Finite Group Databasebeta

← #34174941

Subgroups — #34174941

Order 20224.

338 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 49 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#23764398C211yesmaximal, characteristic
2#23761324C211yesmaximal, characteristic, complemented
3C79×#1289C211yesmaximal, characteristic, complemented
4C22×D632C411yescharacteristic, complemented
5C2×#14579135—21no—
6C2×#14579083—21no—
7C22⋊(C79⋊C16)—21no—
8#19098326C411yescharacteristic, complemented
9C2×C22⋊C8×C79C2211yescharacteristic
10#380×C79—21no—
11C22×D316—21no—
12C22×C22⋊D79—41no—
13C2×D632—41no—
14C2×D632—42no—
15#14579135—42no—
16#14579135—42no—
17C2×C4⋊Dic79—21no—
18C2×(C2×D79)⋊C4—41no—
19C2×C79⋊C16—41no—
20#14579083—42no—
21C23×C4×C79D411yescharacteristic
22C22×C8×C79C2×C411yescharacteristic
23C22×C8×C79D411yescharacteristic
24C2×C4⋊C4×C79—41no—
25C2×C22⋊C4×C79—41no—
26C22⋊C8×C79—41no—
27#95×C79—22no—
28C23×D79—41no—
29C2×D316—22no—
30C2×D316—41no—
31C2×D316—41no—
32C2×D316—41no—
33C2×D316—41no—
34C2×D316—41no—
35C2×C22⋊D79—44no—
36C2×C22⋊D79—82no—
37C2×C22⋊D79—81no—
38C2×C22⋊D79—81no—
39C22×Dic79—41no—
40D632—42no—
41D632—42no—
42C4⋊Dic79—22no—
43C4⋊Dic79—41no—
44(C2×D79)⋊C4—44no—
45C79⋊C16—81no—
46C24×C79D811yescharacteristic
47C22×C4×C79—21no—
48C22×C4×C79—21no—
49C22×C4×C79—41no—
50C22×C4×C79—41no—
51C22×C4×C79—42no—
52C22×C4×C79C22⋊C411yescharacteristic
53C22×C4×C79SD1611yescharacteristic
54C2×C8×C79—21no—
55C2×C8×C79—21no—
56C2×C8×C79—41no—
57C2×C8×C79—41no—
58C4⋊C4×C79—81no—
59C4⋊C4×C79—81no—
60C22⋊C4×C79—81no—
61C22⋊C4×C79—81no—
62C23×C79—21no—
63C23×C79—21no—
64C23×C79—41no—
65C23×C79—41no—
66C23×C79#10211yescharacteristic
67C23×C79D1611yescharacteristic
68C23×C79SD3211yescharacteristic
69C22×D79—42no—
70C22×D79—81no—
71C22×D79—81no—
72C22×D79—81no—
73C22×D79—81no—
74C22×D79—81no—
75C22×D79—81no—
76C2×C4×C79—21no—
77C2×C4×C79—21no—
78C2×C4×C79—21no—
79C2×C4×C79—41no—
80C2×C4×C79—41no—
81C2×C4×C79—41no—
82C2×C4×C79—81no—
83C2×C4×C79—82no—
84C2×C4×C79—82no—
85C2×C4×C79—82no—
86C2×C4×C79#9911yescharacteristic
87C2×C4×C79#10411yescharacteristic
88C22⋊D79—84no—
89C22⋊D79—84no—
90C2×Dic79—42no—
91C2×Dic79—81no—
92C2×Dic79—81no—
93D316—42no—
94D316—42no—
95D316—81no—
96D316—81no—
97C8×C79—41no—
98C8×C79—41no—
99C22×C79—21no—
100C22×C79—21no—
101C22×C79—21no—
102C22×C79—21no—
103C22×C79—41no—
104C22×C79—41no—
105C22×C79—41no—
106C22×C79—41no—
107C22×C79—81no—
108C22×C79#32711yescharacteristic
109C22×C79#35711yescharacteristic
110C22×C79#36011yescharacteristic
111C2×D79—82no—
112C2×D79—82no—
113C2×D79—82no—
114C2×D79—161no—
115C2×D79—161no—
116C2×D79—161no—
117C2×D79—161no—
118C4×C79—21no—
119C4×C79—41no—
120C4×C79—162no—
121C4×C79AΣL(1,𝔽16)11yescharacteristic
122C4×C79#35311yescharacteristic
123Dic79—82no—
124#7076—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—21no—
127C2×C79—41no—
128C2×C79—41no—
129C2×C79#133911yescharacteristic
