Finite Group Databasebeta

← #34181871

Subgroups — #34181871

Order 20224.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#23766373C211yesmaximal, characteristic
2#23763392C211yesmaximal, characteristic
3#23761302C212yesmaximal, complemented
4#23761259C212yesmaximal, complemented
5C79×#1460C211yesmaximal, characteristic
6#19098587C2212yes—
7C2×#14579078—21no—
8#19098466—21no—
9#19098456—22no—
10#19098429C2211yescharacteristic
11#19098426C412yes—
12C4⋊C4×Dic79—21no—
13C4×C4⋊Dic79C2211yescharacteristic
14C4×C22⋊Dic79C412yes—
15#19098332—21no—
16C2×C4×D4×C79C2211yescharacteristic
17#431×C79C2212yes—
18#323×C79—22no—
19C2×#11929608—22no—
20C2×C4⋊Dic79—21no—
21C2×C4⋊Dic79—21no—
22C2×C22⋊Dic79—22no—
23C2×C22⋊Dic79—22no—
24C2×C4×Dic79—21no—
25C2×C4×Dic79—22no—
26C4⋊(C79⋊C8)—22no—
27C4⋊(C79⋊C8)—22no—
28#14579078—42no—
29C79⋊#95—22no—
30C79⋊#95—21no—
31C79⋊#95C2×C411yescharacteristic
32C79⋊#95C2×C412yes—
33C79⋊#95C2×C411yescharacteristic
34C22×D4×C79D411yescharacteristic
35C23×C4×C79C2×C412yes—
36C4×D4×C79—22no—
37C4×D4×C79—22no—
38C2×C4⋊C4×C79D411yescharacteristic
39C2×C22⋊C4×C79D412yes—
40C2×C42×C79C2311yescharacteristic
41C4⋊C8×C79—22no—
42C22⋊C8×C79—22no—
43C4×C8×C79—22no—
44C22×Dic79—21no—
45C22×Dic79—22no—
46C22×Dic79—21no—
47#11929608—42no—
48#11929608—42no—
49C2×C79⋊C8—42no—
50C4⋊Dic79—42no—
51C4⋊Dic79—41no—
52C4⋊Dic79—41no—
53C22⋊Dic79—44no—
54C22⋊Dic79—82no—
55C4×Dic79—42no—
56C4×Dic79—82no—
57C24×C79C22⋊C412yes—
58C2×D4×C79—22no—
59C2×D4×C79—22no—
60C2×D4×C79—21no—
61C2×D4×C79—21no—
62C22×C4×C79—22no—
63C22×C4×C79—22no—
64C22×C4×C79—82no—
65C22×C4×C79C22⋊C412yes—
66C22×C4×C79C22×C411yescharacteristic
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79C2×D411yescharacteristic
69C2×C8×C79—42no—
70C4⋊C4×C79—22no—
71C22⋊C4×C79—24no—
72C42×C79#4112yes—
73C42×C79#4112yes—
74C23×C79—22no—
75C23×C79—22no—
76C23×C79—22no—
77C23×C79—42no—
78C23×C79—42no—
79C23×C79C2×C22⋊C411yescharacteristic
80D4×C79—42no—
81D4×C79—42no—
82C2×C4×C79—22no—
83C2×C4×C79—41no—
84C2×C4×C79—41no—
85C2×C4×C79—42no—
86C2×C4×C79—82no—
87C2×C4×C79—82no—
88C2×C4×C79#11711yescharacteristic
89C2×C4×C79#11711yescharacteristic
90C2×C4×C79#12312yes—
91C2×C4×C79#12312yes—
92C2×C4×C79#13512yes—
93C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
94C2×C4×C79#13711yescharacteristic
95C2×Dic79—22no—
96C2×Dic79—41no—
97C2×Dic79—41no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—41no—
101C2×Dic79—41no—
102C2×Dic79—41no—
103C2×Dic79—42no—
104C8×C79—82no—
105C79⋊C8—82no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79—42no—
109C22×C79—42no—
110C22×C79—42no—
111C22×C79—82no—
112C22×C79—82no—
113C22×C79#38111yescharacteristic
114C22×C79#41611yescharacteristic
115C22×C79#41711yescharacteristic
116C4×C79—22no—
117C4×C79—21no—
118C4×C79—21no—
119C4×C79—42no—
120C4×C79#48212yes—
121C4×C79#48512yes—
122Dic79—42no—
123Dic79—81no—
124Dic79—82no—
125#8771—791nomaximal, Sylow
126C2×C79—21no—
127C2×C79—21no—
