Finite Group Databasebeta

← #34181179

Subgroups — #34181179

Order 20224.

486 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 101 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×(C22⋊D79)⋊C8C211yesmaximal, characteristic
2#23763436C212yesmaximal, complemented
3#23761736C212yesmaximal
4#23761277C211yesmaximal, characteristic, complemented
5C79×#1293C211yesmaximal, characteristic, complemented
6C23×C79⋊C8C2211yescharacteristic
7C22×C8×D79C411yescharacteristic
8C2×C4×C22⋊D79C2211yescharacteristic
9(C22⋊D79)⋊C8—22no—
10(C22⋊D79)⋊C8—22no—
11C2×#14579096—22no—
12C2×Dic79⋊C8C411yescharacteristic
13#19098527C2212yes—
14C2×(C2×D79)⋊C8C411yescharacteristic
15C2×C8×Dic79C411yescharacteristic
16#19098327—22no—
17#19098288—22no—
18#19098286—22no—
19#407×C79C2212yes—
20C2×C22⋊C8×C79C2211yescharacteristic
21C22×C22⋊D79Q811yescharacteristic
22C22×C4×D79C2×C411yescharacteristic
23C2×#11929608—22no—
24C22×C79⋊C8—42no—
25C22×C79⋊C8C2×C411yescharacteristic
26C22×C79⋊C8C2×C411yescharacteristic
27C22×C79⋊C8D412yes—
28C22×C79⋊C8D412yes—
29C2×C8×D79—21no—
30C2×C8×D79—21no—
31C2×C8×D79—42no—
32C4×C22⋊D79—22no—
33C4×C22⋊D79—22no—
34C2×Dic79⋊C4D411yescharacteristic
35C2×C22⋊Dic79Q811yescharacteristic
36C2×(C2×D79)⋊C4D411yescharacteristic
37C2×C4×Dic79C2×C411yescharacteristic
38#14579096—44no—
39Dic79⋊C8—22no—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—22no—
42(C2×D79)⋊C8—22no—
43C8×Dic79—21no—
44C8×Dic79—21no—
45C23×C4×C79C2311yescharacteristic
46C2×M16×C79—22no—
47C22×C8×C79C2×C411yescharacteristic
48C22×C8×C79C2×C411yescharacteristic
49C22⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C22⋊C8×C79—22no—
52C23×D79C4⋊C411yescharacteristic
53C2×C22⋊D79—22no—
54C2×C22⋊D79—22no—
55C2×C22⋊D79—22no—
56C22×Dic79C4⋊C411yescharacteristic
57C22×Dic79C4⋊C411yescharacteristic
58C22×Dic79C4⋊C411yescharacteristic
59C2×C4×D79—21no—
60C2×C4×D79—21no—
61C2×C4×D79—41no—
62C2×C4×D79—41no—
63#11929608—42no—
64#11929608—42no—
65C2×C79⋊C8—24no—
66C2×C79⋊C8—22no—
67C2×C79⋊C8—41no—
68C2×C79⋊C8—42no—
69C2×C79⋊C8—42no—
70C2×C79⋊C8C22⋊C411yescharacteristic
71C2×C79⋊C8C22⋊C412yes—
72C2×C79⋊C8C22⋊C411yescharacteristic
73C8×D79—42no—
74C8×D79—42no—
75Dic79⋊C4—22no—
76C22⋊Dic79—22no—
77(C2×D79)⋊C4—22no—
78C4×Dic79—21no—
79C4×Dic79—21no—
80C24×C79C2×Q811yescharacteristic
81C22×C4×C79—41no—
82C22×C4×C79—41no—
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D412yes—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79#4112yes—
87M16×C79—42no—
88M16×C79—42no—
89C2×C8×C79—21no—
90C2×C8×C79—21no—
91C2×C8×C79—41no—
92C2×C8×C79—41no—
93C2×C8×C79—42no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79C2×C4⋊C411yescharacteristic
99C23×C79C22⋊Q812yes—
100C22×D79—21no—
101C22×D79—21no—
102C22×D79—41no—
103C22×D79—41no—
104C22×D79#10812yescomplemented
105C2×C4×C79—22no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
108C2×C4×C79—21no—
109C2×C4×C79—41no—
110C2×C4×C79—42no—
111C2×C4×C79C2×C22⋊C411yescharacteristic
112C2×C4×C79#11711yescharacteristic
113C2×C4×C79#12112yes—
114C22⋊D79—28no—
115C2×Dic79—21no—
116C2×Dic79—21no—
117C2×Dic79—21no—
118C2×Dic79—21no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79#10812yescomplemented
122C4×D79—42no—
123C4×D79—41no—
124C4×D79—41no—
125C8×C79—41no—
126C8×C79—41no—
127C8×C79—82no—
128C79⋊C8—21no—
129C79⋊C8—22no—
130C79⋊C8—82no—
131C79⋊C8#9912yes—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—22no—
135C22×C79—22no—
136C22×C79—22no—
137C22×C79—42no—
138C22×C79—42no—
139C22×C79#38011yescharacteristic
140C22×C79C2×#10811yescharacteristic
141C22×C79C2×#10811yescharacteristic
142C2×D79—22no—
143C2×D79—42no—
144C2×D79—41no—
145C2×D79—41no—
146C2×D79—42no—
147C4×C79—21no—
148C4×C79—22no—
