Finite Group Databasebeta

← #37510032

Subgroups — #37510032

Order 22912.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19153146C211yesmaximal, characteristic, complemented
2#26039008C211yesmaximal, characteristic
3C2×#19153101C211yesmaximal, characteristic, complemented
4C2×#19153041C211yesmaximal, characteristic, complemented
5#26038447C211yesmaximal, characteristic, complemented
6C2×#19153029C211yesmaximal, characteristic, complemented
7#339×C179C211yesmaximal, characteristic, complemented
8C2×#15668259C2211yescharacteristic
9C22×Dic358C2211yescharacteristic
10C22×D716C2211yescharacteristic
11#19153146—22no—
12#19153146—22no—
13C2×Dic716—21no—
14C2×#15668248—21no—
15C2×#15668243C2211yescharacteristic
16#19153110—21no—
17#19153101—22no—
18#19153101—21no—
19#19153101—22no—
20#19153088—22no—
21#19153083—22no—
22C2×(C2×D179)⋊C4—22no—
23#19153041—22no—
24#19153036—22no—
25#19153029—22no—
26C2×M16×C179C2211yescharacteristic
27#117×C179C2211yescharacteristic
28C2×C42×C179C2211yescharacteristic
29C23×D179D412yes—
30#15668259—21no—
31#15668259—21no—
32#15668259—21no—
33#15668259—21no—
34C2×Dic358—21no—
35C2×Dic358—21no—
36C2×Dic358—21no—
37C2×Dic358—21no—
38C2×Dic358—21no—
39C2×Dic358D411yescharacteristic
40C2×Dic358D412yescomplemented
41C2×D716—24no—
42C2×D716—21no—
43C2×D716—21no—
44C2×D716—21no—
45C2×D716—21no—
46C2×D716—21no—
47C2×D716—21no—
48C2×D716D411yescharacteristic
49C2×C22⋊D179—21no—
50C22×Dic179—21no—
51C2×C4×D179—21no—
52Dic716—24no—
53#15668248—24no—
54#15668243—21no—
55#15668243—21no—
56C2×C179⋊C8—21no—
57(C2×D179)⋊C4—28no—
58(C2×D179)⋊C4—24no—
59C22×C4×C179—21no—
60C22×C4×C179C2311yescharacteristic
61M16×C179—21no—
62M16×C179D412yescomplemented
63C2×C8×C179—21no—
64C4⋊C4×C179D412yes—
65C22⋊C4×C179—21no—
66C42×C179—21no—
67C42×C179—21no—
68C42×C179D412yescomplemented
69C22×D179—22no—
70C22×D179—22no—
71C22×D179—21no—
72C22×D179—22no—
73C22×D179—44no—
74Dic358—22no—
75Dic358—21no—
76Dic358—22no—
77Dic358—21no—
78Dic358—41no—
79Dic358—41no—
80D716—21no—
81D716—21no—
82D716—41no—
83D716—41no—
84D716—41no—
85D716—41no—
86D716—84no—
87C22⋊D179—41no—
88C22⋊D179—41no—
89C2×Dic179—21no—
90C2×Dic179—22no—
91C2×Dic179—41no—
92C2×Dic179—41no—
93C4×D179—41no—
94C4×D179—41no—
95C179⋊C8—41no—
96C23×C179C2×D411yescharacteristic
97C2×C4×C179—22no—
98C2×C4×C179—22no—
99C2×C4×C179—42no—
100C2×C4×C179C2×D411yescharacteristic
101C2×C4×C179C2×D411yescharacteristic
102C2×C4×C179C2×D411yescharacteristic
103C2×C4×C179C2×D411yescharacteristic
104C2×C4×C179C2×D411yescharacteristic
105C2×C4×C179#4111yescharacteristic
106C8×C179—22no—
107C22×C179—22no—
108C22×C179#12011yescharacteristic
109C22×C179#12011yescharacteristic
110C22×C179#12111yescharacteristic
111C2×D179—44no—
112C2×D179—41no—
113C2×D179—42no—
114C2×D179—42no—
115C2×D179—44no—
116C2×D179—82no—
117C4×C179—42no—
118C4×C179—42no—
119C4×C179#12011yescharacteristic
120C4×C179#12111yescharacteristic
121C4×C179#12111yescharacteristic
122C4×C179#12711yescharacteristic
123Dic179—41no—
124Dic179—42no—
125C2×C179—21no—
126C2×C179—21no—
127C2×C179#39211yescharacteristic
128C2×C179#45512yes—
129D179—81no—
130D179—82no—
131C179#200411yescharacteristic, complemented, Sylow
132#2004—1791nomaximal, Sylow
