Finite Group Databasebeta

← #34054053

Subgroups — #34054053

Order 19072.

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

normal subgroups only

This group has 698 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
1C2×#18011958C211yesmaximal, characteristic, complemented
2#23750423C211yesmaximal, characteristic, complemented
3#23750196C211yesmaximal, characteristic, complemented
4#23749806C214yesmaximal, complemented
5#23749796C214yesmaximal, complemented
6#23749642C211yesmaximal, characteristic, complemented
7#23749641C211yesmaximal, characteristic, complemented
8C2×#18011829C211yesmaximal, characteristic, complemented
9#575×C149C211yesmaximal, characteristic, complemented
10#18011958—22no—
11#18011958—22no—
12#18011958—22no—
13#18011958C2214yes—
14C2×#14558736—21no—
15C2×#14558736C2211yescharacteristic
16C2×#14558735—21no—
17C2×#14558735—22no—
18C22×Dic298—21no—
19C22×Dic298C2211yescharacteristic
20C2×Q8×D149—21no—
21#18011942C2212yescomplemented
22#18011941—21no—
23#18011937C2212yescomplemented
24#18011936—21no—
25#18011931—22no—
26C2×#14558720C2211yescharacteristic
27#18011909C2212yescomplemented
28#18011898C2212yescomplemented
29#18011892—22no—
30#18011890—22no—
31#18011870C2211yescharacteristic
32#18011838C2214yescomplemented
33#18011837C2214yescomplemented
34#18011829C2214yescomplemented
35C2×#41×C149C2211yescharacteristic
36C2×#41×C149C2211yescharacteristic
37#137×C149—21no—
38#137×C149C2212yescomplemented
39AGL(1,ℤ/8)×C149—21no—
40AGL(1,ℤ/8)×C149C2212yescomplemented
41#135×C149—22no—
42C2×M16×C149C2211yescharacteristic
43#14558736—24no—
44#14558736—24no—
45#14558736—21no—
46#14558736—21no—
47#14558736D412yes—
48#14558736C2312yes—
49#14558735—28no—
50#14558735—24no—
51#14558735—28no—
52#14558735—24no—
53C2×Dic298—22no—
54C2×Dic298—24no—
55C2×Dic298—24no—
56C2×Dic298—21no—
57C2×Dic298—21no—
58C2×Dic298—21no—
59C2×Dic298—21no—
60C2×Dic298D411yescharacteristic
61C2×Dic298D412yes—
62C2×Dic298C2312yes—
63Q8×D149—24no—
64Q8×D149—24no—
65C2×D596D411yescharacteristic
66C2×C22⋊D149—22no—
67C2×C22⋊D149—21no—
68C22×Dic149—22no—
69C22×Dic149D412yes—
70C2×C4×D149—22no—
71C2×C4×D149—21no—
72#14558726—22no—
73#14558724—22no—
74#14558723—22no—
75#14558721—22no—
76#14558720—21no—
77#14558720C2312yes—
78C2×#11924746—21no—
79C4⋊Dic149—21no—
80#14558715—24no—
81C22⋊Dic149—24no—
82C22⋊Dic149—21no—
83C4×Dic149C2312yes—
84#41×C149—21no—
85#41×C149—21no—
86#41×C149—21no—
87#41×C149—21no—
88#41×C149—21no—
89#41×C149—21no—
90#41×C149D412yes—
91#41×C149C2312yes—
92C2×Q8×C149D411yescharacteristic
93C2×Q8×C149C2311yescharacteristic
94C2×D4×C149—21no—
95C2×D4×C149—21no—
96C2×D4×C149D411yescharacteristic
97C2×D4×C149C2311yescharacteristic
98C22×C4×C149—21no—
99C22×C4×C149—21no—
100C22×C4×C149C2311yescharacteristic
101Q16×C149—22no—
102SD16×C149—22no—
103SD16×C149—22no—
104D8×C149—22no—
105M16×C149—21no—
106M16×C149C2312yes—
107C2×C8×C149—21no—
108C22×D149—21no—
109Dic298—28no—
110Dic298—22no—
