Finite Group Databasebeta

← #13365833

Subgroups — #13365833

Order 1792.

848 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
1C2×#89618C211yesmaximal, characteristic
2#11731765C211yesmaximal, characteristic, complemented
3#11731062C211yesmaximal, characteristic
4C7×#1947C211yesmaximal, characteristic, complemented
5#11728377C212yesmaximal
6#11728365C212yesmaximal, complemented
7Dic7×C4⋊C8C211yesmaximal, characteristic, complemented
8#11727338C211yesmaximal, characteristic
9#11727244C211yesmaximal, characteristic, complemented
10#11726606C212yesmaximal
11#11726568C212yesmaximal, complemented
12C2×#6450—21no—
13C2×C4×D28C2211yescharacteristic
14C2×#6435—21no—
15C2×#6432C2211yescharacteristic
16C2×C7×#107C2212yes—
17C2×C7×C4⋊C8C2211yescharacteristic
18C7×#384C2211yescharacteristic
19C4×C7×C4⋊C4C2211yescharacteristic
20C2×#6395C2212yes—
21C2×#6392C2212yes—
22C2×C8×Dic7C2212yes—
23#89760C2211yescharacteristic
24#89760C2211yescharacteristic
25C4⋊C4×Dic7—21no—
26C4⋊C4×Dic7C2212yes—
27#89746—21no—
28#89745—21no—
29#89745C2212yes—
30#89745C2211yescharacteristic
31#89745C2211yescharacteristic
32C2×#6383C2212yes—
33C4×(C2×D7)⋊C4—21no—
34C42×Dic7C2211yescharacteristic
35C2×C4⋊(C7⋊C8)C2211yescharacteristic
36#89618C418yescomplemented
37#89441—21no—
38#89441—21no—
39C7×#340C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#22110—71nomaximal, Sylow
45C22×D28C2311yescharacteristic
46C22×C4×D7—21no—
47C22×Dic14—21no—
48C22×C7×C8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C42×C7—21no—
55C2×C42×C7C2311yescharacteristic
56C2×(C2×D7)⋊C4—21no—
57C2×(C2×D7)⋊C4—21no—
58C2×(C2×D7)⋊C4—21no—
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—21no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2312yes—
64C2×Dic7⋊C4—21no—
65C2×Dic7⋊C4—21no—
66C2×Dic7⋊C4—22no—
67C2×C4×Dic7—22no—
68C2×C4×Dic7D411yescharacteristic
69C2×C4×Dic7D411yescharacteristic
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7C2311yescharacteristic
73C2×C4×Dic7C2311yescharacteristic
74C22×C7⋊C8C2312yes—
75#6450—44no—
76C4×D28—22no—
77C4×D28C2×C414yes—
78#6435—44no—
79#6432—22no—
80#6432C2×C414yes—
81C7×#107—42no—
82C7×C4⋊C8C2×C414yes—
83C7×#95—21no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88#6395C2×C418yes—
89#6392C2×C418yes—
90C8×Dic7—42no—
91#6383—42no—
92C4⋊(C7⋊C8)—22no—
93C2×#474—71no—
94C4×C4⋊Q8—71no—
95#2293—71no—
96#1988—71no—
97#1947D711yescharacteristic, complemented
98#1942—72no—
99#1934—72no—
100#1925—72no—
101#1918—72no—
102#1802—71no—
103C4×C4⋊C8—71no—
104C23×D7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—22no—
112C22×Dic7—21no—
113C22×Dic7C2×D411yescharacteristic
114C22×Dic7C2×D411yescharacteristic
115C22×Dic7C2×D411yescharacteristic
116C22×Dic7C2×D411yescharacteristic
117C2×D28—21no—
118C2×D28—21no—
119C2×D28—21no—
120C2×D28—21no—
121C2×D28—21no—
122C2×D28C22×C412yes—
123C2×C4×D7—24no—
124C2×C4×D7—42no—
125C2×C4×D7—42no—
126C2×Dic14—24no—
127C2×Dic14—42no—
128C2×Dic14—42no—
129C2×C7×C8—22no—
130C2×C7×C8—22no—
131C2×C7×C8C22×C414yes—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4—42no—
140C42×C7—42no—
141C42×C7C22×C412yes—
142C42×C7#4112yes—
143(C2×D7)⋊C4—24no—
144(C2×D7)⋊C4—42no—
145(C2×D7)⋊C4—42no—
146(C2×D7)⋊C4—42no—
147(C2×D7)⋊C4—42no—
148C4⋊Dic7—24no—
149C4⋊Dic7—22no—
150C4⋊Dic7—21no—
151C4⋊Dic7C22×C414yes—
