Finite Group Databasebeta

← #34198948

Subgroups — #34198948

Order 20224.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#23769166C211yesmaximal, characteristic
2#23763477C211yesmaximal, characteristic
3#23761401C214yesmaximal
4C79×#1464C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic, complemented
6C2×#14579140—21no—
7#19098493—21no—
8#19098493C412yescomplemented
9#19098488—21no—
10#19098488C2212yes—
11#19098469—21no—
12#19098429C411yescharacteristic, complemented
13#19098426—21no—
14#19098330—21no—
15#19098330C2212yes—
16C2×#117×C79C2211yescharacteristic
17#432×C79—21no—
18#324×C79—21no—
19#323×C79C2212yes—
20C22×Dic158—21no—
21#14579161—41no—
22#14579161—41no—
23#14579140—44no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C79⋊#95—21no—
31C79⋊#95—22no—
32C79⋊#95—22no—
33C79⋊#95—41no—
34C79⋊#95—41no—
35C79⋊#95C2×C412yes—
36C23×C4×C79D411yescharacteristic
37#117×C79—41no—
38#117×C79—41no—
39C2×C4⋊C4×C79C2×C411yescharacteristic
40C2×C4⋊C4×C79D411yescharacteristic
41C2×C22⋊C4×C79D411yescharacteristic
42C2×C22⋊C4×C79C2311yescharacteristic
43C2×C42×C79C2×C411yescharacteristic
44C2×C42×C79D411yescharacteristic
45C4⋊C8×C79—41no—
46C22⋊C8×C79—22no—
47#97×C79—41no—
48C2×Dic158—42no—
49C2×Dic158—42no—
50C2×Dic158—41no—
51C2×Dic158—41no—
52C22×Dic79—22no—
53C22×Dic79—22no—
54C22×Dic79—42no—
55C4⋊Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58Dic79⋊C4—44no—
59C22⋊Dic79—42no—
60C22⋊Dic79—44no—
61C22⋊Dic79—42no—
62C24×C79C2×D411yescharacteristic
63C22×C4×C79—22no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—41no—
67C22×C4×C79C22⋊C411yescharacteristic
68C22×C4×C79C22⋊C411yescharacteristic
69C22×C4×C79C22⋊C411yescharacteristic
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22×C411yescharacteristic
72C22×C4×C79C2×D411yescharacteristic
73C2×C8×C79—42no—
74C4⋊C4×C79—41no—
75C4⋊C4×C79—41no—
76C22⋊C4×C79—41no—
77C22⋊C4×C79—41no—
78C42×C79—21no—
79C42×C79—21no—
80C42×C79—21no—
81C42×C79—21no—
82C23×C79—21no—
83C23×C79—21no—
84C23×C79—41no—
85C23×C79—41no—
86C23×C79C2×C22⋊C411yescharacteristic
87C23×C79AGL(1,ℤ/8)12yes—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—21no—
92C2×C4×C79—21no—
93C2×C4×C79—21no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79—41no—
103C2×C4×C79#9911yescharacteristic
104C2×C4×C79#9911yescharacteristic
105C2×Dic79—42no—
106C2×Dic79—42no—
107C2×Dic79—42no—
108C2×Dic79—42no—
109C2×Dic79—42no—
110C2×Dic79—42no—
111C2×Dic79—82no—
112C2×Dic79—82no—
113C2×Dic79—82no—
114Dic158—82no—
115Dic158—82no—
116C8×C79—82no—
117C22×C79—21no—
118C22×C79—21no—
119C22×C79—22no—
120C22×C79—42no—
121C22×C79—41no—
122C22×C79—41no—
123C22×C79—81no—
124C22×C79C2×#9911yescharacteristic
125C22×C79#41811yescharacteristic
126C22×C79#42111yescharacteristic
127C4×C79—21no—
128C4×C79—21no—
129C4×C79—41no—
130C4×C79—41no—
131C4×C79—41no—
132C4×C79—41no—
133C4×C79—41no—
134Dic79—82no—
135Dic79—82no—
136Dic79—162no—
137#11801—791nomaximal, Sylow
138C2×C79—21no—
139C2×C79—21no—
140C2×C79—42no—
