Finite Group Databasebeta

← #13508591

Subgroups — #13508591

Order 1792.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3050C211yesmaximal, characteristic
2C2×#89878C211yesmaximal, characteristic, complemented
3#11729831C212yesmaximal, complemented
4C2×C22⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
5#11728834C212yesmaximal, complemented
6#11728833C211yesmaximal, characteristic, complemented
7#11728832C211yesmaximal, characteristic, complemented
8#11728824C212yesmaximal, complemented
9#11728823C211yesmaximal, characteristic, complemented
10#11728822C211yesmaximal, characteristic, complemented
11#11728402C212yesmaximal, complemented
12C2×C7×#135C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C7×#525—21no—
15C2×#6489—21no—
16C7×#467C2211yescharacteristic
17C7×#467C2211yescharacteristic
18C7×#466C2211yescharacteristic
19C7×#466C2211yescharacteristic
20C7×#465—21no—
21C7×#465—21no—
22C7×#427C2211yescharacteristic
23C7×#416—21no—
24#89878—21no—
25#89878—21no—
26#89878C2212yes—
27#89878C2212yes—
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391—21no—
31C22×C7⋊C16C2212yes—
32#89359C2212yescomplemented
33#89359C2212yescomplemented
34#89357C2212yescomplemented
35#89357C2212yescomplemented
36C22⋊(C7⋊C16)C2211yescharacteristic
37C22⋊(C7⋊C16)C2211yescharacteristic
38C22⋊(C7⋊C16)C2212yes—
39#89285C2212yescomplemented
40#89285C2212yescomplemented
41#89284C2212yescomplemented
42#89284C2212yescomplemented
43#33645—71nomaximal, Sylow
44C2×C7×#41—21no—
45C23×C4×C7D411yescharacteristic
46C7×#135—24no—
47C2×C7×Q16C2311yescharacteristic
48C2×C7×SD16—21no—
49C2×C7×D8C2311yescharacteristic
50C22×C7×C8—24no—
51C22×C7×C8D411yescharacteristic
52C22×C7×C8D411yescharacteristic
53C22×C7×C8D411yescharacteristic
54C22×C7×C8C2311yescharacteristic
55C22×C7×C8C2311yescharacteristic
56C22×C7×C8C2311yescharacteristic
57C7×#123—41no—
58C7×C22⋊Q8—21no—
59C7×C22⋊Q8—21no—
60C7×#121—21no—
61C7×#121—21no—
62C7×#120—41no—
63C4×C7×D4—41no—
64C4×C7×D4—41no—
65C7×#117—21no—
66C2×C22⋊Dic7—41no—
67#6489—22no—
68#6489—22no—
69#6489—41no—
70#6489—41no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—21no—
73C2×C4×Dic7—41no—
74C7×#107C2311yescharacteristic
75C7×#107C2311yescharacteristic
76C7×#106—21no—
77C7×#103—21no—
78C7×#103—21no—
79C7×#102—21no—
80C7×#102—21no—
81#6392—21no—
82#6392—21no—
83#6392C2312yes—
84#6392C2312yes—
85#6391—22no—
86#6391—22no—
87C2×C7⋊C16—24no—
88C2×C7⋊C16C2312yes—
89C2×C7⋊C16C2312yes—
90#3050D711yescharacteristic
91C2×#427—71no—
92#2237—72no—
93#2236—71no—
94#2235—71no—
95#2234—72no—
96#2233—71no—
97#2232—71no—
98#2158—72no—
99#2138—72no—
100C2×C22⋊C16—71no—
101C24×C7D811yescharacteristic
102C7×#41—42no—
103C7×#41—42no—
104C2×C7×Q8—21no—
105C2×C7×D4—21no—
106C2×C7×D4—41no—
107C2×C7×D4—41no—
108C2×C7×D4—41no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—41no—
112C22×C4×C7D811yescharacteristic
113C22×C4×C7D811yescharacteristic
114C22×C4×C7D811yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×Dic7—41no—
119C22×Dic7—41no—
120C7×Q16—22no—
121C7×SD16—42no—
122C7×D8—22no—
123C2×C7×C8—22no—
