Finite Group Databasebeta

← #13251716

Subgroups — #13251716

Order 1792.

956 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2405C211yesmaximal, characteristic
2#11741488C211yesmaximal, characteristic
3#11729884C211yesmaximal, characteristic
4#11729296C212yesmaximal, complemented
5C7⋊#1304C212yesmaximal, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#123C412yescomplemented
8C22×C7×C22⋊C4C2211yescharacteristic
9C22×C22⋊Dic7—21no—
10C7×#394—22no—
11C7×#386C412yescomplemented
12C7×#385—21no—
13C7×#379—21no—
14C7×#379C2211yescharacteristic
15#90019C2212yes—
16#90019C2212yes—
17#89992—24no—
18C2×#6408—22no—
19#89983—21no—
20#89321—24no—
21#9499—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C25×C7C2311yescharacteristic
24C22×C7×D4—22no—
25C22×C7×D4—22no—
26C22×C7×D4—44no—
27C22×C7×D4C2×C412yes—
28C22×C7×D4D412yes—
29C23×C4×C7C2×C411yescharacteristic, complemented
30C23×C4×C7C2×C411yescharacteristic, complemented
31C23×C4×C7D411yescharacteristic
32C23×Dic7—22no—
33C7×#123—44no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4C2×C412yescomplemented
44C2×C22⋊Dic7—22no—
45C2×C22⋊Dic7—22no—
46C2×C22⋊Dic7—22no—
47C2×C22⋊Dic7—22no—
48C2×C22⋊Dic7—22no—
49C2×C22⋊Dic7—22no—
50C2×C22⋊Dic7—44no—
51C2×C22⋊Dic7D414yes—
52C7×#95—22no—
53#6408—44no—
54C7⋊#95—24no—
55#2405D711yescharacteristic
56#2308—71no—
57#1819—72no—
58#1781—71no—
59#1304—72no—
60C24×C7—21no—
61C24×C7—24no—
62C24×C7—21no—
63C24×C7—42no—
64C24×C7—42no—
65C24×C7—42no—
66C24×C7—81no—
67C24×C7—81no—
68C24×C7C22⋊C412yes—
69C24×C7C22×C411yescharacteristic
70C24×C7C2×D411yescharacteristic
71C24×C7C2×D411yescharacteristic
72C24×C7C2×D412yes—
73C24×C7#4112yes—
74C2×C7×D4—24no—
75C2×C7×D4—24no—
76C2×C7×D4—24no—
77C2×C7×D4—24no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—44no—
81C2×C7×D4—44no—
82C2×C7×D4—48no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—22no—
87C22×C4×C7—21no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7—81no—
93C22×C4×C7C22⋊C412yes—
94C22×Dic7—24no—
95C22×Dic7—44no—
96C22×Dic7—42no—
97C22×Dic7—82no—
98C22×Dic7—82no—
99C7×C4⋊C4—44no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—44no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—81no—
107C7×C22⋊C4—82no—
108C22⋊Dic7—44no—
109C22⋊Dic7—44no—
110C22⋊Dic7—44no—
111C22⋊Dic7—44no—
112C22⋊Dic7—44no—
113C22⋊Dic7—82no—
114C22⋊Dic7—82no—
115C23×D4C2×D711yescharacteristic
116C2×#123Dic712yescomplemented
117C22×C22⋊C4—141no—
118C22×C22⋊C4C2×D711yescharacteristic
119C2×#99—142no—
120#394—22no—
121#386—144no—
122#386Dic712yescomplemented
123#385—21no—
124#381—141no—
125#379—21no—
126#379—72no—
127#379—72no—
128#379C2×D711yescharacteristic
129#342—144no—
130C23×C7—22no—
131C23×C7—22no—
132C23×C7—24no—
133C23×C7—22no—
134C23×C7—22no—
135C23×C7—22no—
136C23×C7—22no—
137C23×C7—22no—
138C23×C7—22no—
139C23×C7—24no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—42no—
144C23×C7—42no—
145C23×C7—41no—
146C23×C7—42no—
147C23×C7—42no—
148C23×C7—41no—
149C23×C7—42no—
150C23×C7—42no—
151C23×C7—42no—
152C23×C7—42no—
153C23×C7—42no—
154C23×C7—81no—
155C23×C7—81no—
156C23×C7—88no—
157C23×C7—88no—
158C23×C7#9914yes—
159C23×C7C2×C22⋊C411yescharacteristic
160C23×C7C4×D412yes—
161C23×C7#12011yescharacteristic
162C23×C7#12112yes—
163C23×C7#12311yescharacteristic
164C7×D4—48no—
165C7×D4—48no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—82no—
179C2×C4×C7—82no—
180C2×Dic7—44no—
181C2×Dic7—44no—
182C2×Dic7—44no—
183C2×Dic7—88no—
184C2×Dic7—82no—
185C2×Dic7—82no—
186C2×Dic7—82no—
187C25C22⋊D711yescharacteristic
188C25C22×D711yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—44no—
192C22×D4C2×Dic712yes—
193C22×D4C22⋊D712yes—
194C23×C4—142no—
195C23×C4C2×Dic711yescharacteristic, complemented
