Finite Group Databasebeta

← #13409946

Subgroups — #13409946

Order 1792.

1094 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C211yesmaximal, characteristic
2#11740220C212yesmaximal, complemented
3#11739819C211yesmaximal, characteristic
4#11729649C214yesmaximal, complemented
5#11729455C212yesmaximal, complemented
6#11729411C214yesmaximal
7C4×#6408C211yesmaximal, characteristic
8C7×#554—22no—
9C7×#554—24no—
10C7×D4×Q8—22no—
11C7×#544—21no—
12C7×#543—21no—
13C7×#543C2212yes—
14C7×#533—21no—
15C2×C7×C22⋊Q8—21no—
16C2×C7×C22⋊Q8C2212yes—
17C7×#518C2212yes—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20#90002—22no—
21#89992C2212yes—
22C2×#6408C2212yes—
23#89984—21no—
24#89984C2211yescharacteristic
25C4×C22⋊Dic7C2212yes—
26#89758—21no—
27#89755—24no—
28#89748—22no—
29#89721—21no—
30#89718C2218yes—
31#89718C2212yes—
32#89700—21no—
33#89321C2218yes—
34#89321C2212yes—
35#25853—71nomaximal, Sylow
36C22×C7×Q8—21no—
37C22×C7×Q8D412yes—
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7C2312yes—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—24no—
45C7×#125—22no—
46C7×#125—24no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—24no—
50C7×C22⋊Q8—24no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—22no—
56C7×C22⋊Q8—22no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—24no—
59C4×C7×Q8—22no—
60C4×C7×Q8—22no—
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C4×C7×D4—24no—
64C4×C7×D4—24no—
65C7×#117—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4D412yes—
69C2×C7×C4⋊C4D414yes—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4D411yescharacteristic
73C2×C7×C22⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4C2314yes—
75C2×C42×C7D411yescharacteristic
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7C2312yes—
78C2×C22⋊Dic7C2314yes—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7—22no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—28no—
86C2×C4×Dic7—22no—
87#6408Q818yescomplemented
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95—28no—
91C7⋊#95—21no—
92C7⋊#95—22no—
93#3501D711yescharacteristic
94#2502—72no—
95#2487—71no—
96#2062—74no—
97#1996—72no—
98#1867—74no—
99C4×#99—71no—
100C24×C7C2×D411yescharacteristic
101C24×C7C2411yescharacteristic
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—22no—
105C2×C7×Q8—22no—
106C2×C7×Q8—44no—
107C2×C7×Q8—82no—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4C2×Q814yes—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—42no—
116C22×C4×C7—42no—
117C22×C4×C7C2×D412yes—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D414yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D414yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×Dic7—24no—
127C22×Dic7—22no—
128C22×Dic7—24no—
129C22×Dic7—22no—
130C22×Dic7—82no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—22no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—22no—
147C42×C7—24no—
148C42×C7—21no—
149C42×C7—21no—
150C22⋊Dic7—24no—
151C22⋊Dic7—24no—
152C22⋊Dic7—82no—
153C22⋊Dic7C2×Q818yescomplemented
154C4⋊Dic7—42no—
155C4⋊Dic7—41no—
156C4⋊Dic7—42no—
157C4⋊Dic7—41no—
158Dic7⋊C4—88no—
159Dic7⋊C4—82no—
160C4×Dic7—44no—
161#554—22no—
162#554—24no—
163D4×Q8—22no—
164#544—21no—
165#543—21no—
166#543C2×D712yes—
167#533—21no—
168C2×C22⋊Q8—21no—
169C2×C22⋊Q8C2×D712yes—
170#518C2×D712yes—
171C2×C4×D4C2×D711yescharacteristic
172C22×C4⋊C4—71no—
173C2×#99—72no—
174#400—141no—
175#399—141no—
176#398—144no—
177#391—142no—
178#389—141no—
179#386—72no—
180#385—72no—
181#385—78no—
182#384—142no—
183#380—71no—
184#380—141no—
185C4×C22⋊C4—72no—
186#342—72no—
187#342—78no—
188C23×C7—21no—
189C23×C7—21no—
190C23×C7—41no—
191C23×C7—41no—
192C23×C7—41no—
193C23×C7—41no—
194C23×C7C4⋊Q812yes—
195C23×C7C22×D411yescharacteristic
196C23×C7C22×D412yes—
197C23×C7C22×Q812yes—
198C23×C72−1+412yes—
199C7×Q8—44no—
200C7×Q8—82no—
201C7×Q8—82no—
202C7×Q8—82no—
203C7×D4—28no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—24no—
207C2×C4×C7—22no—
208C2×C4×C7—21no—
209C2×C4×C7—24no—
210C2×C4×C7—24no—
211C2×C4×C7—21no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7—44no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—44no—
219C2×C4×C7—81no—
220C2×C4×C7—82no—
221C2×C4×C7—82no—
222C2×C4×C7#12011yescharacteristic
223C2×C4×C7#12012yes—
