Finite Group Databasebeta

← #34204270

Subgroups — #34204270

Order 20224.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C4⋊Q8×Dic79C211yesmaximal, characteristic
2C2×#19098362C211yesmaximal, characteristic
3#23763394C212yesmaximal
4#23761526C212yesmaximal
5C2×#356×C79C211yesmaximal, characteristic
6C2×Q8×Dic79C411yescharacteristic
7C2×Q8×Dic79C2211yescharacteristic
8#19098586—22no—
9C2×#14579080C2212yes—
10#19098470C411yescharacteristic
11#19098470C2211yescharacteristic
12#19098462—22no—
13#19098433C411yescharacteristic
14C4⋊C4×Dic79—22no—
15C42×Dic79C411yescharacteristic
16#19098362—22no—
17#19098331—22no—
18C2×C4⋊Q8×C79C2211yescharacteristic
19C2×#101×C79C2212yes—
20#356×C79—22no—
21C2×#11929608—22no—
22Q8×Dic79—41no—
23Q8×Dic79—41no—
24Q8×Dic79—41no—
25Q8×Dic79—41no—
26C2×C4⋊Dic79—22no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79D412yes—
31C2×C4×Dic79—21no—
32C2×C4×Dic79—21no—
33C2×C4×Dic79—21no—
34C2×C4×Dic79C2×C411yescharacteristic
35C2×C4×Dic79C2×C411yescharacteristic
36C2×C4×Dic79D412yes—
37Dic79⋊C8—42no—
38Dic79⋊C8—42no—
39#14579080—22no—
40#14579080—22no—
41C79⋊#95—22no—
42C79⋊#95—22no—
43C22×Q8×C79C2×C411yescharacteristic
44C22×Q8×C79C2311yescharacteristic
45C2×M16×C79—22no—
46C4⋊Q8×C79—22no—
47C4⋊Q8×C79D412yes—
48C4⋊Q8×C79Q812yes—
49C2×C4⋊C4×C79—22no—
50C2×C42×C79C2×C411yescharacteristic
51#101×C79—22no—
52#101×C79—22no—
53C22×Dic79—21no—
54C22×Dic79—21no—
55C22×Dic79C22⋊C411yescharacteristic
56C22×Dic79C22⋊C412yes—
57C22×Dic79C22⋊C411yescharacteristic
58#11929608—42no—
59#11929608—42no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—41no—
62C4⋊Dic79—42no—
63C4⋊Dic79—41no—
64C4⋊Dic79—44no—
65C4⋊Dic79—42no—
66C4×Dic79—24no—
67C4×Dic79—22no—
68C4×Dic79—22no—
69C4×Dic79—21no—
70C4×Dic79—21no—
71C4×Dic79—41no—
72C4×Dic79—42no—
73C4×Dic79—41no—
74C4×Dic79—41no—
75C2×Q8×C79—21no—
76C2×Q8×C79—41no—
77C2×Q8×C79—41no—
78C2×Q8×C79—41no—
79C2×Q8×C79—41no—
80C2×Q8×C79C4⋊C412yes—
81C2×Q8×C79C2×D411yescharacteristic
82C2×Q8×C79C2×Q811yescharacteristic
83C2×Q8×C79#4112yes—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79C22×C411yescharacteristic
87C22×C4×C79C2×D412yes—
88M16×C79—42no—
89M16×C79—42no—
90C2×C8×C79—42no—
91C4⋊C4×C79—24no—
92C4⋊C4×C79—24no—
93C42×C79—21no—
94C42×C79C4⋊C412yes—
95C23×C79C2×C22⋊C411yescharacteristic
96Q8×C79—41no—
97Q8×C79—41no—
98Q8×C79—42no—
99Q8×C79—42no—
100Q8×C79—42no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79C2×C4⋊C411yescharacteristic
112C2×C4×C79#11711yescharacteristic
113C2×C4×C79#12112yes—
114C2×C4×C79C22⋊Q812yes—
115C2×Dic79—22no—
116C2×Dic79—22no—
117C2×Dic79—21no—
118C2×Dic79—22no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—41no—
125C2×Dic79#9911yescharacteristic
126C2×Dic79#9911yescharacteristic
127C8×C79—82no—
128C79⋊C8—82no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79#38011yescharacteristic
132C22×C79C2×#9911yescharacteristic
133C22×C79#41011yescharacteristic
134C4×C79—24no—
135C4×C79—21no—
136C4×C79—21no—
137C4×C79—22no—
138C4×C79—22no—
139C4×C79—42no—
140Dic79—21no—
141Dic79—42no—
142Dic79—41no—
143Dic79—42no—
144Dic79—81no—
145Dic79#35612yescomplemented
146#12802—791nomaximal, Sylow
147C2×C79—21no—
148C2×C79—21no—
