Finite Group Databasebeta

← #14535259

Subgroups — #14535259

Order 2176.

554 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587570C211yesmaximal, characteristic
2#11762742C211yesmaximal, characteristic, complemented
3#11762735C211yesmaximal, characteristic, complemented
4#11762731C211yesmaximal, characteristic, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762724C211yesmaximal, characteristic, complemented
7#11762724C211yesmaximal, characteristic, complemented
8#11762711C211yesmaximal, characteristic, complemented
9#11762693C211yesmaximal, characteristic, complemented
10#11762686C211yesmaximal, characteristic, complemented
11#11762686C211yesmaximal, characteristic, complemented
12#11762684C211yesmaximal, characteristic, complemented
13#11762680C211yesmaximal, characteristic, complemented
14#11762666C211yesmaximal, characteristic, complemented
15C2×C17×#126C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C23×Dic17C2211yescharacteristic
19C22×D68C2211yescharacteristic
20C22×C4×D17C2211yescharacteristic
21C17×#126—24no—
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C4⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C42×C17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×C4⋊Dic17C2211yescharacteristic
40C2×C4⋊Dic17C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×#63225C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44#10587570—24no—
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C23×D17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17—22no—
63C2×C22⋊D17—22no—
64C2×C22⋊D17—22no—
65C2×C22⋊D17—22no—
66C2×C22⋊D17—41no—
67C2×C22⋊D17—41no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17—41no—
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C2×D68—22no—
78C2×D68—22no—
79C2×D68—41no—
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—41no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—22no—
86C22⋊C4×C17—22no—
87C22⋊C4×C17—22no—
88C22⋊C4×C17D414yes—
89C42×C17—22no—
90C22⋊Dic17—41no—
91#63227—22no—
92#63227—22no—
93#63227—41no—
94#63227—41no—
95#63227—41no—
96#63227—41no—
97#63227—41no—
98#63227—41no—
99#63227—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102#63225—22no—
103#63225—22no—
104#63225—41no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17C2×D412yes—
112C23×C17C2411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17#4112yes—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
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—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17C2×D412yes—
144C2×C4×C17C2×D412yes—
145C22⋊D17—44no—
146C22⋊D17—44no—
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—44no—
165C2×Dic17#4112yes—
166D68—44no—
167C4×D17—44no—
168#2691—171nomaximal, Sylow
169C22×C17—22no—
170C22×C17—22no—
171C22×C17—22no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17C22×D411yescharacteristic
177C22×C17C2×#4111yescharacteristic
178C22×C17C2×#4111yescharacteristic
179C22×C17H(5,𝔽2)11yescharacteristic
180C22×C17H(5,𝔽2)11yescharacteristic
181C22×C17H(5,𝔽2)11yescharacteristic
182C22×C172−1+411yescharacteristic
183C2×D17—22no—
184C2×D17—22no—
185C2×D17—22no—
186C2×D17—41no—
187C2×D17—41no—
188C2×D17—41no—
189C2×D17—41no—
190C2×D17—41no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C2×D17—41no—
196C2×D17—41no—
197C4×C17—22no—
198C4×C17—22no—
199C4×C17—41no—
200C4×C17—41no—
201C4×C17—41no—
202C4×C17—41no—
203Dic17—22no—
204Dic17—41no—
205Dic17—41no—
206Dic17—41no—
207Dic17—41no—
208Dic17—41no—
209C2×#126D1711yescharacteristic, complemented
210C2×#121—171no—
211#398—171no—
212#397—171no—
213#396—171no—
214#394—171no—
215#394—171no—
216#394—171no—
217#390—171no—
218#387—171no—
219#386—171no—
220#386—171no—
221#385—171no—
222#385—171no—
223#380—171no—
224C2×C17—42no—
225C2×C17#53711yescharacteristic
226C2×C17#53911yescharacteristic
227C2×C17#54211yescharacteristic
228C2×C17#54611yescharacteristic
229C2×C17#54611yescharacteristic
230C2×C17#55311yescharacteristic