130C2×C79#134311yescharacteristic
131C2×C79#134711yescharacteristic
132D79—162no—
133#1999—791no—
134#1314—791no—
135#1289D7911yescharacteristic, complemented
136C79#707611yescharacteristic, complemented, Sylow
137C22×D8—791no—
138C2×#121—1581no—
139C2×#102—1581no—
140C2×C22⋊C8C2×D7911yescharacteristic
141#380—21no—
142C22⋊C16—1581no—
143#340—791no—
144C22×D4—3161no—
145C22×D4—1581no—
146C23×C4C22⋊D7911yescharacteristic
147C2×D8—3161no—
148C2×D8—3162no—
149C22×C8C4×D7911yescharacteristic
150C22×C8D31611yescharacteristic
151#121—3162no—
152#121—3162no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—1581no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—3161no—
157C2×C16—3161no—
158#102—3162no—
159C22⋊C8—41no—
160#95—22no—
161C24—3161no—
162C24#1192960911yescharacteristic
163C2×D4—3161no—
164C2×D4—3161no—
165C2×D4—3161no—
166C2×D4—3164no—
167C2×D4—3161no—
168C2×D4—3161no—
169C2×D4—6322no—
170C2×D4—6321no—
171C2×D4—6321no—
172C2×D4—1582no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—3161no—
179C22×C4(C2×D79)⋊C411yescharacteristic
180C22×C4#1192961111yescharacteristic
181D8—3162no—
182D8—3162no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—3161no—
190C4⋊C4—1582no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C22⋊C4—3164no—
194C16—6321no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23—3162no—
200C23—6321no—
201C23—6321no—
202C23—6321no—
203C23—6321no—
204C23—6321no—
205C23—6321no—
206C23#1457908111yescharacteristic
207C23#1457910211yescharacteristic
208C23#1457910411yescharacteristic
209D4—3162no—
210D4—3162no—
211D4—6321no—
212D4—6324no—
213D4—6324no—
214D4—6321no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—3162no—
226C2×C4—6321no—
227C2×C4—6321no—
228C2×C4#1457907511yescharacteristic
229C2×C4#1457908911yescharacteristic
230C8—41no—
231C8—41no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—6322no—
242C22—6322no—
243C22—6322no—
244C22—12641no—
245C22—12641no—
246C22—12641no—
247C22—12641no—
248C22#1909829611yescharacteristic
249C22#1909836311yescharacteristic
250C22#1909837211yescharacteristic
251C4—21no—
252C4—41no—
253C4—162no—
254C4—6322no—
255C4#1909834911yescharacteristic
256C4#1909835511yescharacteristic
257C2—21no—
258C2—21no—
259C2—41no—
260C2—41no—
261C2—12642no—
262C2#2376138411yescharacteristic
263C2#2376139611yescharacteristic
264C2#2376140811yescharacteristic
subgroupG/H
centerC22#19098296
commutatorC22×C8×C79C2×C4
FrattiniC2×C22⋊C8C2×D79
FittingC79×#1289C2
socleC22×C79#327
prime radical#23764398C2
cosocleC2×C22⋊C8×C79C22
hypercenter#1289D79
hypocenterC79#7076
LCSG ▷ C22×C8×C79 ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #23761324C2semidirect
3. C79×#1289C2semidirect
4. C22×D632C4semidirect
8. #19098326C4semidirect
135. #1289D79semidirect
136. C79#7076semidirect

Maximal subgroups

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

subgroupindex
#237643982
#237613242
C79×#12892
#707679