Finite Group Databasebeta

← #34181183

Subgroups — #34181183

Order 20224.

460 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19098631C211yesmaximal, characteristic
2#23763436C212yesmaximal, complemented
3#23761736C212yesmaximal
4#23761277C211yesmaximal, characteristic, complemented
5C79×#1293C211yesmaximal, characteristic, complemented
6C22×#11929607C411yescharacteristic
7C23×C79⋊C8C2211yescharacteristic
8C2×C4×C22⋊D79C2211yescharacteristic
9#19098631—22no—
10#19098631—22no—
11C2×#14579096—22no—
12C2×Dic79⋊C8C411yescharacteristic
13#19098527C2212yes—
14C2×(C2×D79)⋊C8C411yescharacteristic
15C2×#14579071C411yescharacteristic
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—
24C2×#11929607—21no—
25C2×#11929607—21no—
26C2×#11929607—42no—
27C22×C79⋊C8—42no—
28C22×C79⋊C8C2×C411yescharacteristic
29C22×C79⋊C8C2×C411yescharacteristic
30C22×C79⋊C8D412yes—
31C22×C79⋊C8D412yes—
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—
43#14579071—21no—
44#14579071—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—42no—
55C22×Dic79C4⋊C411yescharacteristic
56C22×Dic79C4⋊C411yescharacteristic
57C22×Dic79C4⋊C411yescharacteristic
58C2×C4×D79—21no—
59C2×C4×D79—21no—
60C2×C4×D79—41no—
61C2×C4×D79—41no—
62#11929608—42no—
63#11929608—42no—
64#11929607—42no—
65#11929607—42no—
66C2×C79⋊C8—24no—
67C2×C79⋊C8—22no—
68C2×C79⋊C8—41no—
69C2×C79⋊C8—42no—
70C2×C79⋊C8—42no—
71C2×C79⋊C8C22⋊C411yescharacteristic
72C2×C79⋊C8C22⋊C412yes—
73C2×C79⋊C8C22⋊C411yescharacteristic
74Dic79⋊C4—22no—
75C22⋊Dic79—22no—
76(C2×D79)⋊C4—22no—
77C4×Dic79—21no—
78C4×Dic79—21no—
79C24×C79C2×Q811yescharacteristic
80C22×C4×C79—41no—
81C22×C4×C79—41no—
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D412yes—
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79#4112yes—
86M16×C79—42no—
87M16×C79—42no—
88C2×C8×C79—21no—
89C2×C8×C79—21no—
90C2×C8×C79—41no—
91C2×C8×C79—41no—
92C2×C8×C79—42no—
93C23×C79—22no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79C2×C4⋊C411yescharacteristic
97C23×C79C22⋊Q812yes—
98C22×D79—21no—
99C22×D79—22no—
100C22×D79—41no—
101C22×D79—41no—
102C2×C4×C79—22no—
103C2×C4×C79—21no—
104C2×C4×C79—22no—
105C2×C4×C79—21no—
106C2×C4×C79—41no—
107C2×C4×C79—42no—
108C2×C4×C79C2×C22⋊C411yescharacteristic
109C2×C4×C79#11711yescharacteristic
110C2×C4×C79#12112yes—
111C22⋊D79—44no—
112C2×Dic79—21no—
113C2×Dic79—21no—
114C2×Dic79—21no—
115C2×Dic79—22no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C4×D79—42no—
119C4×D79—41no—
120C4×D79—41no—
121C8×C79—41no—
122C8×C79—41no—
123C8×C79—82no—
124C79⋊C8—22no—
125C79⋊C8—21no—
126C79⋊C8—82no—
127C79⋊C8#9912yes—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—22no—
131C22×C79—22no—
132C22×C79—22no—
133C22×C79—44no—
134C22×C79#38011yescharacteristic
135C22×C79#43012yes—
136C2×D79—41no—
137C2×D79—41no—
138C2×D79—41no—
139C2×D79—82no—
140C4×C79—21no—
141C4×C79—22no—
142C4×C79C2×#9911yescharacteristic
143C4×C79#41011yescharacteristic
144Dic79—41no—
145Dic79—41no—
146Dic79—41no—
147#8613—791nomaximal, Sylow
148C2×C79—22no—
149C2×C79—42no—
150C2×C79#179911yescharacteristic
