Finite Group Databasebeta

← #13511375

Subgroups — #13511375

Order 1792.

464 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3336C211yesmaximal, characteristic
2#11736858C211yesmaximal, characteristic
3#11729111C211yesmaximal, characteristic, complemented
4#11729108C211yesmaximal, characteristic, complemented
5#11728837C211yesmaximal, characteristic, complemented
6#11728837C211yesmaximal, characteristic, complemented
7#11728834C211yesmaximal, characteristic, complemented
8#11728833C211yesmaximal, characteristic, complemented
9#11728666C211yesmaximal, characteristic, complemented
10C4×#6403C211yesmaximal, characteristic, complemented
11#11728612C211yesmaximal, characteristic
12#11728409C211yesmaximal, characteristic, complemented
13#11728403C211yesmaximal, characteristic, complemented
14#11728357C211yesmaximal, characteristic, complemented
15#11728350C211yesmaximal, characteristic, complemented
16C7×#552—21no—
17#90232—21no—
18C7×#499C2211yescharacteristic
19C7×#485—21no—
20C7×#483—21no—
21C7×#479—21no—
22C7×#467C2211yescharacteristic
23C7×#467C2211yescharacteristic
24C4×C7×Q16C2211yescharacteristic
25C7×C8×D4C2211yescharacteristic
26C7×#427C2211yescharacteristic
27C2×C7×#107C2211yescharacteristic
28C2×#6403C2211yescharacteristic
29#89888C2211yescharacteristic
30#89888C2211yescharacteristic
31#89878C2211yescharacteristic
32C2×#6392C2211yescharacteristic
33C2×#6386C2211yescharacteristic
34C2×#6386C2211yescharacteristic
35#89601—21no—
36#89495—21no—
37#89493—21no—
38C4×Dic28C2211yescharacteristic
39#89455C2211yescharacteristic
40#89359C2211yescharacteristic
41#89359C2211yescharacteristic
42#89359C2211yescharacteristic
43C22⋊(C7⋊C16)C2211yescharacteristic
44C22⋊(C7⋊C16)C2211yescharacteristic
45#89287C2211yescharacteristic
46#89287C2211yescharacteristic
47#89287C2211yescharacteristic
48#89285C2211yescharacteristic
49#89285C2211yescharacteristic
50#89285C2211yescharacteristic
51#89285C2211yescharacteristic
52#89284C2211yescharacteristic
53#89284C2211yescharacteristic
54#89284C2211yescharacteristic
55#89284C2211yescharacteristic
56#89284C2211yescharacteristic
57C4⋊(C7⋊C16)C2211yescharacteristic
58C4×C7⋊C16C2211yescharacteristic
59#33919—71nomaximal, Sylow
60C2×C7×Q16C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C7×#126—41no—
64C7×#125—21no—
65C7×#125—41no—
66C7×#123—41no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×D4D411yescharacteristic
71C7×#117—21no—
72C7×#117—41no—
73C2×C7×C4⋊C4—21no—
74#6498—41no—
75#6489—21no—
76C2×C4⋊Dic7—21no—
77#6487—21no—
78#6487—21no—
79C2×Dic28C2311yescharacteristic
80#6452—41no—
81#6442—41no—
82#6440—41no—
83#6433—21no—
84C4×Dic14—21no—
85C7×#107—21no—
86C7×#107C2311yescharacteristic
87C7×#107C2311yescharacteristic
88C7×#107C2311yescharacteristic
89C7×#106—21no—
90C7×#106—21no—
91C7×C4⋊C8D411yescharacteristic
92C7×#103—21no—
93C7×#103—21no—
94C7×#103—21no—
95C7×C22⋊C8D411yescharacteristic
96C7×C22⋊C8D411yescharacteristic
97C4×C7×C8C2311yescharacteristic
98#6403—22no—
99#6392—21no—
100#6392C2311yescharacteristic
101#6392C2311yescharacteristic
102#6392C2311yescharacteristic
103#6391—21no—
104#6391—21no—
105#6390—21no—
106#6390—21no—
107#6390—21no—
108#6386—22no—
109#6386—22no—
110C2×C7⋊C16C2311yescharacteristic
111C2×C7⋊C16C2311yescharacteristic
112C2×C7⋊C16C2311yescharacteristic
113C2×C7⋊C16C2311yescharacteristic
114#3336—71no—
115#3336D711yescharacteristic
116#2253—71no—
