Finite Group Databasebeta

← #13323007

Subgroups — #13323007

Order 1792.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11736737C211yesmaximal, characteristic
2#11736728C211yesmaximal, characteristic
3#11736495C211yesmaximal, characteristic
4C7×#1574C211yesmaximal, characteristic, complemented
5#11728506C211yesmaximal, characteristic, complemented
6#11728501C211yesmaximal, characteristic, complemented
7#11728247C211yesmaximal, characteristic, complemented
8#11726328C211yesmaximal, characteristic, complemented
9#11726327C211yesmaximal, characteristic, complemented
10#11726325C211yesmaximal, characteristic, complemented
11#11726324C211yesmaximal, characteristic, complemented
12#11726289C211yesmaximal, characteristic, complemented
13#11726286C211yesmaximal, characteristic, complemented
14#11726285C211yesmaximal, characteristic, complemented
15#11726283C211yesmaximal, characteristic, complemented
16C4×#1157C2211yescharacteristic
17C7×#431C2211yescharacteristic
18C7×#431C2211yescharacteristic
19C7×#423C2211yescharacteristic
20#89900—21no—
21#89899—21no—
22#89698C2211yescharacteristic
23#89697C2211yescharacteristic
24C2×C4⋊(C7⋊C8)C2211yescharacteristic
25#89612C412yes—
26#89611C412yes—
27#89609C2211yescharacteristic
28#89609C2211yescharacteristic
29#89609C2211yescharacteristic
30#89608—21no—
31#89607C2211yescharacteristic
32#89607C2211yescharacteristic
33#89607C2211yescharacteristic
34#89605C2211yescharacteristic
35D7×C4⋊C8C2211yescharacteristic
36#89600—21no—
37#89502—21no—
38#89502—21no—
39#89499C412yes—
40#89492C412yes—
41#89460—21no—
42C4×#1131C2211yescharacteristic
43#89457C2211yescharacteristic
44#89457C2211yescharacteristic
45C4×C8×D7C2211yescharacteristic
46#89451—21no—
47C7×#335C2211yescharacteristic
48C7×#335C2211yescharacteristic
49C7×#334C2211yescharacteristic
50C7×#334C2211yescharacteristic
51#89275C2211yescharacteristic
52#89275C2211yescharacteristic
53#89275C2211yescharacteristic
54#89275C2211yescharacteristic
55#89275C2211yescharacteristic
56#89275C2211yescharacteristic
57#89274C2211yescharacteristic
58#89274C2211yescharacteristic
59#89253C2211yescharacteristic
60#89253C2211yescharacteristic
61#89252C2211yescharacteristic
62#89252C2211yescharacteristic
63#17824—71nomaximal, Sylow
64C2×#1157C2×C411yescharacteristic
65C2×C7×M16—21no—
66C2×C42×C7C2311yescharacteristic
67C4×C22⋊D7D411yescharacteristic
68C4×C22⋊D7D411yescharacteristic
69C4×C22⋊D7Q811yescharacteristic
70C4×C22⋊D7Q811yescharacteristic
71#6489C2×C411yescharacteristic
72C22×C7⋊C8—21no—
73C2×#1131—21no—
74C2×#1131—21no—
75C2×#1131—21no—
76C2×#1131C2×C412yes—
77C2×C8×D7—21no—
78C2×C8×D7C2×C412yes—
79C4×D28D411yescharacteristic
80C4×D28Q811yescharacteristic
81#6435C2×C411yescharacteristic
82#6435C2×C411yescharacteristic
83C42×D7C2311yescharacteristic
84C42×D7C2311yescharacteristic
85C4×Dic14D411yescharacteristic
86C4×Dic14Q811yescharacteristic
87C7×C4⋊C8—21no—
88C7×C4⋊C8C2×C412yes—
89C7×C4⋊C8C2311yescharacteristic
90C7×C4⋊C8C2311yescharacteristic
91C7×C4⋊C8C2311yescharacteristic
92C7×C4⋊C8C2311yescharacteristic
