Finite Group Databasebeta

← #13327849

Subgroups — #13327849

Order 1792.

908 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3458C211yesmaximal, characteristic
2#11740768C212yesmaximal, complemented
3#11739707C211yesmaximal, characteristic
4#11729652C214yesmaximal, complemented
5#11729644C214yesmaximal
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×#6487—21no—
20C2×#6432C2211yescharacteristic
21#89853—22no—
22#89851—22no—
23#89815C2218yes—
24#89792—24no—
25#89755—21no—
26#89755C2212yes—
27#89725—21no—
28#89725C2212yes—
29#89708C2211yescharacteristic
30#89701—21no—
31#89697C2212yes—
32#89333C2218yes—
33#89323C2212yes—
34#89320C2212yes—
35#18310—71nomaximal, Sylow
36C23×C4×C7D411yescharacteristic
37C22×Dic14—21no—
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×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2312yes—
80#6487—44no—
81C2×Dic7⋊C4—22no—
82#6432—22no—
83#6432—22no—
84#6409—28no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87C7⋊#95—21no—
88C7⋊#95C2312yes—
89C22⋊(C7⋊C8)C2314yes—
90#6382—28no—
91C4⋊(C7⋊C8)—22no—
92C4×C7⋊C8—22no—
93#3458D711yescharacteristic
94#3229—72no—
95#2621—71no—
96#1611—74no—
97#1608—74no—
98#1518—72no—
99#1458—71no—
100C24×C7C2×D411yescharacteristic
101C2×C7×Q8—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8C2×D414yes—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7SD1614yes—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2411yescharacteristic
117C22×Dic7—22no—
118C22×Dic7—22no—
119C2×Dic14—41no—
120C2×Dic14—41no—
121C2×Dic14—42no—
122C2×Dic14—42no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—24no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—24no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4C2×D414yes—
137C7×C22⋊C4—24no—
138C7×C22⋊C4—24no—
139C7×C22⋊C4—22no—
140C7×C22⋊C4—22no—
141C42×C7—24no—
142C42×C7—24no—
143C42×C7—21no—
144C42×C7—21no—
145C42×C7C2×D412yes—
146C42×C7C2×D412yes—
147C22⋊Dic7—44no—
148C4⋊Dic7—24no—
149C4⋊Dic7—42no—
150C4⋊Dic7—42no—
151Dic7⋊C4—44no—
152C2×C7⋊C8—24no—
153#544—22no—
154#544—22no—
155#544—21no—
156#544C2×D712yes—
157#543—21no—
158#543C2×D712yes—
159#533—24no—
160#533—21no—
161#533C2×D712yes—
162C2×C4⋊Q8—71no—
163C2×C22⋊Q8—141no—
164C2×#117—21no—
165C2×#117C2×D711yescharacteristic
166#498—142no—
167#487—142no—
168#483—78no—
169#475—144no—
170#431—72no—
171#398—72no—
172#398—141no—
173#397—72no—
174#397—141no—
175#391—141no—
176#382—71no—
177#328—78no—
178#324—72no—
179#323—72no—
180C23×C7—21no—
181C23×C7—21no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7C2×SD1612yes—
185C23×C7C22×D411yescharacteristic
186C7×Q8—44no—
187C7×Q8—82no—
188C7×Q8—82no—
189C7×Q8—82no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—24no—
193C2×C4×C7—24no—
194C2×C4×C7—24no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—24no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—82no—
203C2×C4×C7—82no—
204C2×C4×C7#12012yes—
205C2×C4×C7#12011yescharacteristic
206C2×C4×C7#12011yescharacteristic
207C2×C4×C7C2×SD1612yes—
208C2×C4×C7C2×SD1612yes—
209C2×C4×C7#13712yes—
210C2×C4×C7C22×D412yes—
211C2×Dic7—42no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7—42no—
215C2×Dic7—42no—
216C2×Dic7—42no—
217Dic14—82no—
218Dic14—82no—
219C7⋊C8—44no—
220C22×Q8—141no—
221C23×C4C22⋊D711yescharacteristic
222C4⋊Q8—24no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—24no—
226C4⋊Q8—142no—
227C4⋊Q8—142no—
228C4⋊Q8C22⋊D712yes—
229C4⋊Q8C22⋊D712yes—
230#125—24no—
231#125—22no—
232#125—24no—
233#125—22no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—24no—