149C4×C79C2×#9911yescharacteristic
150C4×C79#41011yescharacteristic
151Dic79—22no—
152Dic79—41no—
153Dic79—41no—
154#8612—791nomaximal, Sylow
155C2×C79—21no—
156C2×C79—21no—
157C2×C79—42no—
158C2×C79#179911yescharacteristic
159C2×C79#181011yescharacteristic
160C2×C79#181311yescharacteristic
161D79—42no—
162C2×C8×D4—791no—
163#1810—792no—
164#1465—792no—
165#1293—791no—
166#1293D7911yescharacteristic, complemented
167C79#861211yescharacteristic, complemented, Sylow
168C23×C8—791no—
169C23×C8—791no—
170C2×C4×D4—791no—
171C8×D4—1582no—
172C8×D4—1582no—
173C2×#108—1582no—
174C2×C4⋊C8—791no—
175#407—792no—
176#407C2×D7912yes—
177C2×C22⋊C8—791no—
178C2×C22⋊C8C2×D7911yescharacteristic
179C2×C4×C8—791no—
180#341—1582no—
181#324—1582no—
182#323—1582no—
183C22×D4—791no—
184C23×C4—791no—
185C23×C4C22×D7911yescharacteristic
186C2×M16—22no—
187C2×M16—1582no—
188C22×C8—3162no—
189C22×C8—3162no—
190C22×C8—791no—
191C22×C8—792no—
192C22×C8—791no—
193C22×C8—792no—
194C22×C8—1581no—
195C22×C8—1581no—
196C22×C8C4×D7911yescharacteristic
197C22×C8C4×D7911yescharacteristic
198C4×D4—1582no—
199C4×D4—1582no—
200C2×C4⋊C4—791no—
201C2×C22⋊C4—791no—
202C2×C22⋊C4—791no—
203C2×C42—791no—
204#108—3164no—
205C4⋊C8—1582no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1582no—
210C22⋊C8—1582no—
211C22⋊C8—1582no—
212C4×C8—1581no—
213C4×C8—1581no—
214C24—791no—
215C24Q8×D7911yescharacteristic
216C2×D4—1582no—
217C2×D4—1582no—
218C2×D4—1582no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—3161no—
222C22×C4—3161no—
223C22×C4—791no—
224C22×C4—791no—
225C22×C4—791no—
226C22×C4—1581no—
227C22×C4—1581no—
228C22×C4C2×C4×D7911yescharacteristic
229C22×C4D4×D7912yes—
230C22×C4D4×D7911yescharacteristic
231C22×C4#1192962412yes—
232M16—42no—
233M16—42no—
234M16—3162no—
235M16—3162no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—3162no—
242C2×C8—3162no—
243C2×C8—3162no—
244C2×C8—3162no—
245C2×C8—3161no—
246C2×C8—791no—
247C2×C8—791no—
248C2×C8—792no—
249C2×C8—1584no—
250C2×C8—1582no—
251C4⋊C4—1582no—
252C22⋊C4—1582no—
253C22⋊C4—1582no—
254C42—1581no—
255C42—1581no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—3161no—
261C23—3161no—
262C23—792no—
263C23—1581no—
264C23—1581no—
265C23C4⋊C4×D7911yescharacteristic
266C23#1457913912yes—
267D4—1588no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—3162no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—792no—
280C2×C4—1581no—
281C2×C4—1581no—
282C2×C4—1581no—
283C2×C4—1581no—
284C2×C4C22⋊C4×D7911yescharacteristic
285C2×C4#1457911811yescharacteristic
286C2×C4#1457913412yes—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—792no—
291C8—6322no—
292C8—1581no—
293C8—1582no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—3162no—
302C22—3162no—
303C22—3161no—
304C22—3161no—
305C22—1582no—
306C22#1909842211yescharacteristic
307C22#108×D7911yescharacteristic
308C22#108×D7911yescharacteristic
309C4—22no—
310C4—21no—
311C4—3161no—
312C4—3161no—
313C4—1582no—
314C4#99×D7911yescharacteristic
315C4#1909853511yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—3162no—
320C2#2376338711yescharacteristic
321C2#2376343711yescharacteristic
322C2#2376344911yescharacteristic
subgroupG/H
centerC22#19098422
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1293C2
socleC22×C79#380
prime radicalC22×C22⋊D79Q8
cosocleC23×C4×C79C23
hypercenter#1293D79
hypocenterC79#8612
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #23763436C2semidirect
4. #23761277C2semidirect
5. C79×#1293C2semidirect
104. C22×D79#108semidirect
121. C2×Dic79#108semidirect
166. #1293D79semidirect
167. C79#8612semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D79)⋊C82
#237634362
#237617362
#237612772
C79×#12932
#861279