Finite Group Databasebeta

← #34197572

Subgroups — #34197572

Order 20224.

368 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 83 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#23772099C211yesmaximal, characteristic
2#23765019C212yesmaximal, complemented
3#23761352C212yesmaximal, complemented
4#23761336C211yesmaximal, characteristic, complemented
5C79×#1320C211yesmaximal, characteristic, complemented
6C2×#14579169C2211yescharacteristic
7C22×#11929608C411yescharacteristic
8#19099074C411yescharacteristic
9M32×D79—22no—
10#19098638—22no—
11#19098637—22no—
12#19098591C2211yescharacteristic
13#19098531C411yescharacteristic
14#19098521C411yescharacteristic
15#19098345C2212yes—
16#19098344—22no—
17#407×C79C2211yescharacteristic
18#349×C79C2212yes—
19C2×#11929624C2×C411yescharacteristic
20#14579169—42no—
21C2×#11929608—21no—
22C2×#11929608—21no—
23C2×#11929608—42no—
24C2×#11929607D412yes—
25C22×C79⋊C8C2×C411yescharacteristic
26C22×C79⋊C8C2×C411yescharacteristic
27C2×C8×D79D412yes—
28#14579144C812yes—
29#14579140—21no—
30#14579135—21no—
31#14579130C812yes—
32C4×C22⋊D79—22no—
33#14579120C2×C411yescharacteristic
34#14579101—22no—
35#14579100—42no—
36C16×D79—42no—
37Dic79⋊C8—22no—
38(C2×D79)⋊C8—22no—
39#14579071—21no—
40C8×Dic79—21no—
41C23×C4×C79C2×C411yescharacteristic
42C2×M16×C79C2×C411yescharacteristic
43C2×M16×C79C2311yescharacteristic
44M32×C79—22no—
45C22⋊C8×C79—22no—
46#11929624—41no—
47#11929624—41no—
48C2×Dic158M1611yescharacteristic
49C2×D316M1611yescharacteristic
50C2×C22⋊D79—22no—
51C2×C22⋊D79C2×C812yes—
52C2×C4×D79C22⋊C412yes—
53#11929608—42no—
54#11929608—42no—
55#11929607—42no—
56C2×C79⋊C8—22no—
57C2×C79⋊C8—21no—
58C2×C79⋊C8—41no—
59C2×C79⋊C8C22⋊C411yescharacteristic
60C2×C79⋊C8C22⋊C411yescharacteristic
61C8×D79—22no—
62C8×D79—22no—
63C4⋊Dic79—21no—
64Dic79⋊C4—22no—
65C22⋊Dic79—22no—
66C22⋊Dic79C2×C812yes—
67(C2×D79)⋊C4—22no—
68C4×Dic79—21no—
69C79⋊C16—42no—
70C24×C79C2×C811yescharacteristic
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—41no—
74C22×C4×C79—41no—
75C22×C4×C79C2×C811yescharacteristic
76C22×C4×C79C22×C411yescharacteristic
77M16×C79—41no—
78M16×C79—41no—
79C2×C8×C79—21no—
80C2×C8×C79C2×D412yes—
81C16×C79—42no—
82C23×C79—21no—
83C23×C79—41no—
84C23×C79—81no—
85C23×C79C22×C811yescharacteristic
86C22×D79C22⋊C812yes—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—42no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79C2×C22⋊C411yescharacteristic
95C2×C4×C79C2×M1611yescharacteristic
96C22⋊D79—42no—
97C22⋊D79—82no—
98C2×Dic79—22no—
99C2×Dic79C22⋊C812yes—
100D316—41no—
101C4×D79—22no—
102C4×D79—22no—
103Dic158—41no—
104C8×C79—22no—
105C8×C79—41no—
106C79⋊C8—21no—
107C79⋊C8—41no—
108C79⋊C8#9912yes—
109C22×C79—21no—
110C22×C79—41no—
111C22×C79—41no—
112C22×C79—41no—
113C22×C79—41no—
114C22×C79—82no—
115C22×C79C2×C22⋊C811yescharacteristic
116C2×D79—22no—
117C2×D79—42no—
118C4×C79—21no—
119C4×C79—41no—
120C4×C79C2×#9911yescharacteristic
121C4×C79#41311yescharacteristic
