Finite Group Databasebeta

← #14534832

Subgroups — #14534832

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762751C211yesmaximal, characteristic, complemented
2#11762750C211yesmaximal, characteristic, complemented
3#11762747C211yesmaximal, characteristic, complemented
4#11762742C211yesmaximal, characteristic, complemented
5#11762737C211yesmaximal, characteristic
6#11762737C211yesmaximal, characteristic
7#11762734C211yesmaximal, characteristic
8#11762732C211yesmaximal, characteristic
9#11762704C211yesmaximal, characteristic, complemented
10#11762695C211yesmaximal, characteristic, complemented
11#11762692C211yesmaximal, characteristic, complemented
12#11762691C211yesmaximal, characteristic, complemented
13#11762682C211yesmaximal, characteristic
14C4×C22⋊Dic17C211yesmaximal, characteristic
15C17×#385C211yesmaximal, characteristic
16C23×C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×C4×Dic17C2211yescharacteristic
34C2×C4×Dic17C2211yescharacteristic
35C17×#95C2211yescharacteristic
36C17×#95C2211yescharacteristic
37#10587502C2211yescharacteristic
38#10587502C2211yescharacteristic
39#10587502C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C22×C4×C17—22no—
53C22×C4×C17—41no—
54C22×C4×C17—41no—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C4⋊C4×C17—22no—
70C4⋊C4×C17—41no—
71C22⋊C4×C17—22no—
72C22⋊C4×C17—41no—
73C22⋊Dic17—22no—
74C22⋊Dic17—22no—
75C22⋊Dic17—22no—
76C22⋊Dic17—41no—
77C4⋊Dic17—22no—
78C4⋊Dic17—41no—
79#63225—22no—
80#63225—22no—
81#63225—41no—
82#63225—41no—
83#63225—41no—
84#63225—41no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17#4112yes—
94C23×C17C2411yescharacteristic
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×C17—21no—
112C2×C4×C17—44no—
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×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17#4112yes—
138#2652—171nomaximal, Sylow
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17C2×#4111yescharacteristic
147C22×C17C2×#4111yescharacteristic
148C22×C17C2×#4111yescharacteristic
149C22×C17H(5,𝔽2)11yescharacteristic
150C22×C17H(5,𝔽2)11yescharacteristic
151C22×C172−1+411yescharacteristic
152C22×C172−1+411yescharacteristic
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159Dic17—22no—
160Dic17—41no—
161Dic17—41no—
162Dic17—41no—
163Dic17—41no—
164Dic17—41no—
165Dic17—41no—
166Dic17—41no—
167#400—171no—
168#400—171no—
169#400—171no—
170#398—171no—
171#397—171no—
172#397—171no—
173#397—171no—
174#396—171no—
175#389—171no—
176#387—171no—
177#387—171no—
178#387—171no—
179#385—171no—
180#385D1711yescharacteristic
181C4×C22⋊C4—171no—
182C2×C17—42no—
183C2×C17#52911yescharacteristic
184C2×C17#53811yescharacteristic
185C2×C17#53911yescharacteristic
186C2×C17#53911yescharacteristic
187C2×C17#53911yescharacteristic
188C2×C17#54111yescharacteristic
189C2×C17#54211yescharacteristic
190C23×C4C2×D1711yescharacteristic
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×C22⋊C4—171no—
202C2×C22⋊C4—171no—
203C2×C22⋊C4—171no—
204C2×C22⋊C4—171no—
205C2×C22⋊C4C2×D1711yescharacteristic
206C2×C22⋊C4C2×D1711yescharacteristic
207C2×C42—171no—
208C2×C42—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#95—171no—
219#95—171no—
220#95—171no—
221#95—171no—
222#95—171no—
223#95C2×D1711yescharacteristic
224#95C2×D1711yescharacteristic
225C17#265211yescharacteristic, complemented, Sylow
226C24C22×D1711yescharacteristic
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4C22×D1711yescharacteristic
239C22×C4C22×D1711yescharacteristic
240C22×C4C22×D1711yescharacteristic
241C22×C4C22×D1711yescharacteristic
242C22×C4C22×D1711yescharacteristic
243C22×C4C22×D1711yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—342no—
247C4⋊C4—342no—
248C4⋊C4—342no—
249C4⋊C4—681no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C22⋊C4—22no—
255C22⋊C4—41no—
256C22⋊C4—342no—
257C22⋊C4—342no—
258C22⋊C4—342no—
259C22⋊C4—681no—
260C42—681no—
261C42—681no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#6325212yes—
269C23C23×D1711yescharacteristic
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—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—172no—
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×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4#6325412yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#6325211yescharacteristic
321C22C2×#6325411yescharacteristic
322C22#1058769111yescharacteristic
323C22#1058769111yescharacteristic
324C22#1058769411yescharacteristic
325C22#41×D1711yescharacteristic
326C22#1058769711yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—342no—
334C4—681no—
335C4—681no—
336C4—681no—
337C4—681no—
338C4—681no—
339C4—681no—
340C4—681no—
341C2—42no—
342C2#1176333911yescharacteristic
343C2#1176339411yescharacteristic
344C2#1176340111yescharacteristic
345C2#1176340511yescharacteristic
346C2#1176340611yescharacteristic
347C2#1176341911yescharacteristic
348C2#1176342711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2652
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762750C2semidirect
3. #11762747C2semidirect
4. #11762742C2semidirect
9. #11762704C2semidirect
10. #11762695C2semidirect
11. #11762692C2semidirect
12. #11762691C2semidirect
225. C17#2652semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627502
#117627472
#117627422
#117627372
#117627342
#117627322
#117627042
#117626952
#117626922
#117626912
#117626822
C4×C22⋊Dic172
C17×#3852
#265217