151C2×C79#181111yescharacteristic
152C2×C79#181311yescharacteristic
153D79—81no—
154C2×#435—791no—
155#1810—792no—
156#1465—792no—
157#1293—791no—
158#1293D7911yescharacteristic, complemented
159C79#861311yescharacteristic, complemented, Sylow
160C22×M16—791no—
161C23×C8—791no—
162C2×C4×D4—791no—
163#435—1582no—
164#435—1582no—
165C2×#108—1582no—
166C2×C4⋊C8—791no—
167#407—792no—
168#407C2×D7912yes—
169C2×C22⋊C8—791no—
170C2×C22⋊C8C2×D7911yescharacteristic
171C2×#97—791no—
172#341—1582no—
173#324—1582no—
174#323—1582no—
175C22×D4—791no—
176C23×C4—791no—
177C23×C4C22×D7911yescharacteristic
178C2×M16—22no—
179C2×M16—3162no—
180C2×M16—1581no—
181C2×M16—1581no—
182C2×M16—1582no—
183C22×C8—3162no—
184C22×C8—791no—
185C22×C8—792no—
186C22×C8—791no—
187C22×C8—792no—
188C22×C8C4×D7911yescharacteristic
189C22×C8C4×D7911yescharacteristic
190C4×D4—1582no—
191C4×D4—1582no—
192C2×C4⋊C4—791no—
193C2×C22⋊C4—791no—
194C2×C22⋊C4—791no—
195C2×C42—791no—
196#108—3164no—
197C4⋊C8—1582no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—1582no—
202C22⋊C8—1582no—
203C22⋊C8—1582no—
204#97—1581no—
205#97—1581no—
206C24—791no—
207C24Q8×D7911yescharacteristic
208C2×D4—3162no—
209C2×D4—1582no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—3161no—
213C22×C4—3161no—
214C22×C4—791no—
215C22×C4—791no—
216C22×C4—791no—
217C22×C4—1581no—
218C22×C4—1581no—
219C22×C4C2×C4×D7911yescharacteristic
220C22×C4D4×D7912yes—
221C22×C4D4×D7911yescharacteristic
222C22×C4#1192962412yes—
223M16—42no—
224M16—42no—
225M16—3162no—
226M16—3162no—
227M16—3162no—
228M16—3162no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—3162no—
235C2×C8—3162no—
236C2×C8—3161no—
237C2×C8—791no—
238C2×C8—791no—
239C2×C8—792no—
240C2×C8—1584no—
241C2×C8—1582no—
242C4⋊C4—1582no—
243C22⋊C4—1582no—
244C22⋊C4—1582no—
245C42—1581no—
246C42—1581no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—3161no—
251C23—3161no—
252C23—1581no—
253C23—1582no—
254C23C4⋊C4×D7911yescharacteristic
255C23#1457913912yes—
256D4—3164no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—3162no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4—3161no—
267C2×C4—3161no—
268C2×C4—1581no—
269C2×C4—1581no—
270C2×C4—1581no—
271C2×C4—1582no—
272C2×C4C22⋊C4×D7911yescharacteristic
273C2×C4#1457911811yescharacteristic
274C2×C4#1457913412yes—
275C8—41no—
276C8—41no—
277C8—82no—
278C8—792no—
279C8—6322no—
280C8—1581no—
281C8—1582no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—44no—
288C22—3161no—
289C22—3161no—
290C22—3161no—
291C22—6322no—
292C22#1909842211yescharacteristic
293C22#1909861012yes—
294C4—22no—
295C4—21no—
296C4—3161no—
297C4—3161no—
298C4—3161no—
299C4#99×D7911yescharacteristic
300C4#1909853511yescharacteristic
301C2—22no—
302C2—42no—
303C2—6321no—
304C2#2376338711yescharacteristic
305C2#2376344111yescharacteristic
306C2#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#8613
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
158. #1293D79semidirect
159. C79#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#190986312
#237634362
#237617362
#237612772
C79×#12932
#861379