Finite Group Databasebeta

← #13327788

Subgroups — #13327788

Order 1792.

912 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3458C211yesmaximal, characteristic
2#11740815C212yesmaximal, complemented
3#11739709C211yesmaximal, characteristic
4#11729658C214yesmaximal, complemented
5#11729647C214yesmaximal
6#11729592C212yesmaximal, complemented
7#11729325C211yesmaximal, characteristic
8C7×#544—22no—
9C7×#544—22no—
10C7×#544—21no—
11C7×#544C2212yes—
12C7×#543—21no—
13C7×#543C2212yes—
14C7×#533—24no—
15C7×#533—21no—
16C7×#533C2212yes—
17C2×C7×#117—21no—
18C2×C7×#117C2211yescharacteristic
19C2×#6432C2211yescharacteristic
20#89984—21no—
21#89865—22no—
22#89864—22no—
23#89822C2218yes—
24#89803—24no—
25#89758—21no—
26#89751C2212yes—
27#89721—21no—
28#89719C2212yes—
29#89708C2211yescharacteristic
30#89700—21no—
31#89697C2212yes—
32#89333C2218yes—
33#89323C2212yes—
34#89320C2212yes—
35#18303—71nomaximal, Sylow
36C23×C4×C7D411yescharacteristic
37C22×Dic14C2312yes—
38C7×C4⋊Q8—24no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—24no—
42C7×C4⋊Q8D412yes—
43C7×C4⋊Q8D412yes—
44C7×#125—24no—
45C7×#125—22no—
46C7×#125—24no—
47C7×#125—22no—
48C7×C22⋊Q8—24no—
49C7×C22⋊Q8—24no—
50C7×C22⋊Q8—24no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—24no—
55C7×C22⋊Q8C2314yes—
56C4×C7×Q8—22no—
57C4×C7×Q8—22no—
58C4×C7×Q8—22no—
59C4×C7×Q8D414yes—
60C7×#117—22no—
61C7×#117—22no—
62C7×#117—22no—
63C7×#117—22no—
64C7×#117—22no—
65C7×#117—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4D411yescharacteristic
72C2×C7×C22⋊C4C2311yescharacteristic
73C2×C42×C7—22no—
74C2×C42×C7D411yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—22no—
77C2×#1144—28no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×Dic7⋊C4—22no—
82#6432—22no—
83#6432—22no—
84C7⋊#95—22no—
85C7⋊#95—21no—
86C7⋊#95—22no—
87C7⋊#95C2312yes—
88C22⋊(C7⋊C8)C2314yes—
89#6384—28no—
90C4⋊(C7⋊C8)—22no—
91C4×C7⋊C8—22no—
92#3458D711yescharacteristic
93#3209—72no—
94#2628—71no—
95#1605—74no—
96#1602—74no—
97#1518—72no—
98#1458—71no—
99C24×C7C2×D411yescharacteristic
100C2×C7×Q8—22no—
101C2×C7×Q8—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8C2×D414yes—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7Q1614yescomplemented
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2411yescharacteristic
116C22×Dic7—22no—
117C22×Dic7—22no—
118C2×Dic14—24no—
119C2×Dic14—82no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—24no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—24no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4C2×D414yes—
134C7×C22⋊C4—24no—
135C7×C22⋊C4—24no—
136C7×C22⋊C4—22no—
137C7×C22⋊C4—22no—
138C42×C7—24no—
139C42×C7—24no—
140C42×C7—21no—
141C42×C7—21no—
142C42×C7C2×D412yes—
143C42×C7C2×D412yes—
144C22⋊Dic7—82no—
145#1144—88no—
146C4⋊Dic7—42no—
147C4⋊Dic7—41no—
148C4⋊Dic7—41no—
149C4⋊Dic7—42no—
150Dic7⋊C4—82no—
151C2×C7⋊C8—24no—
152#544—22no—
153#544—22no—
154#544—21no—
155#544C2×D712yes—
156#543—21no—
157#543C2×D712yes—
158#533—24no—
159#533—21no—
160#533C2×D712yes—
161C2×C4⋊Q8—71no—
162C2×#117—21no—
163C2×#117C2×D711yescharacteristic
164#494—142no—
165#487—142no—
166#462—144no—
167#451—78no—
168#431—72no—
169#400—141no—
170#399—141no—
171#395—72no—
172#393—72no—
173#389—141no—
174#382—71no—
175#380—141no—
176#328—78no—
177#324—72no—
178#323—72no—
179C23×C7—21no—
180C23×C7—21no—
181C23×C7—41no—
182C23×C7—41no—
183C23×C7C2×Q1612yes—
184C23×C7C22×D411yescharacteristic
185C7×Q8—44no—
186C7×Q8—82no—
187C7×Q8—82no—
188C7×Q8—82no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—24no—
192C2×C4×C7—24no—
193C2×C4×C7—24no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—24no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—82no—
202C2×C4×C7—82no—
203C2×C4×C7#12012yes—
204C2×C4×C7#12011yescharacteristic
205C2×C4×C7#12011yescharacteristic
206C2×C4×C7C2×Q1612yes—
207C2×C4×C7C2×Q1612yes—
208C2×C4×C7#13712yes—
