Finite Group Databasebeta

← #13337663

Subgroups — #13337663

Order 1792.

650 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3112C211yesmaximal, characteristic, complemented
2#11740736C211yesmaximal, characteristic
3#11739889C211yesmaximal, characteristic, complemented
4C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
5#11729561C211yesmaximal, characteristic, complemented
6#11728904C212yesmaximal, complemented
7#11728903C212yesmaximal, complemented
8#11728776C212yesmaximal, complemented
9#11728775C212yesmaximal, complemented
10#11728570C211yesmaximal, characteristic, complemented
11#11728548C211yesmaximal, characteristic, complemented
12C7×#532C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14#90168C2211yescharacteristic
15C7×#480—21no—
16C7×#477C2212yes—
17C7×#460C2212yes—
18C7×#448—21no—
19C7×#440—22no—
20C7×#432C2211yescharacteristic
21#89999—21no—
22#89860C2211yescharacteristic
23#89850C2211yescharacteristic
24#89845C2211yescharacteristic
25#89842C2211yescharacteristic
26#89810—21no—
27#89810—21no—
28#89799C2212yes—
29#89798—22no—
30Q8×C7⋊C8—21no—
31Q8×C7⋊C8—21no—
32Q8×C7⋊C8C2212yes—
33Q8×C7⋊C8C2212yes—
34#89793C2212yes—
35C4×#1141—21no—
36C4×#1141—21no—
37#89739—21no—
38#89698C2211yescharacteristic
39C2×C4⋊(C7⋊C8)—21no—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8—21no—
42C2×C4×C7⋊C8C2211yescharacteristic
43#89341C2212yes—
44#89339C2212yes—
45#89280C2212yes—
46#89276C2212yes—
47#89275C2212yes—
48#89274C2212yes—
49#19264—71nomaximal, Sylow
50C2×C7×#41—21no—
51C22×C7×Q8D411yescharacteristic
52C2×#1157—21no—
53C2×C7×SD16—22no—
54C7×C4⋊Q8C2311yescharacteristic
55C7×#127C2311yescharacteristic
56C7×#124—21no—
57C7×#121—21no—
58C7×#121—21no—
59C4×C7×Q8—21no—
60C4×C7×Q8—21no—
61C4×C7×Q8D412yes—
62C4×C7×Q8C2312yes—
63C4×C7×D4—21no—
64C4×C7×D4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4D411yescharacteristic
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×#1143—22no—
70C2×#1141—21no—
71C2×#1141—21no—
72#6493—21no—
73#6493—21no—
74C22×C7⋊C8—21no—
75C22×C7⋊C8D411yescharacteristic
76C22×C7⋊C8D411yescharacteristic
77#6438—21no—
78#6437C2311yescharacteristic
79C4×D28—21no—
80C4×D28—21no—
81#6432C2311yescharacteristic
82C7×#107—22no—
83C7×C4⋊C8C2312yes—
84C7×#103—22no—
85C7×#102—22no—
86C7×C22⋊C8—21no—
87C7×#97—21no—
88#6409—22no—
89#6406—21no—
90#6406—21no—
91C22⋊(C7⋊C8)—21no—
92#6383—22no—
93#6383—21no—
94#6383—21no—
95#6381—22no—
96#6381—21no—
97#6381—21no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)—22no—
100C4⋊(C7⋊C8)—22no—
101C4⋊(C7⋊C8)C2312yes—
102C4⋊(C7⋊C8)C2312yes—
103#6377—21no—
104C4×C7⋊C8—22no—
105C4×C7⋊C8—22no—
106C4×C7⋊C8D411yescharacteristic
107C4×C7⋊C8D411yescharacteristic
108C4×C7⋊C8C2311yescharacteristic
109C4×C7⋊C8C2311yescharacteristic
110#3169—71no—
111#3112—71no—
112#3112D711yescharacteristic, complemented
113C2×C8×Q8—71no—
114#1678—72no—
115#1678—72no—
116#1676—72no—
117#1668—72no—
118#1568—71no—
119#1551—71no—
120#1551—71no—
121C7×#41—82no—
122C2×C7×Q8—24no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8C2×D412yes—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—41no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—41no—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7#4111yescharacteristic
136C2×C22⋊D7—41no—
137#1157—82no—
