Finite Group Databasebeta

← #49168581

Subgroups — #49168581

Order 37504.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#34036682C211yesmaximal, characteristic, complemented
2C2×#22627586C211yesmaximal, characteristic
3#34036432C212yesmaximal, complemented
4#34036428C212yesmaximal, complemented
5#34036418C212yesmaximal, complemented
6#34036408C212yesmaximal, complemented
7#34036169C211yesmaximal, characteristic, complemented
8#34036152C211yesmaximal, characteristic, complemented
9#34036135C211yesmaximal, characteristic
10#34036133C211yesmaximal, characteristic
11#428×C293C211yesmaximal, characteristic, complemented
12C22×D1172C2211yescharacteristic
13C22×C4×D293—21no—
14C2×#17989786C2211yescharacteristic
15#22627623C2211yescharacteristic, complemented
16#22627623C2211yescharacteristic, complemented
17#22627615C2211yescharacteristic, complemented
18#22627615C2211yescharacteristic, complemented
19#22627601—21no—
20#22627601—21no—
21C4×Dic586—21no—
22C4×Dic586—21no—
23#22627586—21no—
24#22627586—21no—
25#22627586C2212yes—
26#22627586C2212yes—
27#22627583C2211yescharacteristic
28C2×#17989779—21no—
29C2×C4×Dic293C2211yescharacteristic
30#22627557C2212yescomplemented
31#22627555C2212yescomplemented
32#22627553C2212yescomplemented
33#22627553C2212yescomplemented
34#22627546C2212yes—
35#22627546C2212yes—
36#22627544C2212yes—
37#22627544C2212yes—
38C2×M16×C293C2211yescharacteristic
39#117×C293C2211yescharacteristic
40C2×C4⋊C4×C293C2211yescharacteristic
41#107×C293C2212yescomplemented
42#106×C293C2212yescomplemented
43C23×D293—21no—
44C2×Dic586—21no—
45C2×Dic586—21no—
46C2×D1172—21no—
47C2×D1172—22no—
48C2×D1172Q814yescomplemented
49C2×D1172C2312yes—
50C22×Dic293D411yescharacteristic
51C22×Dic293D411yescharacteristic
52C2×C4×D293—22no—
53C2×C4×D293—22no—
54C2×C4×D293—44no—
55#17989786—22no—
56C2×#14556873C2312yes—
57C4⋊Dic293—21no—
58C4⋊Dic293—21no—
59C4⋊Dic293C2311yescharacteristic
60C4⋊Dic293C2311yescharacteristic
61#17989781—21no—
62#17989781—21no—
63#17989781—21no—
64#17989781—21no—
65C22⋊Dic293—21no—
66#17989779—22no—
67#17989779—22no—
68C4×Dic293—21no—
69C4×Dic293D411yescharacteristic
70C4×Dic293D411yescharacteristic
71C4×Dic293C2311yescharacteristic
72C4×Dic293C2311yescharacteristic
73C22×C4×C293—21no—
74C22×C4×C293C2311yescharacteristic
75M16×C293—22no—
76C2×C8×C293C2312yes—
77C4⋊C4×C293—21no—
78C4⋊C4×C293C2312yes—
79C4⋊C4×C293C2311yescharacteristic
80C4⋊C4×C293C2311yescharacteristic
81C22⋊C4×C293—21no—
82C42×C293—21no—
83C22×D293—24no—
84C22×D293—22no—
85C22×D293—42no—
86C22×D293—42no—
87Dic586—42no—
88Dic586—42no—
89D1172—24no—
90D1172—22no—
91D1172C2×Q814yes—
92C2×Dic293—22no—
93C2×Dic293—21no—
94C2×Dic293—21no—
95C2×Dic293—41no—
96C2×Dic293C2×D411yescharacteristic
97C2×Dic293C2×D411yescharacteristic
98C2×Dic293C2×D411yescharacteristic
99C2×Dic293C2×D411yescharacteristic
100C4×D293—44no—
101C4×D293—44no—
102#14556873—22no—
103C23×C293C2×D411yescharacteristic
104C2×C4×C293—21no—
105C2×C4×C293—21no—
106C2×C4×C293—22no—
107C2×C4×C293—42no—
108C2×C4×C293C2×D411yescharacteristic
109C2×C4×C293C2×Q812yes—
110C2×C4×C293#4112yes—
111C2×C4×C293C2411yescharacteristic
112C8×C293—22no—
113C22×C293—21no—
114C22×C293—21no—
115C22×C293C22⋊Q812yes—
116C22×C293C22×D411yescharacteristic
117C2×D293—24no—
118C2×D293—48no—
119C2×D293—42no—
120C2×D293—42no—
121C4×C293—42no—
122C4×C293—41no—
123C4×C293—41no—
124C4×C293C22⋊Q812yes—
125C4×C293C22×Q811yescharacteristic
126C4×C293C2×#4111yescharacteristic
127Dic293—21no—
128Dic293—21no—
129Dic293—21no—
130Dic293—41no—
131Dic293—41no—
132Dic293AGL(1,ℤ/8)12yes—
133C2×C293—22no—
134C2×C293C2×C22⋊Q811yescharacteristic
135C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C293#57511yescharacteristic
