Finite Group Databasebeta

← #14535992

Subgroups — #14535992

Order 2176.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762744C211yesmaximal, characteristic, complemented
2#11762744C211yesmaximal, characteristic, complemented
3#11762743C211yesmaximal, characteristic, complemented
4#11762741C211yesmaximal, characteristic, complemented
5#11762733C211yesmaximal, characteristic, complemented
6#11762731C211yesmaximal, characteristic, complemented
7#11762711C211yesmaximal, characteristic
8#11762693C211yesmaximal, characteristic, complemented
9#11762687C211yesmaximal, characteristic
10#11762687C211yesmaximal, characteristic
11#11762683C211yesmaximal, characteristic, complemented
12#11762683C211yesmaximal, characteristic, complemented
13#11762668C211yesmaximal, characteristic, complemented
14#11762666C211yesmaximal, characteristic, complemented
15C17×#387C211yesmaximal, characteristic, complemented
16C22×D68C2211yescharacteristic
17C22×C4×D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C42×C17C2211yescharacteristic
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C17×#95C2211yescharacteristic
42C17×#95C2211yescharacteristic
43C17×#95C2211yescharacteristic
44C17×#95C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C23×D17C2311yescharacteristic
52C23×D17C2311yescharacteristic
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C2×D68—41no—
67C2×D68—41no—
68C2×D68Q814yes—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—41no—
72C2×C4×D17—41no—
73C2×C4×D17—41no—
74C2×C4×D17—41no—
75C4⋊C4×C17—22no—
76C4⋊C4×C17—41no—
77C42×C17—41no—
78#63227—22no—
79#63227—22no—
80#63227—41no—
81#63227—41no—
82#63227—41no—
83#63227—41no—
84#63227—41no—
85#63227—41no—
86#63227—41no—
87#63227—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91#63225—22no—
92#63225—41no—
93#63225—41no—
94#63225—41no—
95#63225—41no—
96#63225—41no—
97C23×C17C2411yescharacteristic
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17C2×Q812yes—
111C22×D17C2×Q812yes—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17C2×Q812yes—
132C2×C4×C17#4112yes—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17#4112yes—
151D68—44no—
152C4×D17—44no—
153C4×D17—44no—
154#2748—171nomaximal, Sylow
155C22×C17C22×Q811yescharacteristic
156C22×C17C2×#4111yescharacteristic
157C22×C17C2×#4111yescharacteristic
158C22×C17H(5,𝔽2)11yescharacteristic
159C22×C17H(5,𝔽2)11yescharacteristic
160C22×C17H(5,𝔽2)11yescharacteristic
161C22×C17H(5,𝔽2)11yescharacteristic
162C2×D17—22no—
163C2×D17—22no—
164C2×D17—22no—
165C2×D17—22no—
166C2×D17—22no—
167C2×D17—22no—
168C2×D17—41no—
169C2×D17—41no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C4×C17—22no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183Dic17—22no—
184Dic17—41no—
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189#399—171no—
190#399—171no—
191#398—171no—
192#398—171no—
193#396—171no—
194#396—171no—
195#390—171no—
196#387—171no—
197#387D1711yescharacteristic, complemented
198#386—171no—
199#386—171no—
200#385—171no—
201#385—171no—
202#380—171no—
203#380—171no—
204C2×C17#54011yescharacteristic
205C2×C17#54011yescharacteristic
206C2×C17#54311yescharacteristic
207C2×C17#55111yescharacteristic
208C2×C17#55411yescharacteristic
209C2×C17#55411yescharacteristic
210C2×C17#56011yescharacteristic
211D17—42no—
212D17—42no—
213C22×D4—171no—
214C23×C4—171no—
215C23×C4—171no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4C2×D1711yescharacteristic
226C2×C4⋊C4C2×D1711yescharacteristic
227C2×C22⋊C4—171no—
228C2×C22⋊C4—171no—
229C2×C22⋊C4—171no—
230C2×C22⋊C4—171no—
231C2×C22⋊C4—171no—
232C2×C22⋊C4—171no—
233C2×C22⋊C4—171no—
234C2×C22⋊C4—171no—
235C2×C22⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C42C2×D1711yescharacteristic
238#95—171no—
239#95—171no—
240#95—171no—
241#95—171no—
242#95—171no—
243#95—171no—
244#95C2×D1711yescharacteristic
245#95C2×D1711yescharacteristic
246#95C2×D1711yescharacteristic
247#95C2×D1711yescharacteristic
248C17#274811yescharacteristic, complemented, Sylow
249C24—171no—
250C24—171no—
251C2×D4—174no—
252C2×D4—681no—
253C2×D4—681no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—171no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—342no—
261C22×C4—342no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4C22×D1711yescharacteristic
267C22×C4C22×D1711yescharacteristic
268C22×C4C22×D1711yescharacteristic
269C22×C4C22×D1711yescharacteristic
270C22×C4C22×D1711yescharacteristic
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—342no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C22⋊C4—342no—
285C22⋊C4—342no—
286C22⋊C4—681no—
287C22⋊C4—681no—
288C22⋊C4—681no—
289C22⋊C4—681no—
290C22⋊C4—681no—
291C22⋊C4—681no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C42—41no—
295C23—172no—
296C23—172no—
297C23—341no—
298C23—341no—
299C23—341no—
300C23—341no—
301C23—341no—
302C23—341no—
303C23—341no—
304C23—341no—
305C23—341no—
306C23—341no—
307C23—341no—
308C23—341no—
309C23C23×D1711yescharacteristic
310D4—684no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—172no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—684no—
349C2×C4—684no—
350C2×C4#6325212yes—
351C2×C4Q8×D1712yes—
352C22—342no—
353C22—342no—
354C22—342no—
355C22—342no—
356C22—342no—
357C22—342no—
358C22—681no—
359C22—681no—
360C22—681no—
361C22—681no—
362C22—681no—
363C22—681no—
364C22—681no—
365C22—681no—
366C22C2×#6325211yescharacteristic
367C22#1058769111yescharacteristic
368C22C2×Q8×D1711yescharacteristic
369C22#41×D1711yescharacteristic
370C22#1058769611yescharacteristic
371C22#1058769611yescharacteristic
372C22#1058769611yescharacteristic
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—342no—
381C4—681no—
382C4—681no—
383C4—681no—
384C4—681no—
385C4—681no—
386C2—682no—
387C2—682no—
388C2#1176341011yescharacteristic
389C2#1176341111yescharacteristic
390C2#1176343411yescharacteristic
391C2#1176348311yescharacteristic
392C2#1176351511yescharacteristic
393C2#1176351911yescharacteristic
394C2#1176355011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#387C2
prime radicalC22×D68C22
hypercenter#387D17
hypocenterC17#2748
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. #11762744C2semidirect
2. #11762744C2semidirect
3. #11762743C2semidirect
4. #11762741C2semidirect
5. #11762733C2semidirect
6. #11762731C2semidirect
8. #11762693C2semidirect
11. #11762683C2semidirect
12. #11762683C2semidirect
13. #11762668C2semidirect
14. #11762666C2semidirect
15. C17×#387C2semidirect
197. #387D17semidirect
248. C17#2748semidirect

Maximal subgroups

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

subgroupindex
#117627442
#117627432
#117627412
#117627332
#117627312
#117627112
#117626932
#117626872
#117626832
#117626682
#117626662
C17×#3872
#274817