Finite Group Databasebeta

← #14540671

Subgroups — #14540671

Order 2176.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763499C211yesmaximal, characteristic, complemented
2#11763163C211yesmaximal, characteristic, complemented
3#11763147C211yesmaximal, characteristic, complemented
4#11763120C211yesmaximal, characteristic, complemented
5#11763116C211yesmaximal, characteristic, complemented
6#11763048C211yesmaximal, characteristic, complemented
7#11763036C211yesmaximal, characteristic, complemented
8#11763028C211yesmaximal, characteristic, complemented
9#11762971C211yesmaximal, characteristic, complemented
10#11762965C211yesmaximal, characteristic, complemented
11C4×#63231C211yesmaximal, characteristic, complemented
12D4×#3622C211yesmaximal, characteristic, complemented
13#11762805C211yesmaximal, characteristic, complemented
14C2×#10587506C211yesmaximal, characteristic, complemented
15C17×#548C211yesmaximal, characteristic
16C2×#41×C17C2211yescharacteristic
17C22×Q8×C17C2211yescharacteristic
18C17×#124—21no—
19C17×#124C2211yescharacteristic
20C17×C22⋊Q8—21no—
21C17×C22⋊Q8C2211yescharacteristic
22C17×#121—21no—
23C17×#121C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17—21no—
26C4×D4×C17C2211yescharacteristic
27C2×#63231C2211yescharacteristic
28C2×#63231C2211yescharacteristic
29#10587621—21no—
30C2×#63229C2211yescharacteristic
31#10587612C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33#10587610C2211yescharacteristic
34#10587610C2211yescharacteristic
35C22×#3622C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587575—21no—
38#10587565—21no—
39#10587563—21no—
40#10587556C2211yescharacteristic
41C4×Dic34C2211yescharacteristic
42#10587506C2211yescharacteristic
43#10587506C2211yescharacteristic
44#10587506C2211yescharacteristic
45#10587506C2211yescharacteristic
46#10587506C2211yescharacteristic
47#10587506C2211yescharacteristic
48#10587503C2211yescharacteristic
49#10587503C2211yescharacteristic
50#10587503C2211yescharacteristic
51C22⋊#3622C2211yescharacteristic
52C22⋊#3622C2211yescharacteristic
53#10587481C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587478C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58C4×#3622C2211yescharacteristic
59#41×C17—22no—
60#41×C17—41no—
61C2×Q8×C17—21no—
62C2×Q8×C17—22no—
63C2×Q8×C17—41no—
64C2×Q8×C17C2311yescharacteristic
65C2×Q8×C17C2311yescharacteristic
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17D411yescharacteristic
69C2×D4×C17C2311yescharacteristic
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C22×Dic17—21no—
75C2×Dic34C2311yescharacteristic
76C4⋊C4×C17—21no—
77C4⋊C4×C17—21no—
78C4⋊C4×C17D411yescharacteristic
79C4⋊C4×C17C2311yescharacteristic
80C22⋊C4×C17—21no—
81C22⋊C4×C17—21no—
82C22⋊C4×C17—21no—
83C22⋊C4×C17—21no—
84C22⋊C4×C17D411yescharacteristic
85C22⋊C4×C17D411yescharacteristic
86C42×C17—21no—
87C42×C17C2311yescharacteristic
88C22⋊Dic17—21no—
89C22⋊Dic17—21no—
90C22⋊Dic17—21no—
91#63231—22no—
92#63231—41no—
93#63229—41no—
94C4⋊Dic17—21no—
95C4⋊Dic17C2311yescharacteristic
96C4⋊Dic17C2311yescharacteristic
97C4⋊Dic17C2311yescharacteristic
98#63225—21no—
99#63225—21no—
100#63225—21no—
101#63225—21no—
102#63225—21no—
103C4×Dic17—21no—
104C4×Dic17—21no—
105C2×#3622—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17—21no—
112C23×C17C2×D411yescharacteristic
113C23×C17C2×D411yescharacteristic
114Q8×C17—21no—
115Q8×C17—21no—
116Q8×C17—21no—
117Q8×C17—42no—
118Q8×C17#4112yes—
119D4×C17—22no—
120D4×C17—21no—
121D4×C17—21no—
122D4×C17—42no—
123D4×C17—42no—
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×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17C2×D411yescharacteristic
134C2×C4×C17C2×D411yescharacteristic
135C2×C4×C17C2×D411yescharacteristic
136C2×C4×C17C2×D411yescharacteristic
137C2×C4×C17C2411yescharacteristic
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—42no—
143Dic34—21no—
144Dic34—21no—
145#3622—21no—
146#3622—21no—
147#3622—21no—
148#3622—21no—
149#3300—171nomaximal, Sylow
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17#13512yes—
159C22×C17C22×D411yescharacteristic
