Finite Group Databasebeta

← #34181719

Subgroups — #34181719

Order 20224.

404 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 99 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#23765895C211yesmaximal, characteristic
2#23761712C212yesmaximal
3#23761232C212yesmaximal, complemented
4#23761229C211yesmaximal, characteristic, complemented
5C79×#1274C211yesmaximal, characteristic, complemented
6C2×#14579118C2211yescharacteristic
7#19098616C2212yes—
8#19098452—22no—
9#19098445—22no—
10C4×Dic79⋊C4C412yes—
11C4×(C2×D79)⋊C4C412yes—
12C42×Dic79C2211yescharacteristic
13#19098288—22no—
14#19098286—22no—
15#431×C79C2212yes—
16#376×C79C2211yescharacteristic
17C22×C4×D79C2×C411yescharacteristic
18#14579118—21no—
19#14579118—22no—
20#14579118—21no—
21C2×Dic79⋊C4—22no—
22C2×Dic79⋊C4D412yes—
23C2×(C2×D79)⋊C4—22no—
24C2×(C2×D79)⋊C4D412yes—
25C2×C4×Dic79—22no—
26C2×C4×Dic79C2×C412yes—
27C2×C4×Dic79C2×C411yescharacteristic
28C4⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C4×C79⋊C8—22no—
31C79⋊#95—22no—
32C79⋊#95—22no—
33C2×C42×C79C2×C412yes—
34C2×C42×C79C2311yescharacteristic
35C4⋊C8×C79—22no—
36C22⋊C8×C79—22no—
37C4×C8×C79—22no—
38#95×C79—22no—
39C23×D79C22⋊C411yescharacteristic
40C22×Dic79—22no—
41C22×Dic79C22⋊C411yescharacteristic
42C22×Dic79C22⋊C412yes—
43C2×C4×D79—21no—
44C2×C4×D79—21no—
45C2×C4×D79—22no—
46C2×C4×D79—22no—
47C2×C79⋊C8—42no—
48Dic79⋊C4—22no—
49Dic79⋊C4—22no—
50Dic79⋊C4—44no—
51(C2×D79)⋊C4—22no—
52(C2×D79)⋊C4—22no—
53(C2×D79)⋊C4—44no—
54C4×Dic79—21no—
55C4×Dic79—21no—
56C4×Dic79—22no—
57C4×Dic79—22no—
58C4×Dic79—44no—
59C22×C4×C79—22no—
60C22×C4×C79C22×C411yescharacteristic
61C22×C4×C79C2×D412yes—
62C2×C8×C79—42no—
63C42×C79—22no—
64C42×C79—22no—
65C42×C79#4111yescharacteristic
66C42×C79#4112yes—
67C42×C79#4111yescharacteristic
68C23×C79C2×C22⋊C411yescharacteristic
69C22×D79—21no—
70C22×D79—22no—
71C22×D79—42no—
72C2×C4×C79—42no—
73C2×C4×C79—42no—
74C2×C4×C79—42no—
75C2×C4×C79—42no—
76C2×C4×C79#11711yescharacteristic
77C2×C4×C79#11711yescharacteristic
78C2×C4×C79#12312yes—
79C2×C4×C79#12312yes—
80C2×C4×C79#13112yes—
81C2×C4×C79#13112yes—
82C2×Dic79—24no—
83C2×Dic79—22no—
84C2×Dic79—21no—
85C2×Dic79—22no—
86C2×Dic79—42no—
87C2×Dic79—42no—
88C2×Dic79—42no—
89C4×D79—41no—
90C4×D79—41no—
91C4×D79—42no—
92C8×C79—82no—
93C79⋊C8—82no—
94C22×C79—22no—
95C22×C79#38111yescharacteristic
96C22×C79#40812yes—
97C2×D79—41no—
98C2×D79—42no—
99C2×D79—82no—
100C4×C79—22no—
101C4×C79—22no—
102C4×C79—82no—
103C4×C79#43311yescharacteristic
104C4×C79#43311yescharacteristic
105C4×C79#43312yes—
106Dic79—44no—
107Dic79—41no—
108Dic79—42no—
109#8732—791nomaximal, Sylow
110C2×C79—22no—
111C2×C79#181811yescharacteristic
112C2×C79#182712yes—
113D79—81no—
114#2298—791no—
115#1458—792no—
116#1276—792no—