93C7×C22⋊C8—21no—
94C7×C22⋊C8C2×C412yes—
95C4×C7×C8—21no—
96C4×C7×C8C2311yescharacteristic
97C4×C7×C8C2311yescharacteristic
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)C2×C412yes—
100(C2×D7)⋊C8—21no—
101(C2×D7)⋊C8—21no—
102(C2×D7)⋊C8—21no—
103(C2×D7)⋊C8—21no—
104(C2×D7)⋊C8C2×C412yes—
105(C2×D7)⋊C8C2×C412yes—
106#6389—21no—
107#6389—21no—
108#6389—21no—
109#6389C2×C412yes—
110Dic7⋊C8—21no—
111Dic7⋊C8—21no—
112Dic7⋊C8—21no—
113Dic7⋊C8—21no—
114Dic7⋊C8C2×C412yes—
115Dic7⋊C8C2×C412yes—
116C8×Dic7—21no—
117C8×Dic7C2×C412yes—
118C4⋊(C7⋊C8)—21no—
119C4⋊(C7⋊C8)C2×C412yes—
120C4⋊(C7⋊C8)C2311yescharacteristic
121C4⋊(C7⋊C8)C2311yescharacteristic
122C4⋊(C7⋊C8)C2311yescharacteristic
123C4⋊(C7⋊C8)C2311yescharacteristic
124#6377—21no—
125C4×C7⋊C8C2311yescharacteristic
126C4×C7⋊C8C2311yescharacteristic
127#2989—71no—
128#2989—71no—
129#2922—71no—
130#1574—71no—
131#1574D711yescharacteristic, complemented
132#1573—71no—
133#1573—71no—
134#1572—71no—
135#1572—71no—
136#1570—71no—
137#1570—71no—
138#1569—71no—
139#1569—71no—
140#1568—71no—
141#1566—71no—
142C22×C4×C7C22×C411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×Q811yescharacteristic
145C2×C22⋊D7C4⋊C411yescharacteristic
146C2×C22⋊D7C4⋊C411yescharacteristic
147#1157—42no—
148C2×D28C4⋊C411yescharacteristic
149C2×C4×D7C22×C411yescharacteristic
150C2×C4×D7C22×C411yescharacteristic
151C2×C4×D7C2×D411yescharacteristic
152C2×C4×D7C2×D411yescharacteristic
153C2×C4×D7C2×Q811yescharacteristic
154C2×C4×D7C2×Q811yescharacteristic
155C2×Dic14C4⋊C411yescharacteristic
156C7×M16—81no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8C22×C412yes—
162C2×C7×C8C22×C412yes—
163C42×C7C22×C411yescharacteristic
164C42×C7C2×D411yescharacteristic
165C42×C7C2×Q811yescharacteristic
166C42×C7C2411yescharacteristic
167C22⋊Dic7C4⋊C411yescharacteristic
168C22⋊Dic7C4⋊C411yescharacteristic
169(C2×D7)⋊C4C4⋊C411yescharacteristic
170(C2×D7)⋊C4C4⋊C411yescharacteristic
171(C2×D7)⋊C4C4⋊C411yescharacteristic
172(C2×D7)⋊C4C4⋊C411yescharacteristic
173C4⋊Dic7C4⋊C411yescharacteristic
174C4⋊Dic7C4⋊C411yescharacteristic
175Dic7⋊C4C4⋊C411yescharacteristic
176Dic7⋊C4C4⋊C411yescharacteristic
177Dic7⋊C4C4⋊C411yescharacteristic
178Dic7⋊C4C4⋊C411yescharacteristic
179C4×Dic7C22×C411yescharacteristic
180C4×Dic7C22×C411yescharacteristic
181C4×Dic7C2×D411yescharacteristic
182C4×Dic7C2×D411yescharacteristic
183C4×Dic7C2×Q811yescharacteristic
184C4×Dic7C2×Q811yescharacteristic
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—81no—
190C2×C7⋊C8C22×C412yes—
191C2×C7⋊C8C22×C412yes—
192#1131—42no—
193#1131—81no—
194#1131—81no—
195#1131—81no—
196C8×D7—42no—
197C8×D7—81no—
198C4×#41—71no—
199#446—141no—
200#446—141no—
201#436—141no—
202#436—141no—
203#435—141no—
204#435—141no—
205#435—141no—
206#435—141no—
207#433—72no—
208#433—72no—
209#432—71no—
210#432—71no—
211#432—71no—
212#432—71no—
213#431—71no—
214#431—71no—
215#431—71no—
216#431—71no—
217#431C2×D711yescharacteristic
218#431C2×D711yescharacteristic
219#424—72no—
220#423—71no—
221#423—71no—
222#423—71no—
223#423C2×D711yescharacteristic
