Finite Group Databasebeta

← #49168707

Subgroups — #49168707

Order 37504.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#34036639C211yesmaximal, characteristic
2C2×C4⋊C4×D293C211yesmaximal, characteristic, complemented
3#34036465C211yesmaximal, characteristic, complemented
4#34036464C211yesmaximal, characteristic, complemented
5#34036461C212yesmaximal, complemented
6#34036445C211yesmaximal, characteristic, complemented
7#34036441C212yesmaximal, complemented
8#34036168C212yesmaximal, complemented
9#34036127C212yesmaximal, complemented
10#34036091C211yesmaximal, characteristic, complemented
11#481×C293C211yesmaximal, characteristic, complemented
12C2×#17989801C2211yescharacteristic
13C22×C4×D293—21no—
14C22×C4×D293C2211yescharacteristic
15C2×#17989785C2212yes—
16#22627606—21no—
17#22627604C2211yescharacteristic
18#22627603C2211yescharacteristic
19#22627595C2211yescharacteristic
20#22627592—21no—
21D4×Dic293—21no—
22#22627584—21no—
23#22627582C2211yescharacteristic
24C2×C4⋊Dic293C2211yescharacteristic
25C2×#17989781—21no—
26C4⋊C4×D293—21no—
27C4⋊C4×D293—21no—
28C4⋊C4×D293C2212yes—
29C4⋊C4×D293C2212yes—
30#22627558C2212yescomplemented
31#22627557C2212yescomplemented
32#22627555C2212yescomplemented
33#22627549C2212yescomplemented
34#22627547C2212yescomplemented
35#22627545C2212yescomplemented
36C22⋊#14556873C2211yescharacteristic
37#22627536C2212yes—
38#121×C293C2211yescharacteristic
39C2×C4⋊C4×C293C2211yescharacteristic
40#106×C293C2212yescomplemented
41#102×C293C2212yescomplemented
42C22⋊C8×C293C2211yescharacteristic
43C23×D293D411yescharacteristic
44#17989801—24no—
45C2×Dic586C2311yescharacteristic
46C2×C22⋊D293—21no—
47C2×C22⋊D293—21no—
48C22×Dic293—21no—
49C22×Dic293—21no—
50C22×Dic293D411yescharacteristic
51C2×C4×D293—24no—
52C2×C4×D293—22no—
53C2×C4×D293—22no—
54C2×C4×D293—44no—
55C2×C4×D293D411yescharacteristic
56C2×C4×D293D411yescharacteristic
57C2×C4×D293C2311yescharacteristic
58C2×C4×D293C2311yescharacteristic
59#17989785—24no—
60C2×#14556873C2312yes—
61C4⋊Dic293—21no—
62C4⋊Dic293—21no—
63C4⋊Dic293C2311yescharacteristic
64C4⋊Dic293C2312yes—
65#17989781—22no—
66#17989781—22no—
67#17989781—21no—
68C22⋊Dic293—21no—
69C22⋊Dic293—21no—
70#17989779—21no—
71#17989779—21no—
72C4×Dic293—21no—
73C2×D4×C293—21no—
74C2×D4×C293C2311yescharacteristic
75C22×C4×C293—21no—
76C22×C4×C293C2311yescharacteristic
77C2×C8×C293C2312yes—
78C4⋊C4×C293—21no—
79C4⋊C4×C293C2311yescharacteristic
80C4⋊C4×C293C2312yes—
81C22⋊C4×C293—21no—
82C22×D293—22no—
83C22×D293—22no—
84C22×D293—22no—
85C22×D293C2×D411yescharacteristic
86C22×D293C2×D411yescharacteristic
87Dic586—22no—
88C22⋊D293—42no—
89C22⋊D293—42no—
90C2×Dic293—22no—
91C2×Dic293—21no—
92C2×Dic293—21no—
93C2×Dic293—22no—
94C2×Dic293—42no—
95C2×Dic293—42no—
96C2×Dic293C2×D411yescharacteristic
97C2×Dic293C2×D411yescharacteristic
98C4×D293—22no—
99C4×D293—22no—
100C4×D293—22no—
101C4×D293—44no—
102C4×D293—44no—
103C4×D293#4114yes—
104#14556873—22no—
105C23×C293—21no—
106C23×C293C2×D411yescharacteristic
107D4×C293—22no—
108D4×C293—42no—
109C2×C4×C293—22no—
110C2×C4×C293—21no—
111C2×C4×C293—22no—
112C2×C4×C293—42no—
113C2×C4×C293C2×D411yescharacteristic
114C2×C4×C293C2411yescharacteristic
115C8×C293—22no—
116C22×C293—21no—
117C22×C293—21no—
118C22×C293—41no—
119C22×C293—41no—
120C22×C293—41no—
121C22×C293#13712yes—
122C22×C293C22×D411yescharacteristic
123C2×D293—28no—
124C2×D293—21no—
125C2×D293—21no—
126C2×D293—21no—
127C2×D293—21no—
128C2×D293#12312yes—
129C2×D293AGL(1,ℤ/8)12yes—
130C4×C293—21no—
131C4×C293—42no—
132C4×C293—41no—
133C4×C293C2×#4112yes—
134Dic293—21no—
135Dic293—41no—
136Dic293—42no—
137Dic293—41no—
138Dic293#12312yes—
139C2×C293—22no—
140C2×C293—81no—
141C2×C293C2×#12311yescharacteristic
