Finite Group Databasebeta

← #34180003

Subgroups — #34180003

Order 20224.

738 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19099283C211yesmaximal, characteristic
2#23762073C212yesmaximal, complemented
3#23761828C212yesmaximal, complemented
4#23761721C211yesmaximal, characteristic
5C79×#1800C211yesmaximal, characteristic
6#19099283—24no—
7#19099283—24no—
8C2×#14579161C2211yescharacteristic
9C2×#14579154—22no—
10C2×#14579148—21no—
11C2×#14579140C412yes—
12C2×C4×Dic158C2211yescharacteristic
13C2×#14579126—21no—
14C2×#14579120C412yes—
15#19098616C2212yes—
16#19098310—24no—
17#19098293—24no—
18#19098287—22no—
19C2×C4×D4×C79C2211yescharacteristic
20#423×C79C2212yes—
21C2×#100×C79—21no—
22#343×C79—21no—
23C22×Dic158—21no—
24C22×Dic158C2×C411yescharacteristic
25#14579161—22no—
26#14579161D414yes—
27#14579154—216no—
28#14579148—28no—
29#14579140—24no—
30#14579140—24no—
31C4×Dic158—22no—
32C4×Dic158D414yes—
33#14579126—28no—
34#14579120—24no—
35#14579120—24no—
36C2×C4⋊Dic79—21no—
37C2×C4⋊Dic79C2×C411yescharacteristic
38C2×C4⋊Dic79C2×C412yes—
39C2×Dic79⋊C4—22no—
40C2×Dic79⋊C4—21no—
41C2×C22⋊Dic79—22no—
42C2×C22⋊Dic79—22no—
43C2×C4×Dic79—22no—
44C2×C4×Dic79—21no—
45C4⋊(C79⋊C8)—22no—
46C22⋊(C79⋊C8)—22no—
47C4×C79⋊C8—22no—
48C22×D4×C79D411yescharacteristic
49C23×C4×C79C2×C412yes—
50C2×M16×C79—22no—
51C4×D4×C79—22no—
52C4×D4×C79—22no—
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79D412yes—
55C2×C42×C79C2311yescharacteristic
56C4⋊C8×C79—22no—
57C4⋊C8×C79—22no—
58#100×C79—42no—
59C2×Dic158—21no—
60C2×Dic158—24no—
61C2×Dic158—42no—
62C2×Dic158—42no—
63C2×Dic158—41no—
64C2×Dic158—41no—
65C2×Dic158C22⋊C412yes—
66C22×Dic79—22no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C2×C79⋊C8—42no—
70C4⋊Dic79—21no—
71C4⋊Dic79—22no—
72C4⋊Dic79—24no—
73C4⋊Dic79C22⋊C412yes—
74C4⋊Dic79C22⋊C414yes—
75Dic79⋊C4—28no—
76Dic79⋊C4—24no—
77C22⋊Dic79—28no—
78C22⋊Dic79—28no—
79C4×Dic79—28no—
80C4×Dic79—24no—
81C24×C79C22⋊C412yes—
82C2×D4×C79—22no—
83C2×D4×C79—22no—
84C2×D4×C79—21no—
85C2×D4×C79—21no—
86C22×C4×C79—22no—
87C22×C4×C79—22no—
88C22×C4×C79—44no—
89C22×C4×C79C22⋊C412yes—
90C22×C4×C79C22×C411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93M16×C79—42no—
94M16×C79—42no—
95C2×C8×C79—42no—
96C4⋊C4×C79—22no—
97C22⋊C4×C79—24no—
98C42×C79C2×D412yes—
99C42×C79C2×D412yes—
100C23×C79—22no—
101C23×C79—22no—
102C23×C79—22no—
103C23×C79—42no—
104C23×C79—42no—
105C23×C79C2×C22⋊C411yescharacteristic
106D4×C79—42no—
107D4×C79—42no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—41no—
111C2×C4×C79—44no—
112C2×C4×C79—44no—
113C2×C4×C79—42no—
114C2×C4×C79C2×C22⋊C411yescharacteristic
115C2×C4×C79C2×C22⋊C411yescharacteristic
116C2×C4×C79#12012yes—
117C2×C4×C79#12012yes—
118C2×C4×C79#13512yes—
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C79#13711yescharacteristic
121C2×Dic79—22no—
122C2×Dic79—22no—
123C2×Dic79—24no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128Dic158—24no—
129Dic158—42no—
130Dic158—44no—
131Dic158—44no—
132C8×C79—82no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—42no—
139C22×C79—82no—
140C22×C79—82no—
141C22×C79#37911yescharacteristic
142C22×C79#41611yescharacteristic
143C22×C79#41711yescharacteristic
144C4×C79—22no—
145C4×C79—21no—
146C4×C79—42no—
147C4×C79#41312yes—
148C4×C79#44912yes—
149C4×C79#45212yes—
150Dic79—42no—
151Dic79—42no—
152Dic79—44no—
153#8389—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79—82no—
