Finite Group Databasebeta

← #13405654

Subgroups — #13405654

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2567C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic
3#11739945C211yesmaximal, characteristic
4C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
5#11729541C211yesmaximal, characteristic, complemented
6#11727383C212yesmaximal
7#11727372C212yesmaximal, complemented
8#11727369C212yesmaximal
9#11727364C212yesmaximal, complemented
10C4×#6383C211yesmaximal, characteristic
11#11727221C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×#6509—21no—
15C2×C4×D28C2211yescharacteristic
16C7×#395—22no—
17C7×#391—21no—
18C7×#389C2212yes—
19C7×#389C2212yes—
20C7×#384—21no—
21C4×C7×C4⋊C4C2211yescharacteristic
22#89856—24no—
23Q8×C7⋊C8—24no—
24#89766—22no—
25#89759—21no—
26C2×#6383C2212yes—
27C2×#6383C2211yescharacteristic
28C2×#6383C2211yescharacteristic
29C2×#6382C2212yes—
30C2×#6382C2212yes—
31#89711C2212yes—
32#89707C2211yescharacteristic
33#89700C2212yes—
34C2×C4⋊(C7⋊C8)C2212yes—
35C2×C4⋊(C7⋊C8)C2211yescharacteristic
36C2×C4×C7⋊C8C2212yes—
37C2×C4×C7⋊C8C2211yescharacteristic
38#89324C2211yescharacteristic
39#89324C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89324C2211yescharacteristic
43#25470—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×Q8D411yescharacteristic
46C22×D28C2311yescharacteristic
47C22×C4×D7—21no—
48C7×C4⋊Q8—22no—
49C7×C4⋊Q8—22no—
50C4×C7×Q8—22no—
51C4×C7×Q8—22no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D412yes—
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C42×C7—21no—
61C2×C42×C7C2312yes—
62C2×C42×C7C2311yescharacteristic
63#6509—44no—
64C2×(C2×D7)⋊C4—21no—
65C2×(C2×D7)⋊C4—22no—
66C2×(C2×D7)⋊C4—22no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7C2311yescharacteristic
70C2×C4⋊Dic7C2312yes—
71C2×Dic7⋊C4—22no—
72C22×C7⋊C8C2311yescharacteristic
73C22×C7⋊C8C2312yes—
74C22×C7⋊C8C2311yescharacteristic
75C4×D28—22no—
76C4×D28—22no—
77C7×#95—21no—
78C7×#95—22no—
79C7×#95—22no—
80C7⋊#95—22no—
81#6383—22no—
82#6383—24no—
83#6383Q814yescomplemented
84#6382—24no—
85#6382—24no—
86C4⋊(C7⋊C8)—24no—
87C4⋊(C7⋊C8)D414yes—
88C4×C7⋊C8—22no—
89C4×C7⋊C8—24no—
90C2×#476—71no—
91C2×C8×Q8—71no—
92#2567D711yescharacteristic, complemented
93#2538—71no—
94#2088—72no—
95#2065—72no—
96#1990—71no—
97#1948—71no—
98#1941—72no—
99#1934—72no—
100C4×#102—71no—
101C23×D7—21no—
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—42no—
105C2×C7×Q8—81no—
106C2×C7×Q8—81no—
107C2×C7×Q8SD1614yes—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—22no—
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2411yescharacteristic
116C22×Dic7—21no—
117C22×Dic7—22no—
118C2×D28—21no—
119C2×D28—21no—
120C2×D28—21no—
121C2×D28—21no—
122C2×D28—21no—
123C2×D28C2×Q812yes—
124C2×C4×D7—42no—
125C2×C4×D7—42no—
126C2×C4×D7—42no—
127C7×C4⋊C4—24no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—42no—
134C7×C4⋊C4—82no—
135C7×C4⋊C4—82no—
136C7×C4⋊C4C2×Q812yes—
137C7×C4⋊C4#4112yes—
138C42×C7—22no—
139C42×C7—22no—
140C42×C7—42no—
141C42×C7C2×D412yes—
142C42×C7#4112yes—
143(C2×D7)⋊C4—42no—
144(C2×D7)⋊C4—44no—
145(C2×D7)⋊C4—82no—
146C4⋊Dic7—21no—
147C4⋊Dic7—21no—
148C4⋊Dic7—22no—
149C4⋊Dic7—22no—
150C4⋊Dic7—42no—
151C4⋊Dic7—81no—
152Dic7⋊C4—44no—
