Finite Group Databasebeta

← #49168161

Subgroups — #49168161

Order 37504.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#34036965C211yesmaximal, characteristic
2C2×#22627655C211yesmaximal, characteristic, complemented
3#34036703C211yesmaximal, characteristic, complemented
4#34036310C214yesmaximal, complemented
5#34036305C214yesmaximal, complemented
6#34036146C211yesmaximal, characteristic, complemented
7#34036145C211yesmaximal, characteristic, complemented
8#34036111C211yesmaximal, characteristic, complemented
9C2×#137×C293C211yesmaximal, characteristic, complemented
10#22627672—21no—
11#22627671—21no—
12#22627671C2212yes—
13#22627670—21no—
14#22627670C2212yes—
15#22627669—22no—
16#22627669—21no—
17C2×#17989802—22no—
18C2×#17989802C2211yescharacteristic
19C2×#17989802C2211yescharacteristic
20#41×D293—22no—
21#41×D293—22no—
22#22627655—22no—
23#22627655C2214yescomplemented
24C2×#17989792—21no—
25C2×#17989789—21no—
26C2×#17989786C2211yescharacteristic, complemented
27#22627622C2212yescomplemented
28#22627613C2212yescomplemented
29#22627607—22no—
30#22627600—22no—
31#22627583C2211yescharacteristic, complemented
32#22627551C2214yescomplemented
33#22627550C2214yescomplemented
34#22627542C2212yescomplemented
35#22627540C2212yescomplemented
36C2×#41×C293C2211yescharacteristic
37C22×Q8×C293C2211yescharacteristic, complemented
38#137×C293—22no—
39#137×C293C2214yescomplemented
40C2×Q16×C293—21no—
41C2×SD16×C293—21no—
42C2×M16×C293C2211yescharacteristic, complemented
43#17989803—22no—
44#17989803—22no—
45#17989802—24no—
46#17989802—24no—
47#17989802—22no—
48#17989802—22no—
49#17989802—21no—
50#17989802—21no—
51#17989802—21no—
52#17989802—21no—
53#17989802—21no—
54#17989802—21no—
55#17989802D412yes—
56#17989802C2312yes—
57#17989801—24no—
58#17989801—22no—
59#17989801—24no—
60#17989801—22no—
61C2×Dic586—22no—
62C2×Dic586D411yescharacteristic
63C2×Dic586C2311yescharacteristic
64Q8×D293—22no—
65Q8×D293—22no—
66C2×D1172—22no—
67C2×D1172D411yescharacteristic
68C2×D1172C2311yescharacteristic
69C2×C22⋊D293—24no—
70C2×C22⋊D293—21no—
71C2×C22⋊D293D412yes—
72D4×D293—24no—
73D4×D293—22no—
74D4×D293—24no—
75D4×D293—22no—
76C2×C4×D293—24no—
77C2×C4×D293—21no—
78C2×C4×D293D412yes—
79#17989792—24no—
80#17989789—24no—
81#17989786—21no—
82#17989786C2312yes—
83C2×#14556873—21no—
84C4⋊Dic293—21no—
85#17989781—24no—
86C22⋊Dic293—21no—
87#17989779—24no—
88C4×Dic293C2312yes—
89#41×C293—21no—
90#41×C293—21no—
91#41×C293D412yes—
92#41×C293C2312yes—
93C2×Q8×C293—21no—
94C2×Q8×C293—21no—
95C2×Q8×C293—21no—
96C2×Q8×C293—21no—
97C2×Q8×C293—21no—
98C2×Q8×C293D411yescharacteristic
99C2×Q8×C293C2312yes—
100C2×D4×C293—21no—
101C2×D4×C293D411yescharacteristic
102C22×C4×C293—21no—
103C22×C4×C293—21no—
104C22×C4×C293C2311yescharacteristic
105Q16×C293—24no—
106SD16×C293—24no—
107M16×C293—21no—
108M16×C293C2312yes—
109C2×C8×C293—21no—
110C22×D293—24no—
111C22×D293—21no—
112C22×D293C2×D412yes—
113Dic586—24no—
114Dic586—21no—
115Dic586—21no—
116Dic586—21no—
117Dic586C2×D412yes—
118D1172—24no—
119D1172—21no—
120D1172—21no—
