Finite Group Databasebeta

← #13250819

Subgroups — #13250819

Order 1792.

534 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3193C211yesmaximal, characteristic
2#11740084C211yesmaximal, characteristic
3#11729455C211yesmaximal, characteristic, complemented
4#11729443C211yesmaximal, characteristic, complemented
5#11729427C211yesmaximal, characteristic
6#11725491C211yesmaximal, characteristic, complemented
7#11725488C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C7×#484—21no—
11C7×#480C412yescomplemented
12C7×#468—22no—
13C7×#450—21no—
14C7×#448C412yescomplemented
15C7×#407C2211yescharacteristic
16#89993—21no—
17#89992—21no—
18#89992—21no—
19#89984C2211yescharacteristic
20#89983C2211yescharacteristic
21C4×C22⋊Dic7C2211yescharacteristic
22#89979C2211yescharacteristic
23#89753—21no—
24#89723—21no—
25#89722—21no—
26#89333—21no—
27#89333—21no—
28#89332—22no—
29#89321—21no—
30#89321—21no—
31#9341—71nomaximal, Sylow
32C22×C7×Q8C2×C411yescharacteristic, complemented
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7C2311yescharacteristic
36C23×Dic7—21no—
37C2×C7×SD16—22no—
38C2×C7×M16—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×#121—21no—
44C7×#121—21no—
45C7×#121C2×C412yes—
46C7×#121D412yes—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C22⋊Dic7—21no—
53C2×C22⋊Dic7—21no—
54C2×C22⋊Dic7—21no—
55C2×C22⋊Dic7—21no—
56C2×C22⋊Dic7—21no—
57C2×C22⋊Dic7D411yescharacteristic
58C2×C22⋊Dic7D411yescharacteristic
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4×Dic7—21no—
62C2×C4×Dic7—21no—
63C2×#1136—21no—
64C7×#107—22no—
65C7×#103—22no—
66C7×#102—22no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8C2×C412yescomplemented
69C7⋊#95—21no—
70C7⋊#95—21no—
71C7⋊#95—21no—
72C7⋊#95—21no—
73C7⋊#95—21no—
74C7⋊#95D411yescharacteristic
75C7⋊#95D411yescharacteristic
76C22⋊(C7⋊C8)—21no—
77C22⋊(C7⋊C8)—21no—
78#3193D711yescharacteristic
79#2377—71no—
80#1996—71no—
81#1511—71no—
82#1507—71no—
83#1305—71no—
84#1284—71no—
85C24×C7—21no—
86C24×C7C2×D411yescharacteristic
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—81no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—21no—
93C2×C7×D4—81no—
94C2×C7×D4—81no—
95C2×C7×D4—81no—
96C2×C7×D4C22⋊C412yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—81no—
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4111yescharacteristic
106C22×C4×C7#4111yescharacteristic
107C22×Dic7—21no—
108C22×Dic7—21no—
109C22×Dic7—21no—
110C22×Dic7—21no—
111C22×Dic7—41no—
112C22×Dic7—81no—
113C22×Dic7—81no—
114C22×Dic7—81no—
115C7×SD16—82no—
116C7×M16—81no—
117C2×C7×C8—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4C22⋊C412yes—
124C7×C22⋊C4—22no—
125C7×C22⋊C4—22no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C22⋊Dic7—22no—
129C22⋊Dic7—41no—
130C22⋊Dic7—41no—
131C22⋊Dic7—41no—
132C22⋊Dic7—81no—
133C22⋊Dic7—81no—
134C22⋊Dic7—81no—
135C22⋊Dic7—81no—
136C4⋊Dic7—81no—
137C4⋊Dic7—81no—
138C4×Dic7—81no—
139C4×Dic7—81no—
140#1136—81no—
141C2×C7⋊C8—41no—
142C2×C22⋊Q8C2×D711yescharacteristic
143C2×#121C2×D711yescharacteristic
144#484—21no—
145#480Dic712yescomplemented
146#468—22no—
147#450—21no—
148#448Dic712yescomplemented
149#407—71no—
150#407C2×D711yescharacteristic
151#390—141no—
152#388—141no—
153#387—141no—
154#386—141no—
155#386—141no—
156#385—141no—
157#381—71no—
158#380—71no—
159C4×C22⋊C4—71no—
160#342—141no—
161#342—141no—
162#328—141no—
163#328—141no—
164#327—142no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—22no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7—41no—
176C23×C7C4×D411yescharacteristic
177C23×C7#12011yescharacteristic
178C23×C7#12111yescharacteristic
179C7×Q8—42no—
180C7×Q8—81no—
181C7×D4—42no—
182C7×D4—81no—
183C7×D4—82no—
184C7×D4—82no—
185C2×C4×C7—22no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—81no—
197C2×C4×C7—81no—
198C2×C4×C7C2×C22⋊C411yescharacteristic
199C2×C4×C7C4×D411yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12311yescharacteristic
202C2×Dic7—41no—
203C2×Dic7—41no—
204C2×Dic7—41no—
205C2×Dic7—41no—
206C2×Dic7—41no—
207C2×Dic7—42no—
208C2×Dic7—41no—
209C2×Dic7—41no—
210C2×Dic7—41no—
211C2×Dic7—41no—
212C2×Dic7—41no—
213C2×Dic7—41no—
214C2×Dic7—41no—
215C2×Dic7—81no—
216C2×Dic7—81no—
217C2×Dic7—84no—
218C7×C8—42no—
219C7⋊C8—81no—
220C22×Q8C2×Dic711yescharacteristic, complemented
221C22×D4—21no—
222C22×D4C22⋊D711yescharacteristic
