Finite Group Databasebeta

← #13246776

Subgroups — #13246776

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740361C211yesmaximal, characteristic
2C7×#1513C211yesmaximal, characteristic
3#11729463C211yesmaximal, characteristic
4#11727736C211yesmaximal, characteristic, complemented
5#11727266C211yesmaximal, characteristic, complemented
6#11725491C211yesmaximal, characteristic, complemented
7#11725432C211yesmaximal, characteristic, complemented
8C7×#525C2211yescharacteristic
9C2×#6511C411yescharacteristic, complemented
10C23×C7⋊C8C2211yescharacteristic
11C7×#407C2211yescharacteristic
12C7×#380C2211yescharacteristic
13#90006C411yescharacteristic, complemented
14#89984C2211yescharacteristic
15#89979C2211yescharacteristic
16#89979C2211yescharacteristic
17C2×#6396—21no—
18#89783—21no—
19#89783—21no—
20D4×C7⋊C8—21no—
21D4×C7⋊C8—21no—
22#89771C411yescharacteristic, complemented
23#89769C411yescharacteristic, complemented
24C2×#6381—21no—
25C7×#328—21no—
26C7×#327—21no—
27#89353—21no—
28#89333—21no—
29#89332—21no—
30#89324—21no—
31#8628—71nomaximal, Sylow
32C2×C7×#41C2×C411yescharacteristic
33C23×C4×C7C2311yescharacteristic
34C2×C7×M16—21no—
35C7×#123—21no—
36C7×C22⋊Q8D411yescharacteristic
37C7×C22⋊Q8Q811yescharacteristic
38C7×#121D411yescharacteristic
39C7×#121Q811yescharacteristic
40C7×#120—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C7×#117C2×C411yescharacteristic
44C2×C7×C4⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C22⋊Dic7—21no—
47#6511—42no—
48C2×C4⋊Dic7—21no—
49C2×#1136—21no—
50C2×#1136—21no—
51C2×#1136C2×C411yescharacteristic
52C2×#1136C2×C411yescharacteristic
53C22×C7⋊C8—21no—
54C22×C7⋊C8—42no—
55C22×C7⋊C8C2×C411yescharacteristic
56C22×C7⋊C8C2×C411yescharacteristic
57C22×C7⋊C8D411yescharacteristic
58C22×C7⋊C8D411yescharacteristic
59C22×C7⋊C8D411yescharacteristic
60C22×C7⋊C8D411yescharacteristic
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×#95—21no—
64C7⋊#95—21no—
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70C22⋊(C7⋊C8)—21no—
71#6396—42no—
72#6381—42no—
73C4⋊(C7⋊C8)—21no—
74C4⋊(C7⋊C8)—21no—
75#6377—21no—
76C4×C7⋊C8—21no—
77#2929—71no—
78#1810—71no—
79#1808—71no—
80#1513—71no—
81#1513D711yescharacteristic
82#1305—71no—
83#1305—71no—
84C24×C7#4111yescharacteristic
85C7×#41—41no—
86C7×#41—41no—
87C2×C7×Q8C4⋊C411yescharacteristic
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4C4⋊C411yescharacteristic
92C22×C4×C7—21no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×Q811yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C22×Dic7—41no—
103C7×M16—41no—
104C7×M16—41no—
105C2×C7×C8—41no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4C4⋊C411yescharacteristic
111C7×C4⋊C4C4⋊C411yescharacteristic
112C7×C22⋊C4—21no—
113C7×C22⋊C4—21no—
114C7×C22⋊C4—21no—
115C7×C22⋊C4—21no—
116C7×C22⋊C4—21no—
117C7×C22⋊C4—81no—
118C42×C7—21no—
119C22⋊Dic7—81no—
120C4⋊Dic7—41no—
121C4⋊Dic7—41no—
122#1136—42no—
123#1136—41no—
124#1136—41no—
125#1136—41no—
126#1136—41no—
127C2×C7⋊C8—22no—
128C2×C7⋊C8—22no—
129C2×C7⋊C8—22no—
130C2×C7⋊C8—21no—
131C2×C7⋊C8—21no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×C7⋊C8—41no—
135C2×C7⋊C8—42no—
136C2×C7⋊C8—41no—
137C2×C7⋊C8C22⋊C411yescharacteristic
138C2×C7⋊C8C22⋊C411yescharacteristic
139C2×C7⋊C8C22⋊C411yescharacteristic
140C2×C7⋊C8C22⋊C411yescharacteristic
141C2×#131—71no—
142C23×C8—71no—
143#525C2×D711yescharacteristic
144#435—141no—
145#435—141no—
146C8×D4—141no—
147C8×D4—141no—
148C2×#108—141no—
149C2×#107—141no—
150#424—71no—
151#408—71no—
152#407—71no—
153#407—71no—
154#407C2×D711yescharacteristic
155#405—71no—
156#380—71no—
157#380C2×D711yescharacteristic
158#341—141no—
159#340—141no—
160#328—21no—
161#328—141no—
162#327—21no—
163#327—141no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7#11711yescharacteristic
170C23×C7#12111yescharacteristic
171C23×C7C22⋊Q811yescharacteristic
172C7×Q8—41no—
173C7×D4—41no—
174C7×D4—42no—
175C7×D4—42no—
176C7×D4—41no—
177C7×D4—41no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7—81no—
195C2×C4×C7C2×C22⋊C411yescharacteristic
196C2×C4×C7C2×C4⋊C411yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7C22⋊Q811yescharacteristic
199C2×Dic7—81no—
200C2×Dic7—81no—
201C2×Dic7—81no—
202C7×C8—81no—
