Finite Group Databasebeta

← #37510084

Subgroups — #37510084

Order 22912.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#15668248C211yesmaximal, characteristic
2C22×#15668246C211yesmaximal, characteristic
3C2×#19153111C211yesmaximal, characteristic, complemented
4C2×#19153036C211yesmaximal, characteristic, complemented
5#26038340C211yesmaximal, characteristic
6#26038193C211yesmaximal, characteristic, complemented
7C2×#103×C179C211yesmaximal, characteristic, complemented
8C22×Dic358C2211yescharacteristic
9C22×D716C2211yescharacteristic
10C2×#15668248—22no—
11C2×#15668248—22no—
12C2×#15668248—22no—
13C2×#15668246—22no—
14C2×#15668246—22no—
15C2×#15668246—22no—
16C22×C179⋊C8C2211yescharacteristic
17#19153111—22no—
18#19153111—22no—
19C2×Dic179⋊C4—22no—
20C2×(C2×D179)⋊C4—22no—
21C2×C4×Dic179C2211yescharacteristic
22#19153036—22no—
23C22×Q8×C179C2211yescharacteristic
24C22×C8×C179C2211yescharacteristic
25C2×C4⋊C4×C179C2211yescharacteristic
26#103×C179—22no—
27C23×D179—21no—
28C2×Dic358—21no—
29C2×Dic358—21no—
30C2×Dic358—21no—
31C2×Dic358—21no—
32C2×Dic358—21no—
33C2×Dic358D412yes—
34C2×D716—21no—
35C2×D716—21no—
36C2×D716—21no—
37C2×D716—21no—
38C2×D716—21no—
39C2×D716D412yes—
40C22×Dic179—21no—
41C22×Dic179D412yes—
42#15668248—28no—
43#15668246—28no—
44C2×C179⋊C8—21no—
45C2×C179⋊C8—21no—
46C2×C179⋊C8D412yes—
47Dic179⋊C4—28no—
48(C2×D179)⋊C4—44no—
49C4×Dic179—21no—
50C4×Dic179—21no—
51C2×Q8×C179—21no—
52C2×Q8×C179—21no—
53C2×Q8×C179—21no—
54C2×Q8×C179—21no—
55C2×Q8×C179—21no—
56C2×Q8×C179D412yes—
57C22×C4×C179—21no—
58C22×C4×C179—21no—
59C22×C4×C179C2311yescharacteristic
60C2×C8×C179—21no—
61C2×C8×C179—21no—
62C2×C8×C179D412yes—
63C4⋊C4×C179—21no—
64C4⋊C4×C179—21no—
65C22×D179—22no—
66C22×D179—41no—
67C22×D179—41no—
68C22×D179—42no—
69C22×D179—42no—
70Dic358—22no—
71Dic358—22no—
72Dic358—41no—
73Dic358—41no—
74D716—22no—
75D716—22no—
76D716—41no—
77D716—41no—
78C2×Dic179—22no—
79C2×Dic179—22no—
80C2×Dic179—22no—
81C2×Dic179—41no—
82C2×Dic179—41no—
83C2×Dic179SD1614yescomplemented
84C179⋊C8—22no—
85C23×C179C2×D411yescharacteristic
86Q8×C179—22no—
87Q8×C179—22no—
88Q8×C179—41no—
89Q8×C179—41no—
90C2×C4×C179—22no—
91C2×C4×C179—41no—
92C2×C4×C179—41no—
93C2×C4×C179—41no—
94C2×C4×C179—42no—
95C2×C4×C179C2×D411yescharacteristic
96C2×C4×C179C2×D411yescharacteristic
97C2×C4×C179C2×D411yescharacteristic
98C2×C4×C179C2×D411yescharacteristic
99C2×C4×C179C2×D411yescharacteristic
100C2×C4×C179#4111yescharacteristic
101C8×C179—22no—
102C22×C179#12011yescharacteristic
103C22×C179#12111yescharacteristic
104C22×C179#12711yescharacteristic
105C22×C179C2×SD1612yes—
106C22×C179#13712yes—
107C2×D179—42no—
108C2×D179—44no—
109C2×D179—82no—
110C2×D179—81no—
111C2×D179—81no—
112C4×C179—42no—
113C4×C179—81no—
114C4×C179#12011yescharacteristic
115C4×C179#12011yescharacteristic
116C4×C179#12111yescharacteristic
117C4×C179#12111yescharacteristic
118Dic179—24no—
119Dic179—42no—
120C2×C179#39211yescharacteristic
121C2×C179#44812yes—
122C2×C179#46112yes—
123C2×C179#49211yescharacteristic
124C2×C179#49711yescharacteristic
125D179—82no—
126C179#201311yescharacteristic, complemented, Sylow