138C2×D28—21no—
139C2×D28—21no—
140C2×D28—21no—
141C2×C4×D7—41no—
142C2×Dic14—21no—
143C7×SD16—44no—
144C2×C7×C8—22no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4C2×D412yes—
151C7×C22⋊C4—41no—
152C7×C22⋊C4—41no—
153C42×C7—22no—
154C42×C7—22no—
155C42×C7C2×D411yescharacteristic
156C42×C7C2×D411yescharacteristic
157C42×C7#4111yescharacteristic
158C42×C7C2411yescharacteristic
159#1143—44no—
160#1141—42no—
161#1141—42no—
162(C2×D7)⋊C4—41no—
163(C2×D7)⋊C4—41no—
164C4⋊Dic7—21no—
165C4⋊Dic7—21no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—41no—
170C2×C7⋊C8—42no—
171C2×C7⋊C8C2×D411yescharacteristic
172C2×C7⋊C8C2×D411yescharacteristic
173C2×C7⋊C8C2×D411yescharacteristic
174C2×C7⋊C8C2×D411yescharacteristic
175#532—71no—
176#532C2×D711yescharacteristic
177C2×C4×Q8C2×D711yescharacteristic
178#500—71no—
179#493—71no—
180#486—71no—
181#486—71no—
182#480—21no—
183#480—141no—
184#477—72no—
185#477C2×D712yes—
186#466—141no—
187#466—141no—
188#460—72no—
189#460C2×D712yes—
190#448—21no—
191#448—141no—
192C8×Q8—72no—
193C8×Q8—72no—
194C8×Q8—141no—
195C8×Q8—141no—
196#440—22no—
197#440—142no—
198C4×D8—141no—
199C4×D8—141no—
200#432—71no—
201#432C2×D711yescharacteristic
202C2×C4⋊C8—71no—
203C2×C4⋊C8—141no—
204C2×C4×C8—71no—
205C2×C4×C8—141no—
206#335—72no—
207#334—72no—
208#333—72no—
209#333—72no—
210#331—72no—
211#331—72no—
212C23×C7—41no—
213C23×C7#12111yescharacteristic
214C22×D7—41no—
215C7×Q8—24no—
216C7×Q8—22no—
217C7×Q8—81no—
218C7×Q8#13514yes—
219C7×D4—42no—
220C7×D4—41no—
221C7×D4—41no—
222C7×D4—81no—
223C7×D4—82no—
224C2×C4×C7—24no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—22no—
231C2×C4×C7—42no—
232C2×C4×C7—41no—
233C2×C4×C7—82no—
234C2×C4×C7#12111yescharacteristic
235C2×C4×C7#12111yescharacteristic
236C2×C4×C7#12111yescharacteristic
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C22×D411yescharacteristic
239C2×C4×C7C2×#4111yescharacteristic
240C22⋊D7—82no—
241C2×Dic7—41no—
242D28—42no—
243D28—41no—
244D28—41no—
245D28—81no—
246C4×D7—82no—
247Dic14—81no—
248C7×C8—42no—
249C7⋊C8—21no—
250C7⋊C8—21no—
251C7⋊C8—21no—
252C7⋊C8—42no—
253C7⋊C8—42no—
254C7⋊C8#13712yes—
255C2×#41—21no—
256C2×#41—141no—
257C22×Q8C22⋊D711yescharacteristic
258C2×SD16—22no—
259C2×SD16—142no—
260C2×D8—141no—
261C2×D8—141no—
262C22×C8—71no—
263C22×C8—71no—
264C22×C8—141no—
265C4⋊Q8—71no—
266C4⋊Q8C22×D711yescharacteristic
267#127—71no—
268#127C22×D711yescharacteristic
269#124—21no—
270#124—141no—
271#121—21no—
272#121—21no—
273#121—141no—
274#121—141no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8C22⋊D712yes—
278C4×Q8C22×D712yes—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—141no—
282C4×D4—141no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C22⋊D711yescharacteristic
285C2×C42—21no—
286C2×C42C22×D711yescharacteristic
287#107—22no—
288#107—141no—
289#107—141no—
290#107—142no—
291C4⋊C8—72no—
292C4⋊C8—72no—
293C4⋊C8—142no—
294C4⋊C8—142no—
295C4⋊C8—141no—
296C4⋊C8C22×D712yes—
297#103—22no—
298#103—142no—
299#102—22no—
300#102—141no—
301#102—141no—
302#102—141no—
303#102—141no—
304#102—142no—
305C22⋊C8—21no—
306C22⋊C8—141no—
307#97—21no—
308#97—141no—
309C4×C8—71no—
310C4×C8—71no—
311C4×C8—71no—
312C4×C8—71no—
313C4×C8—142no—
314C4×C8—142no—
315C22×C7—21no—
316C22×C7—21no—
317C22×C7—81no—
318C22×C7—81no—