124C2×C7×C8—22no—
125C2×C7×C8—22no—
126C2×C7×C8—22no—
127C2×C7×C8—22no—
128C2×C7×C8C2×D411yescharacteristic
129C2×C7×C8C2×D411yescharacteristic
130C2×C7×C8C2×D411yescharacteristic
131C2×C7×C8#4114yes—
132C2×C7×C8C2411yescharacteristic
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—41no—
136C7×C4⋊C4—41no—
137C7×C22⋊C4—41no—
138C7×C22⋊C4—41no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C42×C7—41no—
143C22⋊Dic7—42no—
144C22⋊Dic7—42no—
145C4⋊Dic7—22no—
146C4⋊Dic7—22no—
147C4⋊Dic7—41no—
148C4⋊Dic7—41no—
149C4×Dic7—42no—
150C4×Dic7—42no—
151C7⋊C16—22no—
152C7⋊C16—22no—
153C2×#135C2×D711yescharacteristic
154C23×C8C2×D711yescharacteristic
155#525—21no—
156C2×#117—141no—
157C22×C16—72no—
158#467C2×D711yescharacteristic
159#467C2×D711yescharacteristic
160#466C2×D711yescharacteristic
161#466C2×D711yescharacteristic
162#465—21no—
163#465—21no—
164#427—72no—
165#427—72no—
166#427—141no—
167#427—141no—
168#427C2×D711yescharacteristic
169C2×#107—71no—
170C2×#107—71no—
171C2×#106—141no—
172#416—21no—
173#367—72no—
174#367—72no—
175#366—72no—
176#366—72no—
177#358—72no—
178#358—72no—
179#357—72no—
180#357—72no—
181C22⋊C16—71no—
182C22⋊C16—71no—
183C22⋊C16—72no—
184C23×C7—22no—
185C23×C7—22no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7C2×D811yescharacteristic
189C23×C7C2×D811yescharacteristic
190C23×C7C2×D811yescharacteristic
191C7×Q8—41no—
192C7×Q8—41no—
193C7×D4—41no—
194C7×D4—41no—
195C7×D4—82no—
196C7×D4—82no—
197C7×D4—82no—
198C2×C4×C7—22no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—82no—
209C2×C4×C7#12312yes—
210C2×C4×C7C2×D811yescharacteristic
211C2×C4×C7C2×D811yescharacteristic
212C2×C4×C7C2×D811yescharacteristic
213C2×C4×C7#13712yes—
214C2×C4×C7C22×D411yescharacteristic
215C2×Dic7—42no—
216C2×Dic7—42no—
217C2×Dic7—82no—
218C2×Dic7—82no—
219C7×C8—21no—
220C7×C8—21no—
221C7×C8#12312yes—
222C7×C8C2×#4112yes—
223C2×#41—21no—
224C23×C4C22⋊D711yescharacteristic
225#135—24no—
226C2×Q16C22×D711yescharacteristic
227C2×SD16—21no—
228C2×D8C22×D711yescharacteristic
229C22×C8—24no—
230C22×C8C22⋊D711yescharacteristic
231C22×C8C22⋊D711yescharacteristic
232C22×C8C22⋊D711yescharacteristic
233C22×C8C22×D711yescharacteristic
234C22×C8C22×D711yescharacteristic
235C22×C8C22×D711yescharacteristic
236#123—41no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239#121—21no—
240#121—21no—
241#120—41no—
242C4×D4—41no—
243C4×D4—41no—
244#117—21no—
245#117—142no—
246#117—142no—
247#117—281no—
248#117—281no—
249C2×C4⋊C4—141no—
250C2×C4⋊C4—141no—
251C2×C22⋊C4—281no—
252C2×C42—281no—
253C2×C16—72no—
254C2×C16—72no—
255C2×C16—144no—
256#107—72no—
257#107—72no—
258#107—141no—
259#107—141no—
260#107C22×D711yescharacteristic
261#107C22×D711yescharacteristic
262#106—21no—
263#106—142no—
264#106—142no—
265#103—21no—
266#103—21no—
267#102—21no—
268#102—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—44no—
277C22×C7—81no—
278C22×C7—81no—
279C22×C7—81no—
280C22×C7#48012yes—
281C22×C7#50812yes—
282C22×C7C22×D811yescharacteristic
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—81no—
286C4×C7—81no—
287C4×C7—81no—