224C2×C4×C7#12011yescharacteristic
225C2×C4×C7C4⋊Q812yes—
226C2×Dic7—28no—
227C2×Dic7—44no—
228C2×Dic7—44no—
229C2×Dic7—42no—
230C2×Dic7—48no—
231C2×Dic7—42no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—42no—
235C2×Dic7—42no—
236C22×Q8—21no—
237C22×Q8C22⋊D712yes—
238C22×D4C22×D711yescharacteristic
239C23×C4—72no—
240C23×C4C22⋊D711yescharacteristic
241C23×C4C22×D711yescharacteristic
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—24no—
245#125—22no—
246#125—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—24no—
264C4×D4—24no—
265#117—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—141no—
269C2×C4⋊C4—142no—
270C2×C4⋊C4—142no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—148no—
274C2×C4⋊C4—142no—
275C2×C4⋊C4C22⋊D711yescharacteristic
276C2×C4⋊C4C22⋊D712yes—
277C2×C4⋊C4C22⋊D714yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—74no—
280C2×C22⋊C4—72no—
281C2×C22⋊C4—142no—
282C2×C22⋊C4C22⋊D711yescharacteristic
283C2×C22⋊C4C22×D711yescharacteristic
284C2×C22⋊C4C22×D714yes—
285C2×C42—142no—
286C2×C42C22⋊D711yescharacteristic
287#99—78no—
288#95—142no—
289#95—141no—
290#95—142no—
291#95—148no—
292#95—142no—
293C22×C7—22no—
294C22×C7—21no—
295C22×C7—22no—
296C22×C7—21no—
297C22×C7—22no—
298C22×C7—42no—
299C22×C7—42no—
300C22×C7—42no—
301C22×C7—41no—
302C22×C7—41no—
303C22×C7—81no—
304C22×C7C2×#12011yescharacteristic
305C22×C7C2×C4⋊Q811yescharacteristic
306C22×C7#53611yescharacteristic
307C22×C7D4×Q814yes—
308C4×C7—22no—
309C4×C7—21no—
310C4×C7—22no—
311C4×C7—41no—
312C4×C7—44no—
313C4×C7—44no—
314C4×C7—41no—
315C4×C7#45812yes—
316Dic7—48no—
317Dic7—82no—
318Dic7—82no—
319C24C2×C22⋊D711yescharacteristic
320C24C23×D711yescharacteristic
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—44no—
326C2×Q8—82no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4Q8×D714yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—144no—
337C22×C4—142no—
338C22×C4—142no—
339C22×C4—144no—
340C22×C4—562no—
341C22×C4C2×C22⋊D711yescharacteristic
342C22×C4C2×C22⋊D712yes—
343C22×C4C2×C22⋊D711yescharacteristic
344C22×C4C2×C22⋊D714yes—
345C22×C4C2×C22⋊D711yescharacteristic
346C22×C4C2×C22⋊D711yescharacteristic
347C22×C4C2×C22⋊D714yes—
348C22×C4C2×C22⋊D711yescharacteristic
349C22×C4C2×C22⋊D712yes—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—282no—
362C4⋊C4—281no—
363C4⋊C4—282no—
364C4⋊C4—281no—
365C4⋊C4—568no—
366C4⋊C4—562no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—22no—
372C22⋊C4—78no—
373C22⋊C4—144no—
374C22⋊C4—144no—
375C22⋊C4—562no—
376C42—24no—
377C42—21no—
378C42—21no—
379C42—284no—
380C2×C7—24no—
381C2×C7—21no—
382C2×C7—21no—
383C2×C7—42no—
384C2×C7#243411yescharacteristic
385C2×C7C2×#45811yescharacteristic
386C2×C7#302711yescharacteristic
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#650712yes—
394C23C2×Q8×D712yes—
395C23#655112yes—
396C23C22×C22⋊D711yescharacteristic
397C23C22×C22⋊D712yes—
398Q8—44no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402D4—28no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—148no—
422C2×C4—284no—
423C2×C4—284no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—282no—
427C2×C4—288no—
428C2×C4—282no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4#650712yes—
432C2×C4#651611yescharacteristic
433C2×C4#651612yes—
434C2×C4#651611yescharacteristic
435C7#2585311yescharacteristic, complemented, Sylow
436C22—22no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#650711yescharacteristic
448C22Q8×C22⋊D714yes—
449C22#9050711yescharacteristic
450C22C2×#651611yescharacteristic
451C4—22no—
452C4—21no—
453C4—22no—
454C4—41no—
455C4—44no—
456C4—44no—
457C4—41no—
458C4—288no—
459C4—562no—
460C4—562no—
461C4#9001512yes—
462C2—24no—
463C2—21no—
464C2—21no—
465C2—42no—
466C2#1174151011yescharacteristic
467C2C2×#9001511yescharacteristic
468C2#1174157111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC24×C7C24
FrattiniC24C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4C23
hypercenter#3501D7
hypocenterC7#25853
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #11740220C2semidirect
4. #11729649C2semidirect
5. #11729455C2semidirect
87. #6408Q8semidirect
153. C22⋊Dic7C2×Q8semidirect
435. C7#25853semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117402202
#117398192
#117296492
#117294552
#117294112
C4×#64082
#258537