Finite Group Databasebeta

← #13251436

Subgroups — #13251436

Order 1792.

712 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2432C211yesmaximal, characteristic
2#11741519C211yesmaximal, characteristic
3#11729893C211yesmaximal, characteristic
4#11729443C212yesmaximal, complemented
5#11725422C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×C4⋊Q8C411yescharacteristic, complemented
8C2×C7×#124C411yescharacteristic, complemented
9C2×C7×C22⋊Q8C2211yescharacteristic
10C22×#1136—21no—
11C7×#394—21no—
12C7×#394—21no—
13C7×#391C411yescharacteristic, complemented
14C7×#390C411yescharacteristic, complemented
15C7×#385—21no—
16C7×#385—21no—
17C4×C7×C22⋊C4C2211yescharacteristic
18#90006—24no—
19C2×#6408—21no—
20C2×#6408—21no—
21#89983C2212yes—
22#89979—21no—
23#89979C2212yes—
24#89333—24no—
25#9443—71nomaximal, Sylow
26C2×C7×#41—22no—
27C2×C7×#41—22no—
28C2×C7×#41—42no—
29C2×C7×#41—42no—
30C22×C7×Q8C2×C411yescharacteristic
31C22×C7×Q8C2×C411yescharacteristic
32C22×C7×D4C2×C411yescharacteristic
33C22×C7×D4D411yescharacteristic
34C22×C7×D4D411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C7×C4⋊Q8—42no—
39C7×#124—42no—
40C7×C22⋊Q8—22no—
41C7×C22⋊Q8—22no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C42×C7C2×C411yescharacteristic
53C2×C42×C7C2×C411yescharacteristic
54C2×C22⋊Dic7—22no—
55C2×#1136—22no—
56C2×#1136—22no—
57C2×#1136—42no—
58C2×#1136—42no—
59C22×C7⋊C8—22no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63#6408—42no—
64#6408—42no—
65C7⋊#95—22no—
66C7⋊#95—22no—
67C22⋊(C7⋊C8)—24no—
68C22⋊(C7⋊C8)—22no—
69C22⋊(C7⋊C8)D414yes—
70#2879—71no—
71#2432D711yescharacteristic
72#1782—71no—
73#1511—72no—
74#1284—72no—
75C24×C7C22⋊C411yescharacteristic
76C24×C7C22⋊C411yescharacteristic
77C24×C7C2×D411yescharacteristic
78C7×#41—44no—
79C7×#41—44no—
80C7×#41—48no—
81C2×C7×Q8—22no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—41no—
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—22no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7C22⋊C411yescharacteristic
111C22×C4×C7C22⋊C411yescharacteristic
112C22×C4×C7C22×C411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D412yes—
115C22×C4×C7#4112yes—
116C22×Dic7—42no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—42no—
119C7×C4⋊C4—42no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C22⋊C4—42no—
123C7×C22⋊C4—42no—
124C7×C22⋊C4—42no—
125C7×C22⋊C4—42no—
126C7×C22⋊C4—42no—
127C7×C22⋊C4—42no—
128C42×C7—41no—
129C42×C7—41no—
130C22⋊Dic7—42no—
131C22⋊Dic7—42no—
132#1136—44no—
133#1136—42no—
134#1136—42no—
135C2×C7⋊C8—24no—
136C2×C7⋊C8—44no—
137C22×#41C2×D711yescharacteristic
138C22×M16—141no—
139C2×C4⋊Q8Dic711yescharacteristic, complemented
140C2×#124Dic711yescharacteristic, complemented
141C2×C22⋊Q8C2×D711yescharacteristic
142C2×#99—141no—
143C2×#99—141no—
144#408—144no—
145#407—72no—
146#407—141no—
147#394—21no—
148#394—21no—
149#391Dic711yescharacteristic, complemented
150#390Dic711yescharacteristic, complemented
151#385—21no—
152#385—21no—
153#381—72no—
154C4×C22⋊C4C2×D711yescharacteristic
155#328—144no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—21no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7#9912yes—
170C23×C7C2×C22⋊C411yescharacteristic
171C23×C7#12112yes—
172C7×Q8—44no—
173C7×Q8—82no—
174C7×D4—44no—
175C7×D4—44no—
176C7×D4—44no—
177C2×C4×C7—22no—
178C2×C4×C7—22no—
179C2×C4×C7—24no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—22no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—42no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7#9912yes—
206C2×C4×C7C4×D412yes—
207C2×C4×C7#12011yescharacteristic
208C2×C4×C7#12311yescharacteristic
209C2×Dic7—82no—