319C22×C7—81no—
320C22×C7#47212yes—
321C22×C7C2×#12111yescharacteristic
322C2×D7—81no—
323C2×D7—81no—
324C2×D7—81no—
325C4×C7—21no—
326C4×C7—24no—
327C4×C7—21no—
328C4×C7—21no—
329C4×C7—21no—
330C4×C7—42no—
331C4×C7—81no—
332C4×C7C2×#13512yes—
333C4×C7C2×#13711yescharacteristic
334C4×C7#57511yescharacteristic
335Dic7—81no—
336#41—82no—
337#41—562no—
338C2×Q8—24no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—141no—
342C2×Q8C2×C22⋊D712yes—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—141no—
348C2×D4—141no—
349C2×D4—141no—
350C2×D4—281no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—281no—
355C22×C4D4×D711yescharacteristic
356C22×C4#115911yescharacteristic
357C22×C4C2×C22⋊D711yescharacteristic
358SD16—44no—
359SD16—284no—
360D8—282no—
361D8—282no—
362C2×C8—22no—
363C2×C8—71no—
364C2×C8—71no—
365C2×C8—71no—
366C2×C8—71no—
367C2×C8—142no—
368C2×C8—142no—
369C2×C8—142no—
370C2×C8—281no—
371C2×C8—282no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—141no—
378C4⋊C4—141no—
379C4⋊C4C2×C22⋊D712yes—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—281no—
383C22⋊C4—281no—
384C42—22no—
385C42—22no—
386C42D4×D711yescharacteristic
387C42#115911yescharacteristic
388C42C2×C22⋊D711yescharacteristic
389C42C23×D711yescharacteristic
390C2×C7—22no—
391C2×C7—161no—
392C2×C7#303711yescharacteristic
393C2×C7#305011yescharacteristic
394C2×C7C2×#47211yescharacteristic
395D7—161no—
396C23—41no—
397C23—281no—
398C23#650211yescharacteristic
399Q8—24no—
400Q8—22no—
401Q8—81no—
402Q8—561no—
403Q8#651314yes—
404D4—41no—
405D4—41no—
406D4—42no—
407D4—81no—
408D4—82no—
409D4—281no—
410D4—281no—
411D4—282no—
412D4—561no—
413D4—562no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—82no—
424C2×C4—281no—
425C2×C4—562no—
426C2×C4#650211yescharacteristic
427C2×C4#650211yescharacteristic
428C2×C4#650211yescharacteristic
429C2×C4C2×D4×D711yescharacteristic
430C2×C4C2×#115911yescharacteristic
431C2×C4C22×C22⋊D711yescharacteristic
432C8—42no—
433C8—72no—
434C8—141no—
435C8—141no—
436C8—141no—
437C8—282no—
438C8—282no—
439C7#1926411yescharacteristic, complemented, Sylow
440C22—21no—
441C22—21no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—561no—
446C22—561no—
447C22—561no—
448C22#8997112yes—
449C22C2×#650211yescharacteristic
450C4—21no—
451C4—21no—
452C4—24no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—561no—
458C4D7×#13711yescharacteristic
459C4#9046811yescharacteristic
460C4C2×#651312yes—
461C2—22no—
462C2—161no—
463C2—1121no—
464C2#1174036711yescharacteristic
465C2C2×#8997111yescharacteristic
466C2#1174141111yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3112C2
socleC22×C7C2×#121
prime radical#11740736C2
hypercenter#3112D7
hypocenterC7#19264
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. C7×#3112C2semidirect
3. #11739889C2semidirect
4. C2×Q8×C7⋊C8C2semidirect
5. #11729561C2semidirect
6. #11728904C2semidirect
7. #11728903C2semidirect
8. #11728776C2semidirect
9. #11728775C2semidirect
10. #11728570C2semidirect
11. #11728548C2semidirect
112. #3112D7semidirect
439. C7#19264semidirect

Maximal subgroups

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

subgroupindex
C7×#31122
#117407362
#117398892
C2×Q8×C7⋊C82
#117295612
#117289042
#117289032
#117287762
#117287752
#117285702
#117285482
#192647