121D1172—21no—
122D1172C2×D412yes—
123C22⋊D293—24no—
124C22⋊D293—24no—
125C22⋊D293—22no—
126C22⋊D293—22no—
127C22⋊D293—22no—
128C22⋊D293—22no—
129C2×Dic293—24no—
130C2×Dic293—21no—
131C2×Dic293—22no—
132C2×Dic293C2×D412yes—
133C4×D293—24no—
134C4×D293—24no—
135C4×D293—22no—
136C4×D293—22no—
137C4×D293—22no—
138C4×D293—22no—
139#14556873—22no—
140C23×C293—21no—
141C23×C293C2×D411yescharacteristic
142Q8×C293—21no—
143Q8×C293—22no—
144Q8×C293—22no—
145Q8×C293—41no—
146Q8×C293—41no—
147Q8×C293C2×D412yes—
148D4×C293—22no—
149D4×C293—22no—
150D4×C293—21no—
151D4×C293C2×D412yes—
152C2×C4×C293—21no—
153C2×C4×C293—22no—
154C2×C4×C293—22no—
155C2×C4×C293—22no—
156C2×C4×C293—41no—
157C2×C4×C293—41no—
158C2×C4×C293C2×D411yescharacteristic
159C2×C4×C293C2×D411yescharacteristic
160C2×C4×C293C2×D411yescharacteristic
161C2×C4×C293C2×D411yescharacteristic
162C2×C4×C293C2×D411yescharacteristic
163C2×C4×C293C2411yescharacteristic
164C8×C293—22no—
165C22×C293—22no—
166C22×C293—41no—
167C22×C293—81no—
168C22×C293#12011yescharacteristic
169C22×C293#12011yescharacteristic
170C22×C293C22×D411yescharacteristic
171C2×D293—22no—
172C2×D293—22no—
173C2×D293—42no—
174C2×D293—81no—
175C2×D293—84no—
176C4×C293—22no—
177C4×C293—42no—
178C4×C293#12011yescharacteristic
179C4×C293#12011yescharacteristic
180C4×C293C22×D411yescharacteristic
181C4×C293C22×D411yescharacteristic
182Dic293—22no—
183Dic293—22no—
184Dic293—42no—
185C2×C293—21no—
186C2×C293—21no—
187C2×C293—21no—
188C2×C293—42no—
189C2×C293C2×#12011yescharacteristic
190D293—42no—
191D293—42no—
192C293#302211yescharacteristic, complemented, Sylow
193#3022—2931nomaximal, Sylow
194#585—2931no—
195C2×#137—2931no—
196C2×#137D29311yescharacteristic, complemented
197#536—2931no—
198#456—2934no—
199#455—2934no—
200#421—2931no—
201#421—2931no—
202#413—2931no—
2032−1+4—2932no—
2042−1+4—5861no—
2052−1+4—5861no—
206H(5,𝔽2)—2932no—
207H(5,𝔽2)—5862no—
208H(5,𝔽2)—5861no—
209H(5,𝔽2)—5861no—
210C2×#41—2931no—
211C2×#41—2931no—
212C2×#41—5862no—
213C2×#41—5862no—
214C2×#41—5862no—
215C2×#41C2×D29311yescharacteristic
216C22×Q8C2×D29311yescharacteristic, complemented
217#137—22no—
218#137—2934no—
219#137—5862no—
220#137C2×D29314yescomplemented
221C2×Q16—21no—
222C2×Q16—5861no—
223C2×SD16—21no—
224C2×SD16—5861no—
225C2×M16—2931no—
226C2×M16C2×D29311yescharacteristic, complemented
227C4⋊Q8—2932no—
228#124—2932no—
229#123—5862no—
230C22⋊Q8—5862no—
231#117—2931no—
232#104—2934no—
233#104—2934no—
234#101—2932no—
235#100—2932no—
236#41—21no—
237#41—21no—
238#41—2932no—
239#41—2932no—
240#41—5862no—
241#41—5862no—
242#41—5864no—
243#41—5864no—
244#41—5864no—
245#41—5864no—
246#41—5862no—
247#41—5862no—
248#41—5862no—
249#41—5862no—
250#41—5861no—
251#41—5861no—
252#41—5861no—
253#41—5861no—
254#41—5861no—
255#41—5861no—
256#41C22⋊D29312yes—
257#41C22×D29312yes—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—2931no—
264C2×Q8—2931no—
265C2×Q8—5862no—
266C2×Q8—5862no—
267C2×Q8—5862no—
268C2×Q8C22⋊D29311yescharacteristic
269C2×Q8C22×D29312yes—
270C2×D4—21no—
271C2×D4—2932no—
272C2×D4—2931no—
