Finite Group Databasebeta

← #34201492

Subgroups — #34201492

Order 20224.

344 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 71 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#23765600C212yesmaximal, complemented
2#23764828C211yesmaximal, characteristic
3#23761412C212yesmaximal, complemented
4#23761324C211yesmaximal, characteristic
5C79×#3189C211yesmaximal, characteristic
6C2×#14579095C2211yescharacteristic
7#19098423—21no—
8#19098388—22no—
9#19098367—22no—
10C22⋊(C79⋊C16)C2212yes—
11#19098326—21no—
12#19098326C2211yescharacteristic
13C2×C22⋊Q8×C79—21no—
14#484×C79—22no—
15#467×C79—21no—
16#467×C79C2212yes—
17#451×C79—22no—
18C2×C22⋊C8×C79C2211yescharacteristic
19C2×C4⋊Dic79—21no—
20C2×C4⋊Dic79—21no—
21C2×C22⋊Dic79—41no—
22C2×C79⋊C16—22no—
23#14579095—22no—
24C79⋊#95—41no—
25C22×Q8×C79—21no—
26C23×C4×C79D411yescharacteristic
27C2×Q16×C79—21no—
28C2×Q16×C79D412yes—
29C22×C8×C79D411yescharacteristic
30C22×C8×C79C2311yescharacteristic
31C22⋊Q8×C79—22no—
32C22⋊Q8×C79—22no—
33C22⋊Q8×C79—41no—
34C22⋊Q8×C79—41no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—41no—
37C2×C22⋊C4×C79—41no—
38#107×C79—21no—
39#107×C79D412yes—
40#103×C79—22no—
41#103×C79—22no—
42C22⋊C8×C79—22no—
43C22×Dic79—41no—
44C22×Dic79—41no—
45C4⋊Dic79—41no—
46C4⋊Dic79—41no—
47C4⋊Dic79—41no—
48C4⋊Dic79—41no—
49C22⋊Dic79—82no—
50C79⋊C16—42no—
51C24×C79D811yescharacteristic
52C2×Q8×C79—22no—
53C2×Q8×C79—41no—
54C2×Q8×C79—81no—
55C2×Q8×C79—81no—
56C22×C4×C79—21no—
57C22×C4×C79—21no—
58C22×C4×C79—41no—
59C22×C4×C79—41no—
60C22×C4×C79—41no—
61C22×C4×C79—41no—
62C22×C4×C79D811yescharacteristic
63C22×C4×C79C2×D411yescharacteristic
64Q16×C79—42no—
65Q16×C79—81no—
66C2×C8×C79—21no—
67C2×C8×C79—41no—
68C2×C8×C79—41no—
69C2×C8×C79C2×D412yes—
70C4⋊C4×C79—22no—
71C4⋊C4×C79—41no—
72C4⋊C4×C79—42no—
73C4⋊C4×C79—42no—
74C22⋊C4×C79—42no—
75C22⋊C4×C79—42no—
76C23×C79—21no—
77C23×C79—21no—
78C23×C79—41no—
79C23×C79—41no—
80C23×C79Q3212yes—
81C23×C79C2×D811yescharacteristic
82Q8×C79—42no—
83Q8×C79—42no—
84Q8×C79—82no—
85C2×C4×C79—21no—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—41no—
89C2×C4×C79—41no—
90C2×C4×C79—41no—
91C2×C4×C79—42no—
92C2×C4×C79—42no—
93C2×C4×C79—81no—
94C2×C4×C79—82no—
95C2×C4×C79—82no—
96C2×C4×C79#12011yescharacteristic
97C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
98C2×Dic79—81no—
99C2×Dic79—81no—
100C2×Dic79—81no—
101C2×Dic79—81no—
102C2×Dic79—81no—
103C2×Dic79—81no—
104C8×C79—22no—
105C8×C79—41no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79—22no—
109C22×C79—42no—
110C22×C79—41no—
111C22×C79—41no—
112C22×C79—81no—
113C22×C79#44711yescharacteristic
114C22×C79C2×Q3211yescharacteristic
115C22×C79#51011yescharacteristic
116C4×C79—21no—
117C4×C79—41no—
118C4×C79—82no—
119C4×C79—82no—
120C4×C79#45511yescharacteristic
121C4×C79#45811yescharacteristic
122Dic79—161no—
123Dic79—161no—
124#12282—791nomaximal, Sylow
125C2×C79—21no—
126C2×C79—21no—
127C2×C79—42no—
128C2×C79#160711yescharacteristic
129C2×C79#218811yescharacteristic
130C2×C79#219011yescharacteristic
131#3189D7911yescharacteristic
132#2236—792no—
133#2076—791no—
134#1348—792no—
135#1314—791no—
136C79#1228211yescharacteristic, complemented, Sylow
137C2×C22⋊Q8—21no—
138#484—22no—
139#467—21no—
140#467C2×D7912yes—
141#451—22no—
142C2×#107—791no—
143C2×C22⋊C8C2×D7911yescharacteristic
144#380—1581no—
145#366—1582no—
146#358—1582no—
147C22⋊C16—792no—
148#340—791no—
149#340—1581no—
150C22×Q8—21no—
151C23×C4C22⋊D7911yescharacteristic
152C2×Q16—21no—
153C2×Q16C22⋊D7912yes—
154C22×C8C22⋊D7911yescharacteristic
155C22×C8C22×D7911yescharacteristic
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—1581no—
163C2×C4⋊C4—1581no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—3161no—
166C2×C16—1582no—
167#107—21no—
168#107—1582no—
169#107C22⋊D7912yes—
170#103—22no—
171#103—22no—
172C22⋊C8—22no—
173#95—3161no—
174C24#1192960911yescharacteristic
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—3161no—
186C22×C4—3161no—
187C22×C4#1192960911yescharacteristic
188C22×C4C2×C22⋊D7911yescharacteristic
189Q16—42no—
190Q16—81no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8C2×C22⋊D7912yes—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—3161no—
200C4⋊C4—3161no—
201C4⋊C4—3161no—
202C4⋊C4—3161no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—6322no—
206C16—3162no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1457910712yes—
212C23C2×#1192960911yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—6321no—
228C2×C4—6321no—
229C2×C4—6321no—
230C2×C4—6321no—
231C2×C4—6321no—
232C2×C4—6321no—
233C2×C4#1457913011yescharacteristic
234C2×C4#1457918711yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1909869711yescharacteristic
245C22C2×#1457910711yescharacteristic
246C22#1909900411yescharacteristic
247C4—21no—
248C4—41no—
249C4—82no—
250C4—82no—
251C4—12641no—
252C4—12641no—
253C4#1909874111yescharacteristic
254C4#1909875311yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2376238611yescharacteristic
259C2#2376533411yescharacteristic
260C2#2376534611yescharacteristic
subgroupG/H
centerC22#19098697
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#3189C2
socleC22×C79#447
prime radicalC24×C79D8
hypercenter#3189D79
hypocenterC79#12282
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. #23765600C2semidirect
3. #23761412C2semidirect
136. C79#12282semidirect

Maximal subgroups

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

subgroupindex
#237656002
#237648282
#237614122
#237613242
C79×#31892
#1228279