Finite Group Databasebeta

← #13339023

Subgroups — #13339023

Order 1792.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2972C211yesmaximal, characteristic, complemented
2#11739752C211yesmaximal, characteristic
3C2×#89780C211yesmaximal, characteristic, complemented
4#11736714C211yesmaximal, characteristic, complemented
5#11728605C214yesmaximal, complemented
6#11728602C214yesmaximal, complemented
7#11728533C211yesmaximal, 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#89780—22no—
24#89780C2214yes—
25#89738—21no—
26C2×#6382—21no—
27#89696C2211yescharacteristic
28C2×C4⋊(C7⋊C8)C2211yescharacteristic
29#89614—22no—
30#89507—22no—
31#89466C2212yescomplemented
32#89452C2212yescomplemented
33#89281C2212yescomplemented
34#89280C2212yescomplemented
35#89276C2214yescomplemented
36#89252C2214yescomplemented
37#19391—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—
59#6438—21no—
60#6437C2311yescharacteristic
61C4×D28—21no—
62C4×D28—21no—
63#6432—21no—
64#6432—21no—
65#6432—21no—
66#6432C2312yes—
67#6432C2311yescharacteristic
68C7×C4⋊C8D412yes—
69C7×C4⋊C8C2312yes—
70C7×C22⋊C8—21no—
71C7×C22⋊C8—24no—
72C7×C22⋊C8D412yes—
73C4×C7×C8—21no—
74C4×C7×C8C2312yes—
75#6406—24no—
76#6395—24no—
77#6391—24no—
78#6384—22no—
79#6383—22no—
80#6382—24no—
81C4⋊(C7⋊C8)—21no—
82C4⋊(C7⋊C8)—21no—
83C4⋊(C7⋊C8)C2312yes—
84C4⋊(C7⋊C8)C2312yes—
85#3227—71no—
86C2×#476—71no—
87#3036—71no—
88#2972D711yescharacteristic, complemented
89#1677—74no—
90#1671—74no—
91#1570—71no—
92#1548—71no—
93#1540—71no—
94C24×C7—21no—
95C2×C7×D4—21no—
96C2×C7×D4—22no—
97C2×C7×D4SD1614yes—
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—21no—
115C2×Dic14—22no—
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—22no—
135C4⋊Dic7—21no—
136C4⋊Dic7—21no—
137C4⋊Dic7—41no—
138C4⋊Dic7—41no—
139#1136—42no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—22no—
142C2×C7⋊C8—42no—
143C2×AGL(1,ℤ/8)—141no—
144#532—71no—
145C2×C4⋊Q8—71no—
146C2×C4×D4C2×D711yescharacteristic
147#498—72no—
148#486—72no—
149#481—142no—
150#476—74no—
151#476—142no—
152#476—142no—
153#463—142no—
154#461—72no—
155#459—72no—
156C8×D4—22no—
157C8×D4C2×D714yes—
158#431C2×D711yescharacteristic
159#431C2×D711yescharacteristic
160C2×#106—141no—
161#423—71no—
162C2×C4⋊C8—71no—
163#417—141no—
164C2×#102—141no—
165C2×C22⋊C8—22no—
166#334—74no—
167#333—74no—
168#332—72no—
169#331—72no—
170C23×C7—22no—
171C23×C7—24no—
172C23×C7—81no—
173C23×C7—81no—
174C23×C7#12111yescharacteristic
175C22×D7—41no—
176C7×D4—22no—
177C7×D4—24no—
178C7×D4C2×SD1614yes—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—44no—
183C2×C4×C7—82no—
184C2×C4×C7—81no—
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7C2×SD1612yes—
189C2×C4×C7#13512yes—
190C2×C4×C7C22×D411yescharacteristic
191C2×C4×C7C22×D411yescharacteristic
192C2×C4×C7C2×#4111yescharacteristic
193C22⋊D7—82no—
194C2×Dic7—42no—
195C2×Dic7—41no—
196C2×Dic7—82no—
197D28—41no—
198D28—41no—
199D28—42no—
200D28—81no—
201C4×D7—82no—
202Dic14—41no—
203Dic14—41no—
204Dic14—82no—
205Dic14—82no—
206C7×C8—24no—
207C7×C8—42no—
208C7⋊C8—42no—
209C7⋊C8—42no—
210C2×#41—141no—
211C22×Q8—141no—
212C22×D4D2811yescharacteristic
213C23×C4—21no—
214AGL(1,ℤ/8)—284no—
215C2×SD16—142no—
216C2×D8—142no—
217C2×M16—141no—
218C22×C8—24no—
219C22×C8—141no—
220C4⋊Q8—72no—
221C4⋊Q8—71no—
222C4⋊Q8—141no—
223C4⋊Q8—141no—
224C4⋊Q8—141no—
225#127—71no—
226#124—141no—
227#121—141no—
228#121—141no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—141no—
232C4×D4—141no—
233C4×D4D2812yes—
234C4×D4C22×D712yes—
235C2×C4⋊C4—141no—
236C2×C4⋊C4—141no—
237C2×C4⋊C4D2811yescharacteristic
238C2×C22⋊C4—21no—
239C2×C42C22×D711yescharacteristic
240#106—144no—
