Finite Group Databasebeta

← #34197698

Subgroups — #34197698

Order 20224.

330 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 69 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#23772029C211yesmaximal, characteristic
2#23765209C212yesmaximal, complemented
3#23761350C212yesmaximal, complemented
4#23761336C211yesmaximal, characteristic
5C79×#1320C211yesmaximal, characteristic
6C22×C4×Dic79C411yescharacteristic
7#19098617—22no—
8#19098613—22no—
9#19098586C411yescharacteristic
10#19098586C2211yescharacteristic
11#19098527C411yescharacteristic
12M16×Dic79C411yescharacteristic
13M16×Dic79C2211yescharacteristic
14#19098383—22no—
15#19098345C2212yes—
16#19098340—22no—
17#407×C79C2211yescharacteristic
18#349×C79C2212yes—
19C23×Dic79—21no—
20C2×#11929608C2×C411yescharacteristic
21C2×#11929608C2×C411yescharacteristic
22C2×C4×Dic79—21no—
23C2×C4×Dic79—21no—
24C2×C4×Dic79—21no—
25C2×C4×Dic79—21no—
26C2×C4×Dic79—21no—
27C2×C4×Dic79C2×C411yescharacteristic
28C2×C4×Dic79C2×C411yescharacteristic
29#14579101—22no—
30Dic79⋊C8—22no—
31Dic79⋊C8—22no—
32C22⋊(C79⋊C8)—22no—
33#14579071—21no—
34#14579071D412yes—
35C8×Dic79—21no—
36C8×Dic79D412yes—
37C23×C4×C79C2×C411yescharacteristic
38C2×M16×C79C2×C411yescharacteristic
39C2×M16×C79C2311yescharacteristic
40M32×C79—22no—
41C22⋊C8×C79—22no—
42C22×Dic79—21no—
43C22×Dic79—21no—
44C22×Dic79—21no—
45C22×Dic79—21no—
46C22×Dic79—21no—
47C22×Dic79—21no—
48C22×Dic79—44no—
49#11929608—41no—
50#11929608—41no—
51C2×C79⋊C8—21no—
52C2×C79⋊C8C22⋊C412yes—
53C4×Dic79—21no—
54C4×Dic79—21no—
55C4×Dic79—21no—
56C4×Dic79—21no—
57C4×Dic79—21no—
58C4×Dic79—21no—
59C4×Dic79C22⋊C411yescharacteristic
60C4×Dic79C22⋊C411yescharacteristic
61C4×Dic79M1611yescharacteristic
62C4×Dic79M1611yescharacteristic
63C79⋊C16—42no—
64C24×C79M1611yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—41no—
68C22×C4×C79—41no—
69C22×C4×C79M1611yescharacteristic
70C22×C4×C79C22×C411yescharacteristic
71M16×C79—41no—
72M16×C79—41no—
73C2×C8×C79—21no—
74C2×C8×C79C2×D412yes—
75C16×C79—42no—
76C23×C79—21no—
77C23×C79—42no—
78C23×C79—41no—
79C23×C79C2×M1611yescharacteristic
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79—21no—
83C2×C4×C79—42no—
84C2×C4×C79—41no—
85C2×C4×C79—41no—
86C2×C4×C79—41no—
87C2×C4×C79C2×C22⋊C411yescharacteristic
88C2×C4×C79C2×M1611yescharacteristic
89C2×Dic79—24no—
90C2×Dic79—21no—
91C2×Dic79—21no—
92C2×Dic79—21no—
93C2×Dic79—21no—
94C2×Dic79—42no—
95C2×Dic79—42no—
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C8×C79—22no—
100C8×C79—41no—
101C79⋊C8—22no—
102C79⋊C8—41no—
103C22×C79—22no—
104C22×C79—21no—
105C22×C79—41no—
106C22×C79—41no—
107C22×C79—42no—
108C22×C79—42no—
109C22×C79—41no—
110C22×C79#40711yescharacteristic
111C4×C79—21no—
112C4×C79—41no—
113C4×C79#41011yescharacteristic
114C4×C79#41311yescharacteristic
115Dic79—22no—
116Dic79—22no—
117Dic79—41no—
118Dic79—41no—
119#11521—791nomaximal, Sylow
120C2×C79—21no—
121C2×C79—42no—
122C2×C79—41no—
123C2×C79#146411yescharacteristic
124#2920—791no—
125#2165—792no—
126#1327—792no—
127#1320—791no—
128#1320D7911yescharacteristic
129C79#1152111yescharacteristic, complemented, Sylow
130C22×C42—791no—
131#432—1582no—
132#431—1582no—
133#423—791no—
134#423—791no—
135#407—791no—
136#407C2×D7911yescharacteristic
137C4×M16—791no—
138C4×M16—791no—
139#364—1582no—
140#349—792no—
141#349C2×D7912yes—
142#347—1582no—
143C23×C4—1581no—
144C23×C4C4×D7911yescharacteristic
145C2×M16—791no—
146C2×M16—791no—
147C2×M16C4×D7911yescharacteristic
148C2×M16C22×D7911yescharacteristic
149C2×C42—791no—
150C2×C42—791no—
151C2×C42—1581no—
152C2×C42—1581no—
153C2×C42—1581no—
154C2×C42—1581no—
155C2×C42—1581no—
156M32—22no—
157M32—1582no—
158C4⋊C8—1582no—
159C4⋊C8—1582no—
160C22⋊C8—22no—
161C22⋊C8—1582no—
162#97—792no—
163#97—1581no—
164C4×C8—792no—
165C4×C8—1581no—
166C24#1192960711yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—3164no—
172C22×C4—1581no—
173C22×C4—1581no—
174C22×C4—1581no—
175C22×C4—1581no—
176C22×C4—1581no—
177C22×C4—1581no—
178C22×C4#1192960711yescharacteristic
179C22×C4C2×C4×D7911yescharacteristic
180M16—41no—
181M16—41no—
182M16—3161no—
183M16—3161no—
184C2×C8—21no—
185C2×C8—792no—
186C2×C8—1581no—
187C2×C8D4×D7912yes—
188C42—791no—
189C42—791no—
190C42—791no—
191C42—791no—
192C42—1581no—
193C42—1581no—
194C42—1581no—
195C42—1581no—
196C42—1581no—
197C42—1581no—
198C16—42no—
199C16—3162no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×#1192960711yescharacteristic
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—3162no—
212C2×C4—3162no—
213C2×C4—3162no—
214C2×C4—3162no—
215C2×C4—3162no—
216C2×C4—1581no—
217C2×C4—1581no—
218C2×C4—1584no—
219C2×C4—1581no—
220C2×C4—1581no—
221C2×C4C22⋊C4×D7911yescharacteristic
222C2×C4M16×D7911yescharacteristic
223C8—22no—
224C8—41no—
225C8—3161no—
226C8—1582no—
227C22—22no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22#1909852511yescharacteristic
235C4—21no—
236C4—41no—
237C4—3161no—
238C4—3161no—
239C4—1582no—
240C4—1582no—
241C4#1909853511yescharacteristic
242C4#1909854411yescharacteristic
243C2—21no—
244C2—42no—
245C2—41no—
246C2#2376173111yescharacteristic
subgroupG/H
centerC4#19098535
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#1320C2
socleC2×C79#1464
prime radicalC24×C79M16
cosocleC2×M16×C79C23
hypercenter#1320D79
hypocenterC79#11521
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #23765209C2semidirect
3. #23761350C2semidirect
129. C79#11521semidirect

Maximal subgroups

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

subgroupindex
#237720292
#237652092
#237613502
#237613362
C79×#13202
#1152179