Finite Group Databasebeta

← #16829071

Subgroups — #16829071

Order 3712.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11745274C211yesmaximal, characteristic, complemented
2C22×C8×D29C211yesmaximal, characteristic, complemented
3#14271673C212yesmaximal, complemented
4#14271405C211yesmaximal, characteristic, complemented
5#14271404C211yesmaximal, characteristic, complemented
6#14271396C211yesmaximal, characteristic, complemented
7#14271391C212yesmaximal, complemented
8#14271390C212yesmaximal, complemented
9#14271200C211yesmaximal, characteristic, complemented
10#14271156C212yesmaximal, complemented
11C29×#466C211yesmaximal, characteristic, complemented
12C2×#90807C2212yes—
13C22×C4×D29C2211yescharacteristic
14C2×D8×C29C2211yescharacteristic, complemented
15C22×C8×C29C2211yescharacteristic
16C29×#121C2212yes—
17#11745300C2212yes—
18#11745299—22no—
19D4×Dic29—22no—
20C2×#90783C2212yescomplemented
21#11745286C2212yes—
22C22×#6597C2211yescharacteristic
23#11745274—24no—
24C2×Dic116C2211yescharacteristic, complemented
25C2×C8×D29—24no—
26C2×C8×D29C2211yescharacteristic
27C2×C8×D29C2211yescharacteristic
28C2×C8×D29C2211yescharacteristic
29C2×C8×D29C2211yescharacteristic
30#11745259C2212yes—
31#11745254C2212yes—
32#11745245—22no—
33#11745243—22no—
34C29×#107C2211yescharacteristic, complemented
35C29×#102C2212yescomplemented
36#11745180C2212yescomplemented
37#11745166C2211yescharacteristic, complemented
38#11745164C2212yescomplemented
39#11745158C2212yescomplemented
40#11745156C2212yescomplemented
41C23×D29D411yescharacteristic
42C2×D4×C29—22no—
43C2×D4×C29C2312yes—
44C22×C4×C29C2311yescharacteristic
45C2×C22⋊D29—22no—
46C2×C22⋊D29—22no—
47C22×Dic29—22no—
48C22×Dic29D411yescharacteristic
49#90807—24no—
50#90807—24no—
51C2×C4×D29—22no—
52C2×C4×D29—22no—
53C2×C4×D29D411yescharacteristic
54C2×C4×D29D411yescharacteristic
55C2×C4×D29C2311yescharacteristic
56C2×C4×D29C2311yescharacteristic
57C2×Dic58C2312yes—
58D8×C29—22no—
59C2×C8×C29—22no—
60C2×C8×C29C2311yescharacteristic
61C2×C8×C29C2311yescharacteristic
62C4⋊C4×C29C2312yes—
63C22⋊C4×C29—22no—
64C22⋊Dic29—22no—
65C22⋊Dic29—22no—
66#90783—24no—
67#90781—22no—
68#90781—22no—
69C4⋊Dic29—22no—
70C4⋊Dic29C2312yes—
71#90779—22no—
72C4×Dic29—22no—
73C2×#6597—22no—
74C2×#6597C2311yescharacteristic
75C2×#6597C2311yescharacteristic
76Dic116—22no—
77C8×D29—22no—
78C8×D29—22no—
79C8×D29—22no—
80C8×D29D414yescomplemented
81C23×C29—22no—
82C23×C29C2×D411yescharacteristic
83C22×D29—22no—
84C22×D29—22no—
85C22×D29—22no—
86C22×D29C2×D411yescharacteristic
87C22×D29C2×D411yescharacteristic
88D4×C29—22no—
89D4×C29—22no—
90D4×C29—44no—
91C2×C4×C29—21no—
92C2×C4×C29—21no—
93C2×C4×C29—22no—
94C2×C4×C29C2×D411yescharacteristic
95C2×C4×C29C2411yescharacteristic
96C22⋊D29—44no—
97C22⋊D29—44no—
98C2×Dic29—22no—
99C2×Dic29—22no—
100C2×Dic29—22no—
101C2×Dic29—44no—
102C2×Dic29C2×D411yescharacteristic
103C2×Dic29C2×D411yescharacteristic
104C4×D29—22no—
105C4×D29—21no—
106C4×D29—21no—
107C4×D29—22no—
108C4×D29C2×D412yes—
109C4×D29#4112yes—
110Dic58—22no—
111Dic58—22no—
112C8×C29—21no—
113C8×C29C2×D412yes—
114#6597—21no—
115#6597C2×D412yes—
116#3050—291nomaximal, Sylow
117C22×C29—21no—
118C22×C29—21no—
119C22×C29—42no—
120C22×C29—42no—
121C22×C29—42no—
122C22×C29#13512yescomplemented
123C22×C29C22×D411yescharacteristic
124C2×D29—28no—
125C2×D29—21no—
126C2×D29—21no—
127C2×D29—21no—
128C2×D29—21no—
129C2×D29#12112yes—
130C2×D29#13512yescomplemented
131C4×C29—21no—
132C4×C29—42no—
133C4×C29C22×D411yescharacteristic
134C4×C29C2×#4111yescharacteristic
135Dic29—21no—
136Dic29—42no—
137Dic29—42no—
138Dic29#12112yes—
139C2×#135—291no—
140C23×C8—291no—
141#525—292no—
142#467—291no—
143#467—291no—
144#466—291no—
145#466D2911yescharacteristic, complemented
146#465—292no—
147#465—292no—