241#106—144no—
242C4⋊C8—72no—
243C4⋊C8—72no—
244C4⋊C8—141no—
245C4⋊C8—141no—
246C4⋊C8C22⋊D712yes—
247C4⋊C8C22×D712yes—
248#103—142no—
249#102—144no—
250#102—144no—
251#102—142no—
252C22⋊C8—21no—
253C22⋊C8—24no—
254C22⋊C8C22⋊D712yes—
255C4×C8—21no—
256C4×C8C22×D712yes—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—24no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—84no—
263C22×C7#46512yes—
264C22×C7C2×#12111yescharacteristic
265C2×D7—81no—
266C2×D7—81no—
267C2×D7—81no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—42no—
272C4×C7#46512yes—
273C4×C7C22×SD1611yescharacteristic
274C4×C7C2×#13511yescharacteristic
275C4×C7#57712yes—
276Dic7—81no—
277Dic7—82no—
278C24—21no—
279#41—282no—
280#41—282no—
281C2×Q8—141no—
282C2×Q8—142no—
283C2×Q8—281no—
284C2×Q8—562no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—141no—
288C2×D4—141no—
289C2×D4—141no—
290C2×D4—281no—
291C2×D4#113214yes—
292C2×D4C2×D2812yes—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—82no—
297C22×C4—281no—
298C22×C4—281no—
299C22×C4C2×D2811yescharacteristic
300C22×C4#115711yescharacteristic
301C22×C4C2×C22⋊D711yescharacteristic
302SD16—284no—
303D8—284no—
304M16—282no—
305C2×C8—22no—
306C2×C8—84no—
307C2×C8—142no—
308C2×C8—142no—
309C2×C8—282no—
310C2×C8C2×C22⋊D714yes—
311C4⋊C4—21no—
312C4⋊C4—142no—
313C4⋊C4—141no—
314C4⋊C4—142no—
315C4⋊C4—141no—
316C4⋊C4—281no—
317C4⋊C4—281no—
318C4⋊C4C2×D2812yes—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—281no—
322C22⋊C4—281no—
323C42C2×D2811yescharacteristic
324C42#115711yescharacteristic
325C42C2×C22⋊D711yescharacteristic
326C42C23×D711yescharacteristic
327C2×C7—22no—
328C2×C7—44no—
329C2×C7#304211yescharacteristic
330C2×C7C2×#46511yescharacteristic
331C2×C7#306811yescharacteristic
332D7—161no—
333C23—22no—
334C23—24no—
335C23—81no—
336C23—81no—
337C23—281no—
338C23#649311yescharacteristic
339Q8—281no—
340Q8—281no—
341Q8—562no—
342Q8—562no—
343D4—24no—
344D4—22no—
345D4—281no—
346D4—281no—
347D4—282no—
348D4—561no—
349D4—562no—
350D4C2×#113214yes—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—281no—
358C2×C4—282no—
359C2×C4—562no—
360C2×C4—562no—
361C2×C4C2×#113212yes—
362C2×C4#646712yes—
363C2×C4#649311yescharacteristic
364C2×C4#649311yescharacteristic
365C2×C4#649311yescharacteristic
366C2×C4C22×D2811yescharacteristic
367C2×C4C2×#115711yescharacteristic
368C2×C4C22×C22⋊D711yescharacteristic
369C8—24no—
370C8—42no—
371C8—282no—
372C8—282no—
373C7#1939111yescharacteristic, complemented, Sylow
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—561no—
381C22—561no—
382C22—561no—
383C22#8988612yes—
384C22C2×#649311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—561no—
390C4—562no—
391C4#8988612yes—
392C4C22×#113211yescharacteristic
393C4C2×#646711yescharacteristic
394C4#9052712yes—
395C2—22no—
396C2—44no—
397C2—1121no—
398C2#1173998911yescharacteristic
399C2C2×#8988611yescharacteristic
400C2#1174125911yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2972C2
socleC22×C7C2×#121
prime radical#11739752C2
hypercenter#2972D7
hypocenterC7#19391
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×#89780C2semidirect
4. #11736714C2semidirect
5. #11728605C2semidirect
6. #11728602C2semidirect
7. #11728533C2semidirect
8. #11728526C2semidirect
9. #11728503C2semidirect
31. #89466C22semidirect
32. #89452C22semidirect
33. #89281C22semidirect
34. #89280C22semidirect
35. #89276C22semidirect
36. #89252C22semidirect
88. #2972D7semidirect
373. C7#19391semidirect

Maximal subgroups

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

subgroupindex
C7×#29722
#117397522
C2×#897802
#117367142
#117286052
#117286022
#117285332
#117285262
#117285032
#193917