Finite Group Databasebeta

← #37510058

Subgroups — #37510058

Order 22912.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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
2#26039013C211yesmaximal, characteristic, complemented
3#26038455C211yesmaximal, characteristic, complemented
4#26038448C211yesmaximal, characteristic, complemented
5#26038413C211yesmaximal, characteristic, complemented
6#26038211C211yesmaximal, characteristic, complemented
7C2×#137×C179C211yesmaximal, characteristic, complemented
8C2×#15668259C2211yescharacteristic
9#19153144—21no—
10#19153143—21no—
11#19153139—21no—
12#19153138—21no—
13#19153133—22no—
14C2×#15668243C2211yescharacteristic
15#19153111—21no—
16#19153102—21no—
17#19153096—22no—
18#19153089—22no—
19#19153072C2211yescharacteristic
20#19153072C2211yescharacteristic
21#19153039—22no—
22#19153037—22no—
23#19153028—22no—
24C2×#41×C179C2211yescharacteristic
25C22×Q8×C179C2211yescharacteristic
26#137×C179—22no—
27#137×C179—22no—
28C2×Q16×C179—21no—
29C2×SD16×C179—21no—
30C2×M16×C179C2211yescharacteristic
31#15668259—21no—
32#15668259—21no—
33#15668259—21no—
34#15668259—21no—
35C2×Dic358D411yescharacteristic
36C2×D716D411yescharacteristic
37C2×C22⋊D179—21no—
38C2×C4×D179—21no—
39#15668249—22no—
40#15668247—22no—
41#15668246—22no—
42#15668244—22no—
43#15668243—21no—
44#15668243—21no—
45C2×C179⋊C8D412yes—
46C4⋊Dic179—21no—
47C4⋊Dic179—21no—
48Dic179⋊C4—24no—
49C22⋊Dic179—21no—
50C22⋊Dic179D412yes—
51(C2×D179)⋊C4—24no—
52C4×Dic179—21no—
53C4×Dic179—21no—
54#41×C179—21no—
55#41×C179—21no—
56#41×C179—21no—
57#41×C179—21no—
58C2×Q8×C179—21no—
59C2×Q8×C179—21no—
60C2×Q8×C179—21no—
61C2×Q8×C179—21no—
62C2×Q8×C179—21no—
63C2×Q8×C179D411yescharacteristic
64C2×Q8×C179D412yes—
65C2×D4×C179—21no—
66C2×D4×C179D411yescharacteristic
67C22×C4×C179—21no—
68C22×C4×C179—21no—
69C22×C4×C179C2311yescharacteristic
70Q16×C179—24no—
71SD16×C179—24no—
72M16×C179—21no—
73M16×C179D412yes—
74C2×C8×C179—21no—
75C22×D179—21no—
76Dic358—21no—
77Dic358—21no—
78D716—21no—
79D716—21no—
80C22⋊D179—41no—
81C22⋊D179—41no—
82C2×Dic179—22no—
83C2×Dic179—21no—
84C2×Dic179—41no—
85C4×D179—41no—
86C4×D179—41no—
87C179⋊C8—22no—
88C23×C179—21no—
89C23×C179C2×D411yescharacteristic
90Q8×C179—22no—
91Q8×C179—21no—
92Q8×C179—21no—
93Q8×C179—22no—
94Q8×C179—41no—
95Q8×C179—41no—
96D4×C179—21no—
97D4×C179—21no—
98D4×C179—41no—
99D4×C179—41no—
100C2×C4×C179—22no—
101C2×C4×C179—21no—
102C2×C4×C179—41no—
103C2×C4×C179—41no—
104C2×C4×C179—41no—
105C2×C4×C179—41no—
106C2×C4×C179C2×D411yescharacteristic
107C2×C4×C179C2×D411yescharacteristic
108C2×C4×C179C2×D411yescharacteristic
109C2×C4×C179C2×D411yescharacteristic
110C2×C4×C179C2×D411yescharacteristic
111C2×C4×C179#4111yescharacteristic
112C8×C179—22no—
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—41no—
122C4×C179—42no—
123C4×C179#12011yescharacteristic
124C4×C179#12011yescharacteristic
125C4×C179#12111yescharacteristic
126C4×C179#12111yescharacteristic
127Dic179—42no—