117#2237—71no—
118#2237—71no—
119#2236—71no—
120#2236—71no—
121#2229—71no—
122#2229—71no—
123#2219—71no—
124#2219—71no—
125C4×Q32—71no—
126#2169—71no—
127#2159—71no—
128#2159—71no—
129C2×C7×Q8—21no—
130C2×C7×D4D811yescharacteristic
131C22×C4×C7—41no—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×Dic7—41no—
135C2×Dic14—21no—
136C7×Q16—21no—
137C7×Q16—21no—
138C2×C7×C8—21no—
139C2×C7×C8—21no—
140C2×C7×C8—21no—
141C2×C7×C8—21no—
142C2×C7×C8C2×D411yescharacteristic
143C2×C7×C8C2×D411yescharacteristic
144C2×C7×C8C2×D411yescharacteristic
145C2×C7×C8C2411yescharacteristic
146C7×C4⋊C4—21no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—41no—
150C7×C4⋊C4—41no—
151C7×C4⋊C4—41no—
152C7×C4⋊C4—41no—
153C7×C4⋊C4—41no—
154C7×C4⋊C4—41no—
155C7×C4⋊C4D811yescharacteristic
156C7×C22⋊C4—41no—
157C7×C22⋊C4—41no—
158C7×C22⋊C4—41no—
159C7×C22⋊C4D811yescharacteristic
160C7×C22⋊C4D811yescharacteristic
161C42×C7—41no—
162C42×C7—41no—
163C42×C7C2×D411yescharacteristic
164C22⋊Dic7—41no—
165C22⋊Dic7—41no—
166C22⋊Dic7—41no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—41no—
171Dic7⋊C4—41no—
172Dic7⋊C4—41no—
173Dic7⋊C4—41no—
174Dic7⋊C4—41no—
175Dic7⋊C4—41no—
176C4×Dic7—41no—
177C4×Dic7—41no—
178Dic28—21no—
179Dic28—21no—
180C7⋊C16—21no—
181C7⋊C16—21no—
182C7⋊C16—21no—
183C7⋊C16#4112yes—
184#552—21no—
185#552—141no—
186C2×Q32—71no—
187C2×M32—71no—
188C2×M32—71no—
189#499—71no—
190#499C2×D711yescharacteristic
191#485—21no—
192#485—141no—
193#483—21no—
194#483—141no—
195#479—21no—
196#479—141no—
197#467—71no—
198#467—71no—
199#467C2×D711yescharacteristic
200#467C2×D711yescharacteristic
201C4×Q16—71no—
202C4×Q16C2×D711yescharacteristic
203C8×D4C2×D711yescharacteristic
204#427—71no—
205#427C2×D711yescharacteristic
206C2×#107—71no—
207C2×#107C2×D711yescharacteristic
208#367—71no—
209#367—71no—
210#367—71no—
211#367—71no—
212#366—71no—
213#366—71no—
214#366—71no—
215#366—71no—
216#366—71no—
217C4⋊C16—71no—
218#358—71no—
219#358—71no—
220#358—71no—
221#358—71no—
222#358—71no—
223#358—71no—
224C22⋊C16—71no—
225C22⋊C16—71no—
226C4×C16—71no—
227C23×C7C2×D811yescharacteristic
228C23×C7C2×D811yescharacteristic
229C7×Q8—41no—
230C7×Q8—41no—
231C7×D4—41no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—82no—
241C2×C4×C7C2×D811yescharacteristic
242C2×C4×C7C2×D811yescharacteristic
243C2×C4×C7C2×D811yescharacteristic
244C2×C4×C7C2×D811yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—82no—
251Dic14—41no—
252Dic14—41no—
253C7×C8—21no—
254C7×C8—21no—
255C7×C8—21no—
256C7×C8C2×#4111yescharacteristic
257C7×C82−1+411yescharacteristic
258C2×Q16—71no—
259C2×Q16C22×D711yescharacteristic
260C22×C8C22×D711yescharacteristic
261C22×C8C22×D711yescharacteristic
262#126—41no—
263#126—281no—
264#125—21no—
265#125—41no—
266#125—141no—
267#125—281no—
268#123—41no—
269#123—281no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—141no—
273C22⋊Q8—141no—
274C4×Q8—21no—
275C4×Q8—141no—
276C4×D4C22⋊D711yescharacteristic
277#117—21no—
278#117—41no—
279#117—141no—
280#117—281no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—141no—
283Q32—142no—
284M32—142no—
285M32—142no—
286C2×C16—71no—
287C2×C16—71no—
288C2×C16—71no—
289C2×C16—71no—
290#107—21no—
291#107—71no—
292#107—71no—
293#107—71no—
294#107—141no—
295#107C22×D711yescharacteristic
296#107C22×D711yescharacteristic
297#107C22×D711yescharacteristic
298#106—21no—
299#106—21no—
300#106—141no—
