Finite Group Databasebeta

← #13338965

Subgroups — #13338965

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2972C211yesmaximal, characteristic, complemented
2#11739752C211yesmaximal, characteristic
3C2×#89779C211yesmaximal, characteristic, complemented
4#11736757C211yesmaximal, characteristic, complemented
5#11728608C214yesmaximal, complemented
6#11728600C214yesmaximal, complemented
7#11728534C211yesmaximal, characteristic, complemented
8#11728526C211yesmaximal, characteristic, complemented
9#11728503C211yesmaximal, characteristic, complemented
10C2×C4×C7×D4C2211yescharacteristic
11C2×#6495—21no—
12#90168C2211yescharacteristic
13C2×#6432C2211yescharacteristic
14C7×C8×D4—22no—
15C7×C8×D4C2214yes—
16C7×#431C2211yescharacteristic
17C7×#431C2211yescharacteristic
18C2×C7×C22⋊C8—22no—
19C2×#6406—21no—
20#89791C2212yes—
21#89788—22no—
22#89787C2212yes—
23#89779—22no—
24#89779C2214yes—
25#89738—21no—
26C2×#6381—21no—
27#89696C2211yescharacteristic
28C2×C4⋊(C7⋊C8)C2211yescharacteristic
29#89613—22no—
30#89510—22no—
31#89465C2212yescomplemented
32#89453C2212yescomplemented
33#89281C2212yescomplemented
34#89280C2212yescomplemented
35#89277C2214yescomplemented
36#89252C2214yescomplemented
37#19385—71nomaximal, Sylow
38C22×C7×D4D411yescharacteristic
39C23×C4×C7—21no—
40C2×#1157—21no—
41C22×Dic14—21no—
42C22×C7×C8—24no—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C4×C7×D4D412yes—
46C4×C7×D4C2312yes—
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C7×C22⋊C4—21no—
49C2×C42×C7C2311yescharacteristic
50C2×#1142—22no—
51#6495—44no—
52C2×#1141—22no—
53#6493—21no—
54#6493—21no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×#1136—21no—
58C22×C7⋊C8—21no—
59C2×#1132—24no—
60#6438—21no—
61#6437C2311yescharacteristic
62C4×D28—21no—
63C4×D28—21no—
64#6432—21no—
65#6432—21no—
66#6432—21no—
67#6432C2312yes—
68#6432C2311yescharacteristic
69C7×C4⋊C8D412yes—
70C7×C4⋊C8C2312yes—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—24no—
73C7×C22⋊C8D412yes—
74C4×C7×C8—21no—
75C4×C7×C8C2312yes—
76#6406—24no—
77#6390—24no—
78#6384—22no—
79#6383—22no—
80#6381—24no—
81C4⋊(C7⋊C8)—21no—
82C4⋊(C7⋊C8)—21no—
83C4⋊(C7⋊C8)C2312yes—
84C4⋊(C7⋊C8)C2312yes—
85#3206—71no—
86C2×#474—71no—
87#3036—71no—
88#2972D711yescharacteristic, complemented
89#1679—74no—
90#1665—74no—
91#1570—71no—
92#1549—71no—
93#1540—71no—
94C24×C7—21no—
95C2×C7×D4—21no—
96C2×C7×D4—22no—
97C2×C7×D4SD1614yescomplemented
98C2×C7×D4C2×D412yes—
99C22×C4×C7—21no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—82no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4111yescharacteristic
106C2×C22⋊D7—41no—
107C22×Dic7—41no—
108#1157—42no—
109#1157—42no—
110C2×D28—21no—
111C2×D28—21no—
112C2×D28—21no—
113C2×C4×D7—41no—
114C2×Dic14—22no—
115C2×Dic14—21no—
116C2×Dic14—41no—
117C2×Dic14—82no—
118C2×C7×C8—22no—
119C2×C7×C8—84no—
120C2×C7×C8C2×D414yes—
121C7×C4⋊C4—21no—
122C7×C4⋊C4C2×D412yes—
123C7×C22⋊C4—22no—
124C7×C22⋊C4—22no—