210C2×Dic7—82no—
211C2×Dic7—82no—
212C7⋊C8—44no—
213C2×#41—22no—
214C2×#41—22no—
215C2×#41—42no—
216C2×#41—42no—
217C22×Q8C2×Dic711yescharacteristic
218C22×Q8C2×Dic711yescharacteristic
219C22×D4C2×Dic711yescharacteristic
220C22×D4C22⋊D711yescharacteristic
221C22×D4C22⋊D711yescharacteristic
222C23×C4C22⋊D711yescharacteristic
223C23×C4C22⋊D711yescharacteristic
224C23×C4C22×D711yescharacteristic
225C2×M16—142no—
226C2×M16—142no—
227C2×M16—282no—
228C2×M16—282no—
229C22×C8—142no—
230C4⋊Q8—42no—
231#124—42no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C2×Dic711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—142no—
245C2×C42C2×Dic711yescharacteristic
246C2×C42C2×Dic711yescharacteristic
247#99—282no—
248#99—282no—
249C22⋊C8—74no—
250C22⋊C8—144no—
251C22⋊C8—142no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—142no—
256#95—142no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—22no—
260C22×C7—22no—
261C22×C7—22no—
262C22×C7—22no—
263C22×C7—42no—
264C22×C7—42no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7—41no—
268C22×C7—41no—
269C22×C7—41no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7#38611yescharacteristic
275C22×C7C2×#9911yescharacteristic
276C22×C7C2×#9911yescharacteristic
277C4×C7—21no—
278C4×C7—21no—
279C4×C7—22no—
280C4×C7—22no—
281C4×C7—42no—
282C4×C7—41no—
283C4×C7—81no—
284C4×C7—81no—
285C4×C7—81no—
286Dic7—162no—
287C24C22⋊Dic711yescharacteristic
288C24C22⋊Dic711yescharacteristic
289C24C2×C22⋊D711yescharacteristic
290#41—44no—
291#41—44no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—282no—
323C22×C4C22⋊Dic711yescharacteristic
324C22×C4C22⋊Dic711yescharacteristic
325C22×C4D4×D712yes—
326C22×C4#115912yes—
327C22×C4C22×Dic711yescharacteristic
328C22×C4C2×C22⋊D711yescharacteristic
329M16—284no—
330M16—282no—
331M16—282no—
332C2×C8—144no—
333C2×C8—284no—
334C4⋊C4—22no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—282no—
346C22⋊C4—282no—
347C42—41no—
348C42—41no—
349C2×C7—21no—
350C2×C7—21no—
351C2×C7—42no—
352C2×C7—42no—
353C2×C7—81no—
354C2×C7#188411yescharacteristic
355C2×C7#189811yescharacteristic
356C2×C7#190911yescharacteristic
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#640812yes—
371C23#650212yes—
372C23C2×C22⋊Dic711yescharacteristic
373Q8—44no—
374Q8—82no—
375D4—44no—
376D4—44no—
377D4—44no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—562no—
407C2×C4—562no—
408C2×C4—562no—
409C2×C4#640812yes—
410C2×C4D4×Dic712yes—
411C2×C4#649811yescharacteristic
412C2×C4#650011yescharacteristic
413C8—284no—
414C7#944311yescharacteristic, complemented, Sylow
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22C2×#640811yescharacteristic
433C22C2×#640811yescharacteristic
434C22#8999211yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—1122no—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2#1172942411yescharacteristic
451C2#1172957111yescharacteristic
452C2#1172961911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2432C2
socleC22×C7#386
prime radicalC22×C7×#41C22
cosocleC23×C4×C7C23
hypercenter#2432D7
hypocenterC7#9443
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2432

Complements

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

NH ≅ G/Ntype
4. #11729443C2semidirect
5. #11725422C2semidirect
7. C2×C7×C4⋊Q8C4semidirect
8. C2×C7×#124C4semidirect
13. C7×#391C4semidirect
14. C7×#390C4semidirect
139. C2×C4⋊Q8Dic7semidirect
140. C2×#124Dic7semidirect
149. #391Dic7semidirect
150. #390Dic7semidirect
414. C7#9443semidirect

Maximal subgroups

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

subgroupindex
C7×#24322
#117415192
#117298932
#117294432
#117254222
#94437