301#106—141no—
302C4⋊C8C22⋊D711yescharacteristic
303#103—21no—
304#103—21no—
305#103—21no—
306#103—141no—
307#103—141no—
308#103—141no—
309C22⋊C8C22⋊D711yescharacteristic
310C22⋊C8C22⋊D711yescharacteristic
311C4×C8C22×D711yescharacteristic
312C22×C7—21no—
313C22×C7—21no—
314C22×C7—21no—
315C22×C7—21no—
316C22×C7—21no—
317C22×C7—21no—
318C22×C7C22×D811yescharacteristic
319C4×C7—21no—
320C4×C7—21no—
321C4×C7—21no—
322C4×C7—21no—
323C4×C7—81no—
324C4×C7—81no—
325C4×C7—81no—
326C4×C7—81no—
327C4×C7#54711yescharacteristic
328C4×C7#57811yescharacteristic
329Dic7—81no—
330Dic7—81no—
331Dic7—81no—
332Dic7—81no—
333C2×Q8—21no—
334C2×Q8—141no—
335C2×D4#114111yescharacteristic
336C22×C4—41no—
337C22×C4—281no—
338C22×C4C2×C22⋊D711yescharacteristic
339C22×C4C2×C22⋊D711yescharacteristic
340Q16—21no—
341Q16—21no—
342Q16—141no—
343Q16—141no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8C2×C22⋊D711yescharacteristic
349C2×C8C2×C22⋊D711yescharacteristic
350C2×C8C2×C22⋊D711yescharacteristic
351C2×C8C23×D711yescharacteristic
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—141no—
362C4⋊C4—141no—
363C4⋊C4—141no—
364C4⋊C4—281no—
365C4⋊C4—281no—
366C4⋊C4—281no—
367C4⋊C4—281no—
368C4⋊C4—281no—
369C4⋊C4—281no—
370C4⋊C4#114111yescharacteristic
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—281no—
375C22⋊C4—281no—
376C22⋊C4—281no—
377C22⋊C4#114111yescharacteristic
378C22⋊C4#114111yescharacteristic
379C42—41no—
380C42—41no—
381C42—281no—
382C42—281no—
383C42C2×C22⋊D711yescharacteristic
384C16—72no—
385C16—141no—
386C16—141no—
387C16—141no—
388C2×C7—41no—
389C2×C7—41no—
390C2×C7#333411yescharacteristic
391C2×C7#341411yescharacteristic
392C2×C7#341711yescharacteristic
393C23C2×#114111yescharacteristic
394C23C2×#114111yescharacteristic
395Q8—41no—
396Q8—41no—
397Q8—281no—
398Q8—281no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—82no—
409C2×C4—281no—
410C2×C4—281no—
411C2×C4—281no—
412C2×C4—281no—
413C2×C4—562no—
414C2×C4C2×#114111yescharacteristic
415C2×C4C2×#114111yescharacteristic
416C2×C4C2×#114111yescharacteristic
417C2×C4C2×#114111yescharacteristic
418C2×C4C22×C22⋊D711yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8D7×#4111yescharacteristic
423C8#655411yescharacteristic
424C7#3391911yescharacteristic, complemented, Sylow
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22C22×#114111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—561no—
441C4—561no—
442C4—561no—
443C4—561no—
444C4#9051511yescharacteristic
445C4#9052911yescharacteristic
446C2—41no—
447C2—41no—
448C2#1174096111yescharacteristic
449C2#1174133911yescharacteristic
450C2#1174146211yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#3336C2
socleC22×C7C22×D8
prime radicalC2×C7×D4D8
hypercenter#3336D7
hypocenterC7#33919
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
3. #11729111C2semidirect
4. #11729108C2semidirect
5. #11728837C2semidirect
6. #11728837C2semidirect
7. #11728834C2semidirect
8. #11728833C2semidirect
9. #11728666C2semidirect
10. C4×#6403C2semidirect
12. #11728409C2semidirect
13. #11728403C2semidirect
14. #11728357C2semidirect
15. #11728350C2semidirect
424. C7#33919semidirect

Maximal subgroups

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

subgroupindex
C7×#33362
#117368582
#117291112
#117291082
#117288372
#117288342
#117288332
#117286662
C4×#64032
#117286122
#117284092
#117284032
#117283572
#117283502
#339197