Finite Group Databasebeta

← #49168698

Subgroups — #49168698

Order 37504.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#34036640C211yesmaximal, characteristic
2#34036470C212yesmaximal, complemented
3#34036464C211yesmaximal, characteristic, complemented
4#34036459C211yesmaximal, characteristic, complemented
5#34036453C212yesmaximal, complemented
6#34036447C211yesmaximal, characteristic, complemented
7#34036442C211yesmaximal, characteristic, complemented
8#34036165C212yesmaximal, complemented
9#34036123C212yesmaximal, complemented
10C2×C22⋊#14556873C211yesmaximal, characteristic, complemented
11C2×#117×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×#14556873C2212yes—
14#22627607—21no—
15#22627603C2211yescharacteristic
16#22627603C2211yescharacteristic
17#22627598C2211yescharacteristic
18#22627598C2211yescharacteristic
19#22627593—21no—
20C4×C22⋊D293—21no—
21C4×C22⋊D293—21no—
22#22627583C2211yescharacteristic
23#22627557C2212yescomplemented
24#22627557C2212yescomplemented
25#22627555C2212yescomplemented
26#22627555C2212yescomplemented
27#22627547C2212yescomplemented
28#22627547C2212yescomplemented
29#22627544C2212yescomplemented
30#22627544C2212yescomplemented
31C22⋊#14556873C2211yescharacteristic
32C22⋊#14556873C2211yescharacteristic
33C22⋊#14556873C2212yes—
34C23×C4×C293C2211yescharacteristic
35#117×C293—21no—
36#117×C293—21no—
37#117×C293C2212yes—
38#117×C293C2212yes—
39C2×C4⋊C4×C293C2211yescharacteristic
40C2×C4⋊C4×C293C2211yescharacteristic
41C2×C22⋊C4×C293—21no—
42C2×C42×C293—21no—
43#17989802—24no—
44C2×Dic586C2311yescharacteristic
45C2×D1172C2311yescharacteristic
46C2×C22⋊D293—21no—
47C2×C22⋊D293—21no—
48C2×C22⋊D293—21no—
49C2×C4×D293—21no—
50C2×#14556873—24no—
51C2×#14556873C2312yes—
52C2×#14556873C2312yes—
53C4⋊Dic293C2311yescharacteristic
54C4⋊Dic293C2311yescharacteristic
55#17989781—21no—
56#17989781—21no—
57C22⋊Dic293—21no—
58C22⋊Dic293—21no—
59C22⋊Dic293—21no—
60#17989779—21no—
61#17989779—21no—
62C4×Dic293—21no—
63C24×C293D411yescharacteristic
64C22×C4×C293—24no—
65C22×C4×C293—21no—
66C22×C4×C293—21no—
67C22×C4×C293D411yescharacteristic
68C22×C4×C293D411yescharacteristic
69C22×C4×C293D411yescharacteristic
70C22×C4×C293C2311yescharacteristic
71C22×C4×C293C2311yescharacteristic
72C22×C4×C293C2311yescharacteristic
73C4⋊C4×C293—21no—
74C4⋊C4×C293—21no—
75C4⋊C4×C293C2312yes—
76C4⋊C4×C293C2312yes—
77C22⋊C4×C293—22no—
78C22⋊C4×C293—22no—
79C42×C293—22no—
80C42×C293—22no—
81C22×D293—21no—
82Dic586—22no—
83D1172—22no—
84C22⋊D293—42no—
85C22⋊D293—42no—
86C22⋊D293—42no—
87C2×Dic293—21no—
88C2×Dic293—21no—
89C2×Dic293—21no—
90C4×D293—42no—
91#14556873—22no—
92#14556873—22no—
93C23×C293—22no—
94C23×C293—22no—
95C23×C293—41no—
96C23×C293C2×D411yescharacteristic
97C23×C293C2×D411yescharacteristic
98C23×C293C2×D411yescharacteristic
99C2×C4×C293—22no—
100C2×C4×C293—22no—
101C2×C4×C293—22no—
102C2×C4×C293—22no—
103C2×C4×C293—22no—
104C2×C4×C293—22no—
105C2×C4×C293—22no—
106C2×C4×C293—42no—
107C2×C4×C293—42no—
108C2×C4×C293C2×D411yescharacteristic
109C2×C4×C293C2×D411yescharacteristic
110C2×C4×C293C2×D411yescharacteristic
111C2×C4×C293#4114yes—
112C2×C4×C293C2411yescharacteristic
113C22×C293—21no—
114C22×C293—21no—
115C22×C293—21no—
116C22×C293—21no—
117C22×C293—21no—
118C22×C293—21no—
119C22×C293—21no—
120C22×C293—44no—
121C22×C293#12312yes—
122C22×C293#13512yes—
123C22×C293C22×D411yescharacteristic
124C2×D293—41no—
125C2×D293—41no—
126C2×D293—41no—
127C4×C293—21no—
128C4×C293—21no—
129C4×C293—42no—
130C4×C293—42no—
131C4×C293#12312yes—
132C4×C293C2×#4112yes—
133Dic293—41no—
134Dic293—41no—
135Dic293—41no—
136C2×C293—22no—
137C2×C293—22no—
138C2×C293—41no—
139C2×C293C2×#12311yescharacteristic
140C2×C293C2×#13511yescharacteristic