133C2×#137—1791no—
134#535—1791no—
135C2×#124—1791no—
136C2×#104—1791no—
137#419—1791no—
138C2×#100—1791no—
139#339D17911yescharacteristic, complemented
140C2×#41—1791no—
141C22×Q8—1791no—
142C22×D4—1791no—
143#137—3582no—
144#137—3582no—
145C2×Q16—3581no—
146C2×SD16—3581no—
147C2×M16—1791no—
148C2×M16C2×D17911yescharacteristic
149#127—3581no—
150#124—3582no—
151#124—3581no—
152#124—3582no—
153#121—3582no—
154#120—3582no—
155#117C2×D17911yescharacteristic
156C2×C22⋊C4—3582no—
157C2×C42C2×D17911yescharacteristic
158#104—3582no—
159#103—3582no—
160#100—3582no—
161C24—1792no—
162#41—3581no—
163#41—3581no—
164#41—3581no—
165#41—3581no—
166C2×Q8—3581no—
167C2×Q8—3581no—
168C2×Q8—3581no—
169C2×Q8—3581no—
170C2×Q8—3581no—
171C2×Q8—1791no—
172C2×Q8—1792no—
173C2×D4—3584no—
174C2×D4—3581no—
175C2×D4—3581no—
176C2×D4—3581no—
177C2×D4—3581no—
178C2×D4—3581no—
179C2×D4—3581no—
180C2×D4—3581no—
181C2×D4—1791no—
182C22×C4—21no—
183C22×C4—3581no—
184C22×C4—3581no—
185C22×C4C22×D17911yescharacteristic
186Q16—3584no—
187SD16—3584no—
188M16—21no—
189M16—3581no—
190M16—3581no—
191M16D71612yescomplemented
192C2×C8—21no—
193C2×C8—3581no—
194C4⋊C4D71612yes—
195C22⋊C4—21no—
196C22⋊C4—3588no—
197C22⋊C4—3584no—
198C42—21no—
199C42—21no—
200C42D71612yescomplemented
201C23—3582no—
202C23—3582no—
203C23—3582no—
204C23—3581no—
205C23—7164no—
206C23D4×D17911yescharacteristic
207Q8—3582no—
208Q8—3581no—
209Q8—3582no—
210Q8—3581no—
211Q8—7161no—
212Q8—7161no—
213D4—3581no—
214D4—3581no—
215D4—14324no—
216D4—7161no—
217D4—7161no—
218D4—7161no—
219D4—7161no—
220D4—7161no—
221D4—7161no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—3582no—
226C2×C4—3581no—
227C2×C4—7161no—
228C2×C4—7161no—
229C2×C4—7161no—
230C2×C4—7161no—
231C2×C4D4×D17911yescharacteristic
232C2×C4D4×D17911yescharacteristic
233C2×C4C2×D71611yescharacteristic
234C2×C4C2×D71611yescharacteristic
235C2×C4C2×D71611yescharacteristic
236C2×C4#1566826011yescharacteristic
237C8—22no—
238C8—7161no—
239C22—22no—
240C22—14322no—
241C22—7164no—
242C22—7161no—
243C22—7162no—
244C22—7162no—
245C22—7164no—
246C22#1915308311yescharacteristic
247C22#1915308311yescharacteristic
248C22#1915308811yescharacteristic
249C4—42no—
250C4—42no—
251C4—7161no—
252C4—7162no—
253C4#1915308311yescharacteristic
254C4#1915308811yescharacteristic
255C4#1915308811yescharacteristic
256C4#1915311011yescharacteristic
257C2—21no—
258C2—21no—
259C2—14322no—
260C2—14321no—
261C2#2603834911yescharacteristic
262C2#2603861812yes—
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#339×C179C2
socleC22×C179#120
prime radical#26039008C2
hypercenter#339D179
hypocenterC179#2004
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#19153146C2semidirect
3. C2×#19153101C2semidirect
4. C2×#19153041C2semidirect
5. #26038447C2semidirect
6. C2×#19153029C2semidirect
7. #339×C179C2semidirect
40. C2×Dic358D4semidirect
62. M16×C179D4semidirect
68. C42×C179D4semidirect
131. C179#2004semidirect
139. #339D179semidirect
191. M16D716semidirect
200. C42D716semidirect

Maximal subgroups

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

subgroupindex
C2×#191531462
#260390082
C2×#191531012
C2×#191530412
#260384472
C2×#191530292
#339×C1792
#2004179