157C2×C79#178511yescharacteristic
158C2×C79#178811yescharacteristic
159C2×C79#179611yescharacteristic
160C2×#556—791no—
161#1800D7911yescharacteristic
162#1546—792no—
163#1487—792no—
164#1460—791no—
165C79#838911yescharacteristic, complemented, Sylow
166#556—1584no—
167#556—1584no—
168C2×C4⋊Q8—791no—
169C2×#126—1582no—
170C2×#124—1581no—
171C2×C22⋊Q8—792no—
172C2×C4×Q8—791no—
173C2×C4×Q8—1581no—
174C2×C4×D4C2×D7911yescharacteristic
175C2×#117—792no—
176#431—792no—
177#423C2×D7912yes—
178C2×#100—21no—
179#343—21no—
180#332—1584no—
181Q8⋊C8—1584no—
182#323—1582no—
183C22×Q8—791no—
184C22×Q8—1581no—
185C22×D4C22⋊D7911yescharacteristic
186C23×C4C4×D7912yes—
187C2×M16—22no—
188C4⋊Q8—794no—
189C4⋊Q8—1582no—
190#126—15816no—
191#124—1588no—
192C22⋊Q8—1584no—
193C22⋊Q8—1584no—
194C4×Q8—794no—
195C4×Q8—1588no—
196C4×Q8—1582no—
197C4×D4—22no—
198C4×D4—22no—
199#117—1584no—
200#117—1584no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—792no—
203C2×C4⋊C4—1582no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4C22⋊D7911yescharacteristic
207C2×C22⋊C4—1582no—
208C2×C22⋊C4—1582no—
209C2×C22⋊C4D31612yes—
210C2×C42—1582no—
211C2×C42—1581no—
212C2×C42C22×D7911yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—1582no—
216#100—42no—
217C22⋊C8—1582no—
218C4×C8—1582no—
219C24(C2×D79)⋊C412yes—
220C2×Q8—3162no—
221C2×Q8—3162no—
222C2×Q8—3161no—
223C2×Q8—3161no—
224C2×Q8—792no—
225C2×Q8—1581no—
226C2×Q8—1584no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—1581no—
235C22×C4—1582no—
236C22×C4—1581no—
237C22×C4(C2×D79)⋊C412yes—
238C22×C4C2×C4×D7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240C22×C4C2×D31611yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—3162no—
245C4⋊C4—22no—
246C4⋊C4—792no—
247C4⋊C4—794no—
248C4⋊C4—1588no—
249C4⋊C4—1581no—
250C4⋊C4—1582no—
251C4⋊C4—1584no—
252C4⋊C4—1584no—
253C22⋊C4—24no—
254C22⋊C4—1588no—
255C22⋊C4—1588no—
256C42—1588no—
257C42—1584no—
258C42D4×D7912yes—
259C42D4×D7912yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×(C2×D79)⋊C411yescharacteristic
266Q8—3162no—
267Q8—3164no—
268Q8—3164no—
269Q8—1584no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—3162no—
279C2×C4—3162no—
280C2×C4—3162no—
281C2×C4—3162no—
282C2×C4—1582no—
283C2×C4—1584no—
284C2×C4—1582no—
285C2×C4C22⋊C4×D7911yescharacteristic
286C2×C4C22⋊C4×D7911yescharacteristic
287C2×C4#1457912912yes—
288C2×C4#1457913112yes—
289C2×C4#1457918112yes—
290C2×C4#1457918711yescharacteristic
291C2×C4#1457919211yescharacteristic
292C8—82no—
293C8—6322no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#1909841811yescharacteristic
302C22#1909855711yescharacteristic
303C22#1909856211yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—3162no—
308C4—3164no—
309C4—3162no—
310C4#1909854412yes—
311C4#1909870612yes—
312C4#1909872312yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2376332511yescharacteristic
317C2#2376333811yescharacteristic
318C2#2376337611yescharacteristic
subgroupG/H
centerC22#19098418
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1800C2
socleC22×C79#379
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#1800D79
hypocenterC79#8389
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #23762073C2semidirect
3. #23761828C2semidirect
165. C79#8389semidirect

Maximal subgroups

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

subgroupindex
C2×#190992832
#237620732
#237618282
#237617212
C79×#18002
#838979