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128C42×C7C2411yescharacteristic
129#1142—44no—
130#1141—44no—
131(C2×D7)⋊C4—41no—
132(C2×D7)⋊C4—41no—
133C4⋊Dic7—22no—
134C4⋊Dic7—21no—
135C4⋊Dic7—22no—
136C4⋊Dic7—21no—
137C4⋊Dic7—41no—
138C4⋊Dic7—41no—
139#1136—42no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—22no—
142C2×C7⋊C8—42no—
143#1132—84no—
144C2×AGL(1,ℤ/8)—141no—
145#532—71no—
146C2×C4⋊Q8—71no—
147C2×C4×D4C2×D711yescharacteristic
148#492—72no—
149#487—72no—
150#474—74no—
151#474—142no—
152#463—142no—
153#461—72no—
154#461—142no—
155#459—72no—
156#448—142no—
157C8×D4—22no—
158C8×D4C2×D714yes—
159#431C2×D711yescharacteristic
160#431C2×D711yescharacteristic
161C2×#107—141no—
162#423—71no—
163C2×C4⋊C8—71no—
164#417—141no—
165C2×#102—141no—
166C2×C22⋊C8—22no—
167#334—74no—
168#332—72no—
169#332—74no—
170#331—72no—
171C23×C7—22no—
172C23×C7—24no—
173C23×C7—81no—
174C23×C7—81no—
175C23×C7#12111yescharacteristic
176C22×D7—41no—
177C7×D4—22no—
178C7×D4—24no—
179C7×D4C2×SD1614yes—
180C2×C4×C7—22no—
181C2×C4×C7—22no—
182C2×C4×C7—41no—
183C2×C4×C7—44no—
184C2×C4×C7—82no—
185C2×C4×C7—81no—
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7C2×SD1612yes—
190C2×C4×C7#13512yes—
191C2×C4×C7C22×D411yescharacteristic
192C2×C4×C7C22×D411yescharacteristic
193C2×C4×C7C2×#4111yescharacteristic
194C22⋊D7—82no—
195C2×Dic7—42no—
196C2×Dic7—41no—
197C2×Dic7—82no—
198D28—41no—
199D28—41no—
200D28—42no—
201D28—81no—
202C4×D7—82no—
203Dic14—41no—
204Dic14—41no—
205Dic14—44no—
206Dic14—82no—
207C7×C8—24no—
208C7×C8—42no—
209C7⋊C8—42no—
210C7⋊C8—42no—
211C2×#41—141no—
212C22×Q8—141no—
213C22×D4D2811yescharacteristic
214C23×C4—21no—
215AGL(1,ℤ/8)—284no—
216C2×SD16—142no—
217C2×SD16—144no—
218C2×D8—142no—
219C2×M16—141no—
220C22×C8—24no—
221C22×C8—141no—
222C4⋊Q8—72no—
223C4⋊Q8—71no—
224C4⋊Q8—141no—
225C4⋊Q8—141no—
226C4⋊Q8—141no—
227#127—71no—
228#124—141no—
229#121—141no—
230#121—141no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—141no—
234C4×D4—141no—
235C4×D4D2812yes—
236C4×D4C22×D712yes—
237C2×C4⋊C4—141no—
238C2×C4⋊C4—141no—
239C2×C4⋊C4D2811yescharacteristic
240C2×C22⋊C4—21no—
241C2×C42C22×D711yescharacteristic
242#107—144no—
243C4⋊C8—72no—
244C4⋊C8—72no—
245C4⋊C8—141no—
246C4⋊C8—141no—
247C4⋊C8C22⋊D712yes—
248C4⋊C8C22×D712yes—
249#103—144no—
250#103—142no—
251#102—144no—
252#102—142no—
253C22⋊C8—21no—
254C22⋊C8—24no—
255C22⋊C8C22⋊D712yes—
256C4×C8—21no—
257C4×C8C22×D712yes—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—24no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7—84no—
264C22×C7#46512yes—
265C22×C7C2×#12111yescharacteristic
266C2×D7—81no—
267C2×D7—81no—
268C2×D7—81no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—42no—
273C4×C7#46512yes—
274C4×C7C22×SD1611yescharacteristic
275C4×C7C2×#13511yescharacteristic
276C4×C7#57612yes—
277Dic7—82no—
278Dic7—81no—
279C24—21no—
280#41—282no—
281#41—282no—
282C2×Q8—141no—
283C2×Q8—142no—
284C2×Q8—281no—
