Finite Group Databasebeta

← #49169168

Subgroups — #49169168

Order 37504.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#34036685C211yesmaximal, characteristic, complemented
2#124×D293C211yesmaximal, characteristic
3#34036490C212yesmaximal, complemented
4#34036487C212yesmaximal, complemented
5#34036486C211yesmaximal, characteristic, complemented
6#34036484C211yesmaximal, characteristic, complemented
7#34036483C211yesmaximal, characteristic, complemented
8#34036136C212yesmaximal, complemented
9#34036132C212yesmaximal
10C4×#17989785C211yesmaximal, characteristic, complemented
11#488×C293C211yesmaximal, characteristic, complemented
12C2×Q8×D293C2211yescharacteristic
13C2×D4×D293C2211yescharacteristic
14C2×#17989785C2212yes—
15#22627623C2212yescomplemented
16#22627616C2211yescharacteristic, complemented
17#22627613C2211yescharacteristic
18#22627612C2211yescharacteristic
19#22627611C2211yescharacteristic
20#22627610—22no—
21#22627602—22no—
22#22627589—22no—
23#22627582C2212yes—
24C22⋊C4×D293—22no—
25C42×D293C2211yescharacteristic
26#22627549C2212yescomplemented
27#22627548C2212yescomplemented
28#22627547C2212yescomplemented
29#22627546C2212yes—
30#22627545C2212yes—
31#22627544C2212yes—
32#22627534C2212yescomplemented
33C4×#14556873C2211yescharacteristic, complemented
34#125×C293C2211yescharacteristic, complemented
35#124×C293C2211yescharacteristic
36#103×C293C2212yescomplemented
37#102×C293C2212yes—
38C4×C8×C293C2211yescharacteristic, complemented
39C23×D293—21no—
40C2×Dic586—22no—
41C2×Dic586—21no—
42C2×Dic586C2311yescharacteristic
43Q8×D293—24no—
44C2×D1172C2311yescharacteristic
45C2×C22⋊D293—21no—
46D4×D293—24no—
47C2×C4×D293—21no—
48C2×C4×D293D412yes—
49C2×C4×D293C2311yescharacteristic
50#17989785—24no—
51C2×#14556873C2312yes—
52C4⋊Dic293C2312yes—
53#17989781—22no—
54#17989781—22no—
55C22⋊Dic293—22no—
56#17989779—22no—
57#17989779—22no—
58#17989779—22no—
59C4×Dic293—22no—
60C4×Dic293D412yes—
61C4×Dic293C2311yescharacteristic
62C2×Q8×C293C2311yescharacteristic
63C2×D4×C293C2311yescharacteristic
64C2×C8×C293C2312yes—
65C4⋊C4×C293—22no—
66C4⋊C4×C293C2312yes—
67C22⋊C4×C293—22no—
68C42×C293C2311yescharacteristic
69C22×D293—21no—
70C22×D293—44no—
71C22×D293—42no—
72C22×D293C2×D411yescharacteristic
73Dic586—22no—
74Dic586—44no—
75Dic586—42no—
76D1172—22no—
77C22⋊D293—42no—
78C2×Dic293—22no—
79C2×Dic293—21no—
80C2×Dic293C2×D412yes—
81C2×Dic293C2×D411yescharacteristic
82C4×D293—42no—
83C4×D293—42no—
84C4×D293#4114yes—
85#14556873—22no—
86C23×C293—21no—
87Q8×C293—22no—
88D4×C293—22no—
89C2×C4×C293—21no—
90C2×C4×C293—22no—
91C2×C4×C293C2×D412yes—
92C2×C4×C293C2411yescharacteristic
93C8×C293—22no—
94C22×C293—41no—
95C22×C293—42no—
96C22×C293C22×D411yescharacteristic
97C2×D293—22no—
98C2×D293—41no—
99C2×D293—42no—
100C2×D293—84no—
101C2×D293#12412yes—
102C4×C293—22no—
103C4×C293—41no—
104C4×C293—42no—
105C4×C293C2×#4112yes—
106Dic293—22no—
107Dic293—41no—
108Dic293—42no—
109Dic293#12412yes—
110C2×C293—81no—
111C2×C293C2×#12411yescharacteristic
112C2×C293#57512yes—
113D293—22no—
114D293—81no—
115C293#314011yescharacteristic, complemented, Sylow
