Finite Group Databasebeta

← #37510018

Subgroups — #37510018

Order 22912.

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

normal subgroups only

This group has 478 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×#19153141C211yesmaximal, characteristic, complemented
2#26039008C211yesmaximal, characteristic, complemented
3C2×#19153110C211yesmaximal, characteristic, complemented
4C2×#19153041C211yesmaximal, characteristic, complemented
5#26038443C211yesmaximal, characteristic, complemented
6C2×#19153029C211yesmaximal, characteristic, complemented
7#339×C179C211yesmaximal, characteristic, complemented
8C2×#15668259C2211yescharacteristic
9C22×D716—22no—
10C22×D716C2211yescharacteristic
11C22×D716C2211yescharacteristic
12#19153141—22no—
13#19153141—22no—
14C2×#15668248—21no—
15C2×D1432—21no—
16C2×#15668243C2211yescharacteristic
17#19153110—22no—
18#19153110—22no—
19#19153110—21no—
20#19153101—21no—
21#19153088—22no—
22#19153083—22no—
23#19153041—22no—
24#19153033—22no—
25#19153029—22no—
26C2×M16×C179C2211yescharacteristic
27#117×C179C2211yescharacteristic
28C2×C42×C179C2211yescharacteristic
29C23×D179—21no—
30C23×D179D412yes—
31#15668259—21no—
32#15668259—21no—
33#15668259—21no—
34#15668259—21no—
35C2×Dic358D411yescharacteristic, complemented
36C2×D716—28no—
37C2×D716—24no—
38C2×D716—21no—
39C2×D716—21no—
40C2×D716—21no—
41C2×D716—21no—
42C2×D716—21no—
43C2×D716—21no—
44C2×D716—21no—
45C2×D716—21no—
46C2×D716—21no—
47C2×D716—21no—
48C2×D716—21no—
49C2×D716—48no—
50C2×D716D411yescharacteristic, complemented
51C2×D716D412yescomplemented
52C2×C22⋊D179—21no—
53C2×C4×D179—21no—
54#15668248—24no—
55D1432—24no—
56#15668243—21no—
57#15668243—21no—
58C2×C179⋊C8—21no—
59(C2×D179)⋊C4—24no—
60C22×C4×C179—21no—
61C22×C4×C179C2311yescharacteristic
62M16×C179—21no—
63M16×C179D412yescomplemented
64C2×C8×C179—21no—
65C4⋊C4×C179D412yescomplemented
66C22⋊C4×C179—21no—
67C42×C179—21no—
68C42×C179—21no—
69C42×C179D412yescomplemented
70C22×D179—22no—
71C22×D179—21no—
72C22×D179—22no—
73C22×D179—22no—
74C22×D179—22no—
75C22×D179—44no—
76C22×D179—44no—
77C22×D179—41no—
78C22×D179—41no—
79Dic358—21no—
80Dic358—21no—
81D716—21no—
82D716—22no—
83D716—22no—
84D716—21no—
85D716—416no—
86D716—41no—
87D716—41no—
88D716—41no—
89D716—41no—
90D716—41no—
91D716—41no—
92D716—84no—
93C22⋊D179—41no—
94C22⋊D179—41no—
95C2×Dic179—21no—
96C4×D179—41no—
97C4×D179—41no—
98C179⋊C8—41no—
99C23×C179C2×D411yescharacteristic
100C2×C4×C179—24no—
101C2×C4×C179—22no—
102C2×C4×C179—22no—
103C2×C4×C179C2×D411yescharacteristic
104C2×C4×C179C2×D411yescharacteristic
105C2×C4×C179C2×D411yescharacteristic
106C2×C4×C179C2×D411yescharacteristic
107C2×C4×C179C2×D411yescharacteristic
108C2×C4×C179#4111yescharacteristic
109C8×C179—22no—
110C22×C179—22no—
111C22×C179#12011yescharacteristic
112C22×C179#12011yescharacteristic
113C22×C179#12111yescharacteristic
114C2×D179—44no—
115C2×D179—44no—
116C2×D179—41no—
117C2×D179—42no—
118C2×D179—42no—
119C2×D179—42no—
120C2×D179—44no—
121C2×D179—82no—
122C2×D179—82no—
123C2×D179—82no—
124C4×C179—24no—
125C4×C179—42no—
126C4×C179#12011yescharacteristic
127C4×C179#12111yescharacteristic
128C4×C179#12111yescharacteristic
129C4×C179#12711yescharacteristic
130Dic179—41no—
131C2×C179—21no—
132C2×C179—21no—
133C2×C179#39211yescharacteristic
134C2×C179#45312yes—
135D179—81no—
136D179—82no—
137D179—82no—
138C179#200211yescharacteristic, complemented, Sylow
139#2002—1791nomaximal, Sylow