153C2×C7⋊C8—21no—
154C2×C7⋊C8—22no—
155C2×C7⋊C8—22no—
156C2×C7⋊C8—21no—
157C2×C7⋊C8—21no—
158C2×C7⋊C8—21no—
159C2×C7⋊C8—21no—
160C2×C7⋊C8C2×D414yes—
161C2×C7⋊C8C2×Q812yes—
162C2×C4⋊Q8C2×D711yescharacteristic
163C2×C22⋊Q8—141no—
164C2×C4×Q8C2×D711yescharacteristic
165C2×C4×D4—71no—
166#476—144no—
167C8×Q8—144no—
168C2×#106—72no—
169C2×#106—72no—
170C2×C4⋊C8—71no—
171C2×C4⋊C8—72no—
172C2×#102—71no—
173C2×#102—71no—
174C2×#102—72no—
175C2×C4×C8—72no—
176C2×C4×C8—71no—
177#396—142no—
178#395—22no—
179#391—21no—
180#390—72no—
181#389—72no—
182#389C2×D712yes—
183#389C2×D712yes—
184#384—21no—
185#384—71no—
186#380—141no—
187C4×C4⋊C4C2×D711yescharacteristic
188#340—72no—
189#340—71no—
190#340—72no—
191#340—72no—
192#340—71no—
193C23×C7C22×D411yescharacteristic
194C22×D7—22no—
195C22×D7—41no—
196C22×D7—41no—
197C22×D7—41no—
198C22×D7—41no—
199C22×D7—41no—
200C22×D7—41no—
201C7×Q8—28no—
202C7×Q8—84no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7#12112yes—
220C2×C4×C7#12112yes—
221C2×C4×C7C2×SD1614yes—
222C2×C4×C7C2×SD1612yes—
223C2×C4×C7#13512yes—
224C2×C4×C7C22×D411yescharacteristic
225C2×C4×C7C22×Q811yescharacteristic
226C2×C4×C7C2×#4111yescharacteristic
227C2×C4×C7C2×#4111yescharacteristic
228C2×C4×C72−1+411yescharacteristic
229C2×C4×C72−1+411yescharacteristic
230C2×Dic7—41no—
231C2×Dic7—42no—
232C2×Dic7—41no—
233C2×Dic7—42no—
234C2×Dic7—41no—
235C2×Dic7—42no—
236D28—22no—
237D28—22no—
238D28—41no—
239D28—41no—
240C4×D7—84no—
241C7⋊C8—24no—
242C7⋊C8—22no—
243C7⋊C8—41no—
244C22×Q8—21no—
245C22×Q8C22⋊D711yescharacteristic
246C22×D4—71no—
247C23×C4—141no—
248C22×C8—71no—
249C22×C8—72no—
250C22×C8—71no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C22⋊Q8—284no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×D4—142no—
257C4×D4—142no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—71no—
262C2×C4⋊C4—72no—
263C2×C4⋊C4—142no—
264C2×C4⋊C4—141no—
265C2×C4⋊C4—141no—
266C2×C4⋊C4C22⋊D711yescharacteristic
267C2×C4⋊C4C22⋊D712yes—
268C2×C4⋊C4C22×D712yes—
269C2×C4⋊C4C22×D711yescharacteristic
270C2×C4⋊C4C22×D711yescharacteristic
271C2×C22⋊C4—141no—
272C2×C22⋊C4—142no—
273C2×C22⋊C4—142no—
274C2×C42—21no—
275C2×C42C22×D712yes—
276C2×C42C22×D711yescharacteristic
277#106—144no—
278#106—144no—
279C4⋊C8—74no—
280C4⋊C8—144no—
281#102—74no—
282#102—144no—
283#102—142no—
284C4×C8—144no—
285C4×C8—142no—
286#95—21no—
287#95—22no—
288#95—22no—
289#95—142no—
290C22×C7C2×#12111yescharacteristic
291C22×C7#54811yescharacteristic
292C22×C7D4×Q811yescharacteristic
293C22×C7C22×SD1611yescharacteristic
294C22×C7C2×#13511yescharacteristic
295C22×C7#57611yescharacteristic
296C22×C7#57711yescharacteristic
297C2×D7—42no—
298C2×D7—42no—
299C2×D7—42no—
300C2×D7—81no—
301C2×D7—81no—
302C2×D7—81no—
303C2×D7—81no—
304C4×C7—24no—
305C4×C7—22no—
306C4×C7—42no—
307C4×C7—42no—
308C4×C7—42no—
309C4×C7—82no—
310C4×C7#46514yes—
311C4×C7#53311yescharacteristic
312C4×C7#54111yescharacteristic
313C4×C7#54711yescharacteristic
314C4×C7D4×Q811yescharacteristic
315Dic7—82no—
316Dic7—81no—
317C24—141no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—42no—
321C2×Q8—81no—
322C2×Q8—81no—
323C2×Q8#114314yes—
324C2×D4—72no—
325C2×D4—141no—
326C2×D4—141no—
327C2×D4—141no—
328C2×D4—141no—