111Dic298—21no—
112Dic298—22no—
113Dic298—22no—
114Dic298—22no—
115Dic298C2×D412yes—
116D596—21no—
117D596C2×D412yes—
118C22⋊D149—28no—
119C22⋊D149—22no—
120C22⋊D149—22no—
121C2×Dic149—28no—
122C2×Dic149—21no—
123C2×Dic149—22no—
124C2×Dic149—22no—
125C2×Dic149—22no—
126C2×Dic149C2×D414yes—
127C4×D149—28no—
128C4×D149—22no—
129C4×D149—22no—
130#11924746—22no—
131C23×C149—21no—
132C23×C149—21no—
133C23×C149C2×D411yescharacteristic
134Q8×C149—21no—
135Q8×C149—21no—
136Q8×C149—21no—
137Q8×C149C2×D412yes—
138D4×C149—22no—
139D4×C149—22no—
140D4×C149—21no—
141D4×C149—21no—
142D4×C149—21no—
143D4×C149—41no—
144D4×C149—41no—
145D4×C149C2×D412yes—
146C2×C4×C149—21no—
147C2×C4×C149—22no—
148C2×C4×C149—22no—
149C2×C4×C149—21no—
150C2×C4×C149—41no—
151C2×C4×C149—41no—
152C2×C4×C149C2×D411yescharacteristic
153C2×C4×C149C2×D411yescharacteristic
154C2×C4×C149C2×D411yescharacteristic
155C2×C4×C149C2×D411yescharacteristic
156C2×C4×C149C2×D411yescharacteristic
157C2×C4×C149C2411yescharacteristic
158C8×C149—22no—
159C22×C149—22no—
160C22×C149—41no—
161C22×C149—42no—
162C22×C149—42no—
163C22×C149—41no—
164C22×C149#12011yescharacteristic
165C22×C149#12011yescharacteristic
166C22×C149C22×D411yescharacteristic
167C2×D149—22no—
168C2×D149—42no—
169C4×C149—22no—
170C4×C149—41no—
171C4×C149#12011yescharacteristic
172C4×C149#12011yescharacteristic
173C4×C149C22×D411yescharacteristic
174C4×C149C22×D411yescharacteristic
175Dic149—22no—
176Dic149—24no—
177Dic149—42no—
178C2×C149—21no—
179C2×C149—21no—
180C2×C149—21no—
181C2×C149—42no—
182C2×C149—81no—
183C2×C149C2×#12011yescharacteristic
184D149—42no—
185C149#302011yescharacteristic, complemented, Sylow
186#3020—1491nomaximal, Sylow
187C2×2−1+4—1491no—
188#575—1491no—
189#575D14911yescharacteristic, complemented
190#536—1491no—
191#456—1494no—
192#454—1494no—
193#421—1491no—
194#421—1491no—
195C2×#101—1491no—
1962−1+4—2982no—
1972−1+4—2982no—
1982−1+4—2982no—
1992−1+4—1494no—
200C2×#41—2981no—
201C2×#41—2981no—
202C2×#41—2982no—
203C2×#41—1491no—
204C2×#41C2×D14911yescharacteristic
205C2×#41C2×D14911yescharacteristic
206C22×Q8—2981no—
207C22×Q8—2981no—
208C22×Q8—1491no—
209#137—21no—
210#137—2981no—
211#137—1492no—
212#137C2×D14912yescomplemented
213AGL(1,ℤ/8)—21no—
214AGL(1,ℤ/8)—2981no—
215AGL(1,ℤ/8)—1492no—
216AGL(1,ℤ/8)C2×D14912yescomplemented
217#135—22no—
218#135—2982no—
219C2×M16—1491no—
220C2×M16C2×D14911yescharacteristic
221C4⋊Q8—1492no—
222#124—1492no—
223#123—2982no—
224C22⋊Q8—2982no—
225#117—1491no—
226#104—1494no—
227#104—1494no—
228#101—1494no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—21no—
234#41—21no—
235#41—2984no—
236#41—2984no—
237#41—2988no—
238#41—2984no—
239#41—2984no—
240#41—2988no—
241#41—2981no—
242#41—2981no—
243#41—1492no—
244#41—1492no—
245#41C22⋊D14912yes—
246#41C22×D14912yes—
247C2×Q8—2982no—
248C2×Q8—2984no—
249C2×Q8—2984no—
250C2×Q8—2984no—
251C2×Q8—2984no—
252C2×Q8—2981no—
253C2×Q8—2981no—
254C2×Q8—2981no—
