Finite Group Databasebeta

← C2×#23750320 #34058209

Subgroups — C2×#23750320 #34058209

Order 19072.

1252 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750320C2116yesmaximal, direct product factor
2C2×#18011906C211yesmaximal, characteristic, complemented
3C2×#18011904C211yesmaximal, characteristic, complemented
4C2×#18011901C211yesmaximal, characteristic, complemented
5C2×#18011893C211yesmaximal, characteristic, complemented
6C2×#18011884C211yesmaximal, characteristic, complemented
7C2×#18011884C211yesmaximal, characteristic, complemented
8C2×#18011883C211yesmaximal, characteristic, complemented
9C2×#18011882C211yesmaximal, characteristic, complemented
10C2×#18011873C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D149C211yesmaximal, characteristic, complemented
12C2×C4×C22⋊D149C211yesmaximal, characteristic, complemented
13C2×#18011871C211yesmaximal, characteristic, complemented
14C2×#18011867C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D149C211yesmaximal, characteristic, complemented
16C2×#123×C149C211yesmaximal, characteristic, complemented
17C22×D596C2211yescharacteristic
18C22×C22⋊D149C2211yescharacteristic
19C22×C22⋊D149C2211yescharacteristic
20C22×C22⋊D149C2211yescharacteristic
21C22×C22⋊D149C2211yescharacteristic
22C23×Dic149C2211yescharacteristic
23C22×C4×D149C2211yescharacteristic
24C22×C4×D149C2211yescharacteristic
25C22×C4×D149C2211yescharacteristic
26C22×C4×D149C2211yescharacteristic
27#18011906C2218yescomplemented
28#18011904C2218yescomplemented
29#18011901C2218yescomplemented
30#18011893C2218yescomplemented
31#18011884C2218yescomplemented
32#18011884C2218yescomplemented
33#18011883C2218yescomplemented
34#18011882C2218yescomplemented
35#18011873C2218yescomplemented
36C4×C22⋊D149C2218yescomplemented
37C4×C22⋊D149C2218yescomplemented
38#18011871C2218yescomplemented
39#18011867C2218yescomplemented
40C2×C4⋊Dic149C2211yescharacteristic
41C2×C4⋊Dic149C2211yescharacteristic
42C2×#14558715C2211yescharacteristic
43C2×#14558715C2211yescharacteristic
44C2×#14558715C2211yescharacteristic
45C2×#14558715C2211yescharacteristic
46C2×#14558715C2211yescharacteristic
47C2×#14558715C2211yescharacteristic
48C4⋊C4×D149C2218yescomplemented
49C2×C22⋊Dic149C2211yescharacteristic
50C2×C22⋊Dic149C2211yescharacteristic
51C2×C22⋊Dic149C2211yescharacteristic
52C2×#14558713C2211yescharacteristic
53C2×#14558713C2211yescharacteristic
54C2×#14558713C2211yescharacteristic
55C2×#14558713C2211yescharacteristic
56C2×C4×Dic149C2211yescharacteristic
57C2×C4×Dic149C2211yescharacteristic
58C2×C4×Dic149C2211yescharacteristic
59C22×D4×C149C2211yescharacteristic
60C23×C4×C149C2211yescharacteristic
61#123×C149C2218yescomplemented
62C2×C4⋊C4×C149C2211yescharacteristic
63C2×C4⋊C4×C149C2211yescharacteristic
64C2×C22⋊C4×C149C2211yescharacteristic
65C2×C22⋊C4×C149C2211yescharacteristic
66C2×C22⋊C4×C149C2211yescharacteristic
67C23×D149C2311yescharacteristic
68C23×D149C2311yescharacteristic
69C2×D596—24no—
70C2×D596C2314yes—
