Finite Group Databasebeta

← #31783735

Subgroups — #31783735

Order 17536.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984048C211yesmaximal, characteristic
2C2×#17984045C211yesmaximal, characteristic, complemented
3C2×#17984044C211yesmaximal, characteristic, complemented
4#22577303C211yesmaximal, characteristic
5#22577303C211yesmaximal, characteristic
6#22577296C211yesmaximal, characteristic, complemented
7#22577295C211yesmaximal, characteristic, complemented
8#22577293C211yesmaximal, characteristic, complemented
9#22577272C211yesmaximal, characteristic, complemented
10#22577262C211yesmaximal, characteristic, complemented
11#22577261C211yesmaximal, characteristic, complemented
12C4×#14555489C211yesmaximal, characteristic, complemented
13#22577254C211yesmaximal, characteristic, complemented
14#22577254C211yesmaximal, characteristic, complemented
15#388×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C23×Dic137C2211yescharacteristic
18C22×C4×D137C2211yescharacteristic
19#17984048C418yescomplemented
20#17984045—24no—
21#17984044—24no—
22C2×C4⋊Dic137C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×#14555489C2211yescharacteristic
27C2×#14555489C2211yescharacteristic
28C2×#14555489C2211yescharacteristic
29C2×#14555489C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic
31C2×C22⋊Dic137C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C2×#14555487C2211yescharacteristic
37C2×C4×Dic137C2211yescharacteristic
38C2×C4×Dic137C2211yescharacteristic
39C2×C4×Dic137C2211yescharacteristic
40C2×C4×Dic137C2211yescharacteristic
41C2×C4×Dic137C2211yescharacteristic
42C2×C4×Dic137C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47C2×C4⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C42×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53#95×C137C2211yescharacteristic
54C23×D137C2311yescharacteristic
55C2×C22⋊D137—22no—
56C2×C22⋊D137—22no—
57C2×C22⋊D137C2×C414yes—
58C22×Dic137—22no—
59C22×Dic137—22no—
60C22×Dic137C2×C414yes—
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C2×C4×D137—22no—
69C2×C4×D137—22no—
70C2×C4×D137C2×C414yes—
71C4⋊Dic137—41no—
72#14555489—22no—
73#14555489—22no—
74#14555489—22no—
75#14555489—22no—
76#14555489—22no—
77#14555489C2×C414yes—
78#14555489D414yescomplemented
79C22⋊Dic137—22no—
80C22⋊Dic137—41no—
81#14555487—22no—
82#14555487—22no—
83#14555487—41no—
84#14555487—41no—
85#14555487C2×C414yes—
86C4×Dic137—22no—
87C4×Dic137—22no—
88C4×Dic137—22no—
89C4×Dic137—22no—
90C4×Dic137—22no—
91C4×Dic137C2×C414yes—
92C24×C137C2311yescharacteristic
93C22×C4×C137C2311yescharacteristic
94C22×C4×C137C2311yescharacteristic
95C22×C4×C137C2311yescharacteristic
96C22×C4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C4⋊C4×C137—41no—
100C22⋊C4×C137—22no—
101C22⋊C4×C137—41no—
102C22⋊C4×C137C2×C414yes—
103C42×C137—22no—
104C22×D137—21no—
105C22×D137—21no—
106C22×D137—21no—
107C22×D137—21no—
108C22×D137—21no—
109C22×D137—21no—
110C22×D137C22×C412yes—
111C22⋊D137—28no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—28no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137C22×C412yes—
