Finite Group Databasebeta

← #14535186

Subgroups — #14535186

Order 2176.

562 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
1C2×#10587622C211yesmaximal, characteristic, complemented
2#11762737C211yesmaximal, characteristic, complemented
3#11762729C211yesmaximal, characteristic, complemented
4#11762723C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762721C211yesmaximal, characteristic, complemented
8#11762697C211yesmaximal, characteristic, complemented
9#11762690C211yesmaximal, characteristic, complemented
10#11762689C211yesmaximal, characteristic, complemented
11#11762689C211yesmaximal, characteristic, complemented
12#11762682C211yesmaximal, characteristic, complemented
13#11762672C211yesmaximal, characteristic, complemented
14#11762667C211yesmaximal, characteristic, complemented
15C2×C17×#121C211yesmaximal, characteristic
16C22×D4×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C23×C4×C17C2211yescharacteristic
20C23×Dic17C2211yescharacteristic
21C22×Dic34C2211yescharacteristic
22C17×#121—24no—
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic
33C2×C22⋊Dic17C2211yescharacteristic
34C2×C22⋊Dic17C2211yescharacteristic
35C2×C22⋊Dic17C2211yescharacteristic
36C2×C22⋊Dic17C2211yescharacteristic
37#10587622—24no—
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×C4×Dic17C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C24×C17C2311yescharacteristic
55C24×C17C2311yescharacteristic
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—41no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17D414yes—
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—41no—
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C22×C4×C17C2311yescharacteristic
71C22×C4×C17C2311yescharacteristic
72C22×Dic17—22no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22×Dic17C2311yescharacteristic
81C22×Dic17C2311yescharacteristic
82C22×Dic17C2311yescharacteristic
83C2×Dic34—22no—
84C2×Dic34—41no—
85C2×Dic34—41no—
86C4⋊C4×C17—22no—
87C22⋊C4×C17—22no—
88C22⋊C4×C17—22no—
89C22⋊Dic17—22no—
90C22⋊Dic17—22no—
91C22⋊Dic17—22no—
92C22⋊Dic17—22no—
93C22⋊Dic17—22no—
94C22⋊Dic17—22no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97C22⋊Dic17—41no—
98C22⋊Dic17—41no—
99C22⋊Dic17—41no—
100#63225—22no—
101#63225—41no—
102#63225—41no—
103#63225—41no—
104C4×Dic17—22no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17C2×D412yes—
125C23×C17C2×D412yes—
126C23×C17C2411yescharacteristic
127D4×C17—44no—
128D4×C17—44no—
129D4×C17—44no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—44no—
141C2×C4×C17C2×D412yes—
142C2×C4×C17#4112yes—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—21no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—44no—
167C2×Dic17#4112yes—
168Dic34—44no—
169#2686—171nomaximal, Sylow
170C22×C17—22no—
171C22×C17—22no—
172C22×C17—22no—
173C22×C17—22no—
174C22×C17—22no—
175C22×C17—22no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—41no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—41no—
189C22×C17—41no—
190C22×C17—41no—
191C22×C17C22×D411yescharacteristic
192C22×C17C2×#4111yescharacteristic
193C22×C17C2×#4111yescharacteristic
194C22×C17H(5,𝔽2)11yescharacteristic
195C22×C17H(5,𝔽2)11yescharacteristic
196C22×C17H(5,𝔽2)11yescharacteristic
197C22×C172−1+411yescharacteristic
198C4×C17—22no—
199C4×C17—22no—
200C4×C17—41no—
201C4×C17—41no—
202Dic17—22no—
203Dic17—41no—
204Dic17—41no—
205Dic17—41no—
206Dic17—41no—
207Dic17—41no—
208Dic17—41no—
209Dic17—41no—
210C2×#124—171no—
211C2×#121D1711yescharacteristic
212#397—171no—
213#395—171no—
214#394—171no—
215#394—171no—
216#394—171no—
217#393—171no—
218#388—171no—
219#387—171no—
220#386—171no—
221#386—171no—
222#385—171no—
223#381—171no—
224#380—171no—
225C2×C17—42no—
226C2×C17—42no—
227C2×C17—81no—
228C2×C17#53411yescharacteristic
229C2×C17#53811yescharacteristic
230C2×C17#54211yescharacteristic
231C2×C17#54611yescharacteristic