231C2×C17#56011yescharacteristic
232D17—42no—
233D17—81no—
234C22×D4—171no—
235C22×D4—171no—
236C22×D4—171no—
237C23×C4—171no—
238C23×C4—171no—
239#126—24no—
240#121—344no—
241C2×C4⋊C4—171no—
242C2×C4⋊C4—171no—
243C2×C4⋊C4—171no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4C2×D1711yescharacteristic
247C2×C4⋊C4C2×D1711yescharacteristic
248C2×C4⋊C4C2×D1711yescharacteristic
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×C22⋊C4C2×D1711yescharacteristic
262C2×C42C2×D1711yescharacteristic
263#95—171no—
264#95—171no—
265#95—171no—
266#95—171no—
267#95—171no—
268#95—171no—
269#95—171no—
270#95—171no—
271C17#269111yescharacteristic, complemented, Sylow
272C24—171no—
273C24—171no—
274C24C22×D1711yescharacteristic
275C2×D4—342no—
276C2×D4—342no—
277C2×D4—342no—
278C2×D4—342no—
279C2×D4—342no—
280C2×D4—342no—
281C2×D4—681no—
282C2×D4—681no—
283C2×D4—681no—
284C22×C4—171no—
285C22×C4—171no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—171no—
290C22×C4—342no—
291C22×C4—342no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4C22×D1711yescharacteristic
297C22×C4C22×D1711yescharacteristic
298C22×C4C22×D1711yescharacteristic
299C22×C4C22×D1711yescharacteristic
300C22×C4C22×D1711yescharacteristic
301C22×C4C22×D1711yescharacteristic
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—342no—
306C4⋊C4—342no—
307C4⋊C4—681no—
308C4⋊C4—681no—
309C4⋊C4—681no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—342no—
313C22⋊C4—342no—
314C22⋊C4—681no—
315C22⋊C4—681no—
316C22⋊C4—681no—
317C22⋊C4—681no—
318C22⋊C4—681no—
319C22⋊C4—681no—
320C22⋊C4—681no—
321C22⋊C4—681no—
322C22⋊C4C22⋊D1714yes—
323C42—22no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—172no—
331C23—341no—
332C23—341no—
333C23—341no—
334C23—341no—
335C23—341no—
336C23—341no—
337C23—341no—
338C23—341no—
339C23—341no—
340C23—341no—
341C23—341no—
342C23—341no—
343C23—341no—
344C23C2×C22⋊D1712yes—
345C23C23×D1711yescharacteristic
346D4—684no—
347D4—684no—
348D4—684no—
349C2×C4—21no—
350C2×C4—21no—
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—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—172no—
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—684no—
384C2×C4—684no—
385C2×C4C2×C22⋊D1712yes—
386C2×C4C2×C22⋊D1712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—342no—
395C22—342no—
396C22—342no—
397C22—681no—
398C22—681no—
399C22—681no—
400C22—681no—
401C22—681no—
402C22—681no—
403C22—681no—
404C22—681no—
405C22—681no—
406C22—681no—
407C22—681no—
408C22#41×D1711yescharacteristic
409C22#41×D1711yescharacteristic
410C22#1058769611yescharacteristic
411C22#1058769611yescharacteristic
412C22#1058769611yescharacteristic
413C22#1058769711yescharacteristic
414C22C22×C22⋊D1711yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—342no—
422C4—681no—
423C4—681no—
424C4—681no—
425C4—681no—
426C4—681no—
427C2—42no—
428C2—682no—
429C2—1361no—
430C2#1176338711yescharacteristic
431C2#1176340011yescharacteristic
432C2#1176342511yescharacteristic
433C2#1176344711yescharacteristic
434C2#1176344711yescharacteristic
435C2#1176351011yescharacteristic
436C2#1176355011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#126C2
prime radicalC2×#10587570C2
hypercenterC2×#126D17
hypocenterC17#2691
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. #11762742C2semidirect
3. #11762735C2semidirect
4. #11762731C2semidirect
5. #11762726C2semidirect
6. #11762724C2semidirect
7. #11762724C2semidirect
8. #11762711C2semidirect
9. #11762693C2semidirect
10. #11762686C2semidirect
11. #11762686C2semidirect
12. #11762684C2semidirect
13. #11762680C2semidirect
14. #11762666C2semidirect
15. C2×C17×#126C2semidirect
209. C2×#126D17semidirect
271. C17#2691semidirect

Maximal subgroups

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

subgroupindex
C2×#105875702
#117627422
#117627352
#117627312
#117627262
#117627242
#117627112
#117626932
#117626862
#117626842
#117626802
#117626662
C2×C17×#1262
#269117