127#2013—1791nomaximal, Sylow
128C22×SD16—1791no—
129C22×SD16—1791no—
130C2×C4⋊Q8—1791no—
131C2×#103—1791no—
132C2×#103D17911yescharacteristic, complemented
133#390—1791no—
134#336—1791no—
135C22×Q8—1791no—
136C22×Q8C2×D17911yescharacteristic
137C22×D4—1791no—
138C2×SD16—3582no—
139C2×SD16—3582no—
140C2×SD16—3582no—
141C2×SD16—3582no—
142C2×SD16—3582no—
143C2×SD16—3582no—
144C22×C8—1791no—
145C22×C8C2×D17911yescharacteristic
146C4⋊Q8—3582no—
147C4⋊Q8—3582no—
148C2×C4⋊C4—3582no—
149C2×C4⋊C4C2×D17911yescharacteristic
150C2×C22⋊C4—3582no—
151C2×C42—1791no—
152#103—22no—
153#103—3582no—
154C24—3581no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—3581no—
161C2×Q8—3581no—
162C2×Q8—3581no—
163C2×Q8—3581no—
164C2×Q8—3581no—
165C2×Q8—1792no—
166C2×Q8D71612yes—
167C2×D4—3581no—
168C2×D4—3581no—
169C2×D4—3581no—
170C2×D4—3581no—
171C2×D4—3581no—
172C2×D4—1792no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—3581no—
176C22×C4—1792no—
177C22×C4C22×D17911yescharacteristic
178SD16—3588no—
179SD16—3588no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—3581no—
183C2×C8—3581no—
184C2×C8—1792no—
185C2×C8C22⋊D17912yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3588no—
189C22⋊C4—7164no—
190C42—3581no—
191C42—3581no—
192C23—3582no—
193C23—7161no—
194C23—7161no—
195C23—7162no—
196C23—7162no—
197C23D4×D17911yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—3582no—
203Q8—3582no—
204Q8—7161no—
205Q8—7161no—
206D4—3582no—
207D4—3582no—
208D4—7161no—
209D4—7161no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—3582no—
216C2×C4—3582no—
217C2×C4—3582no—
218C2×C4—1794no—
219C2×C4—7161no—
220C2×C4—7161no—
221C2×C4D4×D17911yescharacteristic
222C2×C4D4×D17911yescharacteristic
223C2×C4D4×D17911yescharacteristic
224C2×C4C2×C22⋊D17911yescharacteristic
225C2×C4C2×D71611yescharacteristic
226C2×C4#1566825911yescharacteristic
227C8—22no—
228C8—3582no—
229C22—14321no—
230C22—14321no—
231C22—14322no—
232C22—7162no—
233C22—7164no—
234C22#1915308311yescharacteristic
235C22#1915308611yescharacteristic
236C22#1915310911yescharacteristic
237C22SD16×D17912yes—
238C22#1915314512yes—
239C4—42no—
240C4—81no—
241C4—3584no—
242C4—7162no—
243C4#1915308111yescharacteristic
244C4#1915308311yescharacteristic
245C4#1915308611yescharacteristic
246C4#1915308711yescharacteristic
247C2—14322no—
248C2#2603834811yescharacteristic
249C2#2603857912yes—
250C2#2603864312yes—
251C2#2603881011yescharacteristic
252C2#2603883211yescharacteristic
subgroupG/H
centerC23D4×D179
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
FittingC2×#103×C179C2
socleC23×C179C2×D4
prime radicalC22×D716C22
hypercenterC2×#103D179
hypocenterC179#2013
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#19153111C2semidirect
4. C2×#19153036C2semidirect
6. #26038193C2semidirect
7. C2×#103×C179C2semidirect
83. C2×Dic179SD16semidirect
126. C179#2013semidirect
132. C2×#103D179semidirect

Maximal subgroups

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

subgroupindex
C22×#156682482
C22×#156682462
C2×#191531112
C2×#191530362
#260383402
#260381932
C2×#103×C1792
#2013179