Finite Group Databasebeta

← #49161713

Subgroups — #49161713

Order 37504.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627584C211yesmaximal, characteristic
2C2×#22627580C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D293C211yesmaximal, characteristic, complemented
4#34036022C211yesmaximal, characteristic, complemented
5#34036016C211yesmaximal, characteristic, complemented
6#34036010C211yesmaximal, characteristic, complemented
7#34036005C211yesmaximal, characteristic
8#34036005C211yesmaximal, characteristic
9#34035993C211yesmaximal, characteristic, complemented
10#34035988C211yesmaximal, characteristic, complemented
11#34035987C211yesmaximal, characteristic, complemented
12C4×#17989781C211yesmaximal, characteristic, complemented
13C4×#17989779C211yesmaximal, characteristic, complemented
14#34035976C211yesmaximal, characteristic, complemented
15#388×C293C211yesmaximal, characteristic, complemented
16C22×C22⋊D293C2211yescharacteristic
17C23×Dic293C2211yescharacteristic
18C22×C4×D293C2211yescharacteristic
19C22×C4×D293C2211yescharacteristic
20C22×C4×D293C2211yescharacteristic
21#22627584C418yescomplemented
22#22627580—24no—
23C2×C4⋊Dic293C2211yescharacteristic
24C2×#17989781C2211yescharacteristic
25C2×#17989781C2211yescharacteristic
26C2×#17989781C2211yescharacteristic
27C2×#17989781C2211yescharacteristic
28C2×#17989781C2211yescharacteristic
29C4⋊C4×D293—24no—
30C2×C22⋊Dic293C2211yescharacteristic
31C2×C22⋊Dic293C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×C4×Dic293C2211yescharacteristic
36C2×C4×Dic293C2211yescharacteristic
37C2×C4×Dic293C2211yescharacteristic
38C2×C4×Dic293C2211yescharacteristic
39#22627531C2211yescharacteristic
40#22627531C2211yescharacteristic
41#22627531C2211yescharacteristic
42#22627531C2211yescharacteristic
43#22627531C2211yescharacteristic
44#22627531C2211yescharacteristic
45#22627531C2211yescharacteristic
46#22627531C2211yescharacteristic
47C2×C4⋊C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C22⋊C4×C293C2211yescharacteristic
50C2×C22⋊C4×C293C2211yescharacteristic
51C2×C42×C293C2211yescharacteristic
52#95×C293C2211yescharacteristic
53#95×C293C2211yescharacteristic
54C23×D293C2311yescharacteristic
55C2×C22⋊D293—22no—
56C2×C22⋊D293—22no—
57C2×C22⋊D293C2×C414yes—
58C22×Dic293—22no—
59C22×Dic293—22no—
60C22×Dic293C2×C414yescomplemented
61C22×Dic293C2311yescharacteristic
62C22×Dic293C2311yescharacteristic
63C22×Dic293C2311yescharacteristic
64C22×Dic293C2311yescharacteristic
65C22×Dic293C2311yescharacteristic
66C22×Dic293C2311yescharacteristic
67C22×Dic293C2311yescharacteristic
68C2×C4×D293—22no—
69C2×C4×D293—22no—
70C2×C4×D293—22no—
71C2×C4×D293—22no—
72C2×C4×D293—22no—
73C2×C4×D293—22no—
74C2×C4×D293—41no—
75C2×C4×D293—41no—
76C2×C4×D293C2×C414yes—
77C4⋊Dic293—22no—
78#17989781—22no—
79#17989781—22no—
80#17989781—22no—
81#17989781C2×C414yescomplemented
82#17989781D414yescomplemented
83C22⋊Dic293—22no—
84C22⋊Dic293—41no—
85#17989779—22no—
86#17989779—22no—
87#17989779C2×C414yes—
88C4×Dic293—22no—
89C4×Dic293—22no—
90C4×Dic293—41no—
91C4×Dic293C2×C414yescomplemented
92C24×C293C2311yescharacteristic
93C22×C4×C293C2311yescharacteristic
94C22×C4×C293C2311yescharacteristic
95C22×C4×C293C2311yescharacteristic
96C22×C4×C293C2311yescharacteristic
97C22×C4×C293C2311yescharacteristic
98C22×C4×C293C2311yescharacteristic
99C4⋊C4×C293—22no—
100C22⋊C4×C293—22no—
101C22⋊C4×C293—41no—
102C22⋊C4×C293C2×C414yescomplemented
103C42×C293—41no—
104C22×D293—21no—
