Finite Group Databasebeta

← #13259099

Subgroups — #13259099

Order 1792.

1510 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6489C211yesmaximal, characteristic
2C2×C7×#381C211yesmaximal, characteristic
3C2×#89983C211yesmaximal, characteristic
4C2×#89321C214yesmaximal
5C24×C4×C7C2211yescharacteristic
6C22×C7×C22⋊C4C2212yes—
7C22×C22⋊Dic7C2212yes—
8C22×C22⋊Dic7C2212yes—
9C2×#6489—22no—
10C2×#6489—28no—
11C2×#6489C414yescomplemented
12C22×C4⋊Dic7—21no—
13C22×C4×Dic7—21no—
14C7×#381—21no—
15C7×#381—21no—
16C7×#381—22no—
17C2×C7×#95C414yescomplemented
18#89983—22no—
19#89983C414yescomplemented
20C2×C7⋊#95—22no—
21#89321—28no—
22#10736—71nomaximal, Sylow
23C25×C7C2311yescharacteristic
24C23×C4×C7—24no—
25C23×C4×C7—24no—
26C23×C4×C7C2×C412yes—
27C23×C4×C7C2×C412yes—
28C23×C4×C7C2×C412yes—
29C23×C4×C7C2×C414yes—
30C23×C4×C7D412yes—
31C23×C4×C7D411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7D412yes—
34C23×C4×C7Q812yes—
35C23×Dic7—22no—
36C23×Dic7—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—24no—
40C2×C7×C22⋊C4—24no—
41C2×C22⋊Dic7—22no—
42C2×C22⋊Dic7—24no—
43C2×C22⋊Dic7—24no—
44C2×C22⋊Dic7—24no—
45C2×C22⋊Dic7—22no—
46C2×C22⋊Dic7C2×C414yes—
47C2×C22⋊Dic7C2×C414yes—
48C2×C22⋊Dic7D418yes—
49#6489—216no—
50#6489—216no—
51C2×C4⋊Dic7—28no—
52C2×C4⋊Dic7—24no—
53C2×C4×Dic7—28no—
54C2×C4×Dic7—24no—
55C7×#95—24no—
56C7×#95—24no—
57C7⋊#95—28no—
58C22×#117—71no—
59C2×#381—71no—
60C2×#381D711yescharacteristic
61C2×#342—74no—
62C24×C7—44no—
63C24×C7C22×C411yescharacteristic
64C24×C7C22×C411yescharacteristic
65C24×C7C22×C411yescharacteristic
66C24×C7C2×D411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C24×C7C2×D411yescharacteristic
69C24×C7C2×D414yes—
70C24×C7C2×Q811yescharacteristic
71C24×C7#4114yes—
72C22×C4×C7—24no—
73C22×C4×C7—24no—
74C22×C4×C7—28no—
75C22×C4×C7—28no—
76C22×C4×C7—22no—
77C22×C4×C7—22no—
78C22×C4×C7—24no—
79C22×C4×C7—28no—
80C22×C4×C7—28no—
81C22×C4×C7—24no—
82C22×C4×C7—22no—
83C22×C4×C7—22no—
84C22×C4×C7—24no—
85C22×C4×C7—28no—
86C22×C4×C7—48no—
87C22×C4×C7C4214yes—
88C22×C4×C7C22⋊C414yes—
89C22×C4×C7C22⋊C412yes—
90C22×C4×C7C22⋊C412yes—
91C22×C4×C7C22⋊C412yes—
92C22×C4×C7C22⋊C412yes—
93C22×C4×C7C4⋊C414yes—
94C22×C4×C7C4⋊C412yes—
95C22×C4×C7C4⋊C412yes—
96C22×C4×C7C4⋊C414yes—
97C22×Dic7—24no—
98C22×Dic7—28no—
99C22×Dic7—24no—
100C22×Dic7—48no—
101C22×Dic7—48no—
102C22×Dic7—44no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—44no—
105C7×C22⋊C4—42no—
106C22⋊Dic7—24no—
107C22⋊Dic7—28no—
108C22⋊Dic7—28no—
109C22⋊Dic7—42no—
110C22⋊Dic7—42no—
111C22⋊Dic7C22⋊C418yes—
112C4⋊Dic7—216no—
113C4×Dic7—216no—
114C24×C4C2×D711yescharacteristic
115C2×#117—74no—
116C2×#117—148no—
117C2×#117—142no—
118C22×C4⋊C4—141no—
119C22×C22⋊C4—72no—
120C22×C22⋊C4—72no—
