Finite Group Databasebeta

← #13365903

Subgroups — #13365903

Order 1792.

836 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#89859C211yesmaximal, characteristic
3#11730744C211yesmaximal, characteristic
4#11729534C211yesmaximal, characteristic, complemented
5#11727374C212yesmaximal, complemented
6#11727371C212yesmaximal
7#11727338C211yesmaximal, characteristic
8#11727322C211yesmaximal, characteristic, complemented
9#11727217C212yesmaximal
10#11727215C212yesmaximal, complemented
11#11727145C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8—21no—
14C2×C4×D28C2211yescharacteristic
15C2×#6435—21no—
16C7×#391—21no—
17C7×#389—21no—
18C7×#389C2212yes—
19C7×#384C2211yescharacteristic
20C4×C7×C4⋊C4—21no—
21C4×C7×C4⋊C4C2212yes—
22C43×C7C2211yescharacteristic
23#89859C418yescomplemented
24C2×#6383C2212yes—
25C2×#6383C2212yes—
26C2×#6382C2212yes—
27C2×#6381C2212yes—
28#89712—21no—
29#89712—21no—
30#89711C2211yescharacteristic
31#89711C2211yescharacteristic
32C4×(C2×D7)⋊C4—21no—
33C4×C4⋊Dic7C2211yescharacteristic
34#89700C2211yescharacteristic
35#89700C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C2×#6377C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#22116—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×D28C2311yescharacteristic
47C22×C4×D7—21no—
48C7×C4⋊Q8—22no—
49C7×C4⋊Q8C2×C414yes—
50C4×C7×Q8—44no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2312yes—
57C2×C42×C7—22no—
58C2×C42×C7D411yescharacteristic
59C2×C42×C7D411yescharacteristic
60C2×C42×C7D411yescharacteristic
61C2×C42×C7D411yescharacteristic
62C2×C42×C7C2311yescharacteristic
63C2×C42×C7C2311yescharacteristic
64C2×C42×C7C2311yescharacteristic
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—21no—
68C2×(C2×D7)⋊C4—21no—
69C2×(C2×D7)⋊C4—21no—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2311yescharacteristic
72C2×C4⋊Dic7C2311yescharacteristic
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—21no—
75C2×C4×Dic7—21no—
76C22×C7⋊C8C2312yes—
77C22×C7⋊C8C2312yes—
78C4×D28—22no—
79C4×D28C2×C414yes—
80#6435—44no—
81C7×#95—22no—
82C7×#95—22no—
83C7⋊#95—21no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86#6383—42no—
87#6383C2×C418yes—
88#6382—42no—
89#6381C2×C418yes—
90C4⋊(C7⋊C8)—22no—
91C4⋊(C7⋊C8)C2×C414yes—
92#6377—42no—
93C2×#474—71no—
94C4×C4⋊Q8D711yescharacteristic, complemented
95#2293—71no—
96#1988—71no—
97#1948—71no—
98#1943—72no—
99#1936—72no—
100#1926—72no—
101#1921—72no—
102#1802—71no—
103#1767—71no—
104C23×D7—21no—
105C2×C7×Q8—24no—
106C2×C7×Q8—42no—
107C2×C7×Q8—42no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2411yescharacteristic
117C22×Dic7—21no—
118C22×Dic7—21no—
119C22×Dic7—21no—
120C2×D28—21no—
121C2×D28—21no—
122C2×D28—21no—
123C2×D28—21no—
124C2×D28—21no—
125C2×D28C22×C412yes—
126C2×C4×D7—24no—
127C2×C4×D7—42no—
128C2×C4×D7—42no—
129C7×C4⋊C4—24no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—42no—
134C7×C4⋊C4—44no—
135C7×C4⋊C4—42no—
136C7×C4⋊C4C22×C414yes—
137C42×C7—22no—
138C42×C7—22no—
139C42×C7—22no—
140C42×C7—21no—
141C42×C7—21no—
142C42×C7—21no—
143C42×C7—21no—
144C42×C7—41no—
145C42×C7—44no—
146C42×C7C22×C412yes—
147C42×C7#4112yes—
148(C2×D7)⋊C4—24no—
149(C2×D7)⋊C4—42no—
150(C2×D7)⋊C4—42no—
151(C2×D7)⋊C4—42no—
152(C2×D7)⋊C4—42no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157C4⋊Dic7—21no—
158C4⋊Dic7C22×C412yes—
159Dic7⋊C4—42no—