105C22×D293—21no—
106C22×D293—21no—
107C22×D293—21no—
108C22×D293C22×C412yes—
109C22×D293#4112yes—
110C22×D293#4112yes—
111C22⋊D293—28no—
112C2×Dic293—21no—
113C2×Dic293—21no—
114C2×Dic293—21no—
115C2×Dic293—21no—
116C2×Dic293—21no—
117C2×Dic293—21no—
118C2×Dic293—28no—
119C2×Dic293—21no—
120C2×Dic293—21no—
121C2×Dic293—21no—
122C2×Dic293—21no—
123C2×Dic293—21no—
124C2×Dic293—21no—
125C2×Dic293—21no—
126C2×Dic293—21no—
127C2×Dic293—21no—
128C2×Dic293C22×C412yes—
129C2×Dic293C22×C412yes—
130C2×Dic293C22×C412yes—
131C2×Dic293C2×D412yes—
132C2×Dic293C2×D412yes—
133C2×Dic293#4112yes—
134C4×D293—28no—
135C4×D293—44no—
136C4×D293—44no—
137C23×C293—21no—
138C23×C293—21no—
139C23×C293—21no—
140C23×C293—21no—
141C23×C293—21no—
142C23×C293—21no—
143C23×C293C22×C412yes—
144C23×C293C2411yescharacteristic
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293—21no—
149C2×C4×C293—21no—
150C2×C4×C293—21no—
151C2×C4×C293—21no—
152C2×C4×C293—21no—
153C2×C4×C293—21no—
154C2×C4×C293—21no—
155C2×C4×C293—21no—
156C2×C4×C293—21no—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293C22×C412yes—
161C2×C4×C293C22×C412yes—
162C2×C4×C293C2×D412yes—
163C22×C293—22no—
164C22×C293—22no—
165C22×C293—22no—
166C22×C293—41no—
167C22×C293—41no—
168C22×C293—41no—
169C22×C293—41no—
170C22×C293C23×C411yescharacteristic
171C22×C293C22×D411yescharacteristic
172C22×C293C2×#4111yescharacteristic
173C22×C293C2×#4111yescharacteristic
174C22×C293C2×#4111yescharacteristic
175C22×C293H(5,𝔽2)11yescharacteristic
176C22×C2932−1+411yescharacteristic
177C2×D293—22no—
178C2×D293—22no—
179C2×D293—22no—
180C2×D293—22no—
181C2×D293—22no—
182C2×D293—22no—
183C2×D293#11714yescomplemented
184C4×C293—22no—
185C4×C293—22no—
186C4×C293—22no—
187C4×C293—41no—
188C4×C293—41no—
189C4×C293—41no—
190Dic293—22no—
191Dic293—22no—
192Dic293—22no—
193Dic293—41no—
194Dic293—41no—
195Dic293—41no—
196Dic293C4×D414yes—
197C2×C293—42no—
198C2×C293C2×#11711yescharacteristic
199C2×C293C2×C4×D411yescharacteristic
200C2×C293#52011yescharacteristic
201C2×C293#54611yescharacteristic
202C2×C293#54811yescharacteristic
203C2×C293#55211yescharacteristic
204C2×C293#55311yescharacteristic
205D293—24no—
206C293#235211yescharacteristic, complemented, Sylow
207#2352—2931nomaximal, Sylow
208C2×C4×D4—2931no—
209C2×#117—2931no—
210C22×C4⋊C4—2931no—
211#388—2931no—
212#388D29311yescharacteristic, complemented
213#387—2931no—
214#387—2931no—
215#386—2931no—
216#386—2931no—
217#382—2931no—
218#380—2931no—
219#380—2931no—
220C4×C4⋊C4—2931no—
221C4×C22⋊C4—2931no—
222C4×C22⋊C4—2931no—
223C22×D4—2931no—
224C23×C4—2931no—
225C23×C4—2931no—
226C23×C4—2931no—
227C23×C4—2931no—
228C4×D4—2938no—
229#117—5864no—
230C2×C4⋊C4—2931no—
231C2×C4⋊C4—2931no—
232C2×C4⋊C4—2931no—
233C2×C4⋊C4—2931no—
234C2×C4⋊C4—2931no—
235C2×C4⋊C4—2931no—
236C2×C4⋊C4—5864no—
237C2×C4⋊C4C2×D29311yescharacteristic
238C2×C22⋊C4—2931no—
239C2×C22⋊C4—2931no—
240C2×C22⋊C4—2931no—
241C2×C22⋊C4—2931no—
242C2×C22⋊C4—2931no—
243C2×C22⋊C4C2×D29311yescharacteristic
244C2×C22⋊C4C2×D29311yescharacteristic
245C2×C22⋊C4C2×D29311yescharacteristic
246C2×C42—2931no—
247C2×C42—2931no—
248C2×C42—2931no—
249C2×C42—2931no—
250C2×C42C2×D29311yescharacteristic
251#95—2931no—
252#95—2931no—
253#95—2931no—
254#95—2931no—
255#95—2931no—
256#95—2931no—
257#95—2931no—
258#95—2931no—
259#95C2×D29311yescharacteristic
260#95C2×D29311yescharacteristic
261C24—2931no—
262C24C22×D29311yescharacteristic
263C2×D4—2934no—
264C2×D4—5862no—
265C2×D4—5862no—
266C22×C4—2931no—
