Finite Group Databasebeta

← #13249573

Subgroups — #13249573

Order 1792.

692 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740144C211yesmaximal, characteristic
2C2×#89422C211yesmaximal, characteristic, complemented
3C2×C7×#327C211yesmaximal, characteristic
4C2×#89332C211yesmaximal, characteristic
5#11727664C211yesmaximal, characteristic, complemented
6#11725481C211yesmaximal, characteristic, complemented
7#11725418C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C2×#6498—21no—
10C22×Dic7⋊C4—21no—
11C22×C4×Dic7C2211yescharacteristic
12C2×C7×C22⋊C8C2211yescharacteristic
13C2×C7×#95C2211yescharacteristic
14#89993C412yescomplemented
15#89992—21no—
16#89984C2211yescharacteristic
17C2×C7⋊#95—21no—
18C2×C7⋊#95C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89745C412yescomplemented
21#89716—22no—
22#89422—22no—
23#89422—22no—
24C7×#327—22no—
25#89345—22no—
26#89332—22no—
27#9156—71nomaximal, Sylow
28C22×C7×D4—21no—
29C22×C7×D4C2×C411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7C2311yescharacteristic
32C23×Dic7—21no—
33C23×Dic7D411yescharacteristic
34C23×Dic7D411yescharacteristic
35C22×C7×C8—21no—
36C7×#121—21no—
37C7×#121—21no—
38C7×#121—21no—
39C7×#121—21no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C22⋊Dic7—21no—
43C2×C22⋊Dic7—22no—
44C2×C22⋊Dic7—22no—
45#6498—42no—
46#6498—42no—
47C2×C4⋊Dic7C2×C412yes—
48C2×Dic7⋊C4—24no—
49C2×Dic7⋊C4—22no—
50C2×Dic7⋊C4—42no—
51C2×Dic7⋊C4—42no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7—21no—
54C2×C4×Dic7—21no—
55C2×C4×Dic7—21no—
56C2×C4×Dic7C2×C412yes—
57C2×C4×Dic7D412yes—
58C22×C7⋊C8—21no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#95—21no—
62C7×#95D412yes—
63C7⋊#95—24no—
64C7⋊#95—21no—
65C7⋊#95—21no—
66C7⋊#95—22no—
67C7⋊#95D412yes—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70#2854—71no—
71C2×#382—71no—
72#1817—71no—
73C2×#327—71no—
74C2×#327D711yescharacteristic
75#1294—71no—
76#1294—71no—
77C24×C7—21no—
78C24×C7C2×D411yescharacteristic
79C2×C7×D4—21no—
80C2×C7×D4—21no—
81C2×C7×D4—42no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C22×C4×C7—21no—
89C22×C4×C7—22no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7—42no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7C22×C411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7#4111yescharacteristic
100C22×C4×C7#4111yescharacteristic
101C22×Dic7—21no—
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—21no—
105C22×Dic7—21no—
106C22×Dic7—21no—
107C22×Dic7—22no—
108C22×Dic7—22no—
109C22×Dic7—42no—
110C22×Dic7—42no—
111C22×Dic7—42no—
112C22×Dic7—41no—
113C22×Dic7—41no—
114C22×Dic7C22⋊C412yes—
115C22×Dic7C22⋊C412yes—
116C22×Dic7D812yes—
117C22×Dic7SD1612yes—
118C2×C7×C8—42no—
119C2×C7×C8—41no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C22⋊Dic7—44no—
125C22⋊Dic7—41no—
126C22⋊Dic7—41no—
127C22⋊Dic7—44no—
128C4⋊Dic7—42no—
129Dic7⋊C4—48no—
130Dic7⋊C4—44no—
131C4×Dic7—22no—
132C4×Dic7—22no—
133C4×Dic7—22no—
134C4×Dic7—22no—
135C2×C7⋊C8—42no—
136C2×C7⋊C8—41no—
137C2×#123—141no—
138C2×#121C2×D711yescharacteristic
139C22×C4⋊C4—141no—
140C22×C42—71no—
141C2×C22⋊C8—71no—
142C2×C22⋊C8C2×D711yescharacteristic
143#390—72no—
144#389—72no—
145#388—142no—
146#386—141no—
147#382—142no—
148#382—142no—
149#380—71no—
150C2×#95—71no—
151C2×#95—141no—
152C2×#95C2×D711yescharacteristic
153#336—142no—
154#327—22no—
155#327—142no—
156C23×C7—22no—
157C23×C7—22no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—42no—
163C23×C7C2×C22⋊C411yescharacteristic
164C23×C7#12011yescharacteristic
165C23×C7#12311yescharacteristic
166C23×C7C2×D811yescharacteristic
167C23×C7C2×SD1611yescharacteristic
168C23×C7AGL(1,ℤ/8)11yescharacteristic
169C23×C7#13711yescharacteristic
170C7×D4—81no—
171C7×D4—81no—
172C7×D4—81no—
173C7×D4—81no—
174C7×D4—81no—
175C7×D4—81no—
176C2×C4×C7—22no—
177C2×C4×C7—22no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—82no—
189C2×C4×C7—82no—
190C2×C4×C7C4×D411yescharacteristic
191C2×C4×C7C4×D411yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×Dic7—22no—
195C2×Dic7—22no—
196C2×Dic7—22no—
197C2×Dic7—22no—
198C2×Dic7—22no—
199C2×Dic7—22no—
200C2×Dic7—22no—
201C2×Dic7—22no—
202C2×Dic7—22no—
203C2×Dic7—42no—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C2×Dic7—42no—
210C2×Dic7—42no—
211C2×Dic7—82no—
212C2×Dic7—82no—
213C2×Dic7#10214yes—
214C7×C8—81no—
215C7⋊C8—81no—
216C22×D4—21no—
217C22×D4C4×D711yescharacteristic
218C23×C4—21no—
219C23×C4—71no—
220C23×C4—71no—
221C23×C4—141no—
222C23×C4C22×D711yescharacteristic
