Finite Group Databasebeta

← #37510059

Subgroups — #37510059

Order 22912.

348 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 69 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#26039237C211yesmaximal, characteristic
2C2×#19153143C211yesmaximal, characteristic, complemented
3#26039015C211yesmaximal, characteristic, complemented
4#26038457C211yesmaximal, characteristic, complemented
5#26038447C211yesmaximal, characteristic, complemented
6#26038209C211yesmaximal, characteristic, complemented
7#410×C179C211yesmaximal, characteristic, complemented
8C2×#15668259C2211yescharacteristic
9C22×Dic358C2211yescharacteristic
10#19153144—21no—
11#19153143—22no—
12#19153143—21no—
13#19153143—22no—
14#19153139—21no—
15#19153138—21no—
16#19153133—22no—
17C2×#15668249—21no—
18C2×#15668247—21no—
19C2×#15668243C2211yescharacteristic
20C2×#15668243C2211yescharacteristic
21#19153113—21no—
22#19153101—21no—
23#19153097—22no—
24#19153093—22no—
25#19153041—22no—
26#19153036—22no—
27C2×#41×C179C2211yescharacteristic
28#117×C179C2211yescharacteristic
29#117×C179C2211yescharacteristic
30#99×C179—22no—
31#15668259—21no—
32#15668259—21no—
33#15668259—21no—
34#15668259—21no—
35C2×Dic358—21no—
36C2×Dic358—21no—
37C2×Dic358—21no—
38C2×Dic358—21no—
39C2×Dic358—21no—
40C2×Dic358D412yes—
41C2×Dic358D411yescharacteristic
42C2×D716D411yescharacteristic
43C2×C22⋊D179—21no—
44C22×Dic179—21no—
45C2×C4×D179—21no—
46#15668249—24no—
47#15668249—22no—
48#15668247—24no—
49#15668247—22no—
50#15668246—22no—
51#15668244—22no—
52#15668243—21no—
53#15668243—21no—
54#15668243—21no—
55#15668243D412yes—
56C2×C179⋊C8—21no—
57C2×C179⋊C8D412yes—
58C4⋊Dic179—24no—
59(C2×D179)⋊C4—24no—
60#41×C179—21no—
61#41×C179—21no—
62#41×C179—21no—
63#41×C179—21no—
64C2×Q8×C179D411yescharacteristic
65C2×D4×C179—21no—
66C2×D4×C179D411yescharacteristic
67C22×C4×C179—21no—
68C22×C4×C179C2311yescharacteristic
69C4⋊C4×C179—21no—
70C4⋊C4×C179—21no—
71C22⋊C4×C179—21no—
72C22⋊C4×C179D412yes—
73C42×C179—21no—
74C42×C179—21no—
75C22×D179—21no—
76Dic358—22no—
77Dic358—21no—
78Dic358—22no—
79Dic358—21no—
80Dic358—41no—
81Dic358—41no—
82D716—21no—
83D716—21no—
84C22⋊D179—41no—
85C22⋊D179—41no—
86C2×Dic179—22no—
87C2×Dic179—21no—
88C2×Dic179—41no—
89C2×Dic179—41no—
90C4×D179—41no—
91C4×D179—41no—
92C179⋊C8—22no—
93C179⋊C8—22no—
94C23×C179—21no—
95C23×C179C2×D411yescharacteristic
96Q8×C179—21no—
97Q8×C179—21no—
98D4×C179—21no—
99D4×C179—21no—
100D4×C179—41no—
101D4×C179—41no—
102C2×C4×C179—22no—
103C2×C4×C179—21no—
104C2×C4×C179—41no—
105C2×C4×C179—41no—
106C2×C4×C179—41no—
107C2×C4×C179C2×D411yescharacteristic
108C2×C4×C179C2×D411yescharacteristic
109C2×C4×C179C2×D411yescharacteristic
110C2×C4×C179C2×D411yescharacteristic
111C2×C4×C179C2×D411yescharacteristic
112C2×C4×C179#4111yescharacteristic
113C22×C179—41no—
114C22×C179—41no—
115C22×C179—81no—
116C22×C179#12011yescharacteristic
117C22×C179#12111yescharacteristic
118C22×C179#12711yescharacteristic
119C2×D179—42no—
120C2×D179—41no—
121C4×C179—42no—
122C4×C179—41no—
123C4×C179—81no—
124C4×C179#12011yescharacteristic
125C4×C179#12011yescharacteristic
126C4×C179#12111yescharacteristic
127C4×C179#12111yescharacteristic
128Dic179—41no—
129Dic179—42no—
130C2×C179—21no—
131C2×C179—21no—
132C2×C179—21no—