121C22×C22⋊C4C2×D712yes—
122C22×C42—141no—
123#381—21no—
124#381—22no—
125#381—21no—
126#381—74no—
127#381—142no—
128C2×#95—142no—
129C2×#95Dic714yescomplemented
130#342—148no—
131C23×C7—24no—
132C23×C7—24no—
133C23×C7—24no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—48no—
138C23×C7—42no—
139C23×C7—48no—
140C23×C7C2×C4211yescharacteristic
141C23×C7C2×C22⋊C412yes—
142C23×C7C2×C22⋊C411yescharacteristic
143C23×C7C2×C22⋊C411yescharacteristic
144C23×C7C2×C22⋊C411yescharacteristic
145C23×C7C2×C4⋊C411yescharacteristic
146C23×C7C2×C4⋊C411yescharacteristic
147C23×C7C2×C4⋊C411yescharacteristic
148C23×C7#11712yes—
149C23×C7C4×D414yes—
150C23×C7C4×D414yes—
151C23×C7#12012yes—
152C23×C7#12012yes—
153C23×C7#12114yes—
154C23×C7C22⋊Q814yes—
155C23×C7#12312yes—
156C23×C7#12312yes—
157C2×C4×C7—28no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—28no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—24no—
164C2×C4×C7—24no—
165C2×C4×C7—28no—
166C2×C4×C7—28no—
167C2×C4×C7—24no—
168C2×C4×C7—24no—
169C2×C4×C7—44no—
170C2×C4×C7—48no—
171C2×C4×C7—88no—
172C2×C4×C7#9514yes—
173C2×C4×C7#9514yes—
174C2×Dic7—28no—
175C2×Dic7—416no—
176C2×Dic7—48no—
177C2×Dic7—48no—
178C2×Dic7—44no—
179C2×Dic7—88no—
180C25C22×D711yescharacteristic
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—142no—
184C23×C4—142no—
185C23×C4Dic1412yes—
186C23×C4C4×D712yes—
187C23×C4C4×D712yes—
188C23×C4D2812yes—
189C23×C4C2×Dic712yes—
190C23×C4C2×Dic714yes—
191C23×C4C22⋊D711yescharacteristic
192C23×C4C22⋊D712yes—
193C23×C4C22⋊D711yescharacteristic
194#117—1416no—
195#117—1416no—
196C2×C4⋊C4—148no—
197C2×C4⋊C4—144no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—74no—
203C2×C22⋊C4—78no—
204C2×C22⋊C4—74no—
205C2×C22⋊C4—142no—
206C2×C22⋊C4—144no—
207C2×C22⋊C4—144no—
208C2×C22⋊C4—144no—
209C2×C22⋊C4—142no—
210C2×C42—148no—
211C2×C42—144no—
212#95—24no—
213#95—24no—
214#95—148no—
215C22×C7—24no—
216C22×C7—28no—
217C22×C7—28no—
218C22×C7—24no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—42no—
226C22×C7—42no—
227C22×C7—44no—
228C22×C7—42no—
229C22×C7—44no—
230C22×C7—42no—
231C22×C7C2×#9511yescharacteristic
232C22×C7C4×C22⋊C412yes—
233C22×C7#37911yescharacteristic
234C22×C7#38012yes—
235C22×C7#38111yescharacteristic
236C22×C7#38512yes—
237C22×C7#38512yes—
238C22×C7#38612yes—
239C22×C7#38612yes—
240C22×C7#41014yes—
241C4×C7—24no—
242C4×C7—24no—
243C4×C7—84no—
244Dic7—48no—
245Dic7—84no—
246C24—44no—
247C24C2×Dic1411yescharacteristic
248C24C2×C4×D711yescharacteristic
249C24C2×C4×D711yescharacteristic
250C24C2×D2811yescharacteristic
251C24D4×D714yes—
252C24#115914yes—
253C24C22×Dic711yescharacteristic
254C24C2×C22⋊D711yescharacteristic
255C24C2×C22⋊D711yescharacteristic
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—48no—
