Finite Group Databasebeta

← #14534669

Subgroups — #14534669

Order 2176.

620 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D17C211yesmaximal, characteristic, complemented
2#11762747C211yesmaximal, characteristic, complemented
3#11762744C211yesmaximal, characteristic, complemented
4#11762723C211yesmaximal, characteristic, complemented
5#11762723C212yesmaximal, complemented
6#11762717C212yesmaximal, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762697C212yesmaximal, complemented
9#11762685C212yesmaximal, complemented
10#11762672C211yesmaximal, characteristic, complemented
11C2×C17×#121C211yesmaximal, characteristic, complemented
12C22×D4×C17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C22×C22⋊D17C2212yes—
17C22×C4×D17C2211yescharacteristic
18C17×#121—24no—
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×C22⋊Dic17C2212yes—
26C4×C22⋊D17—24no—
27C2×#63227C2211yescharacteristic
28C2×#63227C2212yes—
29C2×#63225C2211yescharacteristic
30C2×#63225C2212yes—
31C2×C4×Dic17C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2312yes—
38C24×C17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C2×D4×C17—24no—
41C2×D4×C17—22no—
42C2×D4×C17—42no—
43C2×D4×C17—42no—
44C2×D4×C17—42no—
45C22×C4×C17—42no—
46C22×C4×C17D414yescomplemented
47C22×C4×C17C2312yes—
48C22×C4×C17C2311yescharacteristic
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—22no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—24no—
53C2×C22⋊D17—22no—
54C2×C22⋊D17—42no—
55C2×C22⋊D17—42no—
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2312yes—
58C22×Dic17C2312yes—
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C2×C4×D17—22no—
62C2×C4×D17—42no—
63C4⋊C4×C17—22no—
64C22⋊C4×C17—24no—
65C22⋊Dic17—22no—
66C22⋊Dic17—24no—
67C22⋊Dic17—42no—
68C22⋊Dic17—42no—
69C22⋊Dic17—42no—
70#63227—22no—
71#63227—42no—
72#63225—22no—
73#63225—42no—
74C4×Dic17—22no—
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17C2×D412yes—
87C23×C17C2411yescharacteristic
88C22×D17—21no—
89C22×D17—21no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17#4112yes—
93D4×C17—44no—
94D4×C17—48no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—44no—
101C2×C4×C17C2×D412yes—
102C2×C4×C17C2×D412yes—
103C22⋊D17—28no—
104C22⋊D17—48no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17#4112yes—
118C4×D17—44no—
119#2635—171nomaximal, Sylow
120C22×C17—22no—
121C22×C17—22no—
122C22×C17—22no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17C22×D411yescharacteristic
133C22×C17C2×#4111yescharacteristic
134C22×C17C2×#4111yescharacteristic
135C22×C17H(5,𝔽2)12yes—
136C22×C17H(5,𝔽2)12yes—
137C2×D17—22no—
138C2×D17—24no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—42no—
142C4×C17—22no—
143C4×C17—22no—
144C4×C17—42no—
145Dic17—22no—
146Dic17—41no—
147Dic17—41no—
148Dic17—42no—
149Dic17—42no—
150C2×#121D1711yescharacteristic, complemented
151C2×C4×D4—171no—
152#400—171no—
153#399—171no—
154#394—171no—
155#394—172no—
156#392—171no—
157#392—172no—
158#388—172no—
159#386—172no—
160#381—171no—
161C2×C17—42no—
162C2×C17—82no—
163C2×C17#52511yescharacteristic
164C2×C17#53411yescharacteristic
165C2×C17#53511yescharacteristic
166C2×C17#53812yes—
167C2×C17#54012yes—
168D17—42no—
169C22×D4—171no—
170C22×D4—172no—
171C22×D4C2×D1712yes—
172C22×D4C2×D1711yescharacteristic
173C23×C4—171no—
174C23×C4C2×D1711yescharacteristic
175#121—24no—
176C4×D4—344no—
177C2×C4⋊C4—171no—
178C2×C4⋊C4—172no—
179C2×C4⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4—171no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—171no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4—172no—
187C2×C22⋊C4C2×D1712yes—
188C2×C42—171no—
189#95—172no—
190#95—171no—
191#95—171no—
192#95—172no—
193#95—172no—
194C17#263511yescharacteristic, complemented, Sylow
195C24—171no—
196C24C22×D1712yes—
197C24C22×D1711yescharacteristic
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—342no—
204C2×D4—342no—
205C2×D4—342no—
206C2×D4—344no—
207C2×D4—682no—
208C2×D4—682no—
209C22×C4—42no—
210C22×C4—171no—
211C22×C4—172no—
212C22×C4—172no—
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—342no—
216C22×C4—682no—
217C22×C4C22⋊D1714yescomplemented
218C22×C4C22×D1712yes—
219C22×C4C22×D1711yescharacteristic
220C22×C4C22×D1711yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—342no—
223C4⋊C4—682no—
224C22⋊C4—24no—
225C22⋊C4—344no—
226C22⋊C4—342no—
227C22⋊C4—342no—
228C22⋊C4—682no—
229C22⋊C4—682no—
230C22⋊C4—682no—
231C22⋊C4—682no—
232C42—342no—
233C23—22no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—172no—
245C23—342no—
246C23—342no—
247C23—341no—
248C23—341no—
249C23C2×C22⋊D1712yes—
250C23C23×D1711yescharacteristic
251D4—44no—
252D4—48no—
253D4—348no—
254D4—688no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—44no—
261C2×C4—172no—
262C2×C4—341no—
263C2×C4—341no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—341no—
273C2×C4—341no—
274C2×C4—684no—
275C2×C4C2×C22⋊D1712yes—
276C2×C4C2×C22⋊D1712yes—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—342no—
290C22—344no—
291C22—681no—
292C22—681no—
293C22—682no—
294C22#1058769112yes—
295C22#1058769112yes—
296C22#41×D1711yescharacteristic
297C22#41×D1711yescharacteristic
298C22C22×C22⋊D1711yescharacteristic
299C4—22no—
300C4—22no—
301C4—42no—
302C4—342no—
303C4—681no—
304C4—681no—
305C4—682no—
306C4—682no—
307C2—42no—
308C2—82no—
309C2—682no—
310C2#1176331611yescharacteristic
311C2#1176336811yescharacteristic
312C2#1176337611yescharacteristic
313C2#1176339112yes—
314C2#1176340912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2635
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊D17C2semidirect
2. #11762747C2semidirect
3. #11762744C2semidirect
4. #11762723C2semidirect
5. #11762723C2semidirect
6. #11762717C2semidirect
7. #11762717C2semidirect
8. #11762697C2semidirect
9. #11762685C2semidirect
10. #11762672C2semidirect
11. C2×C17×#121C2semidirect
46. C22×C4×C17D4semidirect
150. C2×#121D17semidirect
194. C17#2635semidirect
217. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D172
#117627472
#117627442
#117627232
#117627172
#117626972
#117626852
#117626722
C2×C17×#1212
#263517