223C22×C8—21no—
224C22×C8—141no—
225#123—282no—
226#123—282no—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231C2×C4⋊C4—72no—
232C2×C4⋊C4—142no—
233C2×C4⋊C4—144no—
234C2×C4⋊C4—282no—
235C2×C4⋊C4—282no—
236C2×C4⋊C4C4×D711yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—142no—
239C2×C22⋊C4—142no—
240C2×C22⋊C4—141no—
241C2×C42—72no—
242C2×C42—72no—
243C2×C42—141no—
244C2×C42—141no—
245C2×C42—141no—
246C2×C42—141no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—141no—
250C22⋊C8—141no—
251#95—21no—
252#95—72no—
253#95—142no—
254#95—144no—
255#95—141no—
256#95—141no—
257#95C22⋊D712yes—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—22no—
263C22×C7—22no—
264C22×C7—22no—
265C22×C7—22no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—82no—
270C22×C7—82no—
271C22×C7#38611yescharacteristic
272C22×C7#41012yes—
273C22×C7C2×#10211yescharacteristic
274C22×C7#41711yescharacteristic
275C22×C7#42112yes—
276C22×C7#44711yescharacteristic
277C22×C7#44811yescharacteristic
278C22×C7#48011yescharacteristic
279C22×C7#48111yescharacteristic
280C4×C7—21no—
281C4×C7—21no—
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—82no—
285C4×C7—81no—
286Dic7—24no—
287Dic7—42no—
288Dic7—82no—
289Dic7—82no—
290C24—21no—
291C24D4×D711yescharacteristic
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—72no—
309C22×C4—72no—
310C22×C4—72no—
311C22×C4—72no—
312C22×C4—141no—
313C22×C4—141no—
314C22×C4—141no—
315C22×C4—141no—
316C22×C4—141no—
317C22×C4—141no—
318C22×C4—142no—
319C22×C4—142no—
320C22×C4—282no—
321C22×C4—282no—
322C22×C4—282no—
323C22×C4—281no—
324C22×C4—281no—
325C22×C4C2×C4×D711yescharacteristic
326C22×C4#115711yescharacteristic
327C22×C4D4×D711yescharacteristic
328C22×C4D4×D711yescharacteristic
329C22×C4#115911yescharacteristic
330C22×C4C2×C22⋊D711yescharacteristic
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—282no—
334C2×C8—281no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—282no—
338C4⋊C4—284no—
339C4⋊C4—288no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—284no—
343C22⋊C4—284no—
344C22⋊C4—281no—
345C22⋊C4—281no—
346C42—142no—
347C42—142no—
348C42—142no—
349C42—142no—
350C2×C7—21no—
351C2×C7—21no—
352C2×C7—21no—
353C2×C7—21no—
354C2×C7—161no—
355C2×C7#151212yes—
356C2×C7#151712yes—
357C2×C7#187411yescharacteristic
358C2×C7#189711yescharacteristic
359C2×C7#189911yescharacteristic
360C23—22no—
361C23—22no—
362C23—41no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23D7×C22⋊C411yescharacteristic
368C23#644611yescharacteristic
369C23D7×D811yescharacteristic
370C23#647411yescharacteristic
371C23D7×SD1611yescharacteristic
372C23#647811yescharacteristic
373C23#650011yescharacteristic
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—74no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—142no—
396C2×C4—142no—
397C2×C4—142no—
398C2×C4—142no—
399C2×C4—142no—
400C2×C4—142no—
401C2×C4—142no—
402C2×C4—142no—
403C2×C4—142no—
404C2×C4—282no—
405C2×C4—282no—
406C2×C4—282no—
407C2×C4—282no—
408C2×C4—282no—
409C2×C4—282no—
410C2×C4—282no—
411C2×C4—282no—
412C2×C4—562no—
413C2×C4—562no—
414C2×C4(C22⋊D7)⋊C411yescharacteristic
415C2×C4#644711yescharacteristic
416C2×C4C4×C22⋊D711yescharacteristic
417C2×C4#650111yescharacteristic
418C8—81no—
419C8—561no—
420C7#915611yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—82no—
433C22—82no—
434C22#8951212yes—
435C22D7×#10211yescharacteristic
436C22#8954311yescharacteristic
437C22#8954611yescharacteristic
438C22#8954911yescharacteristic
439C22#8959312yes—
440C22#8972811yescharacteristic
441C22#8992811yescharacteristic
442C22#8994211yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—82no—
448C4—81no—
449C4—144no—
450C4—282no—
451C4—562no—
452C4—562no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—161no—
458C2#1172603412yes—
459C2#1172603612yes—
460C2#1172778311yescharacteristic
461C2#1172795411yescharacteristic
462C2#1172813711yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#327C2
socleC23×C7C2×C22⋊C4
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenterC2×#327D7
hypocenterC7#9156
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#89422C2semidirect
5. #11727664C2semidirect
6. #11725481C2semidirect
7. #11725418C2semidirect
14. #89993C4semidirect
20. #89745C4semidirect
420. C7#9156semidirect

Maximal subgroups

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

subgroupindex
#117401442
C2×#894222
C2×C7×#3272
C2×#893322
#117276642
#117254812
#117254182
#91567