Finite Group Databasebeta

← #13271351

Subgroups — #13271351

Order 1792.

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

normal subgroups only

This group has 694 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
2C2×#90009C211yesmaximal, characteristic
3#11729891C211yesmaximal, characteristic
4C2×#89343C212yesmaximal, complemented
5#11725450C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×C7×#124C411yescharacteristic
9C2×C7×C22⋊Q8C411yescharacteristic
10C22×#1136—21no—
11C7×#394—21no—
12C7×#394—21no—
13C7×#391C2211yescharacteristic
14C7×#390C411yescharacteristic
15C7×#385—21no—
16C7×#385—21no—
17C4×C7×C22⋊C4C411yescharacteristic
18#90009—24no—
19C2×#6410C2212yes—
20C2×#6410C2212yes—
21C2×#6407—21no—
22C2×#6407—21no—
23#89979—21no—
24#89343—24no—
25#13095—71nomaximal, Sylow
26C2×C7×#41—22no—
27C2×C7×#41—22no—
28C2×C7×#41—41no—
29C2×C7×#41—41no—
30C2×C7×#41—42no—
31C22×C7×Q8C2×C411yescharacteristic
32C22×C7×Q8C2311yescharacteristic
33C22×C7×D4C2×C411yescharacteristic
34C22×C7×D4D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7C2×C411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C7×C4⋊Q8—22no—
40C7×C4⋊Q8—22no—
41C7×#124—42no—
42C7×C22⋊Q8—41no—
43C7×C22⋊Q8—41no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C42×C7C2×C411yescharacteristic
55C2×C42×C7C2×C411yescharacteristic
56C2×#1136—24no—
57C2×#1136—22no—
58C2×#1136—22no—
59C2×#1136—42no—
60C2×#1136—41no—
61C22×C7⋊C8—41no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65#6410—22no—
66#6410—22no—
67#6410—22no—
68#6410D414yes—
69#6407—24no—
70#6407—42no—
71C22⋊(C7⋊C8)—41no—
72C22⋊(C7⋊C8)—41no—
73C2×#413—71no—
74#2432D711yescharacteristic
75C2×#356—72no—
76#1785—71no—
77#1394—72no—
78C24×C7C22⋊C411yescharacteristic
79C24×C7C22⋊C411yescharacteristic
80C24×C7C22⋊C411yescharacteristic
81C7×#41—44no—
82C7×#41—44no—
83C7×#41—44no—
84C7×#41—44no—
85C2×C7×Q8—22no—
86C2×C7×Q8—41no—
87C2×C7×Q8—41no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8C2×D412yes—
91C2×C7×Q8#4112yes—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4—21no—
96C2×C7×D4—22no—
97C2×C7×D4—21no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C22×C4×C7—22no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—41no—
118C22×C4×C7—41no—
119C22×C4×C7C22⋊C411yescharacteristic
120C22×C4×C7C22×C411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—42no—
128C7×C4⋊C4—41no—
129C7×C22⋊C4—42no—
130C7×C22⋊C4—42no—
131C7×C22⋊C4—42no—
132C7×C22⋊C4—42no—
133C7×C22⋊C4—42no—
134C7×C22⋊C4—42no—
135C42×C7—22no—
136C42×C7—41no—
137#1136—24no—
138#1136—44no—
139#1136—42no—
140#1136—42no—
141#1136—42no—
142C2×C7⋊C8—44no—
143C2×C7⋊C8—42no—
144C2×C7⋊C8—42no—
145C22×#41C2×D711yescharacteristic
146C22×M16—141no—
147C2×C4⋊Q8C2×D711yescharacteristic
148C2×#124Dic711yescharacteristic
149C2×C22⋊Q8Dic711yescharacteristic
150#413—144no—
151C2×#101—72no—
152C2×#101—72no—
153C2×#100—141no—
154C2×#100—141no—
155#407—141no—
156#394—21no—
157#394—21no—
158#391C2×D711yescharacteristic
159#390Dic711yescharacteristic
160#385—21no—
161#385—21no—
162C4×C22⋊C4Dic711yescharacteristic
163#356—144no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7—21no—
169C23×C7—21no—
170C23×C7—21no—
171C23×C7—21no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7—41no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7#9912yes—
179C23×C7C2×C22⋊C411yescharacteristic
180C7×Q8—42no—
181C7×Q8—42no—
182C7×Q8—82no—
183C7×D4—44no—
184C7×D4—44no—
185C7×D4—42no—
186C7×D4—42no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—22no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—42no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7#9911yescharacteristic
216C2×C4×C7#9911yescharacteristic
217C2×C4×C7C4×D412yes—
218C2×C4×C7#12011yescharacteristic