142C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C293C2×#13711yescharacteristic
144D293—22no—
145D293—22no—
146C293#309211yescharacteristic, complemented, Sylow
147#3092—2931nomaximal, Sylow
148#525—2931no—
149C22×C4⋊C4—2931no—
150#484—2931no—
151#484—2931no—
152#483—2932no—
153#481—2931no—
154#481D29311yescharacteristic, complemented
155#480—2932no—
156#428—2932no—
157#417—2932no—
158#407—2931no—
159C2×#41—2931no—
160C23×C4—2931no—
161C23×C4—5861no—
162C2×M16—2932no—
163#123—5861no—
164C22⋊Q8—2931no—
165C22⋊Q8—2931no—
166#121—2931no—
167#121C2×D29311yescharacteristic
168#120—5861no—
169C4×D4—5861no—
170C4×D4—5861no—
171#117—2931no—
172C2×C4⋊C4—2931no—
173C2×C4⋊C4—2932no—
174C2×C4⋊C4—2932no—
175C2×C4⋊C4—5861no—
176C2×C4⋊C4—5861no—
177C2×C4⋊C4—5861no—
178C2×C4⋊C4C2×D29311yescharacteristic
179#107—2932no—
180#107—2932no—
181#106—2932no—
182#106C2×D29312yescomplemented
183#103—2932no—
184#103—2932no—
185#102—2932no—
186#102C2×D29312yescomplemented
187C22⋊C8—2931no—
188C22⋊C8—2932no—
189C22⋊C8C2×D29311yescharacteristic
190C24—2931no—
191#41—5864no—
192C2×Q8—2931no—
193C2×D4—21no—
194C2×D4—5861no—
195C2×D4—5861no—
196C2×D4C22×D29311yescharacteristic
197C22×C4—21no—
198C22×C4—2931no—
199C22×C4—2931no—
200C22×C4—2931no—
201C22×C4—2931no—
202C22×C4—2931no—
203C22×C4—5864no—
204C22×C4—5862no—
205C22×C4—5862no—
206C22×C4—5861no—
207C22×C4—5861no—
208C22×C4—11724no—
209C22×C4C22×D29311yescharacteristic
210M16—5864no—
211C2×C8—2932no—
212C2×C8C22×D29312yes—
213C4⋊C4—21no—
214C4⋊C4—2932no—
215C4⋊C4—2931no—
216C4⋊C4—5862no—
217C4⋊C4—5862no—
218C4⋊C4—5861no—
219C4⋊C4—5861no—
220C4⋊C4—5861no—
221C4⋊C4C22×D29312yes—
222C4⋊C4C22×D29311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—5861no—
225C22⋊C4—5861no—
226C22⋊C4—5861no—
227C22⋊C4—5861no—
228C42—5861no—
229C23—21no—
230C23—2931no—
231C23—2931no—
232C23—5862no—
233C23—5862no—
234C23—5862no—
235C23D4×D29311yescharacteristic
236Q8—5862no—
237D4—22no—
238D4—42no—
239D4—11722no—
240D4—11722no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—2931no—
246C2×C4—2934no—
247C2×C4—2931no—
248C2×C4—5862no—
249C2×C4—5862no—
250C2×C4—5862no—
251C2×C4—5862no—
252C2×C4—5861no—
253C2×C4—5861no—
254C2×C4—5862no—
255C2×C4—11724no—
256C2×C4—11724no—
257C2×C4—11722no—
258C2×C4—11722no—
259C2×C4D4×D29311yescharacteristic
260C2×C4C23×D29311yescharacteristic
261C8—22no—
262C8—5862no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—2932no—
269C22—2932no—
270C22—5868no—
271C22—5861no—
272C22—5861no—
273C22—5861no—
274C22—5861no—
275C22#2262765312yes—
276C22C2×D4×D29311yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—2932no—
281C4—5861no—
282C4—11721no—
283C4—11722no—
284C4—11721no—
285C4#41×D29312yes—
286C2—22no—
287C2—81no—
288C2—5862no—
289C2—5862no—
290C2#123×D29311yescharacteristic
291C2AGL(1,ℤ/8)×D29311yescharacteristic
292C2C2×#2262765311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#481×C293C2
socleC22×C293C22×D4
prime radical#34036639C2
hypercenter#481D293
hypocenterC293#3092
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D293C2semidirect
3. #34036465C2semidirect
4. #34036464C2semidirect
5. #34036461C2semidirect
6. #34036445C2semidirect
7. #34036441C2semidirect
8. #34036168C2semidirect
9. #34036127C2semidirect
10. #34036091C2semidirect
11. #481×C293C2semidirect
30. #22627558C22semidirect
31. #22627557C22semidirect
32. #22627555C22semidirect
33. #22627549C22semidirect
34. #22627547C22semidirect
35. #22627545C22semidirect
40. #106×C293C22semidirect
41. #102×C293C22semidirect
146. C293#3092semidirect
154. #481D293semidirect
182. #106C2×D293semidirect
186. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366392
C2×C4⋊C4×D2932
#340364652
#340364642
#340364612
#340364452
#340364412
#340361682
#340361272
#340360912
#481×C2932
#3092293