Finite Group Databasebeta

← #34047630

Subgroups — #34047630

Order 19072.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011873C211yesmaximal, characteristic
2#23749517C211yesmaximal, characteristic, complemented
3#23749515C211yesmaximal, characteristic, complemented
4#23749511C211yesmaximal, characteristic, complemented
5#23749503C211yesmaximal, characteristic
6#23749503C211yesmaximal, characteristic
7#23749492C211yesmaximal, characteristic, complemented
8#23749487C211yesmaximal, characteristic, complemented
9#23749482C211yesmaximal, characteristic, complemented
10C4×#14558713C211yesmaximal, characteristic, complemented
11C4×#14558713C211yesmaximal, characteristic, complemented
12#23749472C211yesmaximal, characteristic, complemented
13#23749470C211yesmaximal, characteristic, complemented
14#95×D149C211yesmaximal, characteristic, complemented
15#387×C149C211yesmaximal, characteristic, complemented
16C22×D596C2211yescharacteristic
17C22×C4×D149C2211yescharacteristic
18C22×C4×D149C2211yescharacteristic
19C22×C4×D149C2211yescharacteristic
20C22×C4×D149C2211yescharacteristic
21#18011873C418yescomplemented
22C2×C4⋊Dic149C2211yescharacteristic
23C2×#14558713C2211yescharacteristic
24C2×#14558713C2211yescharacteristic
25C2×#14558713C2211yescharacteristic
26C2×#14558713C2211yescharacteristic
27C2×#14558713C2211yescharacteristic
28C2×#14558713C2211yescharacteristic
29C2×#14558713C2211yescharacteristic
30C2×#14558713C2211yescharacteristic
31C2×C4×Dic149C2211yescharacteristic
32C2×C4×Dic149C2211yescharacteristic
33C2×C4×Dic149C2211yescharacteristic
34C2×C4×Dic149C2211yescharacteristic
35#18011818C2211yescharacteristic
36#18011818C2211yescharacteristic
37#18011818C2211yescharacteristic
38#18011818C2211yescharacteristic
39#18011818C2211yescharacteristic
40#18011818C2211yescharacteristic
41#18011818C2211yescharacteristic
42#18011818C2211yescharacteristic
43#18011818C2211yescharacteristic
44#18011818C2211yescharacteristic
45C2×C4⋊C4×C149C2211yescharacteristic
46C2×C4⋊C4×C149C2211yescharacteristic
47C2×C42×C149C2211yescharacteristic
48#95×C149C2211yescharacteristic
49#95×C149C2211yescharacteristic
50#95×C149C2211yescharacteristic
51#95×C149C2211yescharacteristic
52C23×D149C2311yescharacteristic
53C23×D149C2311yescharacteristic
54C2×D596—22no—
55C2×D596—22no—
56C2×D596C2×C414yes—
57C22×Dic149C2311yescharacteristic
58C22×Dic149C2311yescharacteristic
59C22×Dic149C2311yescharacteristic
60C22×Dic149C2311yescharacteristic
61C22×Dic149C2311yescharacteristic
62C22×Dic149C2311yescharacteristic
63C2×C4×D149—22no—
64C2×C4×D149—22no—
65C2×C4×D149—22no—
66C2×C4×D149—22no—
67C2×C4×D149—41no—
68C2×C4×D149—41no—
69C2×C4×D149—41no—
70C2×C4×D149—41no—
71C2×C4×D149—41no—
72C2×C4×D149—41no—
73C2×C4×D149C2×C414yes—
74C2×C4×D149C2×C414yescomplemented
75C4⋊Dic149—41no—
76#14558713—22no—
77#14558713—22no—
78#14558713—22no—
79#14558713—22no—
80#14558713—41no—
81#14558713—41no—
82#14558713C2×C414yes—
83#14558713C2×C414yescomplemented
84C4×Dic149—22no—
85C4×Dic149—22no—
86C4×Dic149—41no—
87C4×Dic149C2×C414yescomplemented
88C22×C4×C149C2311yescharacteristic
89C22×C4×C149C2311yescharacteristic
90C22×C4×C149C2311yescharacteristic
91C22×C4×C149C2311yescharacteristic
92C22×C4×C149C2311yescharacteristic
93C22×C4×C149C2311yescharacteristic
94C22×C4×C149C2311yescharacteristic
95C4⋊C4×C149—41no—
96C4⋊C4×C149C2×C414yescomplemented
97C42×C149—41no—