219C2×C4×C7#12112yes—
220C2×C4×C7#12311yescharacteristic
221C7⋊C8—44no—
222C7⋊C8—82no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—41no—
226C2×#41—41no—
227C2×#41—42no—
228C22×Q8C2×Dic711yescharacteristic
229C22×Q8C22×D711yescharacteristic
230C22×D4C2×Dic711yescharacteristic
231C22×D4C22⋊D711yescharacteristic
232C22×D4C22⋊D711yescharacteristic
233C23×C4C2×Dic711yescharacteristic
234C23×C4C22⋊D711yescharacteristic
235C23×C4C22⋊D711yescharacteristic
236C2×M16—144no—
237C2×M16—142no—
238C2×M16—142no—
239C2×M16—282no—
240C2×M16—281no—
241C22×C8—281no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244#124—42no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×Dic711yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42C2×Dic711yescharacteristic
258C2×C42C2×Dic711yescharacteristic
259#101—74no—
260#101—142no—
261#101—142no—
262#101—142no—
263#100—144no—
264#100—282no—
265C22⋊C8—281no—
266C22⋊C8—281no—
267#95—21no—
268#95—21no—
269#95—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—22no—
273C22×C7—42no—
274C22×C7—42no—
275C22×C7—41no—
276C22×C7—41no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7—41no—
281C22×C7—41no—
282C22×C7—41no—
283C22×C7—81no—
284C22×C7—81no—
285C22×C7—81no—
286C22×C7—81no—
287C22×C7#38611yescharacteristic
288C22×C7C2×#9911yescharacteristic
289C22×C7C2×#9911yescharacteristic
290C4×C7—21no—
291C4×C7—22no—
292C4×C7—21no—
293C4×C7—21no—
294C4×C7—21no—
295C4×C7—22no—
296C4×C7—42no—
297C4×C7—81no—
298C4×C7—81no—
299C4×C7—81no—
300C24C22⋊Dic711yescharacteristic
301C24C22⋊Dic711yescharacteristic
302C24C22⋊Dic711yescharacteristic
303#41—44no—
304#41—44no—
305#41—44no—
306#41—44no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8D4×D712yes—
313C2×Q8#115912yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4C22⋊Dic711yescharacteristic
342C22×C4C22×Dic711yescharacteristic
343C22×C4C2×C22⋊D711yescharacteristic
344C22×C4C2×C22⋊D711yescharacteristic
345M16—144no—
346M16—284no—
347M16—282no—
348M16—282no—
349M16—282no—
350C2×C8—284no—
351C2×C8—282no—
352C2×C8—282no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C42—22no—
366C42—41no—
367C2×C7—21no—
368C2×C7—21no—
369C2×C7—42no—
370C2×C7—81no—
371C2×C7—81no—
372C2×C7#189811yescharacteristic
373C2×C7#213211yescharacteristic
374C2×C7#213511yescharacteristic
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23#640812yes—
390C23C2×C22⋊Dic711yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—82no—
394D4—44no—
395D4—44no—
396D4—42no—
397D4—42no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4#640811yescharacteristic
427C2×C4#640811yescharacteristic
428C2×C4D4×Dic712yes—
429C2×C4#649811yescharacteristic
430C2×C4#650011yescharacteristic
431C2×C4#650212yes—
432C8—284no—
433C8—562no—
434C7#1309511yescharacteristic, complemented, Sylow
435C22—21no—
436C22—21no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22C2×#640811yescharacteristic
453C22C2×#640811yescharacteristic
454C22#8999211yescharacteristic
455C4—21no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—42no—
462C4—81no—
463C4—81no—
464C4—81no—
465C2—21no—
466C2—21no—
467C2—42no—
468C2—81no—
469C2—81no—
470C2#1172942411yescharacteristic
471C2#1172957411yescharacteristic
472C2#1172962711yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC7×#2432C2
socleC22×C7C2×#99
prime radicalC22×C7×#41C22
cosocleC22×C7×Q8C23
hypercenter#2432D7
hypocenterC7#13095
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2432

Complements

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

NH ≅ G/Ntype
4. C2×#89343C2semidirect
5. #11725450C2semidirect
434. C7#13095semidirect

Maximal subgroups

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

subgroupindex
C7×#24322
C2×#900092
#117298912
C2×#893432
#117254502
#130957