133C2×C179—81no—
134C2×C179#39211yescharacteristic
135D179—81no—
136C179#200911yescharacteristic, complemented, Sylow
137#2009—1791nomaximal, Sylow
138#575—1791no—
139C2×#137—1791no—
140#536—1791no—
141#421—1791no—
142#419—1791no—
143#410D17911yescharacteristic, complemented
144#344—1791no—
145C2×#41—1791no—
146C2×#41C2×D17911yescharacteristic
147C22×Q8—1791no—
148#137—3582no—
149#137—3581no—
150#137—3581no—
151#137—3582no—
152AGL(1,ℤ/8)—3581no—
153AGL(1,ℤ/8)—3581no—
154#135—3582no—
155C2×Q16—3581no—
156C2×SD16—3581no—
157C2×M16—1791no—
158C2×M16—1791no—
159C4⋊Q8—3581no—
160#124—3581no—
161#123—3582no—
162C22⋊Q8—3582no—
163#117C2×D17911yescharacteristic
164#117C2×D17911yescharacteristic
165#104—3582no—
166#103—3582no—
167#99—22no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—3581no—
173#41—3581no—
174#41—3581no—
175#41—3581no—
176C2×Q8—3581no—
177C2×Q8—3581no—
178C2×Q8—3581no—
179C2×Q8—3581no—
180C2×Q8—3581no—
181C2×Q8—1792no—
182C2×Q8—1791no—
183C2×Q8D71611yescharacteristic
184C2×D4—21no—
185C2×D4—3581no—
186C2×D4—1791no—
187C2×D4D71611yescharacteristic
188C22×C4—21no—
189C22×C4—3581no—
190C22×C4—3581no—
191C22×C4C22×D17911yescharacteristic
192Q16—3584no—
193Q16—3582no—
194SD16—3584no—
195SD16—3582no—
196SD16—3582no—
197D8—3582no—
198M16—3581no—
199M16—3581no—
200M16—3581no—
201M16—1792no—
202C2×C8—3581no—
203C2×C8—1792no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—3584no—
207C22⋊C4—21no—
208C22⋊C4—3584no—
209C22⋊C4C22⋊D17912yes—
210C42—21no—
211C42—21no—
212C23—21no—
213C23—3581no—
214C23C2×C22⋊D17911yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—3582no—
218Q8—3581no—
219Q8—3582no—
220Q8—3581no—
221Q8—7161no—
222Q8—7161no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—3581no—
228D4—3581no—
229D4—7161no—
230D4—7161no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—3582no—
237C2×C4—3581no—
238C2×C4—7161no—
239C2×C4—7161no—
240C2×C4—7161no—
241C2×C4—7161no—
242C2×C4D4×D17911yescharacteristic
243C2×C4D4×D17911yescharacteristic
244C2×C4D4×D17911yescharacteristic
245C2×C4D4×D17911yescharacteristic
246C2×C4C2×D71611yescharacteristic
247C2×C4#1566825911yescharacteristic
248C8—3582no—
249C8—3582no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—7162no—
254C22—7161no—
255C22#1915308111yescharacteristic
256C22#1915308711yescharacteristic
257C22#1915310911yescharacteristic
258C4—42no—
259C4—41no—
260C4—81no—
261C4—7161no—
262C4—7162no—
263C4#1915308311yescharacteristic
264C4#1915308311yescharacteristic
265C4#1915308611yescharacteristic
266C4#1915308611yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—14321no—
272C2#2603834811yescharacteristic
subgroupG/H
centerC2#26038348
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#410×C179C2
socleC2×C179#392
prime radical#26039237C2
hypercenter#410D179
hypocenterC179#2009
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. C2×#19153143C2semidirect
3. #26039015C2semidirect
4. #26038457C2semidirect
5. #26038447C2semidirect
6. #26038209C2semidirect
7. #410×C179C2semidirect
136. C179#2009semidirect
143. #410D179semidirect

Maximal subgroups

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

subgroupindex
#260392372
C2×#191531432
#260390152
#260384572
#260384472
#260382092
#410×C1792
#2009179