152C4⋊Dic7C22×C412yes—
153Dic7⋊C4—42no—
154Dic7⋊C4—42no—
155Dic7⋊C4—44no—
156C4×Dic7—22no—
157C4×Dic7—21no—
158C4×Dic7—21no—
159C4×Dic7—21no—
160C4×Dic7—21no—
161C4×Dic7—22no—
162C4×Dic7—41no—
163C4×Dic7—44no—
164C4×Dic7D814yes—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C4⋊Q8—71no—
169C2×C4×Q8—141no—
170C2×C4×D4—71no—
171C2×#117—141no—
172#474—78no—
173C2×#107—72no—
174C2×#107C2×D712yes—
175C2×C4⋊C8—71no—
176C2×C4⋊C8C2×D711yescharacteristic
177C2×#102—72no—
178C2×#102—72no—
179C2×C4×C8—72no—
180#391—141no—
181#390—71no—
182#390—71no—
183#389—72no—
184#389—71no—
185#389—71no—
186#389—141no—
187#388—141no—
188#388—141no—
189#384C2×D711yescharacteristic
190C4×C4⋊C4—72no—
191C4×C4⋊C4—141no—
192C4×C4⋊C4C2×D711yescharacteristic
193C4×C22⋊C4—141no—
194C43—71no—
195#340—72no—
196#340—72no—
197#340—72no—
198#340C2×D712yes—
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—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—42no—
219C2×C4×C7C4×D412yes—
220C2×C4×C7C4×D412yes—
221C2×C4×C7C2×D812yes—
222C2×C4×C7#13512yes—
223C2×C4×C7C23×C411yescharacteristic
224C2×C4×C7C2×#4111yescharacteristic
225C2×C4×C7H(5,𝔽2)11yescharacteristic
226C2×C4×C72−1+411yescharacteristic
227C2×C4×C72−1+411yescharacteristic
228C2×C4×C72−1+411yescharacteristic
229C2×Dic7—21no—
230C2×Dic7—21no—
231C2×Dic7—21no—
232C2×Dic7—21no—
233C2×Dic7—21no—
234C2×Dic7—21no—
235C2×Dic7—24no—
236C2×Dic7—22no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—42no—
241C2×Dic7—41no—
242C2×Dic7—42no—
243C2×Dic7—41no—
244C2×Dic7C2×D812yes—
245C2×Dic7C2×D812yes—
246C2×Dic7#13712yes—
247D28—22no—
248D28—22no—
249D28—41no—
250D28—41no—
251C4×D7—48no—
252Dic14—48no—
253C7×C8—24no—
254C7⋊C8—42no—
255C22×Q8—141no—
256C22×D4—71no—
257C23×C4—141no—
258C22×C8—72no—
259C22×C8C22×D712yes—
260C4⋊Q8—74no—
261C4⋊Q8—142no—
262C4×Q8—284no—
263C4×D4—74no—
264C4×D4—142no—
265#117—284no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—71no—
269C2×C4⋊C4—72no—
270C2×C4⋊C4—141no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—142no—
274C2×C4⋊C4C22×D711yescharacteristic
275C2×C4⋊C4C22×D711yescharacteristic
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—21no—
283C2×C42—71no—
284C2×C42—71no—
285C2×C42—71no—
286C2×C42—71no—
287C2×C42—71no—
288C2×C42—71no—
289C2×C42—142no—
290C2×C42C22×D711yescharacteristic
291#107—42no—
292#107—78no—
293C4⋊C8—142no—
294C4⋊C8C4×D714yes—
295#102—78no—
296#102—282no—
297C4×C8—282no—
298#95—21no—
299#95—141no—
300#95—141no—
301#95—142no—
302#95—142no—
303C22×C7C2×C4×D411yescharacteristic
304C22×C7#53611yescharacteristic
305C22×C7#53711yescharacteristic
306C22×C7C22×D811yescharacteristic
307C22×C7C2×#13511yescharacteristic
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—81no—
319C4×C7—82no—
320C4×C7—81no—
321C4×C7C4×D814yes—
322C4×C7#51911yescharacteristic
323C4×C7#52011yescharacteristic
324C4×C7#54111yescharacteristic
325C4×C7#54211yescharacteristic
326Dic7—22no—
327Dic7—22no—
328Dic7—22no—
329Dic7—44no—
330Dic7—42no—
331Dic7—41no—
332C24—141no—
333C2×Q8—144no—
334C2×Q8—282no—
335C2×Q8—282no—
336C2×D4—72no—
337C2×D4—141no—
338C2×D4—141no—
339C2×D4—141no—
340C2×D4—141no—
341C2×D4—141no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—71no—
346C22×C4—71no—
347C22×C4—71no—
348C22×C4—71no—
349C22×C4—142no—
350C22×C4—141no—