127C2×Dic137C22×C412yes—
128C2×Dic137C22×C412yes—
129C2×Dic137C2×D412yes—
130C2×Dic137C2×D412yes—
131C2×Dic137#4112yes—
132C2×Dic137#4112yes—
133C2×Dic137#4112yes—
134C4×D137—28no—
135C23×C137—21no—
136C23×C137—21no—
137C23×C137—21no—
138C23×C137—21no—
139C23×C137—21no—
140C23×C137—21no—
141C23×C137C22×C412yes—
142C23×C137C2411yescharacteristic
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—21no—
155C2×C4×C137—21no—
156C2×C4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137C22×C412yes—
159C2×C4×C137C22×C412yes—
160C2×C4×C137C2×D412yes—
161C22×C137—22no—
162C22×C137—22no—
163C22×C137—22no—
164C22×C137—41no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137C23×C411yescharacteristic
169C22×C137C22×D411yescharacteristic
170C22×C137C2×#4111yescharacteristic
171C22×C137C2×#4111yescharacteristic
172C22×C137C2×#4111yescharacteristic
173C22×C137H(5,𝔽2)11yescharacteristic
174C22×C1372−1+411yescharacteristic
175C2×D137—22no—
176C2×D137—22no—
177C2×D137—22no—
178C2×D137—41no—
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—41no—
182C4×C137—22no—
183C4×C137—22no—
184C4×C137—22no—
185C4×C137—41no—
186C4×C137—41no—
187C4×C137—41no—
188Dic137—22no—
189Dic137—22no—
190Dic137—41no—
191Dic137—41no—
192Dic137—41no—
193Dic137#11714yescomplemented
194Dic137C4×D414yes—
195C2×C137—42no—
196C2×C137C2×#11711yescharacteristic
197C2×C137C2×C4×D411yescharacteristic
198C2×C137#52011yescharacteristic
199C2×C137#54711yescharacteristic
200C2×C137#55011yescharacteristic
201C2×C137#55511yescharacteristic
202C2×C137#55611yescharacteristic
203D137—42no—
204C137#235411yescharacteristic, complemented, Sylow
205#2354—1371nomaximal, Sylow
206C2×C4×D4—1371no—
207C2×#117—1371no—
208C2×#117—1371no—
209#390—1371no—
210#390—1371no—
211#388—1371no—
212#388—1371no—
213#388—1371no—
214#388D13711yescharacteristic, complemented
215#384—1371no—
216#380—1371no—
217#380—1371no—
218C4×C4⋊C4—1371no—
219C4×C4⋊C4—1371no—
220C4×C4⋊C4—1371no—
221C22×D4—1371no—
222C23×C4—1371no—
223C23×C4—1371no—
224C4×D4—1378no—
225#117—2744no—
226#117—2744no—
227C2×C4⋊C4—1371no—
228C2×C4⋊C4—1371no—
229C2×C4⋊C4—1371no—
230C2×C4⋊C4—1371no—
231C2×C4⋊C4—1371no—
232C2×C4⋊C4—1371no—
233C2×C4⋊C4—1371no—
234C2×C4⋊C4—1371no—
235C2×C4⋊C4C2×D13711yescharacteristic
236C2×C22⋊C4—1371no—
237C2×C22⋊C4—1371no—
238C2×C22⋊C4—1371no—
239C2×C22⋊C4—1371no—
240C2×C22⋊C4—1371no—
241C2×C22⋊C4—1371no—
242C2×C22⋊C4—1371no—
243C2×C22⋊C4C2×D13711yescharacteristic
244C2×C22⋊C4C2×D13711yescharacteristic
245C2×C22⋊C4C2×D13711yescharacteristic
246C2×C42—1371no—
247C2×C42—1371no—
248C2×C42—1371no—
249C2×C42—1371no—
250C2×C42—1371no—
251C2×C42—1371no—
252C2×C42C2×D13711yescharacteristic
253#95—1371no—
254#95—1371no—
255#95—1371no—
256#95—1371no—
257#95C2×D13711yescharacteristic
258#95C2×D13711yescharacteristic
259C24—1371no—
260C24C22×D13711yescharacteristic
261C2×D4—2742no—
262C2×D4—2742no—
263C2×D4—1374no—
264C22×C4—2742no—
265C22×C4—2742no—
266C22×C4—2742no—
267C22×C4—2742no—
268C22×C4—1371no—
269C22×C4—1371no—
270C22×C4—1371no—
271C22×C4—1374no—
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4—1374no—
