Finite Group Databasebeta

← #13264272

Subgroups — #13264272

Order 1792.

700 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2539C211yesmaximal, characteristic
2C2×#89321C211yesmaximal, characteristic, complemented
3#11729293C211yesmaximal, characteristic, complemented
4#11725457C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C411yescharacteristic, complemented
8C2×C7×#120—21no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C22×C22⋊Dic7—21no—
11C7×#399C411yescharacteristic, complemented
12C7×#399C2211yescharacteristic
13C7×#392C411yescharacteristic, complemented
14C7×#392C2211yescharacteristic
15C7×#390—21no—
16C7×#390—21no—
17C4×C7×C22⋊C4C411yescharacteristic, complemented
18#89979—21no—
19#89321—21no—
20#89321C2212yes—
21#89320—21no—
22#89320C2212yes—
23#11757—71nomaximal, Sylow
24C25×C7D411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic, complemented
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×Dic7—41no—
32C7×#123—44no—
33C7×#121—41no—
34C7×#121—41no—
35C7×#120—44no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C2×C42×C7—21no—
52C2×C22⋊Dic7—24no—
53C2×C22⋊Dic7—22no—
54C2×C22⋊Dic7—41no—
55C2×C22⋊Dic7—42no—
56C2×#1136—41no—
57C7×#95C2×C411yescharacteristic, complemented
58C7×#95C2×C411yescharacteristic, complemented
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—41no—
61#2539D711yescharacteristic
62C2×#342—71no—
63#1462—71no—
64#1352—72no—
65#1343—72no—
66C24×C7—21no—
67C24×C7—21no—
68C24×C7—22no—
69C24×C7—21no—
70C24×C7—21no—
71C24×C7—21no—
72C24×C7—42no—
73C24×C7—81no—
74C24×C7C22⋊C411yescharacteristic
75C24×C7C22⋊C411yescharacteristic
76C24×C7C2×D411yescharacteristic
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—41no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—42no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—81no—
88C2×C7×D4—81no—
89C2×C7×D4—81no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—42no—
98C22×C4×C7—81no—
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×Dic7—44no—
104C22×Dic7—42no—
105C22×Dic7—82no—
106C22×Dic7—82no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—42no—
109C7×C4⋊C4—42no—
110C7×C22⋊C4—24no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—42no—
120C42×C7—81no—
121C22⋊Dic7—24no—
122C22⋊Dic7—44no—
123C22⋊Dic7—42no—
124#1136—82no—
125C2×C7⋊C8—42no—
126C2×#123—21no—
127C2×#121Dic711yescharacteristic, complemented
128C2×#120—21no—
129C22×C22⋊C4—141no—
130C22×C22⋊C4C2×D711yescharacteristic
131#407—141no—
132#399Dic711yescharacteristic, complemented
133#399C2×D711yescharacteristic
134#392Dic711yescharacteristic, complemented
135#392C2×D711yescharacteristic
136#390—21no—
137#390—21no—
138C4×C22⋊C4Dic711yescharacteristic, complemented
139#342—72no—
140#342—141no—
141#323—72no—
142#323—141no—
143C23×C7—22no—
144C23×C7—22no—
145C23×C7—22no—
146C23×C7—22no—
147C23×C7—22no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—41no—
155C23×C7—42no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—42no—
165C23×C7—41no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—81no—
169C23×C7—81no—
170C23×C7—84no—
171C23×C7—82no—
172C23×C7—82no—
173C23×C7#9911yescharacteristic
174C23×C7#9911yescharacteristic
175C23×C7#10412yes—
176C23×C7C2×C22⋊C411yescharacteristic
177C23×C7AGL(1,ℤ/8)11yescharacteristic
178C23×C7#13711yescharacteristic
179C7×D4—84no—
180C7×D4—82no—
181C7×D4—82no—
182C7×D4—82no—
183C7×D4—82no—
184C7×D4—82no—
185C2×C4×C7—22no—
186C2×C4×C7—22no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
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—81no—
204C2×C4×C7—81no—
205C2×C4×C7—81no—
206C2×Dic7—44no—
207C2×Dic7—88no—
208C2×Dic7—82no—
209C2×Dic7—82no—
210C7⋊C8—82no—
211C25C22⋊D711yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C2×Dic711yescharacteristic, complemented
216C23×C4—281no—
217C23×C4C2×Dic711yescharacteristic
218C23×C4C22⋊D711yescharacteristic
219C2×M16—281no—
220#123—44no—
221#121—41no—
222#121—41no—
