Finite Group Databasebeta

← #34198917

Subgroups — #34198917

Order 20224.

378 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14579080C211yesmaximal, characteristic
2#23761744C211yesmaximal, characteristic
3#23761427C214yesmaximal
4C79×#2568C211yesmaximal, characteristic
5#19098620—21no—
6C2×#14579080—21no—
7C4×#11929608—21no—
8#19098332C2212yes—
9#19098289—21no—
10#19098289C2212yes—
11C2×C4⋊Q8×C79C411yescharacteristic
12C2×C4×Q8×C79C2211yescharacteristic
13#400×C79C412yes—
14#400×C79C2212yes—
15#391×C79—21no—
16#391×C79—21no—
17#391×C79—21no—
18#382×C79—21no—
19C4×C4⋊C4×C79C411yescharacteristic
20C2×#11929608—22no—
21C4⋊(C79⋊C8)—41no—
22#14579080—24no—
23C22⋊(C79⋊C8)—22no—
24#14579072—41no—
25#14579072—41no—
26C4×C79⋊C8—41no—
27C22×Q8×C79—21no—
28C22×Q8×C79D411yescharacteristic
29C4⋊Q8×C79—41no—
30C4⋊Q8×C79—41no—
31C4×Q8×C79—41no—
32C4×Q8×C79—41no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—21no—
35C2×C4⋊C4×C79C2×C412yes—
36C2×C4⋊C4×C79C2×C411yescharacteristic
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C4⋊C4×C79C2311yescharacteristic
39C2×C42×C79—21no—
40C2×C42×C79C2×C411yescharacteristic
41C2×C42×C79D411yescharacteristic
42C2×C42×C79D411yescharacteristic
43#95×C79—22no—
44#95×C79—22no—
45#95×C79—22no—
46#95×C79—22no—
47#95×C79C2×C412yes—
48#11929608—24no—
49#11929608—42no—
50C2×C79⋊C8—42no—
51C2×C79⋊C8—42no—
52C2×Q8×C79—22no—
53C2×Q8×C79—22no—
54C2×Q8×C79—41no—
55C2×Q8×C79—44no—
56C2×Q8×C79—41no—
57C2×Q8×C79—41no—
58C22×C4×C79—22no—
59C22×C4×C79—22no—
60C22×C4×C79C22⋊C411yescharacteristic
61C22×C4×C79C22⋊C411yescharacteristic
62C22×C4×C79C22⋊C411yescharacteristic
63C22×C4×C79C22⋊C411yescharacteristic
64C22×C4×C79C22×C411yescharacteristic
65C22×C4×C79C2×D411yescharacteristic
66C22×C4×C79C2×D411yescharacteristic
67C4⋊C4×C79—41no—
68C4⋊C4×C79—41no—
69C4⋊C4×C79—41no—
70C4⋊C4×C79—42no—
71C4⋊C4×C79—42no—
72C4⋊C4×C79—42no—
73C42×C79—21no—
74C42×C79—21no—
75C42×C79—21no—
76C42×C79—21no—
77C42×C79—21no—
78C42×C79—21no—
79C42×C79—42no—
80C23×C79C2×C22⋊C411yescharacteristic
81Q8×C79—44no—
82Q8×C79—82no—
83Q8×C79—82no—
84C2×C4×C79—21no—
85C2×C4×C79—21no—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—22no—
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—41no—
96C2×C4×C79—41no—
97C2×C4×C79—41no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—44no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79#9911yescharacteristic
104C2×C4×C79#9911yescharacteristic
105C2×C4×C79#10412yes—
106C2×C4×C79#13712yes—
107C79⋊C8—44no—
108C79⋊C8—82no—
109C22×C79—21no—
110C22×C79—21no—
111C22×C79C2×#9911yescharacteristic
112C22×C79#41911yescharacteristic
113C22×C79C2×#10411yescharacteristic
114C4×C79—21no—
115C4×C79—22no—
116C4×C79—22no—
117C4×C79—41no—
118C4×C79—41no—
119C4×C79—41no—
120C4×C79—41no—
121C4×C79—82no—
122C4×C79—82no—
123C4×C79#35512yes—
124#11794—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—21no—
127C2×C79#150811yescharacteristic
128C2×C79#212311yescharacteristic
129C2×C79C2×#35511yescharacteristic
130#2568D7911yescharacteristic
131C4×#101—791no—
132#1467—791no—
133#1353—794no—
134C79#1179411yescharacteristic, complemented, Sylow
135C2×C4⋊Q8Dic7911yescharacteristic
136C2×C4×Q8C2×D7911yescharacteristic
137#432—1581no—
138C2×#101—1581no—
139C4×M16—1581no—
140#400Dic7912yes—
141#400C2×D7912yes—
142#391—21no—
143#391—21no—
144#391—21no—
145#382—21no—
146C4×C4⋊C4Dic7911yescharacteristic
147#343—792no—
148#324—792no—
149#324—1581no—
150C22×Q8—21no—
151C22×Q8C22⋊D7911yescharacteristic
152C2×M16—1582no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic7912yes—
160C2×C4⋊C4C2×Dic7911yescharacteristic
161C2×C4⋊C4C22⋊D7911yescharacteristic
162C2×C4⋊C4C22×D7911yescharacteristic
163C2×C42—21no—
164C2×C42C2×Dic7911yescharacteristic
165C2×C42C22⋊D7911yescharacteristic
166C2×C42C22⋊D7911yescharacteristic
167C4⋊C8—3161no—
168#101—1584no—
169C22⋊C8—1582no—
170#97—3161no—
171#97—3161no—
172C4×C8—3161no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95C2×Dic7912yes—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—44no—
182C2×Q8—41no—
183C2×Q8—41no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4C22⋊Dic7911yescharacteristic
187C22×C4C22⋊Dic7911yescharacteristic
188C22×C4C22⋊Dic7911yescharacteristic
189C22×C4C22⋊Dic7911yescharacteristic
190C22×C4C22×Dic7911yescharacteristic
191C22×C4C2×C22⋊D7911yescharacteristic
192C22×C4C2×C22⋊D7911yescharacteristic
193M16—3162no—
194M16—1584no—
195C2×C8—3162no—
196C2×C8—3162no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C23C2×C22⋊Dic7911yescharacteristic
211Q8—44no—
212Q8—82no—
213Q8—82no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4#1457907611yescharacteristic
234C2×C4#1457907611yescharacteristic
235C2×C4#1457908812yes—
236C2×C4#1457919212yes—
237C8—3164no—
238C8—6322no—
239C22—21no—
240C22—21no—
241C22C2×#1457907611yescharacteristic
242C22#1909857211yescharacteristic
243C22C2×#1457908811yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—82no—
252C4—82no—
253C4#1909836012yes—
254C2—21no—
255C2—21no—
256C2#2376192011yescharacteristic
257C2#2376505411yescharacteristic
258C2C2×#1909836011yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2568C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2568D79
hypocenterC79#11794
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
134. C79#11794semidirect

Maximal subgroups

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

subgroupindex
C4×#145790802
#237617442
#237614272
C79×#25682
#1179479