275C22×C4—1371no—
276C22×C4—1371no—
277C22×C4C22×D13711yescharacteristic
278C22×C4C22×D13711yescharacteristic
279C22×C4C22×D13711yescharacteristic
280C22×C4C22×D13711yescharacteristic
281C22×C4C22×D13711yescharacteristic
282C22×C4C22×D13711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—2742no—
285C4⋊C4—2742no—
286C4⋊C4—2742no—
287C4⋊C4—2742no—
288C4⋊C4—2742no—
289C4⋊C4—5481no—
290C4⋊C4—1374no—
291C4⋊C4—1374no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—2742no—
295C22⋊C4—2742no—
296C22⋊C4—2742no—
297C22⋊C4—5481no—
298C22⋊C4—5481no—
299C22⋊C4—5481no—
300C22⋊C4—1374no—
301C22⋊C4C4×D13714yes—
302C42—22no—
303C42—2742no—
304C42—2742no—
305C42—2742no—
306C42—2742no—
307C42—2742no—
308C42—1374no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—2741no—
316C23—2741no—
317C23—2741no—
318C23—2741no—
319C23—2741no—
320C23—2741no—
321C23—1372no—
322C23C2×C4×D13712yes—
323C23C23×D13711yescharacteristic
324D4—2748no—
325C2×C4—21no—
326C2×C4—21no—
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—2748no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2748no—
346C2×C4—2741no—
347C2×C4—2741no—
348C2×C4—2741no—
349C2×C4—2741no—
350C2×C4—2741no—
351C2×C4—2741no—
352C2×C4—2741no—
353C2×C4—2741no—
354C2×C4—2741no—
355C2×C4—1372no—
356C2×C4—1372no—
357C2×C4—1372no—
358C2×C4—1372no—
359C2×C4—1372no—
360C2×C4—1372no—
361C2×C4—1372no—
362C2×C4—1372no—
363C2×C4C2×C4×D13712yes—
364C2×C4C2×C4×D13712yes—
365C2×C4D4×D13712yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—2742no—
374C22—2742no—
375C22—2742no—
376C22—5481no—
377C22—5481no—
378C22—5481no—
379C22—5481no—
380C22C22×C4×D13711yescharacteristic
381C22C2×D4×D13711yescharacteristic
382C22#41×D13711yescharacteristic
383C22#41×D13711yescharacteristic
384C22#41×D13711yescharacteristic
385C22#1798413311yescharacteristic
386C22#1798413511yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—2742no—
394C4—2742no—
395C4—5481no—
396C4—5481no—
397C4—5481no—
398C4—1374no—
399C4—1374no—
400C2—42no—
401C2—5482no—
402C2#117×D13711yescharacteristic
403C2C4×D4×D13711yescharacteristic
404C2#2257788211yescharacteristic
405C2#2257805111yescharacteristic
406C2#2257807111yescharacteristic
407C2#2257811511yescharacteristic
408C2#2257812011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#388×C137C2
socle, cosocleC23×C137C24
prime radicalC22×C22⋊D137C22
hypercenter#388D137
hypocenterC137#2354
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#17984045C2semidirect
3. C2×#17984044C2semidirect
6. #22577296C2semidirect
7. #22577295C2semidirect
8. #22577293C2semidirect
9. #22577272C2semidirect
10. #22577262C2semidirect
11. #22577261C2semidirect
12. C4×#14555489C2semidirect
13. #22577254C2semidirect
14. #22577254C2semidirect
15. #388×C137C2semidirect
19. #17984048C4semidirect
78. #14555489D4semidirect
193. Dic137#117semidirect
204. C137#2354semidirect
214. #388D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840482
C2×#179840452
C2×#179840442
#225773032
#225772962
#225772952
#225772932
#225772722
#225772622
#225772612
C4×#145554892
#225772542
#388×C1372
#2354137