160Dic7⋊C4—42no—
161C4×Dic7—42no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—22no—
164C2×C7⋊C8—22no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8C22×C414yes—
168C2×C4⋊Q8C2×D711yescharacteristic
169C2×C4×Q8—21no—
170C2×C4×D4—71no—
171C2×#117—141no—
172#474—78no—
173C2×#107—72no—
174C2×#106—72no—
175C2×C4⋊C8—71no—
176C2×C4⋊C8—71no—
177C2×#102—72no—
178C2×#102—72no—
179C2×#97—72no—
180#391—21no—
181#390—71no—
182#390—71no—
183#389—21no—
184#389—71no—
185#389—71no—
186#389C2×D712yes—
187#388—141no—
188#388—141no—
189#384C2×D711yescharacteristic
190C4×C4⋊C4—21no—
191C4×C4⋊C4—71no—
192C4×C4⋊C4C2×D712yes—
193C4×C22⋊C4—141no—
194C43C2×D711yescharacteristic
195#340—72no—
196#340—72no—
197#340—72no—
198#340—72no—
199#336—72no—
200C23×C7C22×D411yescharacteristic
201C22×D7—22no—
202C22×D7—41no—
203C22×D7—41no—
204C22×D7—41no—
205C22×D7—41no—
206C22×D7—41no—
207C22×D7—41no—
208C7×Q8—48no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—21no—
213C2×C4×C7—24no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—21no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7C4×D412yes—
229C2×C4×C7C4×D412yes—
230C2×C4×C7AGL(1,ℤ/8)12yes—
231C2×C4×C7AGL(1,ℤ/8)12yes—
232C2×C4×C7#13712yes—
233C2×C4×C7C23×C411yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C2×C4×C7H(5,𝔽2)11yescharacteristic
236C2×C4×C72−1+411yescharacteristic
237C2×C4×C72−1+411yescharacteristic
238C2×C4×C72−1+411yescharacteristic
239C2×Dic7—22no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248D28—22no—
249D28—22no—
250D28—41no—
251D28—41no—
252C4×D7—48no—
253C7⋊C8—24no—
254C7⋊C8—42no—
255C22×Q8—21no—
256C22×D4—71no—
257C23×C4—141no—
258C22×C8—72no—
259C22×C8—72no—
260C4⋊Q8—22no—
261C4⋊Q8C4×D714yes—
262C4×Q8—44no—
263C4×D4—74no—
264C4×D4—142no—
265#117—284no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—71no—
273C2×C4⋊C4—141no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4C22×D712yes—
276C2×C4⋊C4C22×D712yes—
277C2×C22⋊C4—141no—
278C2×C22⋊C4—141no—
279C2×C22⋊C4—141no—
280C2×C22⋊C4—141no—
281C2×C22⋊C4—141no—
282C2×C42—22no—
283C2×C42—141no—
284C2×C42C22⋊D711yescharacteristic
285C2×C42C22⋊D711yescharacteristic
286C2×C42C22⋊D711yescharacteristic
287C2×C42C22⋊D711yescharacteristic
288C2×C42C22×D711yescharacteristic
289C2×C42C22×D711yescharacteristic
290C2×C42C22×D711yescharacteristic
291#107—78no—
292#106—282no—
293C4⋊C8—74no—
294C4⋊C8—142no—
295#102—78no—
296#102—282no—
297#97—282no—
298#95—22no—
299#95—22no—
300#95—141no—
301#95—141no—
302#95—141no—
303C22×C7C2×C4×D411yescharacteristic
304C22×C7#53611yescharacteristic
305C22×C7#53711yescharacteristic
306C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C7C2×#13711yescharacteristic
309C22×C7#57511yescharacteristic
310C2×D7—42no—
311C2×D7—42no—
312C2×D7—42no—
313C2×D7—81no—
314C2×D7—81no—
315C2×D7—81no—
316C2×D7—81no—
317C4×C7—22no—
318C4×C7—22no—
319C4×C7—44no—
320C4×C7—41no—
321C4×C7—41no—
322C4×C7—41no—
323C4×C7—82no—
324C4×C7#44214yes—
325C4×C7#51911yescharacteristic
326C4×C7#52011yescharacteristic
327C4×C7#54111yescharacteristic
328C4×C7#54211yescharacteristic
329Dic7—42no—
330Dic7—81no—
331Dic7—81no—
332C24—141no—
333C2×Q8—24no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×D4—72no—
337C2×D4—141no—
338C2×D4—141no—
339C2×D4—141no—
340C2×D4—141no—
341C2×D4—141no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—141no—