137D293—44no—
138C293#307711yescharacteristic, complemented, Sylow
139#3077—2931nomaximal, Sylow
140#533—2931no—
141C2×C4×D4—2931no—
142#478—2932no—
143#478—2932no—
144#476—2932no—
145#474—2932no—
146#428—2931no—
147#428D29311yescharacteristic, complemented
148#423—2931no—
149#418—2931no—
150#418—2931no—
151C22×D4—2931no—
152C23×C4—5861no—
153C2×M16—2931no—
154C2×M16C2×D29311yescharacteristic
155C4⋊Q8—2931no—
156C4⋊Q8—2931no—
157#125—2931no—
158#125—2931no—
159C22⋊Q8—5861no—
160C22⋊Q8—5861no—
161C4×Q8—5861no—
162C4×Q8—5861no—
163C4×D4—2932no—
164C4×D4—2932no—
165C4×D4—5861no—
166C4×D4—5861no—
167#117—2931no—
168#117C2×D29311yescharacteristic
169C2×C4⋊C4C2×D29311yescharacteristic
170C2×C22⋊C4—5861no—
171C2×C42—2931no—
172#107—2932no—
173#107C2×D29312yescomplemented
174#106—2932no—
175#106C2×D29312yescomplemented
176C4⋊C8—2932no—
177C4⋊C8—2932no—
178#102—2932no—
179#102—2932no—
180#102—2932no—
181#102—2932no—
182C24—5861no—
183C2×Q8—5861no—
184C2×Q8—5861no—
185C2×D4—2934no—
186C2×D4—2932no—
187C2×D4—5861no—
188C2×D4—5862no—
189C22×C4—21no—
190C22×C4—2931no—
191C22×C4—2931no—
192C22×C4—5862no—
193C22×C4—5862no—
194C22×C4—11724no—
195C22×C4C22×D29311yescharacteristic
196M16—22no—
197M16—5862no—
198C2×C8—2932no—
199C2×C8C22×D29312yes—
200C4⋊C4—21no—
201C4⋊C4—2931no—
202C4⋊C4—2931no—
203C4⋊C4—5861no—
204C4⋊C4—5861no—
205C4⋊C4—5861no—
206C4⋊C4—5861no—
207C4⋊C4—5861no—
208C4⋊C4—5861no—
209C4⋊C4C22×D29311yescharacteristic
210C4⋊C4C22×D29311yescharacteristic
211C4⋊C4C22×D29312yes—
212C22⋊C4—21no—
213C22⋊C4—5862no—
214C22⋊C4—5862no—
215C22⋊C4—5861no—
216C42—21no—
217C42—2931no—
218C42—2931no—
219C42—2931no—
220C42—2931no—
221C42—5861no—
222C23—5864no—
223C23—5862no—
224C23—11722no—
225C23—11722no—
226C23D4×D29311yescharacteristic
227Q8—11722no—
228Q8—11722no—
229D4—2934no—
230D4—5864no—
231D4—5862no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—2931no—
237C2×C4—2931no—
238C2×C4—2931no—
239C2×C4—2931no—
240C2×C4—5862no—
241C2×C4—5861no—
242C2×C4—5861no—
243C2×C4—11724no—
244C2×C4—11724no—
245C2×C4—11721no—
246C2×C4D4×D29311yescharacteristic
247C2×C4Q8×D29312yes—
248C2×C4#1798980212yes—
249C2×C4C23×D29311yescharacteristic
250C8—22no—
251C8—5862no—
252C22—21no—
253C22—21no—
254C22—5864no—
255C22—11728no—
256C22—11722no—
257C22—11722no—
258C22#2262760212yes—
259C22C2×D4×D29311yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—2932no—
264C4—5861no—
265C4—5861no—
266C4—5861no—
267C4—11721no—
268C4—11721no—
269C4#2262760212yes—
270C4C2×Q8×D29311yescharacteristic
271C4C2×#1798980211yescharacteristic
272C2—22no—
273C2—11724no—
274C2C2×#2262760211yescharacteristic
275C2AGL(1,ℤ/8)×D29311yescharacteristic
276C2#3403692811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#428×C293C2
socleC22×C293C22×D4
prime radicalC22×D1172C22
hypercenter#428D293
hypocenterC293#3077
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #34036682C2semidirect
3. #34036432C2semidirect
4. #34036428C2semidirect
5. #34036418C2semidirect
6. #34036408C2semidirect
7. #34036169C2semidirect
8. #34036152C2semidirect
11. #428×C293C2semidirect
15. #22627623C22semidirect
16. #22627623C22semidirect
17. #22627615C22semidirect
18. #22627615C22semidirect
30. #22627557C22semidirect
31. #22627555C22semidirect
32. #22627553C22semidirect
33. #22627553C22semidirect
41. #107×C293C22semidirect
42. #106×C293C22semidirect
48. C2×D1172Q8semidirect
138. C293#3077semidirect
147. #428D293semidirect
173. #107C2×D293semidirect
175. #106C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366822
C2×#226275862
#340364322
#340364282
#340364182
#340364082
#340361692
#340361522
#340361352
#340361332
#428×C2932
#3077293