255C2×Q8—2981no—
256C2×Q8—1491no—
257C2×Q8—1492no—
258C2×Q8—1492no—
259C2×Q8C22⋊D14911yescharacteristic
260C2×Q8C22×D14911yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—2982no—
264C2×D4—2981no—
265C2×D4—1491no—
266C2×D4C22⋊D14911yescharacteristic
267C2×D4C22×D14911yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—2982no—
271C22×C4—2982no—
272C22×C4—2981no—
273C22×C4—1492no—
274C22×C4C22×D14911yescharacteristic
275Q16—22no—
276Q16—2982no—
277SD16—22no—
278SD16—22no—
279SD16—2982no—
280SD16—2982no—
281D8—22no—
282D8—2982no—
283M16—21no—
284M16—2981no—
285M16—1492no—
286M16C22×D14912yes—
287C2×C8—21no—
288C2×C8—2981no—
289C4⋊C4—2984no—
290C4⋊C4—2981no—
291C22⋊C4—2981no—
292C22⋊C4—2984no—
293C42—1492no—
294C23—21no—
295C23—21no—
296C23—2981no—
297C23D4×D14911yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—21no—
301Q8—2988no—
302Q8—2982no—
303Q8—2982no—
304Q8—2981no—
305Q8—2982no—
306Q8—2982no—
307Q8—1492no—
308Q8C2×C22⋊D14912yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—2988no—
317D4—2982no—
318D4—2982no—
319D4—2981no—
320D4—1492no—
321D4C2×C22⋊D14912yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—2988no—
329C2×C4—2988no—
330C2×C4—2981no—
331C2×C4—2982no—
332C2×C4—2982no—
333C2×C4—2982no—
334C2×C4—2982no—
335C2×C4—2982no—
336C2×C4—1494no—
337C2×C4D4×D14911yescharacteristic
338C2×C4D4×D14911yescharacteristic
339C2×C4D4×D14911yescharacteristic
340C2×C4C2×C22⋊D14911yescharacteristic
341C2×C4C2×C22⋊D14911yescharacteristic
342C2×C4C23×D14911yescharacteristic
343C8—22no—
344C8—2982no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—2982no—
351C22—5962no—
352C22#1801187911yescharacteristic
353C22#1801187911yescharacteristic
354C22C2×D4×D14911yescharacteristic
355C4—22no—
356C4—41no—
357C4—2982no—
358C4—2984no—
359C4—5962no—
360C4#1801187911yescharacteristic
361C4#1801187911yescharacteristic
362C4C2×D4×D14911yescharacteristic
363C4C22×C22⋊D14911yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—5962no—
370C2C2×#1801187911yescharacteristic
subgroupG/H
centerC2C2×#18011879
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#575×C149C2
socleC2×C149C2×#120
hypercenter#575D149
hypocenterC149#3020
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#18011958C2semidirect
2. #23750423C2semidirect
3. #23750196C2semidirect
4. #23749806C2semidirect
5. #23749796C2semidirect
6. #23749642C2semidirect
7. #23749641C2semidirect
8. C2×#18011829C2semidirect
9. #575×C149C2semidirect
21. #18011942C22semidirect
23. #18011937C22semidirect
27. #18011909C22semidirect
28. #18011898C22semidirect
32. #18011838C22semidirect
33. #18011837C22semidirect
34. #18011829C22semidirect
38. #137×C149C22semidirect
40. AGL(1,ℤ/8)×C149C22semidirect
185. C149#3020semidirect
189. #575D149semidirect
212. #137C2×D149semidirect
216. AGL(1,ℤ/8)C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
#237504232
#237501962
#237498062
#237497962
#237496422
#237496412
C2×#180118292
#575×C1492
#3020149