Finite Group Databasebeta

← #13339072

Subgroups — #13339072

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2982C211yesmaximal, characteristic, complemented
2C2×#89799C211yesmaximal, characteristic
3#11739876C211yesmaximal, characteristic, complemented
4#11736750C211yesmaximal, characteristic
5#11728694C214yesmaximal, complemented
6#11728692C214yesmaximal, complemented
7#11728530C211yesmaximal, characteristic, complemented
8#11728528C211yesmaximal, characteristic
9#11728503C211yesmaximal, characteristic, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11C22×#1143—21no—
12C2×#6437C2211yescharacteristic
13#90161C2211yescharacteristic
14C7×C8×Q8—22no—
15C7×C8×Q8C2214yescomplemented
16C7×#431—22no—
17C7×#431C2211yescharacteristic
18C7×#431C2211yescharacteristic
19#89997—21no—
20#89799—22no—
21#89799C2214yes—
22#89794—22no—
23#89793C2212yescomplemented
24#89792C2212yescomplemented
25C2×#6383—21no—
26#89736—21no—
27#89696C2211yescharacteristic
28C2×C4⋊(C7⋊C8)C2211yescharacteristic
29#89620—22no—
30#89503—22no—
31#89466C2212yes—
32#89465C2212yes—
33#89280C2214yes—
34#89277C2212yescomplemented
35#89276C2212yescomplemented
36#89252C2214yescomplemented
37#19396—71nomaximal, Sylow
38C22×C7×Q8D411yescharacteristic
39C22×D28—21no—
40C22×D28—21no—
41C22×D28—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8—21no—
44C4×C7×Q8D412yes—
45C4×C7×Q8C2312yes—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4D411yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7C2311yescharacteristic
50C2×#1143—24no—
51C2×#1143—44no—
52#6489—21no—
53#6487—21no—
54#6487—21no—
55C2×#1136—21no—
56C22×C7⋊C8—21no—
57C2×#1132—24no—
58#6437—21no—
59#6437—21no—
60#6437—21no—
61#6437C2312yes—
62#6433—21no—
63#6432C2311yescharacteristic
64#6432C2311yescharacteristic
65C4×Dic14—21no—
66C4×Dic14—21no—
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—22no—
76#6409—22no—
77#6395—24no—
78#6383—24no—
79#6382—22no—
80#6381—22no—
81C4⋊(C7⋊C8)—21no—
82C4⋊(C7⋊C8)—21no—
83C4⋊(C7⋊C8)C2312yes—
84C4⋊(C7⋊C8)C2312yes—
85#3208—71no—
86#3078—71no—
87C2×#460—71no—
88#2982D711yescharacteristic, complemented
89#1678—74no—
90#1666—74no—
91#1570—71no—
92#1545—71no—
93#1542—71no—
94C23×D7—41no—
95C2×C7×Q8—21no—
96C2×C7×Q8—22no—
97C2×C7×Q8SD1614yescomplemented
98C2×C7×Q8C2×D412yes—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7#4111yescharacteristic
104C2×D28—22no—
105C2×D28—22no—
106C2×D28—22no—
107C2×D28—42no—
108C2×D28—42no—
109C2×D28—41no—
110C2×D28—41no—
111C2×D28—41no—
112C2×D28—82no—
113C2×D28—82no—
114C2×Dic14—21no—
115C2×Dic14—21no—
116C2×C7×C8—22no—
117C2×C7×C8C2×D414yes—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4C2×D412yes—
122C42×C7—22no—
123C42×C7—22no—
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7#4111yescharacteristic
127C42×C7C2411yescharacteristic
128C22⋊Dic7—41no—
129#1143—48no—
130C4⋊Dic7—21no—
131C4⋊Dic7—21no—
132C4⋊Dic7—21no—
133C4⋊Dic7—21no—
134Dic7⋊C4—41no—
135Dic7⋊C4—41no—
136C4×Dic7—41no—
137#1136—42no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—42no—
141#1132—84no—
142C22×SD16—141no—
143#533—71no—
144C2×#127—71no—
145C2×C4×Q8C2×D711yescharacteristic
146#492—72no—
147#486—72no—
148#479—142no—
149#477—72no—
150#475—72no—
151#460—74no—
152#460—142no—
153#460—142no—
154#450—142no—
155C8×Q8—22no—
156C8×Q8C2×D714yescomplemented
157#431—22no—
158#431C2×D711yescharacteristic
159#431C2×D711yescharacteristic
160#428—141no—
161#423—71no—
162C2×C4⋊C8—71no—
163#419—141no—
164C2×#102—141no—
165#334—74no—
166#333—72no—
167#332—72no—
168#331—74no—
169C23×C7#12111yescharacteristic
170C22×D7—42no—
171C22×D7—82no—
172C22×D7—82no—
173C22×D7—82no—
174C7×Q8—22no—
175C7×Q8—24no—
176C7×Q8C2×SD1614yes—
177C2×C4×C7—22no—
178C2×C4×C7—24no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—81no—
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7C2×SD1612yes—
187C2×C4×C7#13512yes—
188C2×C4×C7C22×D411yescharacteristic
189C2×C4×C7C22×D411yescharacteristic
190C2×C4×C7C2×#4111yescharacteristic
191C2×Dic7—41no—
192C2×Dic7—41no—
193D28—42no—
194D28—42no—
195D28—44no—
196D28—82no—
197D28—82no—
198D28—82no—
199D28—82no—
200Dic14—42no—
201Dic14—81no—
202C7×C8—24no—
203C7×C8—42no—
204C7⋊C8—42no—