273C2×D4—2931no—
274C2×D4—5862no—
275C2×D4—5864no—
276C2×D4—5864no—
277C2×D4—5862no—
278C2×D4—5864no—
279C2×D4—5861no—
280C2×D4—5862no—
281C2×D4C22⋊D29311yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—2932no—
285C22×C4—5864no—
286C22×C4—5861no—
287C22×C4C22×D29311yescharacteristic
288Q16—24no—
289Q16—5864no—
290SD16—24no—
291SD16—5864no—
292M16—21no—
293M16—2932no—
294M16—5861no—
295M16C22×D29312yes—
296C2×C8—21no—
297C2×C8—5861no—
298C4⋊C4—5864no—
299C4⋊C4—5861no—
300C22⋊C4—5864no—
301C22⋊C4—5861no—
302C42—2932no—
303C23—21no—
304C23—2932no—
305C23—5864no—
306C23—5861no—
307C23D4×D29311yescharacteristic
308Q8—21no—
309Q8—22no—
310Q8—22no—
311Q8—41no—
312Q8—41no—
313Q8—2932no—
314Q8—5864no—
315Q8—5861no—
316Q8—5861no—
317Q8—5861no—
318Q8C2×C22⋊D29312yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—2932no—
323D4—5864no—
324D4—5864no—
325D4—5864no—
326D4—5862no—
327D4—5862no—
328D4—5862no—
329D4—5862no—
330D4—5861no—
331D4—5861no—
332D4—5861no—
333D4C2×C22⋊D29312yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—2932no—
341C2×C4—5864no—
342C2×C4—5864no—
343C2×C4—5864no—
344C2×C4—5861no—
345C2×C4—5862no—
346C2×C4—5862no—
347C2×C4—5862no—
348C2×C4—5862no—
349C2×C4—5862no—
350C2×C4D4×D29311yescharacteristic
351C2×C4D4×D29311yescharacteristic
352C2×C4D4×D29311yescharacteristic
353C2×C4C2×C22⋊D29311yescharacteristic
354C2×C4C2×C22⋊D29311yescharacteristic
355C2×C4C23×D29311yescharacteristic
356C8—22no—
357C8—5862no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—23441no—
362C22—23444no—
363C22—5862no—
364C22—5862no—
365C22—11722no—
366C22#2262759211yescharacteristic
367C22#2262759211yescharacteristic
368C22C2×D4×D29311yescharacteristic
369C4—22no—
370C4—42no—
371C4—5862no—
372C4—5862no—
373C4—11722no—
374C4#2262759211yescharacteristic
375C4#2262759211yescharacteristic
376C4C2×D4×D29311yescharacteristic
377C4C22×C22⋊D29311yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—11722no—
383C2—11722no—
384C2C2×#2262759211yescharacteristic
subgroupG/H
centerC2C2×#22627592
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×#137×C293C2
socleC2×C293C2×#120
prime radical#34036965C2
hypercenterC2×#137D293
hypocenterC293#3022
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#22627655C2semidirect
3. #34036703C2semidirect
4. #34036310C2semidirect
5. #34036305C2semidirect
6. #34036146C2semidirect
7. #34036145C2semidirect
8. #34036111C2semidirect
9. C2×#137×C293C2semidirect
23. #22627655C22semidirect
26. C2×#17989786C22semidirect
27. #22627622C22semidirect
28. #22627613C22semidirect
31. #22627583C22semidirect
32. #22627551C22semidirect
33. #22627550C22semidirect
34. #22627542C22semidirect
35. #22627540C22semidirect
37. C22×Q8×C293C22semidirect
39. #137×C293C22semidirect
42. C2×M16×C293C22semidirect
192. C293#3022semidirect
196. C2×#137D293semidirect
216. C22×Q8C2×D293semidirect
220. #137C2×D293semidirect
226. C2×M16C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369652
C2×#226276552
#340367032
#340363102
#340363052
#340361462
#340361452
#340361112
C2×#137×C2932
#3022293