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—22no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17C2×#4111yescharacteristic
168C4×C17H(5,𝔽2)11yescharacteristic
169Dic17—41no—
170Dic17—41no—
171Dic17—41no—
172Dic17—41no—
173#552—171no—
174#548D1711yescharacteristic
175#488—171no—
176#485—171no—
177#480—171no—
178#479—171no—
179#467—171no—
180#465—171no—
181#464—171no—
182#452—171no—
183#451—171no—
184C4×Q16—171no—
185C8×D4—171no—
186#416—171no—
187C2×#103—171no—
188C2×C17—22no—
189C2×C17—41no—
190C2×C17—81no—
191C2×C17#54611yescharacteristic
192C2×C17C2×#13511yescharacteristic
193C2×C17#57811yescharacteristic
194C2×#41C2×D1711yescharacteristic
195C22×Q8C2×D1711yescharacteristic
196C2×Q16—171no—
197C2×Q16—171no—
198C2×SD16—171no—
199C22×C8—171no—
200C22×C8—171no—
201#126—341no—
202#125—171no—
203#125—341no—
204#124—21no—
205#124C2×D1711yescharacteristic
206#123—341no—
207C22⋊Q8—21no—
208C22⋊Q8—171no—
209C22⋊Q8—171no—
210C22⋊Q8C2×D1711yescharacteristic
211#121—21no—
212#121C2×D1711yescharacteristic
213C4×Q8—171no—
214C4×Q8C2×D1711yescharacteristic
215C4×D4—21no—
216C4×D4C2×D1711yescharacteristic
217#117—171no—
218#117—341no—
219C2×C4⋊C4—171no—
220#107—171no—
221#107—171no—
222#106—171no—
223C4⋊C8—171no—
224#103—171no—
225#103—171no—
226#103—171no—
227#103—171no—
228#103—171no—
229#103—171no—
230#103—171no—
231#102—171no—
232#102—171no—
233#102—171no—
234C22⋊C8—171no—
235C22⋊C8—171no—
236C4×C8—171no—
237C17#330011yescharacteristic, complemented, Sylow
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—171no—
244C2×Q8C22×D1711yescharacteristic
245C2×Q8C22×D1711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D1711yescharacteristic
249C2×D4C22×D1711yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—341no—
253C22×C4C22×D1711yescharacteristic
254C22×C4C22×D1711yescharacteristic
255Q16—342no—
256Q16—681no—
257SD16—681no—
258C2×C8—171no—
259C2×C8—171no—
260C2×C8—171no—
261C2×C8—171no—
262C2×C8—681no—
263C2×C8—681no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—171no—
267C4⋊C4—171no—
268C4⋊C4—171no—
269C4⋊C4—341no—
270C4⋊C4—341no—
271C4⋊C4—341no—
272C4⋊C4—341no—
273C4⋊C4—341no—
274C4⋊C4—341no—
275C4⋊C4C22⋊D1711yescharacteristic
276C4⋊C4C22×D1711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—341no—
282C22⋊C4—341no—
283C22⋊C4—341no—
284C22⋊C4C22⋊D1711yescharacteristic
285C22⋊C4C22⋊D1711yescharacteristic
286C42—21no—
287C42—341no—
288C42—341no—
289C42C22×D1711yescharacteristic
290C23—21no—
291C23C2×C22⋊D1711yescharacteristic
292C23C2×C22⋊D1711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—341no—
298Q8—341no—
299Q8#6325412yes—
300D4—22no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—682no—
319C2×C4C2×C22⋊D1711yescharacteristic
320C2×C4C2×C22⋊D1711yescharacteristic
321C2×C4C2×C22⋊D1711yescharacteristic
322C2×C4C2×C22⋊D1711yescharacteristic
323C2×C4C23×D1711yescharacteristic
324C8—341no—
325C8—341no—
326C8—341no—
327C8—341no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1058763612yes—
337C22C22×C22⋊D1711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—681no—
346C4—681no—
347C4—681no—
348C4—681no—
349C4C2×#6325411yescharacteristic
350C4#1058769111yescharacteristic
351C2—22no—
352C2—41no—
353C2—81no—
354C2#1176345311yescharacteristic
355C2C2×#1058763611yescharacteristic
356C2#1176362411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#548C2
socleC22×C17C22×D4
hypercenter#548D17
hypocenterC17#3300
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #11763499C2semidirect
2. #11763163C2semidirect
3. #11763147C2semidirect
4. #11763120C2semidirect
5. #11763116C2semidirect
6. #11763048C2semidirect
7. #11763036C2semidirect
8. #11763028C2semidirect
9. #11762971C2semidirect
10. #11762965C2semidirect
11. C4×#63231C2semidirect
12. D4×#3622C2semidirect
13. #11762805C2semidirect
14. C2×#10587506C2semidirect
237. C17#3300semidirect

Maximal subgroups

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

subgroupindex
#117634992
#117631632
#117631472
#117631202
#117631162
#117630482
#117630362
#117630282
#117629712
#117629652
C4×#632312
D4×#36222
#117628052
C2×#105875062
C17×#5482
#330017