117#1274—791no—
118#1274D7911yescharacteristic, complemented
119C79#873211yescharacteristic, complemented, Sylow
120C2×#117—791no—
121#431—792no—
122#431C2×D7912yes—
123#388—1582no—
124#387—1582no—
125C4×C4⋊C4—792no—
126C4×C22⋊C4—792no—
127#376C2×D7911yescharacteristic
128C43—791no—
129#324—1582no—
130#323—1582no—
131C23×C4—791no—
132#117—1581no—
133#117—1582no—
134#117—1581no—
135C2×C4⋊C4—792no—
136C2×C4⋊C4—1582no—
137C2×C22⋊C4—792no—
138C2×C22⋊C4—1582no—
139C2×C42—791no—
140C2×C42—792no—
141C2×C42—1582no—
142C2×C42C4×D7912yes—
143C2×C42C22×D7911yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—1582no—
146C22⋊C8—22no—
147C22⋊C8—1582no—
148C4×C8—22no—
149C4×C8—1582no—
150#95—22no—
151#95—1582no—
152#95—1582no—
153C24—791no—
154C22×C4—22no—
155C22×C4—792no—
156C22×C4—791no—
157C22×C4—1582no—
158C22×C4—1581no—
159C22×C4—1581no—
160C22×C4—1582no—
161C22×C4—1582no—
162C22×C4C2×C4×D7911yescharacteristic
163C22×C4D4×D7912yes—
164C2×C8—42no—
165C2×C8—3162no—
166C4⋊C4—3164no—
167C4⋊C4—1582no—
168C4⋊C4—1582no—
169C22⋊C4—3164no—
170C22⋊C4—1582no—
171C22⋊C4—1582no—
172C42—22no—
173C42—22no—
174C42—3164no—
175C42—1582no—
176C42—1582no—
177C42—1581no—
178C42—1581no—
179C42#1192962311yescharacteristic
180C42#1192962412yes—
181C42#1192962511yescharacteristic
182C23—3162no—
183C23—1581no—
184C23—1582no—
185C23C22⋊C4×D7911yescharacteristic
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—3162no—
191C2×C4—3162no—
192C2×C4—3161no—
193C2×C4—3161no—
194C2×C4—3162no—
195C2×C4—3162no—
196C2×C4—1584no—
197C2×C4—1582no—
198C2×C4—1581no—
199C2×C4—1582no—
200C2×C4#1457911811yescharacteristic
201C2×C4#1457911911yescharacteristic
202C2×C4#1457914312yes—
203C2×C4#1457914612yes—
204C2×C4#1457916712yes—
205C2×C4#1457916912yes—
206C8—82no—
207C8—6322no—
208C22—22no—
209C22—3161no—
210C22—3162no—
211C22—6322no—
212C22#1909842511yescharacteristic
213C22#1909852812yes—
214C4—22no—
215C4—22no—
216C4—82no—
217C4—3164no—
218C4—3161no—
219C4—3162no—
220C4#1909862211yescharacteristic
221C4#1909862312yes—
222C4#1909862511yescharacteristic
223C2—22no—
224C2—6321no—
225C2#2376347011yescharacteristic
226C2#2376350212yes—
subgroupG/H
centerC2×C4#14579118
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1274C2
socleC22×C79#381
prime radicalC23×D79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1274D79
hypocenterC79#8732
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #376 ◁ #1274

Complements

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

NH ≅ G/Ntype
3. #23761232C2semidirect
4. #23761229C2semidirect
5. C79×#1274C2semidirect
118. #1274D79semidirect
119. C79#8732semidirect

Maximal subgroups

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

subgroupindex
#237658952
#237617122
#237612322
#237612292
C79×#12742
#873279