Finite Group Databasebeta

← #13327856

Subgroups — #13327856

Order 1792.

888 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3458C211yesmaximal, characteristic
2#11740769C212yesmaximal, complemented
3#11739709C211yesmaximal, characteristic
4#11729653C214yesmaximal, complemented
5#11729645C214yesmaximal
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#89853—22no—
22#89850—22no—
23#89816C2218yes—
24#89793—24no—
25#89758C2212yes—
26#89751—21no—
27#89721C2212yes—
28#89719—21no—
29#89708C2211yescharacteristic
30#89700—21no—
31#89697C2212yes—
32#89333C2218yes—
33#89323C2212yes—
34#89320C2212yes—
35#18311—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—
80C2×Dic7⋊C4—22no—
81#6432—22no—
82#6432—22no—
83#6409—28no—
84C7⋊#95—22no—
85C7⋊#95—21no—
86C7⋊#95—22no—
87C7⋊#95C2312yes—
88C22⋊(C7⋊C8)C2314yes—
89#6381—28no—
90C4⋊(C7⋊C8)—22no—
91C4×C7⋊C8—22no—
92#3458D711yescharacteristic
93#3230—72no—
94#2628—71no—
95#1617—74no—
96#1614—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×C7Q1614yes—
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—42no—
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—
145C4⋊Dic7—24no—
146C4⋊Dic7—41no—
147C4⋊Dic7—41no—
148C4⋊Dic7—42no—
149Dic7⋊C4—82no—
150C2×C7⋊C8—24no—
151#544—22no—
152#544—22no—
153#544—21no—
154#544C2×D712yes—
155#543—21no—
156#543C2×D712yes—
157#533—24no—
158#533—21no—
159#533C2×D712yes—
160C2×C4⋊Q8—71no—
161C2×#117—21no—
162C2×#117C2×D711yescharacteristic
163#500—142no—
164#487—142no—
165#484—78no—
166#477—144no—
167#431—72no—
168#400—72no—
169#399—72no—
170#395—141no—
171#393—141no—
172#389—141no—
173#382—71no—
174#380—141no—
175#328—78no—
176#324—72no—
177#323—72no—
178C23×C7—21no—
179C23×C7—21no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7C2×Q1612yes—
183C23×C7C22×D411yescharacteristic
184C7×Q8—44no—
185C7×Q8—82no—
186C7×Q8—82no—
187C7×Q8—82no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—24no—
191C2×C4×C7—24no—
192C2×C4×C7—24no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—24no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—82no—
201C2×C4×C7—82no—
202C2×C4×C7#12012yes—
203C2×C4×C7#12011yescharacteristic
204C2×C4×C7#12011yescharacteristic
205C2×C4×C7C2×Q1612yes—
206C2×C4×C7C2×Q1612yes—
207C2×C4×C7#13712yes—
208C2×C4×C7C22×D412yes—
209C2×Dic7—42no—
210C2×Dic7—42no—
211C2×Dic7—42no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7—42no—
215Dic14—84no—
216C7⋊C8—44no—
217C22×Q8—141no—
218C23×C4C22⋊D711yescharacteristic
219C4⋊Q8—24no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—24no—
223C4⋊Q8—142no—
224C4⋊Q8—142no—
225C4⋊Q8C22⋊D712yes—
226C4⋊Q8C22⋊D712yes—
227#125—24no—
228#125—22no—
229#125—24no—
230#125—22no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—24no—
238C22⋊Q8C22×D714yes—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8C22⋊D714yes—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—72no—
252C2×C4⋊C4—141no—
253C2×C4⋊C4—141no—
254C2×C4⋊C4—142no—
255C2×C4⋊C4C22⋊D711yescharacteristic
256C2×C4⋊C4C22×D711yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—142no—
259C2×C22⋊C4C22⋊D711yescharacteristic
260C2×C22⋊C4C22×D711yescharacteristic
261C2×C42—22no—
262C2×C42C22⋊D711yescharacteristic
263C2×C42C22×D711yescharacteristic
264#107—148no—
265C4⋊C8—142no—
266#103—148no—
267C22⋊C8—74no—
268C4×C8—142no—
269#95—72no—
270#95—142no—
271#95—141no—
272#95—142no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—22no—
276C22×C7—42no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7—81no—
280C22×C7C2×#12011yescharacteristic
281C22×C7C22×Q1611yescharacteristic
282C22×C7C2×#13711yescharacteristic
283C4×C7—22no—
284C4×C7—22no—
285C4×C7—22no—
286C4×C7—41no—
287C4×C7—44no—
288C4×C7—41no—
289C4×C7—44no—
290C4×C7#45112yes—
291C4×C7#45112yes—
292Dic7—82no—
293Dic7—82no—
294C24C2×C22⋊D711yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—282no—
299C2×Q8—562no—
300C2×Q8C2×C22⋊D714yes—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—142no—
307C22×C4—142no—
308C22×C4#114414yes—
309C22×C4C2×C22⋊D711yescharacteristic
310C22×C4C2×C22⋊D711yescharacteristic
311C22×C4C2×C22⋊D711yescharacteristic
312C22×C4C2×C22⋊D711yescharacteristic
313C22×C4C2×C22⋊D711yescharacteristic
314C22×C4C23×D711yescharacteristic
315C2×C8—144no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—144no—
330C4⋊C4—282no—
331C4⋊C4—281no—
332C4⋊C4—281no—
333C4⋊C4—562no—
334C4⋊C4C2×C22⋊D714yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—562no—
340C42—24no—
341C42—24no—
342C42—21no—
343C42—21no—
344C42C2×C22⋊D712yes—
345C42C2×C22⋊D712yes—
346C2×C7—21no—
347C2×C7—21no—
348C2×C7—42no—
349C2×C7C2×#45111yescharacteristic
350C2×C7#302111yescharacteristic
351C2×C7#302811yescharacteristic
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23C2×#114412yes—
357C23C22×C22⋊D711yescharacteristic
358Q8—44no—
359Q8—82no—
360Q8—82no—
361Q8—82no—
362Q8—564no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—282no—
378C2×C4—282no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4—282no—
383C2×C4#650512yes—
384C2×C4C2×#114412yes—
385C2×C4C2×#114412yes—
386C2×C4#651612yes—
387C2×C4#651611yescharacteristic
388C2×C4#651611yescharacteristic
389C2×C4C22×C22⋊D712yes—
390C8—284no—
391C7#1831111yescharacteristic, complemented, Sylow
392C22—21no—
393C22—21no—
394C22—22no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22C2×#650511yescharacteristic
400C22C22×#114411yescharacteristic
401C22C2×#651611yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—44no—
407C4—41no—
408C4—44no—
409C4—562no—
410C4—562no—
411C4#9000012yes—
412C4#9000012yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2C2×#9000011yescharacteristic
417C2#1174156711yescharacteristic
418C2#1174156911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3458C2
socleC22×C7C2×#120
prime radicalC24×C7C2×D4
hypercenter#3458D7
hypocenterC7#18311
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. #11740769C2semidirect
4. #11729653C2semidirect
6. #11729592C2semidirect
391. C7#18311semidirect

Maximal subgroups

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

subgroupindex
C7×#34582
#117407692
#117397092
#117296532
#117296452
#117295922
#117293252
#183117