Finite Group Databasebeta

← #13339139

Subgroups — #13339139

Order 1792.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2982C211yesmaximal, characteristic, complemented
2#11739886C211yesmaximal, characteristic
3C2×#89793C211yesmaximal, characteristic, complemented
4#11736714C211yesmaximal, characteristic
5#11728697C214yesmaximal, complemented
6#11728694C214yesmaximal, complemented
7#11728533C211yesmaximal, characteristic, complemented
8#11728526C211yesmaximal, characteristic
9#11728503C211yesmaximal, characteristic, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11C2×#6505—21no—
12#90168C2211yescharacteristic
13C2×#6432C2211yescharacteristic
14C7×C8×Q8—22no—
15C7×C8×Q8C2214yes—
16C7×#431—22no—
17C7×#431C2211yescharacteristic
18C7×#431C2211yescharacteristic
19C2×#6409—21no—
20#89803C2212yes—
21#89800—22no—
22#89799C2212yes—
23#89793—22no—
24#89793C2214yes—
25#89738—21no—
26C2×#6381—21no—
27#89696C2211yescharacteristic
28C2×C4⋊(C7⋊C8)C2211yescharacteristic
29#89614—22no—
30#89507—22no—
31#89466C2212yes—
32#89452C2212yes—
33#89281C2212yes—
34#89280C2212yes—
35#89276C2214yes—
36#89252C2214yes—
37#19403—71nomaximal, Sylow
38C22×C7×Q8D411yescharacteristic
39C2×#1157—21no—
40C22×Dic14—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8D412yes—
44C4×C7×Q8C2312yes—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C42×C7—21no—
48C2×C42×C7C2311yescharacteristic
49C2×#1144—22no—
50#6505—44no—
51C2×#1143—22no—
52#6493—21no—
53#6493—21no—
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7—21no—
56C2×#1136—21no—
57C22×C7⋊C8—21no—
58#6438—21no—
59#6437C2311yescharacteristic
60C4×D28—21no—
61C4×D28—21no—
62#6432—21no—
63#6432—21no—
64#6432—21no—
65#6432C2312yes—
66#6432C2311yescharacteristic
67C7×C4⋊C8—24no—
68C7×C4⋊C8D412yes—
69C7×C4⋊C8C2312yes—
70C7×C22⋊C8—21no—
71C7×C22⋊C8D412yes—
72C4×C7×C8—24no—
73C4×C7×C8—21no—
74C4×C7×C8C2312yes—
75#6409—24no—
76#6395—24no—
77#6391—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#3227—71no—
86C2×#477—71no—
87#3037—71no—
88#2982D711yescharacteristic, complemented
89#1678—74no—
90#1672—74no—
91#1570—71no—
92#1548—71no—
93#1540—71no—
94C2×C7×Q8—21no—
95C2×C7×Q8—22no—
96C2×C7×Q8SD1614yes—
97C2×C7×Q8C2×D412yes—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7#4111yescharacteristic
103C2×C22⋊D7—41no—
104C22×Dic7—41no—
105#1157—42no—
106#1157—42no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×C4×D7—41no—
111C2×Dic14—21no—
112C2×Dic14—22no—
113C2×Dic14—41no—
114C2×Dic14—82no—
115C2×C7×C8—22no—
116C2×C7×C8C2×D414yes—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4C2×D412yes—
121C42×C7—22no—
122C42×C7—22no—
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7#4111yescharacteristic
126C42×C7C2411yescharacteristic
127#1144—44no—
128#1143—44no—
129(C2×D7)⋊C4—41no—
130(C2×D7)⋊C4—41no—
131C4⋊Dic7—22no—
132C4⋊Dic7—22no—
133C4⋊Dic7—21no—
134C4⋊Dic7—21no—
135C4⋊Dic7—41no—
136C4⋊Dic7—41no—
137#1136—42no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—42no—
141C2×#137—141no—
142#532—71no—
143C2×C4⋊Q8—71no—
144C2×C4×Q8C2×D711yescharacteristic
145#498—72no—
146#486—72no—
147#481—142no—
148#477—74no—
149#477—142no—
150#476—142no—
151#464—142no—
152#462—72no—
153#460—72no—
154C8×Q8—22no—
155C8×Q8C2×D714yes—
156#431—22no—
157#431C2×D711yescharacteristic
158#431C2×D711yescharacteristic
159C2×#107—141no—
160#423—71no—
161C2×C4⋊C8—71no—
162#417—141no—
163C2×#103—141no—
164#334—74no—
165#333—74no—
166#332—72no—
167#331—72no—
168C23×C7#12111yescharacteristic
169C22×D7—41no—
170C7×Q8—22no—
171C7×Q8—24no—
172C7×Q8C2×SD1614yes—
173C2×C4×C7—22no—
174C2×C4×C7—24no—
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—41no—
178C2×C4×C7—81no—
179C2×C4×C7#12111yescharacteristic
180C2×C4×C7#12111yescharacteristic
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7C2×SD1612yes—
183C2×C4×C7#13512yes—
184C2×C4×C7C22×D411yescharacteristic
185C2×C4×C7C22×D411yescharacteristic
186C2×C4×C7C2×#4111yescharacteristic
187C22⋊D7—82no—
188C2×Dic7—42no—
189C2×Dic7—41no—
190C2×Dic7—82no—
191D28—41no—
192D28—41no—
193D28—42no—
194D28—81no—