351C22×C4—144no—
352C22×C4—282no—
353C22×C4—282no—
354C22×C4D4×D711yescharacteristic
355C22×C4D4×D711yescharacteristic
356C22×C4C23×D711yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—142no—
360C2×C8—142no—
361C2×C8—142no—
362C2×C8C2×C4×D714yes—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—74no—
372C4⋊C4—72no—
373C4⋊C4—144no—
374C4⋊C4—141no—
375C4⋊C4—142no—
376C4⋊C4—282no—
377C4⋊C4—282no—
378C4⋊C4—284no—
379C22⋊C4—144no—
380C22⋊C4—282no—
381C22⋊C4—282no—
382C22⋊C4—282no—
383C22⋊C4—282no—
384C42—42no—
385C42—74no—
386C42—142no—
387C42—142no—
388C42—141no—
389C42—141no—
390C42—141no—
391C42—141no—
392C42—281no—
393C42—284no—
394C42C2×C4×D712yes—
395C42#116112yes—
396C2×C7#232311yescharacteristic
397C2×C7C2×C4×D811yescharacteristic
398C2×C7C4×#13711yescharacteristic
399C2×C7#322611yescharacteristic
400C2×C7#323311yescharacteristic
401C2×C7#324511yescharacteristic
402C2×C7#324811yescharacteristic
403D7—82no—
404C23—142no—
405C23—281no—
406C23—281no—
407C23—281no—
408C23—281no—
409C23—281no—
410C23—281no—
411C23C2×D4×D711yescharacteristic
412Q8—288no—
413D4—142no—
414D4—142no—
415D4—281no—
416D4—281no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—72no—
429C2×C4—72no—
430C2×C4—72no—
431C2×C4—141no—
432C2×C4—141no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—142no—
438C2×C4—144no—
439C2×C4—141no—
440C2×C4—141no—
441C2×C4—141no—
442C2×C4—282no—
443C2×C4—281no—
444C2×C4—281no—
445C2×C4—282no—
446C2×C4—288no—
447C2×C4#645612yes—
448C2×C4#645612yes—
449C2×C4D7×D812yes—
450C2×C4#648212yes—
451C2×C4C22×C4×D711yescharacteristic
452C2×C4#654811yescharacteristic
453C2×C4C2×#116111yescharacteristic
454C2×C4#655111yescharacteristic
455C2×C4#655411yescharacteristic
456C2×C4#655411yescharacteristic
457C8—24no—
458C8—282no—
459C7#2211011yescharacteristic, complemented, Sylow
460C22—282no—
461C22—282no—
462C22—282no—
463C22—561no—
464C22—561no—
465C22—561no—
466C22—561no—
467C22C2×#645611yescharacteristic
468C22#9029911yescharacteristic
469C22#9031911yescharacteristic
470C22C2×D7×D811yescharacteristic
471C22C2×#648211yescharacteristic
472C22#9046411yescharacteristic
473C22D7×#13711yescharacteristic
474C4—22no—
475C4—81no—
476C4—81no—
477C4—82no—
478C4—142no—
479C4—142no—
480C4—142no—
481C4—282no—
482C4—284no—
483C4—281no—
484C4#8964414yes—
485C4#9023711yescharacteristic
486C4#9026311yescharacteristic
487C4#9038811yescharacteristic
488C4#9042411yescharacteristic
489C2—562no—
490C2#1173364211yescharacteristic
491C2C2×#8964411yescharacteristic
492C2#1173664511yescharacteristic
493C2#1173778711yescharacteristic
494C2#1173794211yescharacteristic
495C2#1173841411yescharacteristic
496C2#1173855811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1947C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1947D7
hypocenterC7#22110
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. #11731765C2semidirect
4. C7×#1947C2semidirect
6. #11728365C2semidirect
7. Dic7×C4⋊C8C2semidirect
9. #11727244C2semidirect
11. #11726568C2semidirect
36. #89618C4semidirect
97. #1947D7semidirect
459. C7#22110semidirect

Maximal subgroups

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

subgroupindex
C2×#896182
#117317652
#117310622
C7×#19472
#117283772
#117283652
Dic7×C4⋊C82
#117273382
#117272442
#117266062
#117265682
#221107