Finite Group Databasebeta

← #14535153

Subgroups — #14535153

Order 2176.

592 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587583C211yesmaximal, characteristic, complemented
2C2×#10587582C211yesmaximal, characteristic, complemented
3C2×#10587564C211yesmaximal, characteristic, complemented
4C2×#10587564C211yesmaximal, characteristic, complemented
5#11762733C211yesmaximal, characteristic, complemented
6#11762732C211yesmaximal, characteristic, complemented
7#11762731C211yesmaximal, characteristic, complemented
8#11762730C211yesmaximal, characteristic, complemented
9#11762729C211yesmaximal, characteristic, complemented
10#11762709C211yesmaximal, characteristic
11#11762705C211yesmaximal, characteristic, complemented
12#11762704C211yesmaximal, characteristic, complemented
13#11762686C211yesmaximal, characteristic
14#11762685C211yesmaximal, characteristic
15C17×#396C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C22×Dic34C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×C4⋊Dic17C2211yescharacteristic
36C2×C4⋊Dic17C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×#63225C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44C2×C4×Dic17C2211yescharacteristic
45#10587583—24no—
46#10587582—24no—
47#10587564—24no—
48#10587564—24no—
49C17×#95C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53#10587502C2211yescharacteristic
54#10587502C2211yescharacteristic
55C24×C17C2311yescharacteristic
56C23×D17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C2×C22⋊D17—41no—
64C2×C22⋊D17—41no—
65C2×C22⋊D17Q814yescomplemented
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—41no—
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×Dic34—22no—
80C2×Dic34—22no—
81C2×Dic34—22no—
82C2×Dic34—22no—
83C2×Dic34—41no—
84C2×Dic34—41no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—41no—
88C22⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—41no—
91C22⋊Dic17—22no—
92C22⋊Dic17—22no—
93#63227—22no—
94#63227—22no—
95#63227—22no—
96#63227—22no—
97#63227—41no—
98C4⋊Dic17—22no—
99C4⋊Dic17—22no—
100C4⋊Dic17D414yescomplemented
101#63225—22no—
102#63225—22no—
103#63225—22no—
104#63225—22no—
105#63225—41no—
106#63225—41no—
107#63225D414yescomplemented
108C4×Dic17—41no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17C2×Q812yes—
116C23×C17C2411yescharacteristic
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17C2×Q812yes—
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—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17C2×D412yes—
141C2×C4×C17C2×D412yes—
142C22⋊D17—44no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—44no—
160C2×Dic17C2×D412yes—
161C2×Dic17C2×D412yes—
162C2×Dic17C2×D412yes—
163C2×Dic17C2×D412yes—
164C2×Dic17C2×Q812yes—
165C4×D17—44no—
166Dic34—44no—
167Dic34—44no—
168#2683—171nomaximal, Sylow
169C22×C17—22no—
170C22×C17—22no—
171C22×C17—22no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—41no—
175C22×C17—41no—
176C22×C17C22×D411yescharacteristic
177C22×C17C22×D411yescharacteristic
178C22×C17C22×Q811yescharacteristic
179C22×C17H(5,𝔽2)11yescharacteristic
180C22×C17H(5,𝔽2)11yescharacteristic
181C22×C172−1+411yescharacteristic
182C22×C172−1+411yescharacteristic
183C2×D17—22no—
184C2×D17—22no—
185C2×D17—22no—
186C2×D17—41no—
187C2×D17—41no—
188C2×D17—41no—
189C2×D17—41no—
190C4×C17—22no—
191C4×C17—22no—
192C4×C17—41no—
193C4×C17—41no—
194C4×C17—41no—
195C4×C17—41no—
196Dic17—22no—
197Dic17—22no—
198Dic17—22no—
199Dic17—22no—
200Dic17—22no—
201Dic17—41no—
202Dic17—41no—
203C2×C22⋊Q8—171no—
204C2×C22⋊Q8—171no—
205C2×C22⋊Q8—171no—
206C2×C22⋊Q8—171no—
207#396—171no—
208#396—171no—
209#396—171no—
210#396D1711yescharacteristic, complemented
211#395—171no—
212#395—171no—
213#390—171no—
214#389—171no—
215#389—171no—
216#386—171no—
217#386—171no—
218C2×C17—42no—
219C2×C17#53711yescharacteristic
220C2×C17#53711yescharacteristic
221C2×C17#54311yescharacteristic
222C2×C17D4211yescharacteristic
223C2×C17D4×Q811yescharacteristic
224C2×C17D4×Q811yescharacteristic
225C2×C17#56311yescharacteristic
226D17—42no—
227C22×Q8—171no—
228C22×Q8—171no—
229C22×D4—171no—
230C23×C4—171no—