140C2×AGL(1,ℤ/8)—1791no—
141#535—1791no—
142C2×#127—1791no—
143C2×#104—1791no—
144#418—1791no—
145C2×#100—1791no—
146#339D17911yescharacteristic, complemented
147C2×#41—1791no—
148C22×D4—3582no—
149C22×D4—1791no—
150C22×D4—1791no—
151AGL(1,ℤ/8)—3582no—
152AGL(1,ℤ/8)—3582no—
153C2×SD16—3581no—
154C2×D8—3581no—
155C2×M16—1791no—
156C2×M16C2×D17911yescharacteristic
157#127—3582no—
158#127—3581no—
159#127—3582no—
160#124—3581no—
161#121—3582no—
162#120—3582no—
163#117C2×D17911yescharacteristic
164C2×C42C2×D17911yescharacteristic
165#104—3582no—
166#102—3582no—
167#100—3582no—
168C24—3581no—
169C24—1792no—
170#41—3581no—
171#41—3581no—
172#41—3581no—
173#41—3581no—
174C2×Q8—1791no—
175C2×D4—3584no—
176C2×D4—3588no—
177C2×D4—3581no—
178C2×D4—3581no—
179C2×D4—3581no—
180C2×D4—3581no—
181C2×D4—3581no—
182C2×D4—3581no—
183C2×D4—3581no—
184C2×D4—3581no—
185C2×D4—3581no—
186C2×D4—3581no—
187C2×D4—3581no—
188C2×D4—3581no—
189C2×D4—1791no—
190C2×D4—1792no—
191C2×D4—7168no—
192C22×C4—21no—
193C22×C4—3581no—
194C22×C4C22×D17911yescharacteristic
195SD16—3584no—
196D8—3584no—
197M16—21no—
198M16—3581no—
199M16—3581no—
200M16D71612yescomplemented
201C2×C8—21no—
202C2×C8—3581no—
203C4⋊C4D71612yescomplemented
204C22⋊C4—21no—
205C22⋊C4—3584no—
206C42—21no—
207C42—21no—
208C42D71612yescomplemented
209C23—3582no—
210C23—3582no—
211C23—3582no—
212C23—3582no—
213C23—3581no—
214C23—7164no—
215C23—7164no—
216C23—7161no—
217C23—7161no—
218C23D4×D17911yescharacteristic
219Q8—3581no—
220Q8—3581no—
221D4—3581no—
222D4—3582no—
223D4—3582no—
224D4—3581no—
225D4—14324no—
226D4—71616no—
227D4—7161no—
228D4—7161no—
229D4—7161no—
230D4—7161no—
231D4—7161no—
232D4—7161no—
233D4—7161no—
234D4—7161no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—3581no—
239C2×C4—7161no—
240C2×C4—7161no—
241C2×C4D4×D17911yescharacteristic
242C2×C4D4×D17911yescharacteristic
243C2×C4C2×D71611yescharacteristic
244C2×C4C2×D71611yescharacteristic
245C2×C4C2×D71611yescharacteristic
246C2×C4#1566826011yescharacteristic
247C8—22no—
248C8—7161no—
249C22—22no—
250C22—14322no—
251C22—14322no—
252C22—14322no—
253C22—7164no—
254C22—7164no—
255C22—7161no—
256C22—7162no—
257C22—7162no—
258C22—7162no—
259C22—7164no—
260C22#1915308311yescharacteristic
261C22#1915308311yescharacteristic
262C22#1915308811yescharacteristic
263C4—24no—
264C4—42no—
265C4—7161no—
266C4#1915308311yescharacteristic
267C4#1915308811yescharacteristic
268C4#1915308811yescharacteristic
269C4#1915311011yescharacteristic
270C2—21no—
271C2—21no—
272C2—14322no—
273C2—14322no—
274C2—14321no—
275C2#2603834911yescharacteristic
276C2#2603860812yes—
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#339×C179C2
socleC22×C179#120
hypercenter#339D179
hypocenterC179#2002
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×#19153141C2semidirect
2. #26039008C2semidirect
3. C2×#19153110C2semidirect
4. C2×#19153041C2semidirect
5. #26038443C2semidirect
6. C2×#19153029C2semidirect
7. #339×C179C2semidirect
35. C2×Dic358D4semidirect
50. C2×D716D4semidirect
51. C2×D716D4semidirect
63. M16×C179D4semidirect
65. C4⋊C4×C179D4semidirect
69. C42×C179D4semidirect
138. C179#2002semidirect
146. #339D179semidirect
200. M16D716semidirect
203. C4⋊C4D716semidirect
208. C42D716semidirect

Maximal subgroups

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

subgroupindex
C2×#191531412
#260390082
C2×#191531102
C2×#191530412
#260384432
C2×#191530292
#339×C1792
#2002179