98C22×D149—21no—
99C22×D149—21no—
100C22×D149—21no—
101C22×D149—21no—
102C22×D149—21no—
103C22×D149—21no—
104C22×D149—21no—
105C22×D149—21no—
106C22×D149—21no—
107C22×D149—21no—
108C22×D149C22×C412yes—
109C22×D149C22×C412yes—
110C22×D149#4112yes—
111C22×D149#4112yes—
112D596—28no—
113C2×Dic149—21no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149—21no—
117C2×Dic149—21no—
118C2×Dic149—21no—
119C2×Dic149—21no—
120C2×Dic149—21no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149—21no—
124C2×Dic149—21no—
125C2×Dic149—21no—
126C2×Dic149—21no—
127C2×Dic149—21no—
128C2×Dic149C22×C412yes—
129C2×Dic149C22×C412yes—
130C2×Dic149#4112yes—
131C4×D149—44no—
132C4×D149—44no—
133C4×D149—44no—
134C4×D149—44no—
135C23×C149C2411yescharacteristic
136C2×C4×C149—21no—
137C2×C4×C149—21no—
138C2×C4×C149—21no—
139C2×C4×C149—21no—
140C2×C4×C149—21no—
141C2×C4×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149—21no—
148C2×C4×C149—21no—
149C2×C4×C149—21no—
150C2×C4×C149—21no—
151C2×C4×C149—21no—
152C2×C4×C149—21no—
153C2×C4×C149C22×C412yes—
154C2×C4×C149C22×C412yes—
155C2×C4×C149C22×C412yes—
156C2×C4×C149#4112yes—
157C22×C149C23×C411yescharacteristic
158C22×C149C2×#4111yescharacteristic
159C22×C149C2×#4111yescharacteristic
160C22×C149C2×#4111yescharacteristic
161C22×C149C2×#4111yescharacteristic
162C22×C149H(5,𝔽2)11yescharacteristic
163C22×C149H(5,𝔽2)11yescharacteristic
164C2×D149—22no—
165C2×D149—22no—
166C2×D149—22no—
167C2×D149—22no—
168C2×D149—22no—
169C2×D149—22no—
170C2×D149—22no—
171C2×D149—22no—
172C2×D149—22no—
173C2×D149—41no—
174C2×D149—41no—
175C2×D149—41no—
176C2×D149—41no—
177C2×D149#11714yes—
178C4×C149—22no—
179C4×C149—22no—
180C4×C149—22no—
181C4×C149—22no—
182C4×C149—41no—
183C4×C149—41no—
184C4×C149—41no—
185Dic149—22no—
186Dic149—22no—
187Dic149—22no—
188Dic149—41no—
189Dic149—41no—
190Dic149—41no—
191C2×C149C2×#11711yescharacteristic
192C2×C149C4×#4111yescharacteristic
193C2×C149#51811yescharacteristic
194C2×C149#55111yescharacteristic
195C2×C149#55111yescharacteristic
196C2×C149#55311yescharacteristic
197C2×C149#55311yescharacteristic
198D149—24no—
199D149—42no—
200C149#235311yescharacteristic, complemented, Sylow
201#2353—1491nomaximal, Sylow
202C2×C4×D4—1491no—
203#388—1491no—
204#388—1491no—
205#387—1491no—
206#387D14911yescharacteristic, complemented
207#386—1491no—
208#386—1491no—
209#383—1491no—
210#381—1491no—
211#380—1491no—
212C4×C22⋊C4—1491no—
213C4×C22⋊C4—1491no—
214C4×C22⋊C4—1491no—
215#376—1491no—
216C2×#95—1491no—
217C22×D4—1491no—
218C23×C4—1491no—
219C23×C4—1491no—
220C23×C4—1491no—
221C23×C4—1491no—
222C4×D4—1498no—
223C2×C4⋊C4—1491no—
224C2×C4⋊C4C2×D14911yescharacteristic
225C2×C4⋊C4C2×D14911yescharacteristic
226C2×C22⋊C4—1491no—
227C2×C22⋊C4—1491no—
228C2×C22⋊C4—1491no—
229C2×C22⋊C4—1491no—
230C2×C22⋊C4—1491no—
231C2×C22⋊C4—1491no—
232C2×C22⋊C4—1491no—
233C2×C22⋊C4—1491no—
234C2×C42—1491no—
235C2×C42—1491no—
236C2×C42—1491no—
237C2×C42—1491no—
238C2×C42C2×D14911yescharacteristic
239#95—1491no—
240#95—1491no—
241#95—1491no—
242#95—1491no—
243#95—1491no—
244#95—1491no—
245#95—1491no—
246#95—1491no—
247#95—1491no—
248#95—1491no—
249#95C2×D14911yescharacteristic
250#95C2×D14911yescharacteristic
251#95C2×D14911yescharacteristic
252#95C2×D14911yescharacteristic
253C24—1491no—
254C24—1491no—
255C2×D4—2982no—
256C2×D4—2982no—
257C2×D4—1494no—
258C22×C4—2982no—
259C22×C4—2982no—