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226#408—72no—
227C4×M16—71no—
228C2×C4×C8—71no—
229#335—71no—
230#335—71no—
231#335—71no—
232#335—71no—
233#335C2×D711yescharacteristic
234#335C2×D711yescharacteristic
235#334—71no—
236#334—71no—
237#334—71no—
238#334—71no—
239#334—71no—
240#334—71no—
241#334—71no—
242#334—71no—
243#334C2×D711yescharacteristic
244#334C2×D711yescharacteristic
245C23×C7C2×C4⋊C411yescharacteristic
246C22×D7C2×C4⋊C411yescharacteristic
247C22×D7C2×C4⋊C411yescharacteristic
248C2×C4×C7—22no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—41no—
252C2×C4×C7C2×C4⋊C411yescharacteristic
253C2×C4×C7C2×C4⋊C411yescharacteristic
254C2×C4×C7C2×C4⋊C411yescharacteristic
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C22×Q811yescharacteristic
258C22⋊D7—41no—
259C22⋊D7—41no—
260C2×Dic7C2×C4⋊C411yescharacteristic
261C2×Dic7C2×C4⋊C411yescharacteristic
262C2×Dic7C2×C4⋊C411yescharacteristic
263C2×Dic7C2×C4⋊C411yescharacteristic
264C2×Dic7C2×C4⋊C411yescharacteristic
265C2×Dic7C2×C4⋊C411yescharacteristic
266D28—41no—
267C4×D7—22no—
268C4×D7—22no—
269C4×D7—21no—
270C4×D7—21no—
271C4×D7—21no—
272C4×D7—21no—
273C4×D7—41no—
274C4×D7—41no—
275Dic14—41no—
276C7×C8—22no—
277C7×C8—22no—
278C7×C8—41no—
279C7×C8—41no—
280C7×C8—41no—
281C7×C8—41no—
282C7⋊C8—22no—
283C7⋊C8—22no—
284C7⋊C8—41no—
285C7⋊C8—41no—
286C7⋊C8—41no—
287C7⋊C8—41no—
288C2×#41—71no—
289C2×M16—21no—
290C2×M16—72no—
291C2×M16—141no—
292C2×M16—141no—
293C2×M16—141no—
294C22×C8—72no—
295C22×C8—141no—
296C22×C8—141no—
297C4×Q8—71no—
298C4×Q8—71no—
299C4×D4—71no—
300C4×D4—71no—
301C4×D4—71no—
302C4×D4—71no—
303C4×D4—71no—
304C4×D4—71no—
305#117—71no—
306#117—71no—
307#117—71no—
308C2×C42—71no—
309C2×C42—71no—
310C2×C42C22×D711yescharacteristic
311C4⋊C8—21no—
312C4⋊C8—72no—
313C4⋊C8—72no—
314C4⋊C8—72no—
315C4⋊C8—71no—
316C4⋊C8—71no—
317C4⋊C8—71no—
318C4⋊C8—71no—
319C4⋊C8—141no—
320C4⋊C8—141no—
321C4⋊C8—141no—
322C4⋊C8—141no—
323C4⋊C8—141no—
324C4⋊C8C4×D712yes—
325C4⋊C8C22×D711yescharacteristic
326C4⋊C8C22×D711yescharacteristic
327C4⋊C8C22×D711yescharacteristic
328C4⋊C8C22×D711yescharacteristic
329C22⋊C8—21no—
330C22⋊C8—72no—
331C22⋊C8—72no—
332C22⋊C8—72no—
333C22⋊C8—141no—
334C22⋊C8—141no—
335C22⋊C8—141no—
336C22⋊C8—141no—
337C22⋊C8—141no—
338C22⋊C8C4×D712yes—
339#97—72no—
340#97—141no—
341#97—141no—
342#97—141no—
343#97—141no—
344C4×C8—21no—
345C4×C8—72no—
346C4×C8—71no—
347C4×C8—71no—
348C4×C8—141no—
349C4×C8C22×D711yescharacteristic
350C4×C8C22×D711yescharacteristic
351C22×C7—21no—
352C22×C7—21no—
353C22×C7—21no—
354C22×C7C22×C4⋊C411yescharacteristic
355C2×D7—21no—
356C2×D7—21no—
357C2×D7—21no—
358C2×D7—21no—
359C2×D7—21no—
360C2×D7—21no—
361C4×C7—21no—
362C4×C7—21no—
363C4×C7—21no—
364C4×C7—21no—
365C4×C7#56812yes—
366C4×C7#57711yescharacteristic
367C4×C7#57711yescharacteristic
368Dic7—21no—
369Dic7—21no—
370Dic7—21no—
371Dic7—21no—
372Dic7—21no—
373Dic7—21no—
374#41—282no—
375C2×Q8—71no—
376C2×D4—71no—
377C2×D4—71no—
378C2×D4—71no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—71no—
382C22×C4—71no—