195C4×D7—82no—
196Dic14—41no—
197Dic14—41no—
198Dic14—82no—
199Dic14—82no—
200C7×C8—24no—
201C7×C8—42no—
202C7⋊C8—42no—
203C7⋊C8—42no—
204C2×#41—141no—
205C22×Q8—141no—
206C22×Q8D2811yescharacteristic
207#137—284no—
208C2×Q16—142no—
209C2×SD16—142no—
210C2×M16—141no—
211C22×C8—141no—
212C4⋊Q8—72no—
213C4⋊Q8—71no—
214C4⋊Q8—141no—
215C4⋊Q8—141no—
216C4⋊Q8—141no—
217#127—71no—
218#124—141no—
219#121—141no—
220#121—141no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8D2812yes—
224C4×Q8C22×D712yes—
225C4×D4—141no—
226C4×D4—141no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—141no—
229C2×C4⋊C4—141no—
230C2×C4⋊C4D2811yescharacteristic
231C2×C42—21no—
232C2×C42C22×D711yescharacteristic
233#107—144no—
234#106—144no—
235C4⋊C8—24no—
236C4⋊C8—72no—
237C4⋊C8—72no—
238C4⋊C8—141no—
239C4⋊C8—141no—
240C4⋊C8C22⋊D712yes—
241C4⋊C8C22×D712yes—
242#103—144no—
243#103—142no—
244#102—144no—
245#102—142no—
246C22⋊C8—21no—
247C22⋊C8C22⋊D712yes—
248C4×C8—24no—
249C4×C8—21no—
250C4×C8C22×D712yes—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7#46512yes—
254C22×C7C2×#12111yescharacteristic
255C2×D7—81no—
256C2×D7—81no—
257C2×D7—81no—
258C4×C7—21no—
259C4×C7—24no—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—42no—
263C4×C7#46512yes—
264C4×C7C22×SD1611yescharacteristic
265C4×C7C2×#13511yescharacteristic
266C4×C7#57812yes—
267Dic7—81no—
268Dic7—82no—
269#41—282no—
270#41—282no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—141no—
274C2×Q8—142no—
275C2×Q8—281no—
276C2×Q8—562no—
277C2×Q8#113214yes—
278C2×Q8C2×D2812yes—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—141no—
282C2×D4—281no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—281no—
286C22×C4—281no—
287C22×C4C2×D2811yescharacteristic
288C22×C4#115711yescharacteristic
289C22×C4C2×C22⋊D711yescharacteristic
290Q16—284no—
291SD16—284no—
292M16—282no—
293C2×C8—22no—
294C2×C8—142no—
295C2×C8—142no—
296C2×C8—282no—
297C2×C8C2×C22⋊D714yes—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—142no—
302C4⋊C4—141no—
303C4⋊C4—142no—
304C4⋊C4—141no—
305C4⋊C4—281no—
306C4⋊C4—281no—
307C4⋊C4C2×D2812yes—
308C22⋊C4—281no—
309C22⋊C4—281no—
310C42—22no—
311C42—22no—
312C42C2×D2811yescharacteristic
313C42#115711yescharacteristic
314C42C2×C22⋊D711yescharacteristic
315C42C23×D711yescharacteristic
316C2×C7—22no—
317C2×C7#304411yescharacteristic
318C2×C7C2×#46511yescharacteristic
319C2×C7#306811yescharacteristic
320D7—161no—
321C23—281no—
322C23#649311yescharacteristic
323Q8—24no—
324Q8—22no—
325Q8—281no—
326Q8—281no—
327Q8—562no—
328Q8—562no—
329Q8C2×#113214yes—
330D4—281no—
331D4—281no—
332D4—282no—
333D4—561no—
334D4—562no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—281no—
342C2×C4—282no—
343C2×C4—562no—
344C2×C4—562no—
345C2×C4C2×#113212yes—
346C2×C4#646712yes—
347C2×C4#649311yescharacteristic
348C2×C4#649311yescharacteristic
349C2×C4#649311yescharacteristic
350C2×C4C22×D2811yescharacteristic
351C2×C4C2×#115711yescharacteristic
352C2×C4C22×C22⋊D711yescharacteristic
353C8—24no—
354C8—42no—
355C8—282no—
356C8—282no—
357C7#1940311yescharacteristic, complemented, Sylow
358C22—21no—
359C22—21no—
360C22—561no—
361C22—561no—
362C22—561no—
363C22#8988612yes—
364C22C2×#649311yescharacteristic
365C4—24no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—561no—
371C4—562no—
372C4#8988612yes—
373C4C22×#113211yescharacteristic
374C4C2×#646711yescharacteristic
375C4#9052912yes—
376C2—22no—
377C2—1121no—
378C2#1173999911yescharacteristic
379C2C2×#8988611yescharacteristic
380C2#1174125911yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2982C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenter#2982D7
hypocenterC7#19403
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C7×#2982C2semidirect
3. C2×#89793C2semidirect
5. #11728697C2semidirect
6. #11728694C2semidirect
7. #11728533C2semidirect
9. #11728503C2semidirect
88. #2982D7semidirect
357. C7#19403semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
#117398862
C2×#897932
#117367142
#117286972
#117286942
#117285332
#117285262
#117285032
#194037