116#3140—2931nomaximal, Sylow
117#533—2931no—
118C2×#124—2931no—
119#489—2932no—
120#489—2932no—
121#488—2931no—
122#488—2931no—
123#488—2931no—
124#488D29311yescharacteristic, complemented
125#419—2932no—
126#418—2932no—
127C4×M16—2931no—
128C22×Q8—2931no—
129C22×D4—2931no—
130C2×M16—2932no—
131C4⋊Q8—2932no—
132#125—2931no—
133#125C2×D29311yescharacteristic, complemented
134#124—2931no—
135#124—2931no—
136#124—2931no—
137#124—5862no—
138#124C2×D29311yescharacteristic
139C22⋊Q8—5862no—
140C4×Q8—5862no—
141#117—2932no—
142C2×C22⋊C4—5862no—
143C2×C42—2931no—
144#103—2932no—
145#103—2932no—
146#103—2932no—
147#103C2×D29312yescomplemented
148#102—2932no—
149#102—2932no—
150#102—2932no—
151#102C2×D29312yes—
152#97—2932no—
153C4×C8—2931no—
154C4×C8C2×D29311yescharacteristic, complemented
155C24—5861no—
156C2×Q8—2931no—
157C2×Q8—5862no—
158C2×Q8—5861no—
159C2×Q8—5864no—
160C2×Q8C22×D29311yescharacteristic
161C2×D4—2931no—
162C2×D4—5864no—
163C2×D4—5861no—
164C2×D4C22×D29311yescharacteristic
165C22×C4—2931no—
166C22×C4—2932no—
167C22×C4—5861no—
168M16—5864no—
169C2×C8—2932no—
170C2×C8C22×D29312yes—
171C4⋊C4—22no—
172C4⋊C4—2932no—
173C4⋊C4—5862no—
174C4⋊C4—5862no—
175C4⋊C4C22×D29312yes—
176C22⋊C4—22no—
177C22⋊C4—5862no—
178C22⋊C4—5862no—
179C22⋊C4—5862no—
180C22⋊C4—5862no—
181C42—2932no—
182C42—2931no—
183C42—5862no—
184C42C22×D29311yescharacteristic
185C23—21no—
186C23—2931no—
187C23—5861no—
188C23—11722no—
189C23—11724no—
190Q8—22no—
191Q8—5862no—
192Q8—11722no—
193Q8—11724no—
194D4—22no—
195D4—5862no—
196D4—11722no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2931no—
200C2×C4—2934no—
201C2×C4—2932no—
202C2×C4—5862no—
203C2×C4—5861no—
204C2×C4—11722no—
205C2×C4—11722no—
206C2×C4D4×D29312yes—
207C2×C4C23×D29311yescharacteristic
208C8—22no—
209C8—5862no—
210C22—41no—
211C22—42no—
212C22—2932no—
213C22—23444no—
214C22—5862no—
215C22—11721no—
216C22—11722no—
217C22C2×D4×D29311yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—2932no—
222C4—5862no—
223C4—11721no—
224C4—11722no—
225C4#41×D29312yes—
226C2—81no—
227C2—23441no—
228C2—5862no—
229C2#124×D29311yescharacteristic
230C2#3403692612yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#488×C293C2
socleC22×C293C22×D4
prime radicalC2×D4×D293C22
hypercenter#488D293
hypocenterC293#3140
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #34036685C2semidirect
3. #34036490C2semidirect
4. #34036487C2semidirect
5. #34036486C2semidirect
6. #34036484C2semidirect
7. #34036483C2semidirect
8. #34036136C2semidirect
10. C4×#17989785C2semidirect
11. #488×C293C2semidirect
15. #22627623C22semidirect
16. #22627616C22semidirect
26. #22627549C22semidirect
27. #22627548C22semidirect
28. #22627547C22semidirect
32. #22627534C22semidirect
33. C4×#14556873C22semidirect
34. #125×C293C22semidirect
36. #103×C293C22semidirect
38. C4×C8×C293C22semidirect
115. C293#3140semidirect
124. #488D293semidirect
133. #125C2×D293semidirect
147. #103C2×D293semidirect
154. C4×C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366852
#124×D2932
#340364902
#340364872
#340364862
#340364842
#340364832
#340361362
#340361322
C4×#179897852
#488×C2932
#3140293