128Dic179—41no—
129Dic179—81no—
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×#137D17911yescharacteristic, complemented
140#536—1791no—
141#421—1791no—
142#419—1791no—
143#410—1791no—
144#344—1791no—
145C2×#41—1791no—
146C2×#41C2×D17911yescharacteristic
147C22×Q8C2×D17911yescharacteristic
148#137—22no—
149#137—22no—
150#137—3581no—
151#137—3581no—
152AGL(1,ℤ/8)—3581no—
153AGL(1,ℤ/8)—3581no—
154#135—3582no—
155C2×Q16—21no—
156C2×SD16—21no—
157C2×M16—1791no—
158C2×M16C2×D17911yescharacteristic
159C4⋊Q8—3581no—
160#124—3581no—
161#123—3582no—
162C22⋊Q8—3582no—
163#117—1791no—
164#117—1791no—
165#104—3582no—
166#103—3582no—
167#99—3582no—
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—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—1791no—
182C2×Q8C22⋊D17912yes—
183C2×Q8C22⋊D17911yescharacteristic
184C2×D4—21no—
185C2×D4—3581no—
186C2×D4—1791no—
187C2×D4C22⋊D17911yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—3581no—
191C22×C4C22×D17911yescharacteristic
192Q16—24no—
193Q16—3582no—
194SD16—24no—
195SD16—3582no—
196SD16—3582no—
197D8—3582no—
198M16—21no—
199M16—3581no—
200M16—3581no—
201M16C22⋊D17912yes—
202C2×C8—21no—
203C2×C8—1792no—
204C4⋊C4—3584no—
205C4⋊C4—3581no—
206C4⋊C4—3581no—
207C22⋊C4—3584no—
208C22⋊C4—3581no—
209C22⋊C4—1792no—
210C42—3581no—
211C42—3581no—
212C23—21no—
213C23—3581no—
214C23D4×D17911yescharacteristic
215Q8—21no—
216Q8—22no—
217Q8—21no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—3581no—
222Q8—3581no—
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—41no—
237C2×C4—3582no—
238C2×C4—3581no—
239C2×C4—7161no—
240C2×C4—7161no—
241C2×C4—7161no—
242C2×C4D4×D17911yescharacteristic
243C2×C4D4×D17911yescharacteristic
244C2×C4C2×C22⋊D17911yescharacteristic
245C2×C4C2×C22⋊D17911yescharacteristic
246C2×C4C2×C22⋊D17911yescharacteristic
247C2×C4#1566825811yescharacteristic
248C8—22no—
249C8—3582no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—7162no—
254C22—7161no—
255C22#1915308111yescharacteristic
256C22#1915308511yescharacteristic
257C22#1915310911yescharacteristic
258C4—41no—
259C4—42no—
260C4—14321no—
261C4—7162no—
262C4—7161no—
263C4#1915308111yescharacteristic
264C4#1915308211yescharacteristic
265C4#1915308411yescharacteristic
266C4#1915308511yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—14321no—
272C2#2603834711yescharacteristic
subgroupG/H
centerC2#26038347
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
FittingC2×#137×C179C2
socleC2×C179#392
prime radical#26039237C2
hypercenterC2×#137D179
hypocenterC179#2009
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #26039013C2semidirect
3. #26038455C2semidirect
4. #26038448C2semidirect
5. #26038413C2semidirect
6. #26038211C2semidirect
7. C2×#137×C179C2semidirect
136. C179#2009semidirect
139. C2×#137D179semidirect

Maximal subgroups

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

subgroupindex
#260392372
#260390132
#260384552
#260384482
#260384132
#260382112
C2×#137×C1792
#2009179