271C22×C4—144no—
272C22×C4—148no—
273C22×C4—144no—
274C22×C4—288no—
275C22×C4—288no—
276C22×C4—284no—
277C22×C4C4×Dic714yes—
278C22×C4Dic7⋊C414yes—
279C22×C4Dic7⋊C412yes—
280C22×C4Dic7⋊C412yes—
281C22×C4C4⋊Dic714yes—
282C22×C4(C2×D7)⋊C414yes—
283C22×C4(C2×D7)⋊C412yes—
284C22×C4(C2×D7)⋊C412yes—
285C22×C4C22⋊Dic712yes—
286C22×C4C22⋊Dic712yes—
287C4⋊C4—1416no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—44no—
291C22⋊C4—78no—
292C22⋊C4—144no—
293C22⋊C4—148no—
294C22⋊C4—148no—
295C22⋊C4—282no—
296C22⋊C4—282no—
297C42—1416no—
298C2×C7—21no—
299C2×C7—22no—
300C2×C7—22no—
301C2×C7—22no—
302C2×C7—21no—
303C2×C7—22no—
304C2×C7—22no—
305C2×C7C22⋊#9511yescharacteristic
306C2×C7#174012yes—
307C2×C7#189914yes—
308C23—24no—
309C23—24no—
310C23—24no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—48no—
316C23—48no—
317C23#644012yes—
318C23#644114yes—
319C23D7×C22⋊C412yes—
320C23(C22⋊D7)⋊C414yes—
321C23#644512yes—
322C23#644912yes—
323C23C2×C4×Dic711yescharacteristic
324C23C2×Dic7⋊C411yescharacteristic
325C23C2×Dic7⋊C411yescharacteristic
326C23C2×C4⋊Dic711yescharacteristic
327C23C2×(C2×D7)⋊C411yescharacteristic
328C23C2×(C2×D7)⋊C411yescharacteristic
329C23D4×Dic714yes—
330C23#649812yes—
331C23#650012yes—
332C23#650214yes—
333C23C2×C22⋊Dic711yescharacteristic
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—88no—
349C2×C4—148no—
350C2×C4—2816no—
351C2×C4—288no—
352C2×C4—288no—
353C2×C4—284no—
354C2×C4—568no—
355C2×C4C7⋊#9514yes—
356C2×C4C7⋊#9514yes—
357C7#1073611yescharacteristic, complemented, Sylow
358C22—28no—
359C22—24no—
360C22—28no—
361C22—24no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—42no—
369C22—44no—
370C22—42no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22C22⋊C4×Dic712yes—
375C22#8971412yes—
376C22#8971511yescharacteristic
377C22#8971812yes—
378C22#8972212yes—
379C22#8972711yescharacteristic
380C22#8973012yes—
381C22C2×C7⋊#9511yescharacteristic
382C22#8999212yes—
383C22#9001214yes—
384C4—24no—
385C4—24no—
386C4—84no—
387C4—288no—
388C4—564no—
389C2—21no—
390C2—22no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2—22no—
395C2—22no—
396C2#1172929811yescharacteristic
397C2#1172934612yes—
398C2#1172945914yes—
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC23×C7C2×C42
FrattiniC25C22×D7
FittingC2×C7×#381C2
socleC23×C7C2×C22⋊C4
prime radical, cosocleC25×C7C23
hypercenterC2×#381D7
hypocenterC7#10736
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
11. C2×#6489C4semidirect
17. C2×C7×#95C4semidirect
19. #89983C4semidirect
129. C2×#95Dic7semidirect
357. C7#10736semidirect

Maximal subgroups

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

subgroupindex
C22×#64892
C2×C7×#3812
C2×#899832
C2×#893212
#107367