288C4×C7#48012yes—
289C4×C7C2×#12311yescharacteristic
290C4×C7C2×#13711yescharacteristic
291Dic7—82no—
292Dic7—82no—
293C24#114111yescharacteristic
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—281no—
305C22×C4—281no—
306C22×C4#114111yescharacteristic
307C22×C4#114111yescharacteristic
308C22×C4#114111yescharacteristic
309C22×C4C2×C22⋊D711yescharacteristic
310C22×C4C2×C22⋊D711yescharacteristic
311C22×C4C2×C22⋊D711yescharacteristic
312Q16—22no—
313SD16—42no—
314D8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8#115914yes—
321C2×C8C2×C22⋊D711yescharacteristic
322C2×C8C2×C22⋊D711yescharacteristic
323C2×C8C2×C22⋊D711yescharacteristic
324C2×C8C23×D711yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—142no—
330C4⋊C4—142no—
331C4⋊C4—281no—
332C4⋊C4—281no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—282no—
339C22⋊C4—282no—
340C42—41no—
341C42—282no—
342C42—282no—
343C16—142no—
344C16—142no—
345C2×C7—22no—
346C2×C7—22no—
347C2×C7—41no—
348C2×C7—161no—
349C2×C7C2×#48011yescharacteristic
350C2×C7C2×#50811yescharacteristic
351C2×C7#341411yescharacteristic
352C23—22no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23C2×#114111yescharacteristic
357C23C2×#114111yescharacteristic
358C23C2×#114111yescharacteristic
359Q8—41no—
360Q8—41no—
361D4—41no—
362D4—41no—
363D4—82no—
364D4—82no—
365D4—82no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—282no—
378C2×C4—282no—
379C2×C4—562no—
380C2×C4—562no—
381C2×C4C2×#114111yescharacteristic
382C2×C4C2×#114111yescharacteristic
383C2×C4C2×#114111yescharacteristic
384C2×C4#649812yes—
385C2×C4#651412yes—
386C2×C4C22×C22⋊D711yescharacteristic
387C8—21no—
388C8—21no—
389C8#649812yes—
390C8C2×#115912yes—
391C7#3364511yescharacteristic, complemented, Sylow
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#8980512yes—
404C22#8996612yes—
405C22C22×#114111yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—562no—
412C4—562no—
413C4#8980512yes—
414C4C2×#649811yescharacteristic
415C4C2×#651411yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—161no—
420C2C2×#8980511yescharacteristic
421C2C2×#8996611yescharacteristic
422C2#1174133911yescharacteristic
subgroupG/H
centerC2×C4C2×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
Fitting, prime radicalC7×#3050C2
socleC22×C7C22×D8
hypercenter#3050D7
hypocenterC7#33645
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
2. C2×#89878C2semidirect
3. #11729831C2semidirect
4. C2×C22⋊(C7⋊C16)C2semidirect
5. #11728834C2semidirect
6. #11728833C2semidirect
7. #11728832C2semidirect
8. #11728824C2semidirect
9. #11728823C2semidirect
10. #11728822C2semidirect
11. #11728402C2semidirect
32. #89359C22semidirect
33. #89359C22semidirect
34. #89357C22semidirect
35. #89357C22semidirect
39. #89285C22semidirect
40. #89285C22semidirect
41. #89284C22semidirect
42. #89284C22semidirect
391. C7#33645semidirect

Maximal subgroups

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

subgroupindex
C7×#30502
C2×#898782
#117298312
C2×C22⋊(C7⋊C16)2
#117288342
#117288332
#117288322
#117288242
#117288232
#117288222
#117284022
#336457