Finite Group Databasebeta

← #34047366

Subgroups — #34047366

Order 19072.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011888C211yesmaximal, characteristic
2C2×#18011876C211yesmaximal, characteristic, complemented
3C2×#18011867C211yesmaximal, characteristic
4#23749529C211yesmaximal, characteristic, complemented
5#23749523C211yesmaximal, characteristic, complemented
6#23749522C211yesmaximal, characteristic, complemented
7#23749520C211yesmaximal, characteristic, complemented
8#23749519C211yesmaximal, characteristic
9#23749519C211yesmaximal, characteristic
10#23749518C211yesmaximal, characteristic
11#23749508C211yesmaximal, characteristic, complemented
12#23749507C211yesmaximal, characteristic, complemented
13#23749483C211yesmaximal, characteristic
14C4×#14558715C211yesmaximal, characteristic, complemented
15#388×C149C211yesmaximal, characteristic
16C22×Dic298C2211yescharacteristic
17C23×Dic149C2211yescharacteristic
18#18011888C418yescomplemented
19#18011876—24no—
20#18011867—24no—
21C2×C4⋊Dic149C2211yescharacteristic
22C2×C4⋊Dic149C2211yescharacteristic
23C2×#14558715C2211yescharacteristic
24C2×#14558715C2211yescharacteristic
25C2×#14558715C2211yescharacteristic
26C2×#14558715C2211yescharacteristic
27C2×#14558715C2211yescharacteristic
28C2×#14558715C2211yescharacteristic
29C2×#14558715C2211yescharacteristic
30C2×#14558715C2211yescharacteristic
31C2×C22⋊Dic149C2211yescharacteristic
32C2×C22⋊Dic149C2211yescharacteristic
33C2×C22⋊Dic149C2211yescharacteristic
34C2×C4×Dic149C2211yescharacteristic
35C2×C4×Dic149C2211yescharacteristic
36C2×C4×Dic149C2211yescharacteristic
37C2×C4×Dic149C2211yescharacteristic
38C2×C4×Dic149C2211yescharacteristic
39#18011818C2211yescharacteristic
40#18011818C2211yescharacteristic
41#18011818C2211yescharacteristic
42#18011818C2211yescharacteristic
43#18011818C2211yescharacteristic
44#18011818C2211yescharacteristic
45#18011818C2211yescharacteristic
46#18011818C2211yescharacteristic
47C2×C4⋊C4×C149C2211yescharacteristic
48C2×C22⋊C4×C149C2211yescharacteristic
49C2×C22⋊C4×C149C2211yescharacteristic
50C2×C22⋊C4×C149C2211yescharacteristic
51C2×C42×C149C2211yescharacteristic
52#95×C149C2211yescharacteristic
53#95×C149C2211yescharacteristic
54C2×Dic298—22no—
55C2×Dic298—22no—
56C2×Dic298C2×C414yes—
57C22×Dic149—22no—
58C22×Dic149—22no—
59C22×Dic149C2×C414yes—
60C22×Dic149C2311yescharacteristic
61C22×Dic149C2311yescharacteristic
62C22×Dic149C2311yescharacteristic
63C22×Dic149C2311yescharacteristic
64C22×Dic149C2311yescharacteristic
65C22×Dic149C2311yescharacteristic
66C22×Dic149C2311yescharacteristic
67C22×Dic149C2311yescharacteristic
68C4⋊Dic149—41no—
69C4⋊Dic149C2×C414yes—
70#14558715—22no—
71#14558715—22no—
72#14558715—22no—
73#14558715—22no—
74#14558715—22no—
75#14558715—41no—
76#14558715C2×C414yes—
77#14558715C2×C414yes—
78C22⋊Dic149—22no—
79C22⋊Dic149—41no—
80C22⋊Dic149C2×C414yes—
81C4×Dic149—22no—
82C4×Dic149—22no—
83C4×Dic149—22no—
84C4×Dic149—41no—
85C4×Dic149D414yes—
86C24×C149C2311yescharacteristic
87C22×C4×C149C2311yescharacteristic
88C22×C4×C149C2311yescharacteristic