122Dic79—22no—
123Dic79—42no—
124#11484—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—41no—
127C2×C79—81no—
128C2×C79#145911yescharacteristic
129D79—42no—
130#2930—791no—
131#2115—792no—
132#1328—792no—
133#1320—791no—
134#1320D7911yescharacteristic, complemented
135C79#1148411yescharacteristic, complemented, Sylow
136C2×#131—791no—
137C22×M16—791no—
138#525—791no—
139C2×M32—1582no—
140#436—1582no—
141#435—1582no—
142#424—791no—
143#408—791no—
144#407C2×D7911yescharacteristic
145#405—791no—
146#349—792no—
147#349—1582no—
148#349C2×D7912yes—
149C2×#41—791no—
150C23×C4C4×D7911yescharacteristic
151#131—3162no—
152C2×M16—3162no—
153C2×M16—792no—
154C2×M16—1581no—
155C2×M16—1581no—
156C2×M16C4×D7911yescharacteristic
157C2×M16C22×D7911yescharacteristic
158C22×C8—791no—
159C22×C8—791no—
160C22×C8—792no—
161#123—792no—
162C22⋊Q8—1581no—
163#121—1581no—
164#120—792no—
165C4×D4—1582no—
166#117—791no—
167M32—22no—
168M32—3162no—
169M32—1582no—
170C2×C16—3162no—
171C4⋊C8—1582no—
172C22⋊C8—22no—
173C22⋊C8—1582no—
174#97—1581no—
175C4×C8—1581no—
176C24C8×D7911yescharacteristic
177#41—3161no—
178#41—3161no—
179C2×Q8—791no—
180C2×D4—792no—
181C2×D4—791no—
182C2×D4—1582no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—792no—
188C22×C4C8×D7911yescharacteristic
189C22×C4C2×C4×D7911yescharacteristic
190M16—41no—
191M16—41no—
192M16—3162no—
193M16—3162no—
194M16—3162no—
195C2×C8—21no—
196C2×C8—3161no—
197C2×C8—791no—
198C2×C8—791no—
199C2×C8—1582no—
200C2×C8—1582no—
201C2×C8—1581no—
202C2×C8—1582no—
203C2×C8D4×D7912yes—
204C4⋊C4—1582no—
205C4⋊C4—1581no—
206C22⋊C4—792no—
207C22⋊C4—1582no—
208C22⋊C4—1582no—
209C42—1581no—
210C16—42no—
211C16—3162no—
212C23—21no—
213C23—41no—
214C23—81no—
215C23—792no—
216C23C2×C8×D7911yescharacteristic
217Q8—3161no—
218D4—3161no—
219D4—3162no—
220D4—6322no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—792no—
229C2×C4—1582no—
230C2×C4—1582no—
231C2×C4—1582no—
232C2×C4C22⋊C4×D7911yescharacteristic
233C2×C4M16×D7911yescharacteristic
234C8—22no—
235C8—41no—
236C8—3161no—
237C8—792no—
238C8—1581no—
239C22—21no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—82no—
245C22—3162no—
246C22—1582no—
247C22C22⋊C8×D7911yescharacteristic
248C4—21no—
249C4—41no—
250C4—3162no—
251C4—1582no—
252C4#99×D7911yescharacteristic
253C4#1909854411yescharacteristic
254C2—21no—
255C2—41no—
256C2—81no—
257C2—3162no—
258C2#2376171511yescharacteristic
subgroupG/H
centerC4#99×D79
commutatorC23×C79C22×C8
FrattiniC2×M16C22×D79
FittingC79×#1320C2
socleC2×C79#1459
prime radical#19099074C4
cosocleC2×M16×C79C23
hypercenter#1320D79
hypocenterC79#11484
LCSG ▷ C23×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. #23765019C2semidirect
3. #23761352C2semidirect
4. #23761336C2semidirect
5. C79×#1320C2semidirect
134. #1320D79semidirect
135. C79#11484semidirect

Maximal subgroups

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

subgroupindex
#237720992
#237650192
#237613522
#237613362
C79×#13202
#1148479