Finite Group Databasebeta

← #13363318

Subgroups — #13363318

Order 1792.

1054 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic
3#11731198C211yesmaximal, characteristic
4C7×#1988C211yesmaximal, characteristic, complemented
5#11729534C211yesmaximal, characteristic, complemented
6#11728213C211yesmaximal, characteristic, complemented
7#11727243C211yesmaximal, characteristic, complemented
8#11726567C214yesmaximal
9#11726559C214yesmaximal, complemented
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×#6509—21no—
12C2×Q8×Dic7—21no—
13#90419—24no—
14#90419C418yescomplemented
15C2×#6460—21no—
16C2×#6455C2212yes—
17C2×#6453C2212yes—
18C2×#6450—21no—
19C2×#6435—21no—
20C2×C42×D7C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C8C2212yes—
25C2×#6389C2212yes—
26#89759C2212yes—
27#89748C2212yes—
28C4×C4⋊Dic7C2211yescharacteristic
29C4×Dic7⋊C4—21no—
30C2×C4⋊(C7⋊C8)C2211yescharacteristic
31#89607—24no—
32#89439—21no—
33#89422—21no—
34C7×#340C2214yes—
35#89346C2214yes—
36#89324C2214yes—
37#89324C2214yes—
38#21839—71nomaximal, Sylow
39C22×C7×Q8—21no—
40C22×C4×D7D411yescharacteristic
41C22×C4×D7D411yescharacteristic
42C22×C4×D7C2311yescharacteristic
43C22×Dic14—21no—
44C22×C7×C8C2312yes—
45C7×C4⋊Q8—22no—
46C7×C4⋊Q8C2×C414yes—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2312yes—
50C2×C42×C7—21no—
51C2×C42×C7C2311yescharacteristic
52#6509—28no—
53Q8×Dic7—28no—
54C2×(C2×D7)⋊C4—22no—
55C2×(C2×D7)⋊C4—22no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—22no—
58C2×C4⋊Dic7C2312yes—
59C2×C4⋊Dic7C2312yes—
60C2×Dic7⋊C4—22no—
61C2×Dic7⋊C4—22no—
62C2×Dic7⋊C4—22no—
63C2×C4×Dic7—22no—
64C2×C4×Dic7—21no—
65C2×C4×Dic7D411yescharacteristic
66C2×C4×Dic7D411yescharacteristic
67C2×C4×Dic7C2311yescharacteristic
68C22×C7⋊C8C2312yes—
69#6460—28no—
70#6455—24no—
71#6455C2×C418yes—
72#6453—24no—
73#6453C2×C418yes—
74#6450—28no—
75#6435—44no—
76C42×D7C2×C414yes—
77C42×D7D414yes—
78#6432—22no—
79#6432C2×C414yes—
80C7×C4⋊C8—22no—
81C7×#95—21no—
82C7⋊#95—21no—
83C7⋊#95—22no—
84(C2×D7)⋊C8—24no—
85#6389—24no—
86C4⋊(C7⋊C8)—22no—
87C2×#533—71no—
88C2×#432—71no—
89#2285—71no—
90#1992—74no—
91#1988—71no—
92#1988D711yescharacteristic, complemented
93#1987—71no—
94#1987—71no—
95#1825—74no—
96C23×D7C2×D411yescharacteristic
97C2×C7×Q8—24no—
98C2×C7×Q8—42no—
99C2×C7×Q8—42no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2411yescharacteristic
105C22×Dic7—22no—
106C22×Dic7—22no—
107C22×Dic7C2×D411yescharacteristic
108C22×Dic7C2×D411yescharacteristic
109C22×Dic7C2×D411yescharacteristic
110C2×C4×D7—22no—
111C2×C4×D7—22no—
112C2×C4×D7—22no—
113C2×C4×D7—22no—
114C2×C4×D7—42no—
115C2×C4×D7C22⋊C414yes—
116C2×C4×D7C22⋊C414yes—
117C2×C4×D7C22×C412yes—
118C2×C4×D7C2×D412yes—
119C2×Dic14—24no—
120C2×Dic14—44no—
121C2×C7×C8—22no—
122C2×C7×C8—24no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4C22×C414yes—
128C42×C7—42no—
129C42×C7C22×C412yes—
130C42×C7C2×D412yes—
131(C2×D7)⋊C4—28no—
132(C2×D7)⋊C4—44no—
133C4⋊Dic7—24no—
