Finite Group Databasebeta

← #14535558

Subgroups — #14535558

Order 2176.

432 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587630C211yesmaximal, characteristic
2C2×#10587576C211yesmaximal, characteristic
3#11762751C211yesmaximal, characteristic
4#11762749C211yesmaximal, characteristic
5#11762742C211yesmaximal, characteristic
6#11762740C211yesmaximal, characteristic
7#11762729C211yesmaximal, characteristic
8#11762716C211yesmaximal, characteristic
9#11762713C211yesmaximal, characteristic
10#11762704C211yesmaximal, characteristic
11#11762696C211yesmaximal, characteristic
12#11762695C211yesmaximal, characteristic
13#11762691C211yesmaximal, characteristic
14#11762691C211yesmaximal, characteristic
15C17×#395C211yesmaximal, characteristic
16C22×Q8×C17C2211yescharacteristic
17C22×Dic34C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21#10587630—24no—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×C4×Dic17C2211yescharacteristic
34C2×C4×Dic17C2211yescharacteristic
35C2×C4×Dic17C2211yescharacteristic
36#10587576—24no—
37C17×#95C2211yescharacteristic
38C17×#95C2211yescharacteristic
39C17×#95C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C2×Q8×C17—22no—
54C2×Q8×C17—22no—
55C2×Q8×C17—41no—
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C2×Dic34—22no—
72C2×Dic34—22no—
73C2×Dic34—41no—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—22no—
76C4⋊C4×C17—41no—
77C4⋊Dic17—22no—
78C4⋊Dic17—22no—
79C4⋊Dic17—22no—
80#63225—22no—
81#63225—22no—
82#63225—22no—
83#63225—22no—
84#63225—41no—
85#63225—41no—
86#63225D414yes—
87#63225Q814yescomplemented
88C4×Dic17—22no—
89C4×Dic17—22no—
90C4×Dic17—41no—
91C23×C17C2411yescharacteristic
92Q8×C17—44no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17C2×D412yes—
112C2×C4×C17C2×Q812yes—
113C2×C4×C17#4112yes—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17C2×D412yes—
135C2×Dic17C2×D412yes—
136C2×Dic17C2×Q812yes—
137C2×Dic17C2×Q812yes—
138Dic34—44no—
139#2713—171nomaximal, Sylow
140C22×C17C22×D411yescharacteristic
141C22×C17C22×Q811yescharacteristic
142C22×C17C2×#4111yescharacteristic
143C22×C17H(5,𝔽2)11yescharacteristic
144C22×C172−1+411yescharacteristic
145C22×C172−1+411yescharacteristic
146C22×C172−1+411yescharacteristic
147C4×C17—22no—
148C4×C17—22no—
149C4×C17—22no—
150C4×C17—41no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—41no—
154Dic17—22no—
155Dic17—22no—
156Dic17—22no—
157Dic17—22no—
158Dic17—41no—
159Dic17—41no—
160Dic17—41no—
161Dic17—41no—
162C2×C4⋊Q8—171no—
163C2×#125—171no—
164#400—171no—
165#400—171no—
166#398—171no—
167#398—171no—
168#395—171no—
169#395D1711yescharacteristic
170#391—171no—
171#391—171no—
172#389—171no—
173#387—171no—
174#387—171no—
175#387—171no—
176#387—171no—
177C2×C17#53611yescharacteristic
178C2×C17#54111yescharacteristic
179C2×C17#54411yescharacteristic
180C2×C17#54611yescharacteristic
181C2×C17D4×Q811yescharacteristic
182C2×C17#55711yescharacteristic
183C2×C17#56311yescharacteristic
184C22×Q8—171no—
185C22×Q8C2×D1711yescharacteristic
186C4⋊Q8—344no—
187#125—344no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4—171no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4C2×D1711yescharacteristic
200C2×C4⋊C4C2×D1711yescharacteristic
201C2×C4⋊C4C2×D1711yescharacteristic
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—171no—
205#95—171no—
206#95—171no—
207#95—171no—
208#95—171no—
209#95—171no—
210#95—171no—
211#95—171no—
212#95—171no—
213#95—171no—
214#95—171no—
215#95—171no—
216#95—171no—
217#95—171no—
218#95C2×D1711yescharacteristic
219#95C2×D1711yescharacteristic
220#95C2×D1711yescharacteristic
221C17#271311yescharacteristic, complemented, Sylow
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—342no—
226C2×Q8—342no—
227C2×Q8—681no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4C22×D1711yescharacteristic
237C22×C4C22×D1711yescharacteristic
238C22×C4C22×D1711yescharacteristic
239C22×C4C22×D1711yescharacteristic
240C22×C4C22×D1711yescharacteristic
241C22×C4C22×D1711yescharacteristic
242C22×C4C22×D1711yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—174no—
247C4⋊C4—174no—
248C4⋊C4—342no—
249C4⋊C4—342no—
250C4⋊C4—342no—
251C4⋊C4—342no—
252C4⋊C4—342no—
253C4⋊C4—342no—
254C4⋊C4—342no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C42—342no—
258C42—342no—
259C42—681no—
260C23C23×D1711yescharacteristic
261Q8—44no—
262Q8—684no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—172no—
282C2×C4—172no—
283C2×C4—172no—
284C2×C4—172no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4D4×D1712yes—
306C2×C4#6325412yes—
307C2×C4Q8×D1712yes—
308C22C2×D4×D1711yescharacteristic
309C22C2×#6325411yescharacteristic
310C22#1058769111yescharacteristic
311C22C2×Q8×D1711yescharacteristic
312C22#1058769411yescharacteristic
313C22#1058769411yescharacteristic
314C22#1058769711yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—342no—
323C4—342no—
324C4—342no—
325C4—342no—
326C4—681no—
327C4—681no—
328C4—681no—
329C4—681no—
330C2#1176338411yescharacteristic
331C2#1176341911yescharacteristic
332C2#1176343611yescharacteristic
333C2#1176344611yescharacteristic
334C2#1176347211yescharacteristic
335C2#1176353411yescharacteristic
336C2#1176356011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2713
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
87. #63225Q8semidirect
221. C17#2713semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875762
#117627512
#117627492
#117627422
#117627402
#117627292
#117627162
#117627132
#117627042
#117626962
#117626952
#117626912
C17×#3952
#271317