71C2×C22⋊D149—24no—
72C2×C22⋊D149—24no—
73C2×C22⋊D149—24no—
74C2×C22⋊D149—42no—
75C2×C22⋊D149C2314yes—
76C2×C22⋊D149C2314yes—
77C2×C22⋊D149C2314yes—
78C2×C22⋊D149C2314yes—
79C22×Dic149—24no—
80C22×Dic149C2311yescharacteristic
81C22×Dic149C2311yescharacteristic
82C22×Dic149C2311yescharacteristic
83C22×Dic149C2314yes—
84C22×Dic149C2311yescharacteristic
85C22×Dic149C2311yescharacteristic
86C22×Dic149C2311yescharacteristic
87C2×C4×D149—24no—
88C2×C4×D149—42no—
89C2×C4×D149—42no—
90C2×C4×D149—42no—
91C2×C4×D149C2314yes—
92C2×C4×D149C2314yes—
93C2×C4×D149C2314yes—
94C2×C4×D149C2314yes—
95C4⋊Dic149C2314yes—
96C4⋊Dic149C2314yes—
97#14558715C2314yes—
98#14558715C2314yes—
99#14558715C2314yes—
100#14558715C2314yes—
101#14558715C2314yes—
102#14558715C2314yes—
103C22⋊Dic149C2314yes—
104C22⋊Dic149C2314yes—
105C22⋊Dic149C2314yes—
106#14558713C2314yes—
107#14558713C2314yes—
108#14558713C2314yes—
109#14558713C2314yes—
110C4×Dic149C2314yes—
111C4×Dic149C2314yes—
112C4×Dic149C2314yes—
113C24×C149C2311yescharacteristic
114C24×C149C2311yescharacteristic
115C2×D4×C149—42no—
116C2×D4×C149C2314yes—
117C22×C4×C149—42no—
118C22×C4×C149C2311yescharacteristic
119C22×C4×C149C2311yescharacteristic
120C22×C4×C149C2311yescharacteristic
121C22×C4×C149C2311yescharacteristic
122C22×C4×C149C2311yescharacteristic
123C22×C4×C149C2314yes—
124C4⋊C4×C149C2314yes—
125C4⋊C4×C149C2314yes—
126C22⋊C4×C149C2314yes—
127C22⋊C4×C149C2314yes—
128C22⋊C4×C149C2314yes—
129C22×D149—22no—
130C22×D149—22no—
131C22×D149—22no—
132C22×D149—22no—
133C22×D149—22no—
134C22×D149#4114yes—
135C22×D149C2412yes—
136C22×D149C2412yes—
137D596—28no—
138C22⋊D149—28no—
139C22⋊D149—28no—
140C22⋊D149—28no—
141C22⋊D149—44no—
142C2×Dic149—22no—
143C2×Dic149—22no—
144C2×Dic149—22no—
145C2×Dic149—22no—
146C2×Dic149—22no—
147C2×Dic149—28no—
148C2×Dic149#4114yes—
149C2×Dic149C2412yes—
150C2×Dic149C2412yes—
151C2×Dic149C2412yes—
152C2×Dic149C2412yes—
153C2×Dic149C2412yes—
154C2×Dic149C2412yes—
155C4×D149—28no—
156C4×D149—44no—
157C4×D149—44no—
158C4×D149—44no—
159C23×C149—22no—
160C23×C149—22no—
161C23×C149—22no—
162C23×C149—22no—
163C23×C149—22no—
164C23×C149—22no—
165C23×C149C2412yes—
166C23×C149C2412yes—
167C23×C149C2411yescharacteristic
168D4×C149—44no—
169C2×C4×C149—22no—
170C2×C4×C149—22no—
171C2×C4×C149—22no—
172C2×C4×C149—22no—
173C2×C4×C149—22no—
174C2×C4×C149—44no—
175C2×C4×C149C2412yes—
176C2×C4×C149C2412yes—
177C2×C4×C149C2412yes—
178C2×C4×C149C2412yes—
179C2×C4×C149C2412yes—
180C22×C149—22no—
181C22×C149—22no—
182C22×C149—22no—
183C22×C149—22no—
184C22×C149—22no—
185C22×C149—22no—
186C22×C149—44no—
187C22×C149—44no—
188C22×C149C2×#4112yes—
189C22×C149C2×#4112yes—
190C22×C149H(5,𝔽2)12yes—
191C22×C149C2511yescharacteristic
192C2×D149—28no—
193C2×D149—22no—
194C2×D149—22no—
195C2×D149—22no—
196C2×D149—22no—
197C2×D149—22no—
198C2×D149—44no—
199C2×D149C2×#4114yes—
200C4×C149—22no—
201C4×C149—22no—
202C4×C149—22no—
203C4×C149—22no—
204C4×C149—22no—
205Dic149—22no—
206Dic149—22no—