141C2×C79#151011yescharacteristic
142C2×C79#212311yescharacteristic
143C2×C79#212911yescharacteristic
144#2621—791no—
145#1820—791no—
146#1464D7911yescharacteristic
147#1344—794no—
148C79#1180111yescharacteristic, complemented, Sylow
149C2×C4⋊Q8—791no—
150C2×C22⋊Q8—1581no—
151C2×#117C2×D7911yescharacteristic
152#432—21no—
153#398—792no—
154#398—1581no—
155#397—792no—
156#397—1581no—
157#391—1581no—
158#382—791no—
159#381—1581no—
160#342—792no—
161#342—1581no—
162#324—21no—
163#323C2×D7912yes—
164C22×Q8—1581no—
165C23×C4D31611yescharacteristic
166C4⋊Q8—3161no—
167C4⋊Q8—3161no—
168C22⋊Q8—3164no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—792no—
172C2×C4⋊C4—1582no—
173C2×C4⋊C4—1581no—
174C2×C4⋊C4—1581no—
175C2×C4⋊C4C4×D7911yescharacteristic
176C2×C4⋊C4C22⋊D7911yescharacteristic
177C2×C22⋊C4—1582no—
178C2×C22⋊C4—1582no—
179C2×C22⋊C4D31611yescharacteristic
180C2×C22⋊C4C22×D7911yescharacteristic
181C2×C42C4×D7911yescharacteristic
182C2×C42C22⋊D7911yescharacteristic
183C4⋊C8—41no—
184C22⋊C8—22no—
185#97—41no—
186#95—3161no—
187#95—3161no—
188#95—792no—
189#95—1581no—
190#95—1582no—
191#95—1582no—
192C24C2×D31611yescharacteristic
193C2×Q8—3162no—
194C2×Q8—3162no—
195C2×Q8—3161no—
196C2×Q8—3161no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—3162no—
202C22×C4—1582no—
203C22×C4—1582no—
204C22×C4(C2×D79)⋊C411yescharacteristic
205C22×C4(C2×D79)⋊C411yescharacteristic
206C22×C4(C2×D79)⋊C411yescharacteristic
207C22×C4(C2×D79)⋊C411yescharacteristic
208C22×C4C2×C4×D7911yescharacteristic
209C22×C4C2×C22⋊D7911yescharacteristic
210C2×C8—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—3162no—
214C4⋊C4—3164no—
215C4⋊C4—3162no—
216C4⋊C4—3162no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—3164no—
220C22⋊C4—3162no—
221C22⋊C4—3162no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×(C2×D79)⋊C411yescharacteristic
231C23#1457918912yes—
232Q8—6322no—
233Q8—6322no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—3162no—
250C2×C4—3162no—
251C2×C4—3162no—
252C2×C4—3162no—
253C2×C4—3162no—
254C2×C4—3162no—
255C2×C4—6322no—
256C2×C4—6322no—
257C2×C4—6322no—
258C2×C4#1457907511yescharacteristic
259C2×C4#1457907511yescharacteristic
260C8—82no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1457907511yescharacteristic
269C22#1909856911yescharacteristic
270C22#1909857911yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—6322no—
279C4—6322no—
280C4—12642no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2376193011yescharacteristic
285C2#2376505511yescharacteristic
286C2#2376507411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1464C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1464D79
hypocenterC79#11801
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
5. C2×#14579161C4semidirect
8. #19098493C4semidirect
12. #19098429C4semidirect
148. C79#11801semidirect

Maximal subgroups

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

subgroupindex
#237691662
#237634772
#237614012
C79×#14642
#1180179