89C22×C4×C149C2311yescharacteristic
90C22×C4×C149C2311yescharacteristic
91C22×C4×C149C2311yescharacteristic
92C22×C4×C149C2311yescharacteristic
93C4⋊C4×C149—22no—
94C22⋊C4×C149—22no—
95C22⋊C4×C149—41no—
96C22⋊C4×C149C2×C414yes—
97C42×C149—41no—
98Dic298—28no—
99C2×Dic149—21no—
100C2×Dic149—21no—
101C2×Dic149—21no—
102C2×Dic149—21no—
103C2×Dic149—21no—
104C2×Dic149—21no—
105C2×Dic149—21no—
106C2×Dic149—21no—
107C2×Dic149—28no—
108C2×Dic149—21no—
109C2×Dic149—21no—
110C2×Dic149—21no—
111C2×Dic149—21no—
112C2×Dic149—21no—
113C2×Dic149—21no—
114C2×Dic149—21no—
115C2×Dic149—21no—
116C2×Dic149—21no—
117C2×Dic149C22×C412yes—
118C2×Dic149C22×C412yes—
119C2×Dic149C22×C412yes—
120C2×Dic149C22×C412yes—
121C2×Dic149C2×D412yes—
122C2×Dic149C2×D412yes—
123C2×Dic149#4112yes—
124C23×C149—21no—
125C23×C149—21no—
126C23×C149—21no—
127C23×C149—21no—
128C23×C149—21no—
129C23×C149—21no—
130C23×C149C22×C412yes—
131C23×C149C2411yescharacteristic
132C2×C4×C149—21no—
133C2×C4×C149—21no—
134C2×C4×C149—21no—
135C2×C4×C149—21no—
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×C149C22×C412yes—
148C2×C4×C149C22×C412yes—
149C2×C4×C149C2×D412yes—
150C22×C149—22no—
151C22×C149—22no—
152C22×C149—22no—
153C22×C149—41no—
154C22×C149—41no—
155C22×C149—41no—
156C22×C149—41no—
157C22×C149C23×C411yescharacteristic
158C22×C149C22×D411yescharacteristic
159C22×C149C2×#4111yescharacteristic
160C22×C149H(5,𝔽2)11yescharacteristic
161C22×C1492−1+411yescharacteristic
162C22×C1492−1+411yescharacteristic
163C22×C1492−1+411yescharacteristic
164C4×C149—22no—
165C4×C149—22no—
166C4×C149—22no—
167C4×C149—41no—
168C4×C149—41no—
169C4×C149—41no—
170Dic149—22no—
171Dic149—22no—
172Dic149—22no—
173Dic149—22no—
174Dic149—41no—
175Dic149—41no—
176Dic149—41no—
177Dic149C4×D414yes—
178C2×C149—42no—
179C2×C149C2×C4×D411yescharacteristic
180C2×C149#51911yescharacteristic
181C2×C149#52011yescharacteristic
182C2×C149#53611yescharacteristic
183C2×C149#53711yescharacteristic
184C2×C149#54111yescharacteristic
185C2×C149#54211yescharacteristic
186C149#232311yescharacteristic, complemented, Sylow
187#2323—1491nomaximal, Sylow
188C2×C22⋊Q8—1491no—
189C2×C4×Q8—1491no—
190C2×#117—1491no—
191#391—1491no—
192#389—1491no—
193#389—1491no—
194#389—1491no—
195#388—1491no—
196#388—1491no—
197#388—1491no—
198#388D14911yescharacteristic
199#387—1491no—
200#387—1491no—
201#380—1491no—
202C4×C4⋊C4—1491no—
203C22×Q8—1491no—
204C23×C4—1491no—
205C22⋊Q8—1498no—
206C4×Q8—2984no—
207#117—2984no—
208C2×C4⋊C4—1491no—
209C2×C4⋊C4—1491no—
210C2×C4⋊C4—1491no—
211C2×C4⋊C4—1491no—
212C2×C4⋊C4—1491no—
213C2×C4⋊C4—1491no—
214C2×C4⋊C4—1491no—
215C2×C4⋊C4—1491no—
216C2×C4⋊C4—1491no—
217C2×C4⋊C4—1491no—
218C2×C4⋊C4C2×D14911yescharacteristic
219C2×C22⋊C4—1491no—
220C2×C22⋊C4—1491no—
221C2×C22⋊C4—1491no—
222C2×C22⋊C4C2×D14911yescharacteristic
223C2×C22⋊C4C2×D14911yescharacteristic
224C2×C22⋊C4C2×D14911yescharacteristic
225C2×C42—1491no—
226C2×C42—1491no—
227C2×C42—1491no—
228C2×C42—1491no—
229C2×C42—1491no—
230C2×C42C2×D14911yescharacteristic
231#95—1491no—
232#95—1491no—