232C2×C17#54711yescharacteristic
233C2×C17#55511yescharacteristic
234C2×C17#56211yescharacteristic
235C22×Q8—171no—
236C22×D4C2×D1711yescharacteristic
237C22×D4C2×D1711yescharacteristic
238C22×D4C2×D1711yescharacteristic
239C23×C4—171no—
240C23×C4C2×D1711yescharacteristic
241#124—344no—
242#121—24no—
243C2×C4⋊C4—171no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—171no—
247C2×C4⋊C4C2×D1711yescharacteristic
248C2×C22⋊C4—171no—
249C2×C22⋊C4—171no—
250C2×C22⋊C4—171no—
251C2×C22⋊C4—171no—
252C2×C22⋊C4—171no—
253C2×C22⋊C4—171no—
254C2×C22⋊C4—171no—
255C2×C22⋊C4—171no—
256C2×C22⋊C4—171no—
257C2×C22⋊C4—171no—
258C2×C22⋊C4—171no—
259C2×C22⋊C4C2×D1711yescharacteristic
260C2×C22⋊C4C2×D1711yescharacteristic
261C2×C42—171no—
262#95—171no—
263#95—171no—
264#95—171no—
265#95—171no—
266#95—171no—
267#95—171no—
268#95—171no—
269#95—171no—
270#95—171no—
271#95—171no—
272C17#268611yescharacteristic, complemented, Sylow
273C24C22×D1711yescharacteristic
274C24C22×D1711yescharacteristic
275C24C22×D1711yescharacteristic
276C2×Q8—342no—
277C2×Q8—681no—
278C2×Q8—681no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4C22⋊D1714yes—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—171no—
292C22×C4—171no—
293C22×C4—171no—
294C22×C4—171no—
295C22×C4—171no—
296C22×C4—171no—
297C22×C4—171no—
298C22×C4—171no—
299C22×C4—342no—
300C22×C4—681no—
301C22×C4—681no—
302C22×C4C22×D1711yescharacteristic
303C22×C4C22×D1711yescharacteristic
304C22×C4C22×D1711yescharacteristic
305C22×C4C22×D1711yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—342no—
308C4⋊C4—681no—
309C4⋊C4—681no—
310C4⋊C4—681no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—342no—
314C22⋊C4—342no—
315C22⋊C4—342no—
316C22⋊C4—342no—
317C22⋊C4—342no—
318C22⋊C4—342no—
319C22⋊C4—681no—
320C22⋊C4—681no—
321C22⋊C4—681no—
322C22⋊C4—681no—
323C22⋊C4—681no—
324C42—342no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23C2×C22⋊D1712yes—
345C23C2×C22⋊D1712yes—
346C23C23×D1711yescharacteristic
347Q8—684no—
348D4—44no—
349D4—44no—
350D4—44no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—172no—
363C2×C4—341no—
364C2×C4—341no—
365C2×C4—341no—
366C2×C4—341no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4—341no—
370C2×C4—341no—
371C2×C4—341no—
372C2×C4—341no—
373C2×C4—341no—
374C2×C4—341no—
375C2×C4—341no—
376C2×C4—341no—
377C2×C4—341no—
378C2×C4—341no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—341no—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—341no—
385C2×C4—341no—
386C2×C4—684no—
387C2×C4#6325412yes—
388C2×C4C2×C22⋊D1712yes—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22C2×#6325411yescharacteristic
411C22#1058769111yescharacteristic
412C22#1058769111yescharacteristic
413C22#1058769111yescharacteristic
414C22#41×D1711yescharacteristic
415C22#1058769711yescharacteristic
416C22C22×C22⋊D1711yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—342no—
422C4—681no—
423C4—681no—
424C4—681no—
425C4—681no—
426C4—681no—
427C4—681no—
428C4—681no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2#1176336811yescharacteristic
433C2#1176339111yescharacteristic
434C2#1176342611yescharacteristic
435C2#1176345311yescharacteristic
436C2#1176345811yescharacteristic
437C2#1176352311yescharacteristic
438C2#1176355411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2686
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587622C2semidirect
2. #11762737C2semidirect
3. #11762729C2semidirect
4. #11762723C2semidirect
5. #11762723C2semidirect
6. #11762723C2semidirect
7. #11762721C2semidirect
8. #11762697C2semidirect
9. #11762690C2semidirect
10. #11762689C2semidirect
11. #11762689C2semidirect
12. #11762682C2semidirect
13. #11762672C2semidirect
14. #11762667C2semidirect
272. C17#2686semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
#117627372
#117627292
#117627232
#117627212
#117626972
#117626902
#117626892
#117626822
#117626722
#117626672
C2×C17×#1212
#268617