203C7⋊C8—21no—
204C7⋊C8—21no—
205C7⋊C8—21no—
206C7⋊C8—41no—
207C7⋊C8—41no—
208C7⋊C8—81no—
209C7⋊C8#9912yes—
210C2×#41C4×D711yescharacteristic
211C23×C4C22×D711yescharacteristic
212#131—282no—
213C2×M16—21no—
214C2×M16—71no—
215C2×M16—71no—
216C2×M16—141no—
217C2×M16—141no—
218C22×C8—71no—
219C22×C8—71no—
220C22×C8—71no—
221C22×C8—71no—
222C22×C8—71no—
223C22×C8—71no—
224C22×C8—141no—
225C22×C8—282no—
226#123—21no—
227C22⋊Q8Dic1411yescharacteristic
228C22⋊Q8C22⋊D711yescharacteristic
229#121Dic1411yescharacteristic
230#121C22⋊D711yescharacteristic
231#120—21no—
232C4×D4—21no—
233C4×D4—21no—
234#117C4×D711yescharacteristic
235C2×C4⋊C4—21no—
236C2×C4⋊C4—141no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—141no—
239#108—282no—
240#107—282no—
241C4⋊C8—141no—
242C4⋊C8—141no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—141no—
246C22⋊C8—141no—
247C22⋊C8—141no—
248C22⋊C8—141no—
249C22⋊C8—141no—
250C22⋊C8—141no—
251#97—141no—
252C4×C8—141no—
253#95—21no—
254#95—141no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—42no—
264C22×C7—42no—
265C22×C7—41no—
266C22×C7—41no—
267C22×C7—41no—
268C22×C7#38011yescharacteristic
269C22×C7#47211yescharacteristic
270C22×C7#47311yescharacteristic
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—161no—
278C4×C7C2×#9911yescharacteristic
279C4×C7#41011yescharacteristic
280Dic7—161no—
281C24#115911yescharacteristic
282#41—41no—
283#41—41no—
284C2×Q8Dic7⋊C411yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4Dic7⋊C411yescharacteristic
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—281no—
294C22×C4C2×Dic1411yescharacteristic
295C22×C4C2×C4×D711yescharacteristic
296C22×C4D4×D711yescharacteristic
297C22×C4D4×D711yescharacteristic
298C22×C4#115911yescharacteristic
299C22×C4C2×C22⋊D711yescharacteristic
300M16—41no—
301M16—41no—
302M16—282no—
303M16—281no—
304M16—281no—
305M16—281no—
306M16—281no—
307C2×C8—41no—
308C2×C8—71no—
309C2×C8—71no—
310C2×C8—71no—
311C2×C8—71no—
312C2×C8—142no—
313C2×C8—142no—
314C2×C8—142no—
315C2×C8—141no—
316C2×C8—141no—
317C2×C8—281no—
318C2×C8—281no—
319C2×C8—281no—
320C2×C8—282no—
321C2×C8—281no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—281no—
327C4⋊C4—281no—
328C4⋊C4Dic7⋊C411yescharacteristic
329C4⋊C4Dic7⋊C411yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—81no—
336C22⋊C4—561no—
337C42—21no—
338C2×C7—21no—
339C2×C7—21no—
340C2×C7—41no—
341C2×C7—41no—
342C2×C7—81no—
343C2×C7#179911yescharacteristic
344C2×C7#181411yescharacteristic
345C2×C7#181611yescharacteristic
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23#644011yescharacteristic
352C23#644111yescharacteristic
353C23#650211yescharacteristic
354Q8—41no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—41no—
359D4—41no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4—561no—
380C2×C4#644111yescharacteristic
381C2×C4D7×C22⋊C411yescharacteristic
382C2×C4C2×Dic7⋊C411yescharacteristic
383C2×C4#650211yescharacteristic
384C8—72no—
385C8—81no—
386C8—141no—
387C8—141no—
388C8—141no—
389C8—281no—
390C8—281no—
391C8—561no—
392C7#862811yescharacteristic, complemented, Sylow
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22#8971411yescharacteristic
407C22#8997111yescharacteristic
408C22#8997611yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—161no—
416C4—1121no—
417C4#8951211yescharacteristic
418C4D7×#9911yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—81no—
424C2#1172778211yescharacteristic
425C2#1172805411yescharacteristic
426C2#1172815011yescharacteristic
subgroupG/H
centerC22#89714
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1513C2
socleC22×C7#380
prime radicalC7×#525C22
cosocleC23×C4×C7C23
hypercenter#1513D7
hypocenterC7#8628
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1513

Complements

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

NH ≅ G/Ntype
4. #11727736C2semidirect
5. #11727266C2semidirect
6. #11725491C2semidirect
7. #11725432C2semidirect
9. C2×#6511C4semidirect
13. #90006C4semidirect
22. #89771C4semidirect
23. #89769C4semidirect
392. C7#8628semidirect

Maximal subgroups

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

subgroupindex
#117403612
C7×#15132
#117294632
#117277362
#117272662
#117254912
#117254322
#86287