134C4⋊Dic7—22no—
135C4⋊Dic7—28no—
136C4⋊Dic7—22no—
137C4⋊Dic7—22no—
138C4⋊Dic7C22×C414yes—
139Dic7⋊C4—28no—
140Dic7⋊C4—44no—
141Dic7⋊C4—44no—
142C4×Dic7—28no—
143C4×Dic7—42no—
144C4×Dic7C22⋊C414yes—
145C4×Dic7C22⋊C414yes—
146C4×Dic7C22×C412yes—
147C4×Dic7C2×D412yes—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—24no—
150#533—78no—
151#533—144no—
152C2×C4⋊Q8—71no—
153C2×C4⋊Q8C2×D711yescharacteristic
154C2×#125—72no—
155C2×C22⋊Q8—141no—
156C2×C22⋊Q8—141no—
157C2×C4×Q8—141no—
158C2×C4×Q8—141no—
159C2×#117—72no—
160C2×#117—141no—
161C22×C42—71no—
162#432—144no—
163C2×C4⋊C8—71no—
164C2×C4⋊C8C2×D711yescharacteristic
165C2×C22⋊C8—72no—
166C2×#97—72no—
167#384—72no—
168#382—141no—
169#380—72no—
170#380—141no—
171C4×C4⋊C4—71no—
172C4×C4⋊C4—141no—
173C4×C4⋊C4C2×D711yescharacteristic
174#340—74no—
175#340—74no—
176#340—74no—
177#340C2×D714yes—
178C23×C7C22×D411yescharacteristic
179C22×D7—22no—
180C22×D7—21no—
181C22×D7—21no—
182C22×D7—22no—
183C22×D7C2×C22⋊C412yes—
184C7×Q8—48no—
185C2×C4×C7—22no—
186C2×C4×C7—24no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7C2×C22⋊C412yes—
193C2×C4×C7C2×C22⋊C412yes—
194C2×C4×C7AGL(1,ℤ/8)12yes—
195C2×C4×C7#13712yes—
196C2×C4×C7C23×C411yescharacteristic
197C2×C4×C7C22×D411yescharacteristic
198C2×C4×C72−1+412yes—
199C2×C4×C72−1+412yes—
200C2×Dic7—22no—
201C2×Dic7—21no—
202C2×Dic7—21no—
203C2×Dic7—22no—
204C2×Dic7—24no—
205C2×Dic7—44no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C2×Dic7C2×C22⋊C412yes—
210C2×Dic7C2×C22⋊C412yes—
211C2×Dic7C2×C22⋊C412yes—
212C4×D7—24no—
213C4×D7—22no—
214C4×D7—22no—
215C4×D7—28no—
216C4×D7—44no—
217Dic14—48no—
218C7×C8—42no—
219C7⋊C8—42no—
220C22×Q8—21no—
221C22×Q8—141no—
222C23×C4—71no—
223C23×C4—71no—
224C23×C4—71no—
225C22×C8—72no—
226C22×C8C22×D712yes—
227C4⋊Q8—22no—
228C4⋊Q8—74no—
229C4⋊Q8—142no—
230C4⋊Q8C4×D714yes—
231#125—78no—
232#125—144no—
233C22⋊Q8—148no—
234C22⋊Q8—148no—
235C4×Q8—148no—
236C4×Q8—148no—
237#117—78no—
238#117—144no—
239#117—284no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—72no—
242C2×C4⋊C4—72no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4—142no—
245C2×C4⋊C4—141no—
246C2×C4⋊C4—142no—
247C2×C4⋊C4—142no—
248C2×C4⋊C4C22×D712yes—
249C2×C4⋊C4C22×D712yes—
250C2×C22⋊C4—142no—
251C2×C22⋊C4—142no—
252C2×C42—21no—
253C2×C42—71no—
254C2×C42—71no—
255C2×C42—71no—
256C2×C42—74no—
257C2×C42—74no—
258C2×C42—142no—
259C2×C42—141no—
260C2×C42C22×D711yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—142no—
263C22⋊C8—144no—
264#97—144no—
265#95—21no—
266#95—142no—
267#95—141no—
268C22×C7C22×C22⋊C411yescharacteristic
269C22×C7#53612yes—
270C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C7C2×#13711yescharacteristic
272C22×C7#57512yes—
273C2×D7—22no—
274C2×D7—22no—
275C2×D7—22no—
276C2×D7—42no—
277C2×D7—42no—
278C4×C7—22no—
279C4×C7—44no—
280C4×C7—82no—
281C4×C7#41714yes—
282C4×C7#51912yes—
283C4×C7#53611yescharacteristic
284C4×C7#53611yescharacteristic
285Dic7—22no—
286Dic7—22no—