241C22⋊Q8—284no—
242C22⋊Q8C22×D714yes—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8C22⋊D714yes—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—72no—
256C2×C4⋊C4—142no—
257C2×C4⋊C4—141no—
258C2×C4⋊C4—141no—
259C2×C4⋊C4C22⋊D711yescharacteristic
260C2×C4⋊C4C22×D711yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—142no—
263C2×C22⋊C4C22⋊D711yescharacteristic
264C2×C22⋊C4C22×D711yescharacteristic
265C2×C42—22no—
266C2×C42C22⋊D711yescharacteristic
267C2×C42C22×D711yescharacteristic
268#106—148no—
269C4⋊C8—142no—
270#103—148no—
271C22⋊C8—74no—
272C4×C8—142no—
273#95—72no—
274#95—141no—
275#95—142no—
276#95—142no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—22no—
280C22×C7—42no—
281C22×C7—41no—
282C22×C7—41no—
283C22×C7—81no—
284C22×C7C2×#12011yescharacteristic
285C22×C7C22×SD1611yescharacteristic
286C22×C7C2×#13711yescharacteristic
287C4×C7—22no—
288C4×C7—22no—
289C4×C7—22no—
290C4×C7—41no—
291C4×C7—44no—
292C4×C7—41no—
293C4×C7—44no—
294C4×C7#44812yes—
295C4×C7#44812yes—
296Dic7—82no—
297Dic7—82no—
298C24C2×C22⋊D711yescharacteristic
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—282no—
303C2×Q8—282no—
304C2×Q8—281no—
305C2×Q8—281no—
306C2×Q8C2×C22⋊D714yes—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—142no—
313C22×C4—142no—
314C22×C4#114314yes—
315C22×C4C2×C22⋊D711yescharacteristic
316C22×C4C2×C22⋊D711yescharacteristic
317C22×C4C2×C22⋊D711yescharacteristic
318C22×C4C2×C22⋊D711yescharacteristic
319C22×C4C2×C22⋊D711yescharacteristic
320C22×C4C23×D711yescharacteristic
321C2×C8—144no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—144no—
336C4⋊C4—284no—
337C4⋊C4—282no—
338C4⋊C4—282no—
339C4⋊C4C2×C22⋊D714yes—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—284no—
345C42—24no—
346C42—24no—
347C42—21no—
348C42—21no—
349C42C2×C22⋊D712yes—
350C42C2×C22⋊D712yes—
351C2×C7—21no—
352C2×C7—21no—
353C2×C7—42no—
354C2×C7C2×#44811yescharacteristic
355C2×C7#302211yescharacteristic
356C2×C7#302811yescharacteristic
357C23—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23C2×#114312yes—
362C23C22×C22⋊D711yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—82no—
366Q8—82no—
367Q8—562no—
368Q8—562no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—282no—
384C2×C4—282no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—282no—
388C2×C4—282no—
389C2×C4C2×#114312yes—
390C2×C4C2×#114312yes—
391C2×C4#650512yes—
392C2×C4#651612yes—
393C2×C4#651611yescharacteristic
394C2×C4#651611yescharacteristic
395C2×C4C22×C22⋊D712yes—
396C8—284no—
397C7#1831011yescharacteristic, complemented, Sylow
398C22—21no—
399C22—21no—
400C22—22no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22C22×#114311yescharacteristic
406C22C2×#650511yescharacteristic
407C22C2×#651611yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—41no—
412C4—44no—
413C4—41no—
414C4—44no—
415C4—562no—
416C4—562no—
417C4#8999912yes—
418C4#8999912yes—
419C2—21no—
420C2—21no—
421C2—42no—
422C2C2×#8999911yescharacteristic
423C2#1174156811yescharacteristic
424C2#1174156911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3458C2
socleC22×C7C2×#120
prime radicalC24×C7C2×D4
hypercenter#3458D7
hypocenterC7#18310
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. #11740768C2semidirect
4. #11729652C2semidirect
6. #11729592C2semidirect
397. C7#18310semidirect

Maximal subgroups

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

subgroupindex
C7×#34582
#117407682
#117397072
#117296522
#117296442
#117295922
#117293252
#183107