223C23×C4—141no—
224C23×C4C22×D711yescharacteristic
225C2×SD16—22no—
226C2×M16—21no—
227C2×M16—141no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234#121C2×Dic712yes—
235#121C22⋊D712yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—141no—
238C2×C4⋊C4—141no—
239C2×C4⋊C4C2×Dic711yescharacteristic, complemented
240C2×C4⋊C4C22⋊D711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—71no—
244C2×C22⋊C4—71no—
245C2×C22⋊C4—141no—
246C2×C22⋊C4—141no—
247C2×C22⋊C4—141no—
248C2×C22⋊C4—141no—
249C2×C22⋊C4—141no—
250C2×C42—141no—
251C2×C42—141no—
252#107—22no—
253#103—22no—
254#102—22no—
255C22⋊C8—21no—
256C22⋊C8—141no—
257C22⋊C8—141no—
258C22⋊C8C2×Dic712yescomplemented
259#95—71no—
260#95—71no—
261#95—141no—
262#95—141no—
263#95—141no—
264#95—141no—
265#95—141no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—42no—
275C22×C7—42no—
276C22×C7—42no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7—81no—
280C22×C7—81no—
281C22×C7—81no—
282C22×C7—81no—
283C22×C7#38611yescharacteristic
284C22×C7#45511yescharacteristic
285C22×C7#45611yescharacteristic
286C4×C7—21no—
287C4×C7—21no—
288C4×C7—21no—
289C4×C7—21no—
290C4×C7—42no—
291C4×C7—81no—
292C4×C7—81no—
293Dic7—81no—
294Dic7—81no—
295Dic7—81no—
296Dic7—81no—
297Dic7—82no—
298C24—21no—
299C24D4×D711yescharacteristic
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—81no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4C22⋊Dic712yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—81no—
314C22×C4—141no—
315C22×C4—141no—
316C22×C4—141no—
317C22×C4—141no—
318C22×C4—281no—
319C22×C4—561no—
320C22×C4—561no—
321C22×C4—561no—
322C22×C4D4×D711yescharacteristic
323C22×C4#115911yescharacteristic
324C22×C4#115911yescharacteristic
325C22×C4C22×Dic711yescharacteristic
326C22×C4C2×C22⋊D711yescharacteristic
327C22×C4C2×C22⋊D711yescharacteristic
328SD16—82no—
329M16—81no—
330M16—561no—
331C2×C8—22no—
332C2×C8—281no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—561no—
339C4⋊C4—561no—
340C4⋊C4C22⋊Dic712yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—142no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C22⋊C4—281no—
349C22⋊C4—561no—
350C22⋊C4—561no—
351C22⋊C4—561no—
352C22⋊C4—561no—
353C42—561no—
354C42—561no—
355C2×C7—21no—
356C2×C7—21no—
357C2×C7—41no—
358C2×C7—41no—
359C2×C7—82no—
360C2×C7#188211yescharacteristic
361C2×C7#190411yescharacteristic
362C2×C7#191311yescharacteristic
363C23—21no—
364C23—21no—
365C23—22no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23D4×Dic711yescharacteristic
375C23#650011yescharacteristic
376C23#650211yescharacteristic
377Q8—42no—
378Q8—81no—
379D4—42no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—282no—
399C2×C4—281no—
400C2×C4—281no—
401C2×C4—281no—
402C2×C4—281no—
403C2×C4—281no—
404C2×C4—281no—
405C2×C4—281no—
406C2×C4—281no—
407C2×C4—281no—
408C2×C4—281no—
409C2×C4—561no—
410C2×C4—561no—
411C2×C4—564no—
412C2×C4D4×Dic711yescharacteristic
413C2×C4#649811yescharacteristic
414C2×C4#650211yescharacteristic
415C2×C4C2×C22⋊Dic711yescharacteristic
416C8—42no—
417C8—561no—
418C7#934111yescharacteristic, complemented, Sylow
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#8997411yescharacteristic
437C22#8997711yescharacteristic
438C22#8999211yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—561no—
447C4—561no—
448C4—561no—
449C4—561no—
450C4—562no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2#1172960211yescharacteristic
457C2#1172984311yescharacteristic
458C2#1172985411yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#3193C2
socleC22×C7#386
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenter#3193D7
hypocenterC7#9341
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3193

Complements

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

NH ≅ G/Ntype
3. #11729455C2semidirect
4. #11729443C2semidirect
6. #11725491C2semidirect
7. #11725488C2semidirect
11. C7×#480C4semidirect
14. C7×#448C4semidirect
32. C22×C7×Q8C2×C4semidirect
48. C2×C7×C4⋊C4C2×C4semidirect
68. C7×C22⋊C8C2×C4semidirect
145. #480Dic7semidirect
148. #448Dic7semidirect
220. C22×Q8C2×Dic7semidirect
239. C2×C4⋊C4C2×Dic7semidirect
258. C22⋊C8C2×Dic7semidirect
418. C7#9341semidirect

Maximal subgroups

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

subgroupindex
C7×#31932
#117400842
#117294552
#117294432
#117294272
#117254912
#117254882
#93417