141C2×C293#57511yescharacteristic
142D293—81no—
143C293#309111yescharacteristic, complemented, Sylow
144#3091—2931nomaximal, Sylow
145#525—2931no—
146C2×#117D29311yescharacteristic, complemented
147#485—2932no—
148#484—2931no—
149#483—2931no—
150#482—2932no—
151#481—2931no—
152#480—2931no—
153#427—2932no—
154#416—2932no—
155C2×C22⋊C8—2931no—
156C2×#41—2931no—
157C23×C4C2×D29311yescharacteristic
158C22×C8—2932no—
159#123—5861no—
160C22⋊Q8—2931no—
161C22⋊Q8—2931no—
162#121—2931no—
163#121—2931no—
164#120—5861no—
165C4×D4—5861no—
166C4×D4—5861no—
167#117—21no—
168#117—21no—
169#117—2931no—
170#117C2×D29312yes—
171#117C2×D29312yes—
172C2×C4⋊C4C2×D29311yescharacteristic
173C2×C4⋊C4C2×D29311yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—2932no—
177#107—2932no—
178#106—2932no—
179#106—2932no—
180#103—2932no—
181#103—2932no—
182#102—2932no—
183#102—2932no—
184C22⋊C8—2931no—
185C22⋊C8—2931no—
186C22⋊C8—2932no—
187C24C22⋊D29311yescharacteristic
188#41—5864no—
189C2×Q8—2931no—
190C2×D4—2931no—
191C2×D4—5861no—
192C2×D4—5861no—
193C2×D4—5861no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—5861no—
198C22×C4C22⋊D29311yescharacteristic
199C22×C4C22⋊D29311yescharacteristic
200C22×C4C22⋊D29311yescharacteristic
201C22×C4C22×D29311yescharacteristic
202C22×C4C22×D29311yescharacteristic
203C22×C4C22×D29311yescharacteristic
204C2×C8—2932no—
205C2×C8—2932no—
206C2×C8—5864no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—2931no—
210C4⋊C4—2931no—
211C4⋊C4—5861no—
212C4⋊C4—5861no—
213C4⋊C4C22×D29312yes—
214C4⋊C4C22×D29312yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—5861no—
218C22⋊C4—5861no—
219C22⋊C4—5861no—
220C22⋊C4—5861no—
221C22⋊C4—5861no—
222C42—22no—
223C42—22no—
224C42—5861no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5861no—
229C23C2×C22⋊D29311yescharacteristic
230C23C2×C22⋊D29311yescharacteristic
231C23C2×C22⋊D29311yescharacteristic
232Q8—5862no—
233D4—5862no—
234D4—11722no—
235D4—11722no—
236D4—11722no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—5861no—
247C2×C4—5861no—
248C2×C4—5861no—
249C2×C4—11722no—
250C2×C4C2×C22⋊D29311yescharacteristic
251C2×C4C2×C22⋊D29311yescharacteristic
252C2×C4C2×C22⋊D29311yescharacteristic
253C2×C4#1798980214yes—
254C2×C4C23×D29311yescharacteristic
255C8—5862no—
256C8—5862no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—11721no—
266C22—11721no—
267C22—11721no—
268C22#2262760712yes—
269C22#2262764412yes—
270C22C22×C22⋊D29311yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—11721no—
276C4—11721no—
277C4—11721no—
278C4#2262760712yes—
279C4C2×#1798980212yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—23441no—
284C2C2×#2262760711yescharacteristic
285C2C2×#2262764411yescharacteristic
286C2#3403692711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×#117×C293C2
socleC22×C293C22×D4
prime radical#34036640C2
hypercenterC2×#117D293
hypocenterC293#3091
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #34036470C2semidirect
3. #34036464C2semidirect
4. #34036459C2semidirect
5. #34036453C2semidirect
6. #34036447C2semidirect
7. #34036442C2semidirect
8. #34036165C2semidirect
9. #34036123C2semidirect
10. C2×C22⋊#14556873C2semidirect
11. C2×#117×C293C2semidirect
23. #22627557C22semidirect
24. #22627557C22semidirect
25. #22627555C22semidirect
26. #22627555C22semidirect
27. #22627547C22semidirect
28. #22627547C22semidirect
29. #22627544C22semidirect
30. #22627544C22semidirect
143. C293#3091semidirect
146. C2×#117D293semidirect

Maximal subgroups

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

subgroupindex
#340366402
#340364702
#340364642
#340364592
#340364532
#340364472
#340364422
#340361652
#340361232
C2×C22⋊#145568732
C2×#117×C2932
#3091293