383C22×C4—71no—
384C22×C4—71no—
385C22×C4C2×C4×D711yescharacteristic
386C22×C4D4×D711yescharacteristic
387C22×C4Q8×D711yescharacteristic
388M16—81no—
389M16—282no—
390M16—561no—
391M16—561no—
392M16—561no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—72no—
398C2×C8—72no—
399C2×C8—141no—
400C2×C8—141no—
401C2×C8—141no—
402C2×C8—141no—
403C2×C8—282no—
404C2×C8—561no—
405C2×C8—561no—
406C2×C8C2×C4×D712yes—
407C2×C8C2×C4×D712yes—
408C4⋊C4—71no—
409C4⋊C4—71no—
410C4⋊C4—71no—
411C4⋊C4—71no—
412C4⋊C4—71no—
413C4⋊C4—71no—
414C22⋊C4—71no—
415C22⋊C4—71no—
416C22⋊C4—71no—
417C22⋊C4—71no—
418C22⋊C4—71no—
419C22⋊C4—71no—
420C42—71no—
421C42—71no—
422C42—71no—
423C42—71no—
424C42—71no—
425C42—71no—
426C42C2×C4×D711yescharacteristic
427C42D4×D711yescharacteristic
428C42Q8×D711yescharacteristic
429C42C23×D711yescharacteristic
430C2×C7—41no—
431C2×C7#290511yescharacteristic
432C2×C7#291211yescharacteristic
433C2×C7#291611yescharacteristic
434D7—41no—
435D7—41no—
436C23—71no—
437C23—71no—
438C23D7×C4⋊C411yescharacteristic
439Q8—281no—
440D4—281no—
441D4—281no—
442D4—281no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—71no—
448C2×C4—71no—
449C2×C4—71no—
450C2×C4—71no—
451C2×C4—71no—
452C2×C4—71no—
453C2×C4—142no—
454C2×C4—142no—
455C2×C4—141no—
456C2×C4—141no—
457C2×C4—141no—
458C2×C4—141no—
459C2×C4—281no—
460C2×C4—281no—
461C2×C4D7×C4⋊C411yescharacteristic
462C2×C4D7×C4⋊C411yescharacteristic
463C2×C4D7×C4⋊C411yescharacteristic
464C2×C4C22×C4×D711yescharacteristic
465C2×C4C2×D4×D711yescharacteristic
466C2×C4C2×Q8×D711yescharacteristic
467C8—22no—
468C8—22no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—142no—
474C8—142no—
475C8—281no—
476C8—281no—
477C8—281no—
478C8—281no—
479C7#1782411yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—21no—
483C22—141no—
484C22—141no—
485C22—141no—
486C22—141no—
487C22—141no—
488C22—141no—
489C22C2×D7×C4⋊C411yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—141no—
495C4—141no—
496C4—141no—
497C4—141no—
498C4—141no—
499C4—141no—
500C4#9044112yes—
501C4#9046611yescharacteristic
502C4#9046911yescharacteristic
503C2—41no—
504C2—281no—
505C2—281no—
506C2#1173652511yescharacteristic
507C2#1173665411yescharacteristic
508C2#1173669711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1574C2
socleC22×C7C22×C4⋊C4
prime radicalC2×#1157C2×C4
cosocleC42×C7C24
hypercenter#1574D7
hypocenterC7#17824
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
4. C7×#1574C2semidirect
5. #11728506C2semidirect
6. #11728501C2semidirect
7. #11728247C2semidirect
8. #11726328C2semidirect
9. #11726327C2semidirect
10. #11726325C2semidirect
11. #11726324C2semidirect
12. #11726289C2semidirect
13. #11726286C2semidirect
14. #11726285C2semidirect
15. #11726283C2semidirect
131. #1574D7semidirect
479. C7#17824semidirect

Maximal subgroups

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

subgroupindex
#117367372
#117367282
#117364952
C7×#15742
#117285062
#117285012
#117282472
#117263282
#117263272
#117263252
#117263242
#117262892
#117262862
#117262852
#117262832
#178247