285C2×Q8—562no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—141no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—281no—
292C2×D4#113214yescomplemented
293C2×D4C2×D2812yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—82no—
298C22×C4—281no—
299C22×C4—281no—
300C22×C4C2×D2811yescharacteristic
301C22×C4#115711yescharacteristic
302C22×C4C2×C22⋊D711yescharacteristic
303SD16—284no—
304SD16—564no—
305D8—284no—
306M16—282no—
307C2×C8—22no—
308C2×C8—84no—
309C2×C8—142no—
310C2×C8—142no—
311C2×C8—282no—
312C2×C8C2×C22⋊D714yes—
313C4⋊C4—21no—
314C4⋊C4—142no—
315C4⋊C4—142no—
316C4⋊C4—141no—
317C4⋊C4—141no—
318C4⋊C4—281no—
319C4⋊C4—281no—
320C4⋊C4C2×D2812yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—281no—
324C22⋊C4—281no—
325C42C2×D2811yescharacteristic
326C42#115711yescharacteristic
327C42C2×C22⋊D711yescharacteristic
328C42C23×D711yescharacteristic
329C2×C7—22no—
330C2×C7—44no—
331C2×C7#304111yescharacteristic
332C2×C7C2×#46511yescharacteristic
333C2×C7#306911yescharacteristic
334D7—161no—
335C23—22no—
336C23—24no—
337C23—81no—
338C23—81no—
339C23—281no—
340C23#649311yescharacteristic
341Q8—281no—
342Q8—281no—
343Q8—284no—
344Q8—562no—
345D4—24no—
346D4—22no—
347D4—281no—
348D4—281no—
349D4—282no—
350D4—561no—
351D4—562no—
352D4C2×#113214yes—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—281no—
360C2×C4—282no—
361C2×C4—562no—
362C2×C4—562no—
363C2×C4C2×#113212yes—
364C2×C4#646712yes—
365C2×C4#649311yescharacteristic
366C2×C4#649311yescharacteristic
367C2×C4#649311yescharacteristic
368C2×C4C22×D2811yescharacteristic
369C2×C4C2×#115711yescharacteristic
370C2×C4C22×C22⋊D711yescharacteristic
371C8—24no—
372C8—42no—
373C8—282no—
374C8—282no—
375C7#1938511yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—561no—
383C22—561no—
384C22—561no—
385C22#8988612yes—
386C22C2×#649311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—561no—
392C4—562no—
393C4#8988612yes—
394C4C22×#113211yescharacteristic
395C4C2×#646711yescharacteristic
396C4#9052612yes—
397C2—22no—
398C2—44no—
399C2—1121no—
400C2#1173998511yescharacteristic
401C2C2×#8988611yescharacteristic
402C2#1174126111yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2972C2
socleC22×C7C2×#121
prime radical#11739752C2
hypercenter#2972D7
hypocenterC7#19385
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. C7×#2972C2semidirect
3. C2×#89779C2semidirect
4. #11736757C2semidirect
5. #11728608C2semidirect
6. #11728600C2semidirect
7. #11728534C2semidirect
8. #11728526C2semidirect
9. #11728503C2semidirect
31. #89465C22semidirect
32. #89453C22semidirect
33. #89281C22semidirect
34. #89280C22semidirect
35. #89277C22semidirect
36. #89252C22semidirect
88. #2972D7semidirect
97. C2×C7×D4SD16semidirect
292. C2×D4#1132semidirect
375. C7#19385semidirect

Maximal subgroups

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

subgroupindex
C7×#29722
#117397522
C2×#897792
#117367572
#117286082
#117286002
#117285342
#117285262
#117285032
#193857