207Dic149—22no—
208Dic149—22no—
209Dic149—22no—
210Dic149C2×#4114yes—
211C2×C149—42no—
212C2×C149—42no—
213C2×C149#55314yescomplemented
214C2×C149C22×#4111yescharacteristic
215C2×C149C22×#4111yescharacteristic
216C2×C149C2×H(5,𝔽2)11yescharacteristic
217D149—24no—
218D149—42no—
219C149C2×#55311yescharacteristic, complemented, Sylow
220C2×#553—1491nomaximal, Sylow
221#553—14916no—
222C2×#126—1491no—
223C2×#126—1491no—
224C2×#125—1491no—
225C2×#123—1491no—
226C2×#123D14911yescharacteristic, complemented
227C2×#121—1491no—
228C2×#121—1491no—
229C2×#121—1491no—
230C2×#121—1491no—
231C2×C4×D4—1491no—
232C2×C4×D4—1491no—
233C2×C4×D4—1491no—
234C2×C4×D4—1491no—
235C2×#117—1491no—
236C22×C4⋊C4—1491no—
237C22×D4—1491no—
238C22×D4—1491no—
239C22×D4—1491no—
240C22×D4—1491no—
241C22×D4—1491no—
242C22×D4C2×D14911yescharacteristic
243C23×C4—1491no—
244C23×C4—1491no—
245C23×C4—1491no—
246C23×C4—1491no—
247C23×C4—1491no—
248C23×C4C2×D14911yescharacteristic
249#126—1498no—
250#126—1498no—
251#125—1498no—
252#123—1498no—
253#123C2×D14918yescomplemented
254#121—1498no—
255#121—1498no—
256#121—1498no—
257#121—1498no—
258C4×D4—1498no—
259C4×D4—1498no—
260C4×D4—1498no—
261C4×D4—1498no—
262#117—1498no—
263C2×C4⋊C4—1491no—
264C2×C4⋊C4—1491no—
265C2×C4⋊C4—1491no—
266C2×C4⋊C4—1491no—
267C2×C4⋊C4—1491no—
268C2×C4⋊C4—1491no—
269C2×C4⋊C4—1491no—
270C2×C4⋊C4—1491no—
271C2×C4⋊C4—1498no—
272C2×C4⋊C4C2×D14911yescharacteristic
273C2×C4⋊C4C2×D14911yescharacteristic
274C2×C22⋊C4—1491no—
275C2×C22⋊C4—1491no—
276C2×C22⋊C4—1491no—
277C2×C22⋊C4—1491no—
278C2×C22⋊C4—1491no—
279C2×C22⋊C4—1491no—
280C2×C22⋊C4—1491no—
281C2×C22⋊C4C2×D14911yescharacteristic
282C2×C22⋊C4C2×D14911yescharacteristic
283C2×C22⋊C4C2×D14911yescharacteristic
284C2×C42—1491no—
285C2×C42—1491no—
286C2×C42—1491no—
287C24—1491no—
288C24—1491no—
289C24C22×D14911yescharacteristic
290C24C22×D14911yescharacteristic
291C2×D4—42no—
292C2×D4—2984no—
293C2×D4—2984no—
294C2×D4—2984no—
295C2×D4—2984no—
296C2×D4—5962no—
297C2×D4—1494no—
298C2×D4—1494no—
299C2×D4—1494no—
300C2×D4—1494no—
301C2×D4—1494no—
302C2×D4C22×D14914yes—
303C22×C4—42no—
304C22×C4—2984no—
305C22×C4—2984no—
306C22×C4—5962no—
307C22×C4—5962no—
308C22×C4—5962no—
309C22×C4—1494no—
310C22×C4—1491no—
311C22×C4—1491no—
312C22×C4—1491no—
313C22×C4—1491no—
314C22×C4—1491no—
315C22×C4—1494no—
316C22×C4—1494no—
317C22×C4—1494no—
318C22×C4—1494no—
319C22×C4—1491no—
320C22×C4C22×D14911yescharacteristic
321C22×C4C22×D14911yescharacteristic
322C22×C4C22×D14911yescharacteristic
323C22×C4C22×D14911yescharacteristic
324C22×C4C22×D14911yescharacteristic
325C22×C4C22×D14914yes—
326C4⋊C4—1494no—
327C4⋊C4—1494no—
328C4⋊C4—1494no—
329C4⋊C4—1494no—
330C4⋊C4—1494no—
331C4⋊C4—1494no—
332C4⋊C4—1494no—
333C4⋊C4—1494no—
334C4⋊C4C22×D14914yes—
335C4⋊C4C22×D14914yes—
336C22⋊C4—1494no—
337C22⋊C4—1494no—
338C22⋊C4—1494no—
339C22⋊C4—1494no—
340C22⋊C4—1494no—
341C22⋊C4—1494no—
342C22⋊C4—1494no—
343C22⋊C4C22×D14914yes—
344C22⋊C4C22×D14914yes—