209C2×C4×C7C22×D412yes—
210C2×Dic7—42no—
211C2×Dic7—42no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7—42no—
215C2×Dic7—42no—
216Dic14—48no—
217C7⋊C8—44no—
218C22×Q8—72no—
219C23×C4C22⋊D711yescharacteristic
220C2×Q16—148no—
221C4⋊Q8—24no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—24no—
225C4⋊Q8—142no—
226C4⋊Q8—142no—
227C4⋊Q8C22⋊D712yes—
228C4⋊Q8C22⋊D712yes—
229#125—24no—
230#125—22no—
231#125—24no—
232#125—22no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C22⋊Q8—24no—
240C22⋊Q8C22×D714yes—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8C22⋊D714yes—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—141no—
254C2×C4⋊C4—141no—
255C2×C4⋊C4—141no—
256C2×C4⋊C4—142no—
257C2×C4⋊C4C22⋊D711yescharacteristic
258C2×C4⋊C4C22×D711yescharacteristic
259C2×C22⋊C4—22no—
260C2×C22⋊C4—142no—
261C2×C22⋊C4C22⋊D711yescharacteristic
262C2×C22⋊C4C22×D711yescharacteristic
263C2×C42—22no—
264C2×C42C22⋊D711yescharacteristic
265C2×C42C22×D711yescharacteristic
266C4⋊C8—142no—
267#103—148no—
268C22⋊C8—74no—
269C4×C8—142no—
270#95—72no—
271#95—142no—
272#95—141no—
273#95—142no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—22no—
277C22×C7—42no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7—81no—
281C22×C7C2×#12011yescharacteristic
282C22×C7C22×Q1611yescharacteristic
283C22×C7C2×#13711yescharacteristic
284C4×C7—22no—
285C4×C7—22no—
286C4×C7—22no—
287C4×C7—41no—
288C4×C7—44no—
289C4×C7—41no—
290C4×C7—44no—
291C4×C7#45112yes—
292C4×C7#45112yes—
293Dic7—82no—
294Dic7—82no—
295C24C2×C22⋊D711yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—144no—
300C2×Q8—562no—
301C2×Q8C2×C22⋊D714yes—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—142no—
308C22×C4—142no—
309C22×C4#114414yescomplemented
310C22×C4C2×C22⋊D711yescharacteristic
311C22×C4C2×C22⋊D711yescharacteristic
312C22×C4C2×C22⋊D711yescharacteristic
313C22×C4C2×C22⋊D711yescharacteristic
314C22×C4C2×C22⋊D711yescharacteristic
315C22×C4C23×D711yescharacteristic
316Q16—568no—
317C2×C8—144no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—282no—
332C4⋊C4—282no—
333C4⋊C4—281no—
334C4⋊C4—281no—
335C4⋊C4—562no—
336C4⋊C4C2×C22⋊D714yes—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—562no—
342C42—24no—
343C42—24no—
344C42—21no—
345C42—21no—
346C42C2×C22⋊D712yes—
347C42C2×C22⋊D712yes—
348C2×C7—21no—
349C2×C7—21no—
350C2×C7—42no—
351C2×C7C2×#45111yescharacteristic
352C2×C7#302211yescharacteristic
353C2×C7#302711yescharacteristic
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23C2×#114412yes—
359C23C22×C22⋊D711yescharacteristic
360Q8—44no—
361Q8—82no—
362Q8—82no—
363Q8—82no—
364Q8—288no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4—282no—
383C2×C4—282no—
384C2×C4—282no—
385C2×C4#650512yes—
386C2×C4C2×#114412yes—
387C2×C4C2×#114412yes—
388C2×C4#651612yes—
389C2×C4#651611yescharacteristic
390C2×C4#651611yescharacteristic
391C2×C4C22×C22⋊D712yes—
392C8—284no—
393C7#1830311yescharacteristic, complemented, Sylow
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C2×#650511yescharacteristic
402C22C22×#114411yescharacteristic
403C22C2×#651611yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—44no—
409C4—41no—
410C4—44no—
411C4—562no—
412C4—562no—
413C4#9000012yes—
414C4#9000012yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2C2×#9000011yescharacteristic
419C2#1174156811yescharacteristic
420C2#1174157111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3458C2
socleC22×C7C2×#120
prime radicalC24×C7C2×D4
hypercenter#3458D7
hypocenterC7#18303
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #11740815C2semidirect
4. #11729658C2semidirect
6. #11729592C2semidirect
109. C22×C4×C7Q16semidirect
309. C22×C4#1144semidirect
393. C7#18303semidirect

Maximal subgroups

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

subgroupindex
C7×#34582
#117408152
#117397092
#117296582
#117296472
#117295922
#117293252
#183037