205C7⋊C8—42no—
206C22×Q8D2811yescharacteristic
207C22×D4—141no—
208C22×D4—141no—
209C22×D4—141no—
210C2×SD16—144no—
211C2×SD16—144no—
212C2×SD16—284no—
213C2×M16—141no—
214C22×C8—141no—
215C4⋊Q8—71no—
216C4⋊Q8—71no—
217#127—72no—
218#127—141no—
219#127—141no—
220#127—141no—
221#125—141no—
222C22⋊Q8—141no—
223C22⋊Q8—141no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—141no—
227C4×Q8—141no—
228C4×Q8D2812yes—
229C4×Q8C22×D712yes—
230#117—141no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4D2811yescharacteristic
233C2×C42—21no—
234C2×C42C22×D711yescharacteristic
235#107—142no—
236#106—142no—
237C4⋊C8—24no—
238C4⋊C8—72no—
239C4⋊C8—72no—
240C4⋊C8—141no—
241C4⋊C8—141no—
242C4⋊C8C22⋊D712yes—
243C4⋊C8C22×D712yes—
244#103—142no—
245#103—142no—
246#102—144no—
247#102—144no—
248C22⋊C8—21no—
249C22⋊C8C22⋊D712yes—
250C4×C8—24no—
251C4×C8—21no—
252C4×C8C22×D712yes—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7#46512yes—
256C22×C7C2×#12111yescharacteristic
257C2×D7—82no—
258C2×D7—82no—
259C2×D7—82no—
260C2×D7—164no—
261C4×C7—21no—
262C4×C7—24no—
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—42no—
266C4×C7#46512yes—
267C4×C7C22×SD1611yescharacteristic
268C4×C7C2×#13511yescharacteristic
269C4×C7#57712yes—
270Dic7—81no—
271Dic7—81no—
272C24—281no—
273C2×Q8—22no—
274C2×Q8—21no—
275C2×Q8—141no—
276C2×Q8—141no—
277C2×Q8#113214yescomplemented
278C2×Q8C2×D2812yes—
279C2×D4—142no—
280C2×D4—142no—
281C2×D4—142no—
282C2×D4—282no—
283C2×D4—282no—
284C2×D4—281no—
285C2×D4—281no—
286C2×D4—281no—
287C2×D4—562no—
288C2×D4—562no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4C2×D2811yescharacteristic
292C22×C4#115711yescharacteristic
293C22×C4C2×C22⋊D711yescharacteristic
294SD16—288no—
295SD16—564no—
296M16—282no—
297C2×C8—22no—
298C2×C8—142no—
299C2×C8—142no—
300C2×C8—282no—
301C2×C8C2×C22⋊D714yes—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—141no—
306C4⋊C4—141no—
307C4⋊C4—141no—
308C4⋊C4—141no—
309C4⋊C4—281no—
310C4⋊C4—281no—
311C4⋊C4C2×D2812yes—
312C22⋊C4—281no—
313C42—22no—
314C42—22no—
315C42—281no—
316C42C2×D2811yescharacteristic
317C42#115711yescharacteristic
318C42C2×C22⋊D711yescharacteristic
319C42C23×D711yescharacteristic
320C2×C7—22no—
321C2×C7#304311yescharacteristic
322C2×C7C2×#46511yescharacteristic
323C2×C7#306811yescharacteristic
324D7—162no—
325C23—282no—
326C23—562no—
327C23—562no—
328C23—562no—
329C23#649311yescharacteristic
330Q8—24no—
331Q8—22no—
332Q8—282no—
333Q8—561no—
334Q8C2×#113214yes—
335D4—282no—
336D4—282no—
337D4—284no—
338D4—562no—
339D4—562no—
340D4—562no—
341D4—562no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—281no—
349C2×C4—281no—
350C2×C4C2×#113212yes—
351C2×C4#646712yes—
352C2×C4#649311yescharacteristic
353C2×C4#649311yescharacteristic
354C2×C4#649311yescharacteristic
355C2×C4C22×D2811yescharacteristic
356C2×C4C2×#115711yescharacteristic
357C2×C4C22×C22⋊D711yescharacteristic
358C8—24no—
359C8—42no—
360C8—282no—
361C8—282no—
362C7#1939611yescharacteristic, complemented, Sylow
363C22—21no—
364C22—21no—
365C22—562no—
366C22—562no—
367C22—562no—
368C22—1124no—
369C22#8988612yes—
370C22C2×#649311yescharacteristic
371C4—24no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—561no—
377C4—561no—
378C4#8988612yes—
379C4C22×#113211yescharacteristic
380C4C2×#646711yescharacteristic
381C4#9052812yes—
382C2—22no—
383C2—1122no—
384C2#1173999611yescharacteristic
385C2C2×#8988611yescharacteristic
386C2#1174125911yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2982C2
socleC22×C7C2×#121
prime radicalC2×#6437C22
hypercenter#2982D7
hypocenterC7#19396
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. #11739876C2semidirect
5. #11728694C2semidirect
6. #11728692C2semidirect
7. #11728530C2semidirect
9. #11728503C2semidirect
15. C7×C8×Q8C22semidirect
23. #89793C22semidirect
24. #89792C22semidirect
34. #89277C22semidirect
35. #89276C22semidirect
36. #89252C22semidirect
88. #2982D7semidirect
97. C2×C7×Q8SD16semidirect
156. C8×Q8C2×D7semidirect
277. C2×Q8#1132semidirect
362. C7#19396semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
C2×#897992
#117398762
#117367502
#117286942
#117286922
#117285302
#117285282
#117285032
#193967