233#95—1491no—
234#95—1491no—
235#95—1491no—
236#95—1491no—
237#95—1491no—
238#95—1491no—
239#95C2×D14911yescharacteristic
240#95C2×D14911yescharacteristic
241C24C22×D14911yescharacteristic
242C2×Q8—2982no—
243C2×Q8—2982no—
244C2×Q8—1494no—
245C22×C4—2982no—
246C22×C4—2982no—
247C22×C4—1491no—
248C22×C4—1491no—
249C22×C4—1491no—
250C22×C4—1494no—
251C22×C4—1491no—
252C22×C4—1491no—
253C22×C4—1491no—
254C22×C4—1491no—
255C22×C4—1491no—
256C22×C4C22×D14911yescharacteristic
257C22×C4C22×D14911yescharacteristic
258C22×C4C22×D14911yescharacteristic
259C22×C4C22×D14911yescharacteristic
260C22×C4C22×D14911yescharacteristic
261C22×C4C22×D14911yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—2982no—
264C4⋊C4—2982no—
265C4⋊C4—2982no—
266C4⋊C4—2982no—
267C4⋊C4—2982no—
268C4⋊C4—5961no—
269C4⋊C4—5961no—
270C4⋊C4—1494no—
271C4⋊C4—1494no—
272C4⋊C4—1494no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—2982no—
276C22⋊C4—5961no—
277C22⋊C4—1494no—
278C22⋊C4C4×D14914yes—
279C42—41no—
280C42—2982no—
281C42—2982no—
282C42—2982no—
283C42—5961no—
284C42—1494no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D14912yes—
292C23C23×D14911yescharacteristic
293Q8—2988no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2981no—
310C2×C4—2981no—
311C2×C4—2981no—
312C2×C4—2981no—
313C2×C4—2981no—
314C2×C4—2981no—
315C2×C4—2981no—
316C2×C4—2981no—
317C2×C4—2988no—
318C2×C4—2981no—
319C2×C4—2981no—
320C2×C4—2981no—
321C2×C4—2981no—
322C2×C4—2981no—
323C2×C4—2981no—
324C2×C4—2981no—
325C2×C4—2981no—
326C2×C4—2981no—
327C2×C4—1492no—
328C2×C4—1492no—
329C2×C4—1492no—
330C2×C4—1492no—
331C2×C4—1492no—
332C2×C4—1492no—
333C2×C4—1492no—
334C2×C4C2×C4×D14912yes—
335C2×C4C2×C4×D14912yes—
336C2×C4D4×D14912yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D14911yescharacteristic
345C22C2×D4×D14911yescharacteristic
346C22#41×D14911yescharacteristic
347C22#1801195611yescharacteristic
348C22#1801195811yescharacteristic
349C22#1801195811yescharacteristic
350C22#1801195811yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2982no—
358C4—2982no—
359C4—2982no—
360C4—2982no—
361C4—5961no—
362C4—5961no—
363C4—5961no—
364C4—1494no—
365C2—42no—
366C2C4×D4×D14911yescharacteristic
367C2#2375010211yescharacteristic
368C2#2375010411yescharacteristic
369C2#2375019511yescharacteristic
370C2#2375020211yescharacteristic
371C2#2375023211yescharacteristic
372C2#2375024311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#388×C149C2
socle, cosocleC23×C149C24
prime radicalC24×C149C23
hypercenter#388D149
hypocenterC149#2323
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#18011876C2semidirect
4. #23749529C2semidirect
5. #23749523C2semidirect
6. #23749522C2semidirect
7. #23749520C2semidirect
11. #23749508C2semidirect
12. #23749507C2semidirect
14. C4×#14558715C2semidirect
18. #18011888C4semidirect
186. C149#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#180118882
C2×#180118762
C2×#180118672
#237495292
#237495232
#237495222
#237495202
#237495192
#237495182
#237495082
#237495072
#237494832
C4×#145587152
#388×C1492
#2323149