148#427—291no—
149#416—292no—
150C2×C29—22no—
151C2×C29—82no—
152C2×C29C2×#12111yescharacteristic
153C2×C29C2×#13511yescharacteristic
154C2×C29C2×#13511yescharacteristic
155D29—22no—
156D29—22no—
157C2×#41—292no—
158C23×C4—291no—
159#135—584no—
160C2×Q16—291no—
161C2×SD16—292no—
162C2×D8C2×D2911yescharacteristic, complemented
163C22×C8—291no—
164C22×C8—291no—
165C22×C8—291no—
166C22×C8—291no—
167C22×C8—291no—
168C22×C8—584no—
169C22×C8C2×D2911yescharacteristic
170#123—582no—
171C22⋊Q8—292no—
172C22⋊Q8—292no—
173#121—292no—
174#121C2×D2912yes—
175#120—582no—
176C4×D4—582no—
177C4×D4—582no—
178#117—292no—
179#107—291no—
180#107C2×D2911yescharacteristic, complemented
181#106—292no—
182#103—292no—
183#103—292no—
184#102—292no—
185#102C2×D2912yescomplemented
186C29#305011yescharacteristic, complemented, Sylow
187C24—291no—
188#41—584no—
189#41—584no—
190C2×Q8—292no—
191C2×D4—22no—
192C2×D4—582no—
193C2×D4—582no—
194C2×D4C22×D2912yes—
195C22×C4—291no—
196C22×C4—291no—
197C22×C4—291no—
198C22×C4—291no—
199C22×C4—291no—
200C22×C4—582no—
201C22×C4—582no—
202C22×C4—582no—
203C22×C4C22×D2911yescharacteristic
204Q16—582no—
205SD16—584no—
206D8—22no—
207C2×C8—22no—
208C2×C8—291no—
209C2×C8—294no—
210C2×C8—291no—
211C2×C8—582no—
212C2×C8—582no—
213C2×C8—582no—
214C2×C8—582no—
215C2×C8C22×D2911yescharacteristic
216C2×C8C22×D2911yescharacteristic
217C4⋊C4—292no—
218C4⋊C4—582no—
219C4⋊C4—582no—
220C4⋊C4C22×D2912yes—
221C22⋊C4—22no—
222C22⋊C4—582no—
223C22⋊C4—582no—
224C22⋊C4—582no—
225C22⋊C4—582no—
226C42—582no—
227C23—22no—
228C23—291no—
229C23—291no—
230C23—582no—
231C23—582no—
232C23—582no—
233C23D4×D2911yescharacteristic
234Q8—582no—
235Q8—582no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—1164no—
240D4—1164no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—291no—
245C2×C4—292no—
246C2×C4—292no—
247C2×C4—291no—
248C2×C4—582no—
249C2×C4—582no—
250C2×C4—582no—
251C2×C4—581no—
252C2×C4—581no—
253C2×C4—582no—
254C2×C4—582no—
255C2×C4—1164no—
256C2×C4D4×D2911yescharacteristic
257C2×C4C23×D2911yescharacteristic
258C8—21no—
259C8—292no—
260C8—581no—
261C8D4×D2912yes—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—292no—
268C22—292no—
269C22—588no—
270C22—581no—
271C22—581no—
272C22—581no—
273C22—581no—
274C22#1174527412yescomplemented
275C22C2×D4×D2911yescharacteristic
276C4—21no—
277C4—42no—
278C4—292no—
279C4—581no—
280C4—1162no—
281C4—1162no—
282C4C2×D4×D2911yescharacteristic
283C4#41×D2911yescharacteristic
284C2—22no—
285C2—82no—
286C2—582no—
287C2—582no—
288C2#121×D2911yescharacteristic
289C2#135×D2911yescharacteristic
290C2C2×#1174527411yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#466C2
socleC22×C29C22×D4
hypercenter#466D29
hypocenterC29#3050
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#11745274C2semidirect
2. C22×C8×D29C2semidirect
3. #14271673C2semidirect
4. #14271405C2semidirect
5. #14271404C2semidirect
6. #14271396C2semidirect
7. #14271391C2semidirect
8. #14271390C2semidirect
9. #14271200C2semidirect
10. #14271156C2semidirect
11. C29×#466C2semidirect
14. C2×D8×C29C22semidirect
20. C2×#90783C22semidirect
24. C2×Dic116C22semidirect
34. C29×#107C22semidirect
35. C29×#102C22semidirect
36. #11745180C22semidirect
37. #11745166C22semidirect
38. #11745164C22semidirect
39. #11745158C22semidirect
40. #11745156C22semidirect
80. C8×D29D4semidirect
122. C22×C29#135semidirect
130. C2×D29#135semidirect
145. #466D29semidirect
162. C2×D8C2×D29semidirect
180. #107C2×D29semidirect
185. #102C2×D29semidirect
186. C29#3050semidirect
274. C22#11745274semidirect

Maximal subgroups

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

subgroupindex
C2×#117452742
C22×C8×D292
#142716732
#142714052
#142714042
#142713962
#142713912
#142713902
#142712002
#142711562
C29×#4662
#305029