267C22×C4—2934no—
268C22×C4—2931no—
269C22×C4—2931no—
270C22×C4—2931no—
271C22×C4—2934no—
272C22×C4—2931no—
273C22×C4—2931no—
274C22×C4—2931no—
275C22×C4—5862no—
276C22×C4—5862no—
277C22×C4—5862no—
278C22×C4—5862no—
279C22×C4—5862no—
280C22×C4—5862no—
281C22×C4—5862no—
282C22×C4—5862no—
283C22×C4—11721no—
284C22×C4—11721no—
285C22×C4C22×D29311yescharacteristic
286C22×C4C22×D29311yescharacteristic
287C22×C4C22×D29311yescharacteristic
288C22×C4C22×D29311yescharacteristic
289C22×C4C22×D29311yescharacteristic
290C22×C4C22×D29311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—2934no—
293C4⋊C4—2934no—
294C4⋊C4—5862no—
295C4⋊C4—5862no—
296C4⋊C4—5862no—
297C4⋊C4—5862no—
298C22⋊C4—22no—
299C22⋊C4—41no—
300C22⋊C4—2934no—
301C22⋊C4—5862no—
302C22⋊C4—5862no—
303C22⋊C4—5862no—
304C22⋊C4—11721no—
305C22⋊C4C4×D29314yescomplemented
306C42—41no—
307C42—2934no—
308C42—5862no—
309C42—5862no—
310C42—11721no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—2932no—
318C23—2932no—
319C23—2932no—
320C23—5861no—
321C23—5861no—
322C23—5861no—
323C23—5861no—
324C23C2×C4×D29312yes—
325C23C23×D29311yescharacteristic
326D4—5868no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2932no—
343C2×C4—2932no—
344C2×C4—2932no—
345C2×C4—2932no—
346C2×C4—2932no—
347C2×C4—2932no—
348C2×C4—5868no—
349C2×C4—5868no—
350C2×C4—5861no—
351C2×C4—5861no—
352C2×C4—5861no—
353C2×C4—5861no—
354C2×C4—5861no—
355C2×C4—5861no—
356C2×C4—5861no—
357C2×C4—5861no—
358C2×C4—5861no—
359C2×C4—5861no—
360C2×C4—5861no—
361C2×C4—5861no—
362C2×C4—5861no—
363C2×C4—5861no—
364C2×C4—5861no—
365C2×C4—11724no—
366C2×C4—11724no—
367C2×C4C2×C4×D29312yes—
368C2×C4C2×C4×D29312yes—
369C2×C4D4×D29312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2934no—
378C22—5862no—
379C22—5862no—
380C22—5862no—
381C22—5862no—
382C22—5862no—
383C22—5862no—
384C22C22×C4×D29311yescharacteristic
385C22C2×D4×D29311yescharacteristic
386C22#41×D29311yescharacteristic
387C22#41×D29311yescharacteristic
388C22#41×D29311yescharacteristic
389C22#2262766911yescharacteristic
390C22#2262767111yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—2934no—
398C4—5862no—
399C4—5862no—
400C4—5862no—
401C4—11721no—
402C4—11721no—
403C4—11721no—
404C2—42no—
405C2—5864no—
406C2#117×D29311yescharacteristic
407C2C4×D4×D29311yescharacteristic
408C2#3403660811yescharacteristic
409C2#3403676811yescharacteristic
410C2#3403678711yescharacteristic
411C2#3403681511yescharacteristic
412C2#3403682411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#388×C293C2
socle, cosocleC23×C293C24
prime radicalC22×C22⋊D293C22
hypercenter#388D293
hypocenterC293#2352
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22627580C2semidirect
3. C2×C4⋊C4×D293C2semidirect
4. #34036022C2semidirect
5. #34036016C2semidirect
6. #34036010C2semidirect
9. #34035993C2semidirect
10. #34035988C2semidirect
11. #34035987C2semidirect
12. C4×#17989781C2semidirect
13. C4×#17989779C2semidirect
14. #34035976C2semidirect
15. #388×C293C2semidirect
21. #22627584C4semidirect
60. C22×Dic293C2×C4semidirect
81. #17989781C2×C4semidirect
82. #17989781D4semidirect
91. C4×Dic293C2×C4semidirect
102. C22⋊C4×C293C2×C4semidirect
183. C2×D293#117semidirect
206. C293#2352semidirect
212. #388D293semidirect
305. C22⋊C4C4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226275842
C2×#226275802
C2×C4⋊C4×D2932
#340360222
#340360162
#340360102
#340360052
#340359932
#340359882
#340359872
C4×#179897812
C4×#179897792
#340359762
#388×C2932
#2352293