287Dic7—22no—
288Dic7—44no—
289Dic7—82no—
290C24—71no—
291C2×Q8—24no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—144no—
295C2×Q8—284no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—71no—
299C22×C4—71no—
300C22×C4—71no—
301C22×C4—74no—
302C22×C4—72no—
303C22×C4—72no—
304C22×C4—74no—
305C22×C4—142no—
306C22×C4—142no—
307C22×C4—142no—
308C22×C4—142no—
309C22×C4—142no—
310C22×C4—142no—
311C22×C4—282no—
312C22×C4D4×D711yescharacteristic
313C22×C4D4×D711yescharacteristic
314C22×C4C23×D711yescharacteristic
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—142no—
318C2×C8—144no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—74no—
324C4⋊C4—148no—
325C4⋊C4—148no—
326C4⋊C4—144no—
327C4⋊C4—142no—
328C4⋊C4—142no—
329C4⋊C4—142no—
330C4⋊C4—284no—
331C4⋊C4—284no—
332C4⋊C4C2×C4×D714yes—
333C22⋊C4—148no—
334C22⋊C4—284no—
335C42—42no—
336C42—74no—
337C42—74no—
338C42—72no—
339C42—72no—
340C42—148no—
341C42—282no—
342C42C2×C4×D712yes—
343C42D4×D712yes—
344C2×C7#231011yescharacteristic
345C2×C7C2×#41711yescharacteristic
346C2×C7#289611yescharacteristic
347C2×C7#323512yes—
348C2×C7#323712yes—
349D7—42no—
350C23—72no—
351C23—141no—
352C23—141no—
353C23—142no—
354C23—142no—
355C23C2×D4×D711yescharacteristic
356Q8—48no—
357Q8—288no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—72no—
366C2×C4—72no—
367C2×C4—72no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—142no—
371C2×C4—142no—
372C2×C4—142no—
373C2×C4—144no—
374C2×C4—148no—
375C2×C4—142no—
376C2×C4—144no—
377C2×C4—284no—
378C2×C4—284no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4D7×C22⋊C412yes—
383C2×C4D7×C22⋊C412yes—
384C2×C4#647412yes—
385C2×C4#647812yes—
386C2×C4C22×C4×D711yescharacteristic
387C2×C4C2×D4×D711yescharacteristic
388C2×C4#655112yes—
389C2×C4#655412yes—
390C8—42no—
391C8—282no—
392C7#2183911yescharacteristic, complemented, Sylow
393C22—142no—
394C22—142no—
395C22—142no—
396C22—282no—
397C22—282no—
398C22C2×D7×C22⋊C411yescharacteristic
399C22#9031912yes—
400C22C2×#647411yescharacteristic
401C22C2×#647811yescharacteristic
402C22#9046812yes—
403C4—22no—
404C4—44no—
405C4—82no—
406C4—142no—
407C4—142no—
408C4—142no—
409C4—284no—
410C4—562no—
411C4#8954314yes—
412C4#9026312yes—
413C4#9036611yescharacteristic
414C4#9041511yescharacteristic
415C2—282no—
416C2#1173405511yescharacteristic
417C2C2×#8954311yescharacteristic
418C2#1173640111yescharacteristic
419C2#1173841612yes—
420C2#1173846212yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1988C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1988D7
hypocenterC7#21839
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. C7×#1988C2semidirect
5. #11729534C2semidirect
6. #11728213C2semidirect
7. #11727243C2semidirect
9. #11726559C2semidirect
14. #90419C4semidirect
92. #1988D7semidirect
392. C7#21839semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
C2×#896072
#117311982
C7×#19882
#117295342
#117282132
#117272432
#117265672
#117265592
#218397