128C2×C79—82no—
129C2×C79#182511yescharacteristic
130C2×C79#182711yescharacteristic
131C2×C79#182911yescharacteristic
132#2357—791no—
133#1800—791no—
134#1460D7911yescharacteristic
135#1303—792no—
136#1286—792no—
137C79#877111yescharacteristic, complemented, Sylow
138C2×C4×D4C2×D7911yescharacteristic
139#431C2×D7912yes—
140#423—792no—
141C2×#100—1581no—
142#390—1581no—
143#388—1582no—
144#382—791no—
145#381—792no—
146C4×C4⋊C4—791no—
147C4×C4⋊C4—1581no—
148C4×C22⋊C4—792no—
149#343—1581no—
150#323—22no—
151C22×D4D31611yescharacteristic
152C23×C4C4×D7912yes—
153C2×M16—1582no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—1581no—
157C2×C4⋊C4—1581no—
158C2×C4⋊C4D31611yescharacteristic
159C2×C22⋊C4—1582no—
160C2×C22⋊C4—1582no—
161C2×C22⋊C4C22⋊D7912yes—
162C2×C42—1581no—
163C2×C42—1582no—
164C2×C42C22×D7911yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—1582no—
167C4⋊C8—1582no—
168#100—3162no—
169C22⋊C8—22no—
170C4×C8—22no—
171#95—791no—
172#95—792no—
173#95—791no—
174#95—1582no—
175#95—1581no—
176C24(C2×D79)⋊C412yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—82no—
184C22×C4—1582no—
185C22×C4—1581no—
186C22×C4—1581no—
187C22×C4(C2×D79)⋊C412yes—
188C22×C4C2×C4×D7911yescharacteristic
189C22×C4C2×C22⋊D7911yescharacteristic
190C22×C4C2×D31611yescharacteristic
191M16—3162no—
192M16—3162no—
193C2×C8—42no—
194C2×C8—3162no—
195C4⋊C4—22no—
196C4⋊C4—3162no—
197C4⋊C4—3161no—
198C4⋊C4—3161no—
199C22⋊C4—24no—
200C22⋊C4—3164no—
201C22⋊C4—6322no—
202C42—3162no—
203C42—6322no—
204C42#1192962312yes—
205C42#1192962312yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×(C2×D79)⋊C411yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—3161no—
221C2×C4—3161no—
222C2×C4—3162no—
223C2×C4—3162no—
224C2×C4—3161no—
225C2×C4—3161no—
226C2×C4—3161no—
227C2×C4—3162no—
228C2×C4—1582no—
229C2×C4#1457911711yescharacteristic
230C2×C4#1457911711yescharacteristic
231C2×C4#1457914212yes—
232C2×C4#1457914512yes—
233C2×C4#1457918312yes—
234C2×C4#1457918911yescharacteristic
235C2×C4#1457919411yescharacteristic
236C8—82no—
237C8—6322no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1909842411yescharacteristic
246C22#1909856011yescharacteristic
247C22#1909856511yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—3162no—
253C4—6321no—
254C4—6322no—
255C4#1909888812yes—
256C4#1909890512yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2376349611yescharacteristic
261C2#2376350111yescharacteristic
262C2#2376350911yescharacteristic
subgroupG/H
centerC22#19098424
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1460C2
socleC22×C79#381
prime radicalC22×D4×C79D4
cosocleC2×C42×C79C23
hypercenter#1460D79
hypocenterC79#8771
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #23761302C2semidirect
4. #23761259C2semidirect
137. C79#8771semidirect

Maximal subgroups

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

subgroupindex
#237663732
#237633922
#237613022
#237612592
C79×#14602
#877179