329C2×D4—141no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—141no—
334C22×C4—142no—
335C22×C4—282no—
336C22×C4—282no—
337C22×C4—282no—
338C22×C4C2×C22⋊D712yes—
339C22×C4C2×C22⋊D711yescharacteristic
340C22×C4C2×C22⋊D712yes—
341C22×C4C2×C22⋊D711yescharacteristic
342C22×C4C23×D711yescharacteristic
343C2×C8—74no—
344C2×C8—72no—
345C2×C8—141no—
346C2×C8—142no—
347C2×C8—142no—
348C2×C8—141no—
349C2×C8—141no—
350C2×C8—141no—
351C2×C8—141no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—141no—
362C4⋊C4—141no—
363C4⋊C4—142no—
364C4⋊C4—142no—
365C4⋊C4—284no—
366C4⋊C4—282no—
367C4⋊C4—561no—
368C4⋊C4Q8×D712yes—
369C4⋊C4#116112yes—
370C22⋊C4—282no—
371C22⋊C4—284no—
372C22⋊C4—562no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42D4×D712yes—
377C42#115912yes—
378C2×C7#242911yescharacteristic
379C2×C7C2×#46511yescharacteristic
380C2×C7#306211yescharacteristic
381C2×C7#336011yescharacteristic
382C2×C7#336211yescharacteristic
383C2×C7Q8×SD1611yescharacteristic
384C2×C7#338011yescharacteristic
385D7—82no—
386C23—142no—
387C23—281no—
388C23—281no—
389C23—281no—
390C23—281no—
391C23—281no—
392C23—281no—
393C23C22×C22⋊D711yescharacteristic
394Q8—28no—
395Q8—84no—
396D4—142no—
397D4—142no—
398D4—281no—
399D4—281no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—281no—
417C2×C4—282no—
418C2×C4—281no—
419C2×C4—282no—
420C2×C4—281no—
421C2×C4—282no—
422C2×C4—564no—
423C2×C4#650212yes—
424C2×C4#650212yes—
425C2×C4C2×#114314yes—
426C2×C4C2×#114312yes—
427C2×C4#651312yes—
428C2×C4C2×D4×D711yescharacteristic
429C2×C4C2×#115911yescharacteristic
430C2×C4C2×Q8×D711yescharacteristic
431C2×C4C2×#116111yescharacteristic
432C2×C4#655111yescharacteristic
433C2×C4#655111yescharacteristic
434C8—144no—
435C8—142no—
436C8—281no—
437C7#2547011yescharacteristic, complemented, Sylow
438C22—282no—
439C22—282no—
440C22—282no—
441C22—561no—
442C22—561no—
443C22—561no—
444C22—561no—
445C22C2×#650211yescharacteristic
446C22C22×#114311yescharacteristic
447C22Q8×C22⋊D711yescharacteristic
448C22#9050811yescharacteristic
449C22C2×#651311yescharacteristic
450C22#9052611yescharacteristic
451C22#9052711yescharacteristic
452C4—24no—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—82no—
458C4—561no—
459C4—562no—
460C4#8982014yes—
461C4#9041711yescharacteristic
462C4#9041811yescharacteristic
463C4#9041911yescharacteristic
464C4#9042211yescharacteristic
465C2—562no—
466C2C2×#8982011yescharacteristic
467C2#1174069911yescharacteristic
468C2#1174082411yescharacteristic
469C2#1174107311yescharacteristic
470C2#1174108211yescharacteristic
471C2#1174114811yescharacteristic
472C2Q8×#114311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2567C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2567D7
hypocenterC7#25470
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
1. C7×#2567C2semidirect
4. C2×Q8×C7⋊C8C2semidirect
5. #11729541C2semidirect
7. #11727372C2semidirect
9. #11727364C2semidirect
11. #11727221C2semidirect
83. #6383Q8semidirect
92. #2567D7semidirect
437. C7#25470semidirect

Maximal subgroups

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

subgroupindex
C7×#25672
C2×#898562
#117399452
C2×Q8×C7⋊C82
#117295412
#117273832
#117273722
#117273692
#117273642
C4×#63832
#117272212
#254707