C22⋊C4×C79—41no—
86C22⋊C4×C79—42no—
87C42×C79—21no—
88C42×C79—21no—
89C42×C79—82no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79C2×C22⋊C411yescharacteristic
99C23×C79C4×Q812yes—
100C23×C79C22⋊Q812yes—
101D4×C79—42no—
102D4×C79—42no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—44no—
116C2×C4×C79—81no—
117C2×C4×C79—82no—
118C2×C4×C79—82no—
119C2×C4×C79#9911yescharacteristic
120C2×C4×C79#9911yescharacteristic
121C2×C4×C79#12411yescharacteristic
122C2×C4×C79C4⋊Q811yescharacteristic
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—84no—
130C79⋊C8—82no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—22no—
134C22×C79—22no—
135C22×C79—42no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—81no—
143C22×C79#39111yescharacteristic
144C22×C79C2×#9911yescharacteristic
145C22×C79#41011yescharacteristic
146C4×C79—21no—
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—41no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—84no—
153C4×C79#35212yes—
154Dic79—82no—
155Dic79—82no—
156#13295—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—42no—
160C2×C79—42no—
161C2×C79#199611yescharacteristic
162C2×C79C2×#35211yescharacteristic
163C2×C79#212211yescharacteristic
164#2487—791no—
165#1800—791no—
166C4×#99D7911yescharacteristic
167#1390—794no—
168C79#1329511yescharacteristic, complemented, Sylow
169C2×C4×D4C2×D7911yescharacteristic
170C22×C4⋊C4—791no—
171#423—1581no—
172C2×#100—792no—
173C2×#99C2×D7912yes—
174#400—1581no—
175#399—1581no—
176#389—1581no—
177#386—792no—
178#385—792no—
179#380—791no—
180#380—1581no—
181C4×C22⋊C4—21no—
182#343—1581no—
183#342—21no—
184C22×D4C22×D7911yescharacteristic
185C23×C4—792no—
186C23×C4C4×D7911yescharacteristic
187C23×C4C22⋊D7911yescharacteristic
188C2×M16—1582no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—3161no—
192C2×C4⋊C4—3161no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4D31611yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—792no—
200C2×C22⋊C4—1582no—
201C2×C22⋊C4C4×D7911yescharacteristic
202C2×C22⋊C4C22⋊D7911yescharacteristic
203C2×C42—41no—
204C2×C42D31611yescharacteristic
205C4⋊C8—3161no—
206C4⋊C8—3161no—
207#100—1584no—
208#99—22no—
209#99Dic15814yes—
210#95—41no—
211#95—1581no—
212#95—1582no—
213#95—1582no—
214C24C2×C4×D7911yescharacteristic
215C24C2×C22⋊D7911yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4C2×Dic15812yes—
219C2×D4#1192962412yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—3162no—
228C22×C4—6322no—
229C22×C4—1582no—
230C22×C4—1582no—
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4(C2×D79)⋊C411yescharacteristic
233C22×C4(C2×D79)⋊C411yescharacteristic
234C22×C4(C2×D79)⋊C411yescharacteristic
235C22×C4C2×D31611yescharacteristic
236M16—3162no—
237M16—3162no—
238C2×C8—3162no—
239C4⋊C4—41no—
240C4⋊C4—3162no—
241C4⋊C4—3162no—
242C4⋊C4—3161no—
243C4⋊C4—3161no—
244C4⋊C4—6322no—
245C22⋊C4—24no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—3162no—
250C22⋊C4—6322no—
251C42—21no—
252C42—21no—
253C42—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×(C2×D79)⋊C411yescharacteristic
263C23C4×Dic15812yes—
264C23#1457914012yes—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—3162no—
284C2×C4—3162no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3162no—
289C2×C4—6324no—
290C2×C4#1457907511yescharacteristic
291C2×C4#1457907511yescharacteristic
292C2×C4#1457914911yescharacteristic
293C2×C4#1457916111yescharacteristic
294C8—6322no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1909846911yescharacteristic
308C22C2×#1457907511yescharacteristic
309C22#1909853611yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—84no—
317C4—6322no—
318C4—6322no—
319C4#1909835212yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2376438511yescharacteristic
325C2C2×#1909835211yescharacteristic
326C2#2376504911yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC4×#99×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#99D79
hypocenterC79#13295
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Dic79C4semidirect
12. #19098435C4semidirect
14. #19098423C4semidirect
168. C79#13295semidirect

Maximal subgroups

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

subgroupindex
#237677172
#237633922
#237615172
C4×#99×C792
#1329579