196C23×C4C2×Dic711yescharacteristic, complemented
197C23×C4C22⋊D711yescharacteristic
198#123—44no—
199C2×C4⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—74no—
209C2×C22⋊C4—142no—
210C2×C22⋊C4—142no—
211C2×C22⋊C4—142no—
212C2×C22⋊C4—142no—
213C2×C22⋊C4—142no—
214C2×C22⋊C4—142no—
215C2×C22⋊C4—284no—
216C2×C22⋊C4C2×Dic712yescomplemented
217#99—284no—
218#95—22no—
219#95—144no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—22no—
225C22×C7—21no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—21no—
229C22×C7—22no—
230C22×C7—24no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—21no—
234C22×C7—44no—
235C22×C7—44no—
236C22×C7—44no—
237C22×C7—44no—
238C22×C7—44no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—84no—
242C22×C7—82no—
243C22×C7—82no—
244C22×C7—84no—
245C22×C7—84no—
246C22×C7#38611yescharacteristic
247C22×C7C2×#9912yes—
248C22×C7#45812yes—
249C4×C7—42no—
250C4×C7—82no—
251C4×C7—81no—
252C4×C7—81no—
253Dic7—84no—
254Dic7—162no—
255C24—21no—
256C24—24no—
257C24—21no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—81no—
262C24—81no—
263C24C22⋊Dic712yes—
264C24D4×D712yes—
265C24#115912yes—
266C24C22×Dic711yescharacteristic
267C24C2×C22⋊D711yescharacteristic
268C24C2×C22⋊D711yescharacteristic
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—48no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—81no—
288C22×C4—144no—
289C22×C4—284no—
290C22×C4—282no—
291C22×C4—562no—
292C22×C4—562no—
293C22×C4C22⋊Dic712yes—
294C4⋊C4—44no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—81no—
302C22⋊C4—82no—
303C22⋊C4—284no—
304C22⋊C4—284no—
305C22⋊C4—284no—
306C22⋊C4—284no—
307C22⋊C4—284no—
308C22⋊C4—562no—
309C22⋊C4—562no—
310C2×C7—22no—
311C2×C7—22no—
312C2×C7—42no—
313C2×C7—42no—
314C2×C7—41no—
315C2×C7—44no—
316C2×C7#189011yescharacteristic
317C2×C7#189811yescharacteristic
318C2×C7#191311yescharacteristic
319C23—22no—
320C23—22no—
321C23—24no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—88no—
346C23—88no—
347C23#640814yes—
348C23D4×Dic712yes—
349C23#649811yescharacteristic
350C23#650011yescharacteristic
351C23#650212yes—
352C23C2×C22⋊Dic711yescharacteristic
353D4—48no—
354D4—48no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—284no—
370C2×C4—284no—
371C2×C4—284no—
372C2×C4—568no—
373C2×C4—562no—
374C2×C4—562no—
375C2×C4—562no—
376C7#949911yescharacteristic, complemented, Sylow
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—24no—
385C22—22no—
386C22—21no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22—42no—
397C22—42no—
398C22—82no—
399C22—82no—
400C22—84no—
401C22—84no—
402C22—84no—
403C22#8982512yes—
404C22C2×#640812yes—
405C22#8999211yescharacteristic
406C4—42no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4—564no—
411C4—1122no—
412C2—22no—
413C2—22no—
414C2—42no—
415C2—41no—
416C2—42no—
417C2—44no—
418C2#1172942411yescharacteristic
419C2#1172945811yescharacteristic
420C2#1172960211yescharacteristic
subgroupG/H
centerC22#89992
commutatorC24×C7C22×C4
FrattiniC25C22×D7
FittingC7×#2405C2
socleC22×C7#386
prime radicalC23×C7×D4C22
cosocleC25×C7C23
hypercenter#2405D7
hypocenterC7#9499
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C25 ◁ #2405

Complements

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

NH ≅ G/Ntype
4. #11729296C2semidirect
5. C7⋊#1304C2semidirect
7. C2×C7×#123C4semidirect
11. C7×#386C4semidirect
29. C23×C4×C7C2×C4semidirect
30. C23×C4×C7C2×C4semidirect
43. C2×C7×C22⋊C4C2×C4semidirect
116. C2×#123Dic7semidirect
122. #386Dic7semidirect
195. C23×C4C2×Dic7semidirect
196. C23×C4C2×Dic7semidirect
216. C2×C22⋊C4C2×Dic7semidirect
376. C7#9499semidirect

Maximal subgroups

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

subgroupindex
C7×#24052
#117414882
#117298842
#117292962
C7⋊#13042
#94997