260C22×C4—2982no—
261C22×C4—2982no—
262C22×C4—5961no—
263C22×C4—5961no—
264C22×C4—5961no—
265C22×C4—5961no—
266C22×C4—5961no—
267C22×C4—5961no—
268C22×C4—1491no—
269C22×C4—1491no—
270C22×C4—1491no—
271C22×C4—1491no—
272C22×C4—1491no—
273C22×C4—1491no—
274C22×C4—1494no—
275C22×C4—1494no—
276C22×C4C22×D14911yescharacteristic
277C22×C4C22×D14911yescharacteristic
278C22×C4C22×D14911yescharacteristic
279C22×C4C22×D14911yescharacteristic
280C22×C4C22×D14911yescharacteristic
281C22×C4C22×D14911yescharacteristic
282C22×C4C22×D14911yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—5961no—
285C4⋊C4C4×D14914yescomplemented
286C22⋊C4—2982no—
287C22⋊C4—2982no—
288C22⋊C4—2982no—
289C22⋊C4—2982no—
290C22⋊C4—5961no—
291C22⋊C4—5961no—
292C22⋊C4—1494no—
293C22⋊C4—1494no—
294C42—41no—
295C42—2982no—
296C42—2982no—
297C42—5961no—
298C42—1494no—
299C23—2981no—
300C23—2981no—
301C23—2981no—
302C23—2981no—
303C23—2981no—
304C23—2981no—
305C23—2981no—
306C23—2981no—
307C23—2981no—
308C23—2981no—
309C23—1492no—
310C23—1492no—
311C23—1492no—
312C23—1492no—
313C23C23×D14911yescharacteristic
314D4—2988no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2981no—
333C2×C4—2981no—
334C2×C4—2981no—
335C2×C4—2981no—
336C2×C4—2981no—
337C2×C4—2981no—
338C2×C4—2981no—
339C2×C4—2981no—
340C2×C4—2981no—
341C2×C4—2981no—
342C2×C4—2981no—
343C2×C4—2981no—
344C2×C4—2981no—
345C2×C4—2981no—
346C2×C4—2981no—
347C2×C4—5964no—
348C2×C4—5964no—
349C2×C4—5964no—
350C2×C4—5964no—
351C2×C4—1492no—
352C2×C4—1492no—
353C2×C4—1492no—
354C2×C4C2×C4×D14912yes—
355C2×C4C2×C4×D14912yes—
356C2×C4C2×C4×D14912yes—
357C2×C4#1455873712yes—
358C22—2982no—
359C22—2982no—
360C22—2982no—
361C22—2982no—
362C22—2982no—
363C22—2982no—
364C22—2982no—
365C22—2982no—
366C22—2982no—
367C22—5961no—
368C22—5961no—
369C22—5961no—
370C22—5961no—
371C22—1494no—
372C22C22×C4×D14911yescharacteristic
373C22#41×D14911yescharacteristic
374C22#41×D14911yescharacteristic
375C22#41×D14911yescharacteristic
376C22C2×#1455873711yescharacteristic
377C22#1801195611yescharacteristic
378C22#1801195711yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—2982no—
387C4—2982no—
388C4—2982no—
389C4—5961no—
390C4—5961no—
391C4—5961no—
392C2—2984no—
393C2—5962no—
394C2#117×D14911yescharacteristic
395C2C4×#1455873711yescharacteristic
396C2#2375009711yescharacteristic
397C2#2375030011yescharacteristic
398C2#2375030211yescharacteristic
399C2#2375032011yescharacteristic
400C2#2375032811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#387×C149C2
socle, cosocleC23×C149C24
prime radicalC22×D596C22
hypercenter#387D149
hypocenterC149#2353
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #23749517C2semidirect
3. #23749515C2semidirect
4. #23749511C2semidirect
7. #23749492C2semidirect
8. #23749487C2semidirect
9. #23749482C2semidirect
10. C4×#14558713C2semidirect
11. C4×#14558713C2semidirect
12. #23749472C2semidirect
13. #23749470C2semidirect
14. #95×D149C2semidirect
15. #387×C149C2semidirect
21. #18011873C4semidirect
74. C2×C4×D149C2×C4semidirect
83. #14558713C2×C4semidirect
87. C4×Dic149C2×C4semidirect
96. C4⋊C4×C149C2×C4semidirect
200. C149#2353semidirect
206. #387D149semidirect
285. C4⋊C4C4×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118732
#237495172
#237495152
#237495112
#237495032
#237494922
#237494872
#237494822
C4×#145587132
#237494722
#237494702
#95×D1492
#387×C1492
#2353149