345C22⋊C4C22×D14914yes—
346C42—1494no—
347C42—1494no—
348C42—1494no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—2982no—
356C23—2982no—
357C23—2982no—
358C23—2982no—
359C23—2982no—
360C23—1492no—
361C23—1494no—
362C23—1492no—
363C23C23×D14912yes—
364C23C23×D14912yes—
365C23C23×D14911yescharacteristic
366D4—44no—
367D4—2988no—
368D4—2988no—
369D4—2988no—
370D4—2988no—
371D4—5964no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—2988no—
379C2×C4—2982no—
380C2×C4—2982no—
381C2×C4—2982no—
382C2×C4—2982no—
383C2×C4—2982no—
384C2×C4—2988no—
385C2×C4—5964no—
386C2×C4—5964no—
387C2×C4—5964no—
388C2×C4—1492no—
389C2×C4—1492no—
390C2×C4—1492no—
391C2×C4—1492no—
392C2×C4—1492no—
393C2×C4—1494no—
394C2×C4—1492no—
395C2×C4C23×D14912yes—
396C2×C4C23×D14912yes—
397C2×C4C23×D14912yes—
398C2×C4C23×D14912yes—
399C2×C4C23×D14912yes—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—44no—
407C22—44no—
408C22—2982no—
409C22—2982no—
410C22—2988no—
411C22—2982no—
412C22—2982no—
413C22—2982no—
414C22—5964no—
415C22—1494no—
416C22#41×D14912yes—
417C22#41×D14912yes—
418C22#1801195612yes—
419C22C24×D14911yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—2982no—
426C4—2982no—
427C4—2982no—
428C4—2982no—
429C4—2982no—
430C4—1494no—
431C2—42no—
432C2—42no—
433C2—2984no—
434C2—5962no—
435C2#2375032014yesdirect product factor
436C2C2×#41×D14911yescharacteristic
437C2C2×#41×D14911yescharacteristic
438C2C2×#1801195611yescharacteristic
subgroupG/H
centerC23C23×D149
commutator, cosocleC22×C149C25
FrattiniC22C24×D149
FittingC2×#123×C149C2
socleC23×C149C24
hypercenterC2×#123D149
hypocenterC149C2×#553
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #23750320C2direct (N × H)
2. C2×#18011906C2semidirect
3. C2×#18011904C2semidirect
4. C2×#18011901C2semidirect
5. C2×#18011893C2semidirect
6. C2×#18011884C2semidirect
7. C2×#18011884C2semidirect
8. C2×#18011883C2semidirect
9. C2×#18011882C2semidirect
10. C2×#18011873C2semidirect
11. C2×C4×C22⋊D149C2semidirect
12. C2×C4×C22⋊D149C2semidirect
13. C2×#18011871C2semidirect
14. C2×#18011867C2semidirect
15. C2×C4⋊C4×D149C2semidirect
16. C2×#123×C149C2semidirect
27. #18011906C22semidirect
28. #18011904C22semidirect
29. #18011901C22semidirect
30. #18011893C22semidirect
31. #18011884C22semidirect
32. #18011884C22semidirect
33. #18011883C22semidirect
34. #18011882C22semidirect
35. #18011873C22semidirect
36. C4×C22⋊D149C22semidirect
37. C4×C22⋊D149C22semidirect
38. #18011871C22semidirect
39. #18011867C22semidirect
48. C4⋊C4×D149C22semidirect
61. #123×C149C22semidirect
213. C2×C149#553semidirect
219. C149C2×#553semidirect
226. C2×#123D149semidirect
253. #123C2×D149semidirect
435. C2#23750320direct (N × H)

Maximal subgroups

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

subgroupindex
#237503202
C2×#180119062
C2×#180119042
C2×#180119012
C2×#180118932
C2×#180118842
C2×#180118832
C2×#180118822
C2×#180118732
C2×C4×C22⋊D1492
C2×#180118712
C2×#180118672
C2×C4⋊C4×D1492
C2×#123×C1492
C2×#553149