231C23×C4—171no—
232C22⋊Q8—344no—
233C22⋊Q8—344no—
234C22⋊Q8—344no—
235C22⋊Q8—344no—
236C2×C4⋊C4—171no—
237C2×C4⋊C4—171no—
238C2×C4⋊C4—171no—
239C2×C4⋊C4—171no—
240C2×C4⋊C4—171no—
241C2×C4⋊C4—171no—
242C2×C4⋊C4—171no—
243C2×C4⋊C4—171no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4C2×D1711yescharacteristic
247C2×C4⋊C4C2×D1711yescharacteristic
248C2×C4⋊C4C2×D1711yescharacteristic
249C2×C22⋊C4—171no—
250C2×C22⋊C4—171no—
251C2×C22⋊C4—171no—
252C2×C22⋊C4—171no—
253C2×C22⋊C4—171no—
254C2×C22⋊C4—171no—
255C2×C22⋊C4—171no—
256C2×C22⋊C4C2×D1711yescharacteristic
257C2×C22⋊C4C2×D1711yescharacteristic
258C2×C22⋊C4C2×D1711yescharacteristic
259C2×C42—171no—
260#95—171no—
261#95—171no—
262#95—171no—
263#95—171no—
264#95—171no—
265#95C2×D1711yescharacteristic
266C17#268311yescharacteristic, complemented, Sylow
267C24—171no—
268C24C22×D1711yescharacteristic
269C2×Q8—342no—
270C2×Q8—342no—
271C2×Q8—342no—
272C2×Q8—342no—
273C2×Q8—681no—
274C2×Q8—681no—
275C2×D4—174no—
276C2×D4—681no—
277C2×D4—681no—
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—171no—
281C22×C4—171no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—171no—
285C22×C4—342no—
286C22×C4—342no—
287C22×C4—342no—
288C22×C4—342no—
289C22×C4—681no—
290C22×C4—681no—
291C22×C4C22×D1711yescharacteristic
292C22×C4C22×D1711yescharacteristic
293C22×C4C22×D1711yescharacteristic
294C22×C4C22×D1711yescharacteristic
295C22×C4C22×D1711yescharacteristic
296C22×C4C22×D1711yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—174no—
301C4⋊C4—174no—
302C4⋊C4—342no—
303C4⋊C4—342no—
304C4⋊C4—342no—
305C4⋊C4—342no—
306C4⋊C4—342no—
307C4⋊C4—342no—
308C4⋊C4—681no—
309C4⋊C4—681no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—342no—
314C22⋊C4—342no—
315C22⋊C4—342no—
316C22⋊C4—342no—
317C22⋊C4—342no—
318C22⋊C4—342no—
319C22⋊C4—681no—
320C42—681no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—172no—
328C23—341no—
329C23—341no—
330C23—341no—
331C23—341no—
332C23—341no—
333C23—341no—
334C23Q8×D1712yes—
335C23C23×D1711yescharacteristic
336Q8—684no—
337Q8—684no—
338D4—684no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—172no—
356C2×C4—172no—
357C2×C4—172no—
358C2×C4—172no—
359C2×C4—172no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—341no—
365C2×C4—341no—
366C2×C4—341no—
367C2×C4—341no—
368C2×C4—341no—
369C2×C4—341no—
370C2×C4—341no—
371C2×C4—341no—
372C2×C4—341no—
373C2×C4—341no—
374C2×C4—341no—
375C2×C4—341no—
376C2×C4—684no—
377C2×C4—684no—
378C2×C4D4×D1712yes—
379C2×C4D4×D1712yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—342no—
388C22—342no—
389C22—342no—
390C22—681no—
391C22—681no—
392C22—681no—
393C22—681no—
394C22C2×D4×D1711yescharacteristic
395C22C2×D4×D1711yescharacteristic
396C22#1058769111yescharacteristic
397C22C2×Q8×D1711yescharacteristic
398C22#1058769611yescharacteristic
399C22#1058769711yescharacteristic
400C22#1058769711yescharacteristic
401C4—22no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—342no—
408C4—342no—
409C4—342no—
410C4—342no—
411C4—342no—
412C4—681no—
413C4—681no—
414C2—42no—
415C2—682no—
416C2#1176338611yescharacteristic
417C2#1176338811yescharacteristic
418C2#1176343411yescharacteristic
419C2#1176344311yescharacteristic
420C2#1176347211yescharacteristic
421C2#1176347211yescharacteristic
422C2#1176356111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#396C2
prime radicalC22×C22⋊D17C22
hypercenter#396D17
hypocenterC17#2683
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#10587583C2semidirect
2. C2×#10587582C2semidirect
3. C2×#10587564C2semidirect
4. C2×#10587564C2semidirect
5. #11762733C2semidirect
6. #11762732C2semidirect
7. #11762731C2semidirect
8. #11762730C2semidirect
9. #11762729C2semidirect
11. #11762705C2semidirect
12. #11762704C2semidirect
15. C17×#396C2semidirect
65. C2×C22⋊D17Q8semidirect
100. C4⋊Dic17D4semidirect
107. #63225D4semidirect
210. #396D17semidirect
266. C17#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×#105875822
C2×#105875642
#117627332
#117627322
#117627312
#117627302
#117627292
#117627092
#117627052
#117627042
#117626862
#117626852
C17×#3962
#268317