149C2×C79#179911yescharacteristic
150C2×C79C2×#35611yescharacteristic
151C2×C79#213211yescharacteristic
152C4×C4⋊Q8—791no—
153C2×#356—791no—
154C2×#356D7911yescharacteristic
155#1801—792no—
156#1394—792no—
157C79#1280211yescharacteristic, complemented, Sylow
158C2×C4⋊Q8C2×D7911yescharacteristic
159C2×C4×Q8—791no—
160C2×C4×Q8—791no—
161#423—1582no—
162C2×#101—792no—
163C2×#101C2×D7912yes—
164#391—791no—
165#391—791no—
166#389—1582no—
167#384—791no—
168C4×C4⋊C4—1582no—
169C43—791no—
170#356—22no—
171#356—1582no—
172#343—1582no—
173C22×Q8C4×D7911yescharacteristic
174C22×Q8C22×D7911yescharacteristic
175C2×M16—22no—
176C2×M16—1582no—
177C4⋊Q8—22no—
178C4⋊Q8Dic15812yes—
179C4⋊Q8C22⋊D7912yes—
180C4×Q8—3161no—
181C4×Q8—3161no—
182C4×Q8—3161no—
183C4×Q8—3161no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—791no—
186C2×C4⋊C4—792no—
187C2×C4⋊C4—791no—
188C2×C4⋊C4—1581no—
189C2×C4⋊C4—1582no—
190C2×C42—791no—
191C2×C42—791no—
192C2×C42—792no—
193C2×C42—1581no—
194C2×C42—1581no—
195C2×C42—1581no—
196C2×C42C4×D7911yescharacteristic
197C4⋊C8—3162no—
198C4⋊C8—3162no—
199#101—22no—
200#101—22no—
201#101—1582no—
202#101—1582no—
203#95—1582no—
204#95—1582no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8Dic79⋊C412yes—
211C2×Q8C2×C22⋊D7911yescharacteristic
212C2×Q8C2×Dic15811yescharacteristic
213C2×Q8#1192962312yes—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—792no—
217C22×C4—791no—
218C22×C4—791no—
219C22×C4—1581no—
220C22×C4—1581no—
221C22×C4C2×C4×D7911yescharacteristic
222C22×C4D4×D7912yes—
223M16—42no—
224M16—42no—
225M16—3162no—
226M16—3162no—
227C2×C8—42no—
228C2×C8—3162no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—3161no—
232C4⋊C4—3164no—
233C4⋊C4—3161no—
234C4⋊C4—3162no—
235C4⋊C4—3162no—
236C42—21no—
237C42—3161no—
238C42—3162no—
239C42—3161no—
240C42—3161no—
241C42—1582no—
242C42—1582no—
243C42—1581no—
244C42—1581no—
245C42—1584no—
246C42Dic79⋊C412yes—
247C23C22⋊C4×D7911yescharacteristic
248Q8—41no—
249Q8—41no—
250Q8—42no—
251Q8—42no—
252Q8—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—3161no—
264C2×C4—3161no—
265C2×C4—3161no—
266C2×C4—3162no—
267C2×C4—3162no—
268C2×C4—3161no—
269C2×C4—791no—
270C2×C4—791no—
271C2×C4—1582no—
272C2×C4—1581no—
273C2×C4—1582no—
274C2×C4—1582no—
275C2×C4C2×Dic79⋊C411yescharacteristic
276C2×C4#1457911711yescharacteristic
277C2×C4#1457913312yes—
278C2×C4#1457913812yes—
279C8—82no—
280C8—6322no—
281C22—21no—
282C22—21no—
283C22#1909842111yescharacteristic
284C22#99×D7911yescharacteristic
285C22#1909853511yescharacteristic
286C4—24no—
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—3162no—
293C4—3161no—
294C4—3162no—
295C4—792no—
296C4—6321no—
297C4—1581no—
298C2—21no—
299C2—21no—
300C2#2376338611yescharacteristic
301C2#356×D7911yescharacteristic
302C2#2376508411yescharacteristic
subgroupG/H
centerC22#99×D79
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC2×#356×C79C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC22×Q8×C79C23
hypercenterC2×#356D79
hypocenterC79#12802
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
145. Dic79#356semidirect
157. C79#12802semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×Dic792
C2×#190983622
#237633942
#237615262
C2×#356×C792
#1280279