Finite Group Databasebeta

← #34205098

Subgroups — #34205098

Order 20224.

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

normal subgroups only

This group has 424 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
1C2×#19098362C211yesmaximal, characteristic
2#23763794C211yesmaximal, characteristic
3#23763614C212yesmaximal
4#23761527C212yesmaximal
5C79×#2381C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098603—22no—
8#19098572—21no—
9C2×#14579080C2211yescharacteristic
10C2×#14579080C2212yes—
11#19098362C414yescomplemented
12#19098323—21no—
13#19098323—22no—
14C2×C4⋊Q8×C79C2211yescharacteristic
15#391×C79—21no—
16#391×C79C2212yes—
17#387×C79—22no—
18#387×C79—22no—
19#383×C79—21no—
20C2×#11929608—22no—
21C2×#11929608—21no—
22Q8×Dic79—42no—
23#14579120—21no—
24#14579120—22no—
25#14579094—42no—
26#14579092—42no—
27#14579085—42no—
28#14579080—22no—
29#14579080—22no—
30#14579080C2×C414yes—
31C22×Q8×C79D411yescharacteristic
32C22×Q8×C79C2311yescharacteristic
33C4⋊Q8×C79—22no—
34C4⋊Q8×C79C2×C412yes—
35C4⋊Q8×C79Q812yes—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79D412yes—
39C2×C42×C79—21no—
40C2×C42×C79D411yescharacteristic
41C2×C42×C79D412yes—
42#95×C79—21no—
43#95×C79—22no—
44#95×C79—21no—
45#95×C79—22no—
46#95×C79—22no—
47#95×C79—22no—
48#11929608—24no—
49#11929608—42no—
50#11929608—41no—
51#11929608—41no—
52C2×C79⋊C8—42no—
53C2×C79⋊C8—41no—
54C4⋊Dic79—22no—
55C4⋊Dic79—22no—
56C4⋊Dic79—22no—
57C22⋊Dic79—41no—
58C4×Dic79—42no—
59C4×Dic79—41no—
60C2×Q8×C79—21no—
61C2×Q8×C79—41no—
62C2×Q8×C79—41no—
63C2×Q8×C79—41no—
64C2×Q8×C79—41no—
65C2×Q8×C79C4⋊C412yes—
66C2×Q8×C79C22×C411yescharacteristic
67C2×Q8×C79C2×Q811yescharacteristic
68C2×Q8×C79#4112yes—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—22no—
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79C2×D412yes—
75C4⋊C4×C79—24no—
76C4⋊C4×C79—24no—
77C4⋊C4×C79—42no—
78C4⋊C4×C79—81no—
79C42×C79—21no—
80C42×C79—42no—
81C42×C79—81no—
82C42×C79C4⋊C412yes—
83C23×C79#12011yescharacteristic
84Q8×C79—41no—
85Q8×C79—41no—
86Q8×C79—42no—
87Q8×C79—42no—
88Q8×C79—81no—
89C2×C4×C79—22no—
90C2×C4×C79—21no—
91C2×C4×C79—22no—
92C2×C4×C79—21no—
93C2×C4×C79—22no—
94C2×C4×C79—22no—
95C2×C4×C79—42no—
96C2×C4×C79—41no—
97C2×C4×C79—41no—
98C2×C4×C79—44no—
99C2×C4×C79—44no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79C2×C4⋊C411yescharacteristic
103C2×C4×C79C4×D412yes—
104C2×C4×C79C22⋊Q812yes—
105C2×C4×C79#12311yescharacteristic
106C2×Dic79—42no—
107C79⋊C8—44no—
108C79⋊C8—81no—
109C22×C79—22no—
110C22×C79#38511yescharacteristic
111C22×C79B(4,𝔽2)11yescharacteristic
112C22×C79#45811yescharacteristic
113C4×C79—21no—
114C4×C79—22no—
115C4×C79—22no—
116C4×C79—21no—
117C4×C79—42no—
118C4×C79—42no—
119C4×C79—82no—
120C4×C79—82no—
121Dic79—82no—
122#12971—791nomaximal, Sylow
123C2×C79—21no—
124C2×C79—21no—
125C2×C79#186711yescharacteristic
126C2×C79#220312yes—
127#2381D7911yescharacteristic
128C2×#356—791no—
129#1886—791no—
130#1857—792no—
131#1394—792no—
132C79#1297111yescharacteristic, complemented, Sylow
133C2×C4⋊Q8C2×D7911yescharacteristic
134#519—791no—
135#428—1582no—
136#419—1581no—
137C2×#101—792no—
138C2×#101—791no—
139#391—21no—
140#391C2×D7912yes—
141#387—22no—
142#387—22no—
143#383—21no—
144#356—794no—
145#339—1582no—
146#339—1581no—
147C22×Q8C22⋊D7911yescharacteristic
148C22×Q8C22×D7911yescharacteristic
149C2×M16—1581no—
150C2×M16—1582no—
151C4⋊Q8—22no—
152C4⋊Q8Dic15812yes—
153C4⋊Q8C4×D7912yes—
154C4×Q8—3162no—
155#117—1581no—
156#117—1582no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C22⋊D7912yes—
160C2×C42—21no—
161C2×C42C22⋊D7911yescharacteristic
162C2×C42C22⋊D7912yes—
163#107—3162no—
164#106—3162no—
165#103—3162no—
166#101—794no—
167#101—1582no—
168#101—1582no—
169#95—21no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95—22no—
174#95—22no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8Dic79⋊C412yes—
181C2×Q8C2×C4×D7911yescharacteristic
182C2×Q8C2×Dic15811yescharacteristic
183C2×Q8#1192962412yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4C2×C22⋊D7911yescharacteristic
189C22×C4C2×C22⋊D7912yes—
190M16—3162no—
191M16—3161no—
192M16—3161no—
193M16—1584no—
194C2×C8—3162no—
195C2×C8—3161no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—1582no—
201C4⋊C4—1582no—
202C4⋊C4—1582no—
203C22⋊C4—3161no—
204C42—21no—
205C42—42no—
206C42—81no—
207C42—3162no—
208C42—3161no—
209C42Dic79⋊C412yes—
210C23#1457913011yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—42no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—3162no—
230C2×C4C2×Dic79⋊C411yescharacteristic
231C2×C4C4×C22⋊D7912yes—
232C2×C4#1457914012yes—
233C2×C4#1457914411yescharacteristic
234C8—3164no—
235C8—6321no—
236C22—22no—
237C22#1909843611yescharacteristic
238C22#1909875011yescharacteristic
239C22#1909875311yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4—6322no—
249C2—21no—
250C2—21no—
251C2#2376368111yescharacteristic
252C2#2376542112yes—
subgroupG/H
centerC22#19098750
commutatorC2×Q8×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#2381C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
cosocleC22×Q8×C79C23
hypercenter#2381D79
hypocenterC79#12971
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2381

Complements

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

NH ≅ G/Ntype
11. #19098362C4semidirect
132. C79#12971semidirect

Maximal subgroups

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

subgroupindex
C2×#190983622
#237637942
#237636142
#237615272
C79×#23812
#1297179