345C22×C4—144no—
346C22×C4—141no—
347C22×C4—141no—
348C22×C4—282no—
349C22×C4—282no—
350C22×C4C2×C22⋊D711yescharacteristic
351C22×C4C2×C22⋊D711yescharacteristic
352C22×C4C2×C22⋊D711yescharacteristic
353C22×C4C2×C22⋊D711yescharacteristic
354C22×C4C2×C22⋊D711yescharacteristic
355C22×C4C2×C22⋊D711yescharacteristic
356C22×C4C23×D711yescharacteristic
357C2×C8—74no—
358C2×C8—142no—
359C2×C8—142no—
360C2×C8—142no—
361C2×C8—142no—
362C2×C8—142no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—72no—
371C4⋊C4—141no—
372C4⋊C4—141no—
373C4⋊C4—141no—
374C4⋊C4—141no—
375C4⋊C4—141no—
376C4⋊C4—282no—
377C4⋊C4—282no—
378C4⋊C4C2×C4×D714yes—
379C22⋊C4—144no—
380C22⋊C4—282no—
381C22⋊C4—282no—
382C22⋊C4—282no—
383C22⋊C4—282no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—41no—
392C42—44no—
393C42—282no—
394C42C2×C4×D712yes—
395C42#115712yes—
396C2×C7#232311yescharacteristic
397C2×C7C2×#44211yescharacteristic
398C2×C7C4×#13711yescharacteristic
399C2×C7#323511yescharacteristic
400C2×C7#323911yescharacteristic
401C2×C7#325111yescharacteristic
402C2×C7#325411yescharacteristic
403D7—82no—
404C23—142no—
405C23—281no—
406C23—281no—
407C23—281no—
408C23—281no—
409C23—281no—
410C23—281no—
411C23C22×C22⋊D711yescharacteristic
412Q8—48no—
413D4—142no—
414D4—142no—
415D4—281no—
416D4—281no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—142no—
437C2×C4—281no—
438C2×C4—281no—
439C2×C4—288no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—281no—
446C2×C4C4×C22⋊D712yes—
447C2×C4C4×C22⋊D712yes—
448C2×C4#649512yes—
449C2×C4#650512yes—
450C2×C4#651212yes—
451C2×C4C22×C4×D711yescharacteristic
452C2×C4C2×#115711yescharacteristic
453C2×C4#655111yescharacteristic
454C2×C4#655111yescharacteristic
455C2×C4#655311yescharacteristic
456C2×C4#655411yescharacteristic
457C8—144no—
458C8—282no—
459C7#2211611yescharacteristic, complemented, Sylow
460C22—282no—
461C22—282no—
462C22—282no—
463C22—561no—
464C22—561no—
465C22—561no—
466C22—561no—
467C22C2×C4×C22⋊D711yescharacteristic
468C22C2×#649511yescharacteristic
469C22C2×#650511yescharacteristic
470C22#9050711yescharacteristic
471C22C2×#651211yescharacteristic
472C22#9051311yescharacteristic
473C22#9052411yescharacteristic
474C4—22no—
475C4—22no—
476C4—41no—
477C4—44no—
478C4—41no—
479C4—41no—
480C4—82no—
481C4—282no—
482C4—561no—
483C4—561no—
484C4#8978614yes—
485C4#9026311yescharacteristic
486C4#9026511yescharacteristic
487C4#9027311yescharacteristic
488C4#9027411yescharacteristic
489C2—562no—
490C2C2×#8978611yescharacteristic
491C2C4×#650511yescharacteristic
492C2#1173993311yescharacteristic
493C2#1174066311yescharacteristic
494C2#1174075711yescharacteristic
495C2#1174080611yescharacteristic
496C2#1174081111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×C4⋊Q8C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenterC4×C4⋊Q8D7
hypocenterC7#22116
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C4×C7×C4⋊Q8C2semidirect
4. #11729534C2semidirect
5. #11727374C2semidirect
8. #11727322C2semidirect
10. #11727215C2semidirect
11. #11727145C2semidirect
23. #89859C4semidirect
94. C4×C4⋊Q8D7semidirect
459. C7#22116semidirect

Maximal subgroups

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

subgroupindex
C4×C7×C4⋊Q82
C2×#898592
#117307442
#117295342
#117273742
#117273712
#117273382
#117273222
#117272172
#117272152
#117271452
#221167