223#120—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×Dic711yescharacteristic, complemented
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—144no—
236C2×C22⋊C4—142no—
237C2×C22⋊C4—281no—
238C2×C22⋊C4—282no—
239C2×C22⋊C4C2×Dic711yescharacteristic
240C2×C22⋊C4C22⋊D711yescharacteristic
241C2×C22⋊C4C22⋊D711yescharacteristic
242C2×C22⋊C4C22×D711yescharacteristic
243C2×C42—21no—
244C22⋊C8—142no—
245C22⋊C8—281no—
246#95C2×Dic711yescharacteristic, complemented
247#95C2×Dic711yescharacteristic, complemented
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—22no—
257C22×C7—21no—
258C22×C7—42no—
259C22×C7—42no—
260C22×C7—42no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—44no—
264C22×C7—44no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7—41no—
268C22×C7—41no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7—81no—
275C22×C7—82no—
276C22×C7—82no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7—81no—
280C22×C7#35312yes—
281C22×C7C2×#9911yescharacteristic
282C22×C7#41711yescharacteristic
283C22×C7C2×#10411yescharacteristic
284C4×C7—42no—
285C4×C7—42no—
286C4×C7—41no—
287C4×C7—41no—
288C4×C7—41no—
289C4×C7—41no—
290C4×C7—81no—
291Dic7—84no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—42no—
299C24—81no—
300C24C22⋊Dic711yescharacteristic
301C24C22⋊Dic711yescharacteristic
302C24C2×C22⋊D711yescharacteristic
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4—81no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—81no—
325C22×C4—284no—
326C22×C4—282no—
327C22×C4—562no—
328C22×C4—562no—
329C22×C4C22⋊Dic711yescharacteristic
330C22×C4C22⋊Dic711yescharacteristic
331C22×C4C22×Dic711yescharacteristic
332C22×C4C2×C22⋊D711yescharacteristic
333M16—562no—
334C2×C8—282no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—24no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—144no—
349C22⋊C4—284no—
350C22⋊C4—282no—
351C42—81no—
352C2×C7—21no—
353C2×C7—21no—
354C2×C7—22no—
355C2×C7—41no—
356C2×C7—41no—
357C2×C7—42no—
358C2×C7—41no—
359C2×C7—161no—
360C2×C7#150411yescharacteristic
361C2×C7#212111yescharacteristic
362C2×C7C2×#35311yescharacteristic
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—41no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—81no—
389C23—81no—
390C23—84no—
391C23—82no—
392C23—82no—
393C23#640811yescharacteristic
394C23#640811yescharacteristic
395C23#641112yes—
396C23#651211yescharacteristic
397C23#651411yescharacteristic
398C23C2×C22⋊Dic711yescharacteristic
399D4—84no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—284no—
427C2×C4—568no—
428C2×C4—562no—
429C2×C4—562no—
430C8—562no—
431C7#1175711yescharacteristic, complemented, Sylow
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#8934012yes—
465C22C2×#640811yescharacteristic
466C22#9000411yescharacteristic
467C22C2×#641111yescharacteristic
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—564no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—161no—
484C2#1172946211yescharacteristic
485C2C2×#8934011yescharacteristic
486C2#1172960111yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2539C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2539D7
hypocenterC7#11757
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
2. C2×#89321C2semidirect
3. #11729293C2semidirect
4. #11725457C2semidirect
5. #11725451C2semidirect
7. C2×C7×#121C4semidirect
11. C7×#399C4semidirect
13. C7×#392C4semidirect
17. C4×C7×C22⋊C4C4semidirect
28. C22×C7×D4C2×C4semidirect
38. C2×C7×C4⋊C4C2×C4semidirect
57. C7×#95C2×C4semidirect
58. C7×#95C2×C4semidirect
127. C2×#121Dic7semidirect
132. #399Dic7semidirect
134. #392Dic7semidirect
138. C4×C22⋊C4Dic7semidirect
215. C22×D4C2×Dic7semidirect
226. C2×C4⋊C4C2×Dic7semidirect
246. #95C2×Dic7semidirect
247. #95C2×Dic7semidirect
431. C7#11757semidirect

Maximal subgroups

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

subgroupindex
C7×#25392
C2×#893212
#117292932
#117254572
#117254512
#117577