Finite Group Databasebeta

← #13371433

Subgroups — #13371433

Order 1792.

778 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
1C2×#89878C211yesmaximal, characteristic
2#11731327C211yesmaximal, characteristic
3#11731326C211yesmaximal, characteristic
4C7×#1807C211yesmaximal, characteristic
5C8×C22⋊Dic7C211yesmaximal, characteristic
6#11728313C212yesmaximal, complemented
7Dic7×#106C212yesmaximal, complemented
8#11727257C211yesmaximal, characteristic
9#11727257C211yesmaximal, characteristic
10#11726574C212yesmaximal, complemented
11#11726568C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C2×#6489C2211yescharacteristic
14C2×#6489C2211yescharacteristic
15C2×C7×#106C2212yes—
16C7×#380C2211yescharacteristic
17C7×#380C2211yescharacteristic
18C4×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89878C418yescomplemented
21C2×#6392C2211yescharacteristic
22C2×#6392C2211yescharacteristic
23C2×#6391C2211yescharacteristic
24C2×#6391C2211yescharacteristic
25C2×C8×Dic7C2212yes—
26C4⋊C4×Dic7C2212yes—
27C4⋊C4×Dic7C2212yes—
28#89745C2212yes—
29#89745C2212yes—
30#89717—22no—
31#89716—22no—
32#89423—21no—
33#89422—21no—
34#89421—21no—
35#89420—21no—
36C7×#340C2212yes—
37#89345C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#22610—71nomaximal, Sylow
43C23×C4×C7C2311yescharacteristic
44C22×C7×C8—24no—
45C22×C7×C8C2×C414yes—
46C22×C7×C8C2311yescharacteristic
47C22×C7×C8C2311yescharacteristic
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2312yes—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C22⋊Dic7—21no—
53C2×C22⋊Dic7—21no—
54C2×C22⋊Dic7D412yes—
55#6489—22no—
56#6489—22no—
57#6489C2×C414yes—
58#6489C2×C414yes—
59C2×C4⋊Dic7C2311yescharacteristic
60C2×C4⋊Dic7C2311yescharacteristic
61C2×C4⋊Dic7C2311yescharacteristic
62C2×C4⋊Dic7C2311yescharacteristic
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×C4×Dic7—21no—
66C2×C4×Dic7—21no—
67C2×C4×Dic7—22no—
68C2×C4×Dic7—22no—
69C2×C4×Dic7C2312yes—
70C22×C7⋊C8C2312yes—
71C7×#106—24no—
72C7×#95—21no—
73C7×#95—21no—
74C7⋊#95—22no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95D411yescharacteristic
79C7⋊#95D411yescharacteristic
80C22⋊(C7⋊C8)—22no—
81#6392C2×C414yes—
82#6392C2×C414yes—
83#6391C2×C414yes—
84#6391C2×C414yes—
85C8×Dic7—24no—
86C2×#427—71no—
87#2347—71no—
88#2346—71no—
89#1941—72no—
90#1921—72no—
91#1918—72no—
92#1825—71no—
93#1825—71no—
94#1807D711yescharacteristic
95C4×#106—72no—
96C8×C22⋊C4—71no—
97C24×C7C2×D411yescharacteristic
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7C22×C412yes—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7#4112yes—
105C22×C4×C7C2411yescharacteristic
106C22×Dic7—21no—
107C22×Dic7—21no—
108C22×Dic7—21no—
109C22×Dic7—21no—
110C22×Dic7C2×D412yes—
111C22×Dic7C2×D412yes—
112C2×C7×C8—22no—
113C2×C7×C8—28no—
114C2×C7×C8C22×C412yes—
115C2×C7×C8C22×C412yes—
116C2×C7×C8#4114yes—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C22⋊C4—81no—
122C7×C22⋊C4—81no—
123C22⋊Dic7—24no—
124C22⋊Dic7—24no—
125C22⋊Dic7—24no—
126C4⋊Dic7—21no—
127C4⋊Dic7—21no—
128C4⋊Dic7—21no—
129C4⋊Dic7—21no—
130C4⋊Dic7C22×C412yes—
131C4⋊Dic7C22×C412yes—
132C4⋊Dic7C22×C412yes—
133C4⋊Dic7C22×C412yes—
134Dic7⋊C4—44no—
135Dic7⋊C4—82no—
136C4×Dic7—24no—
137C4×Dic7—24no—
138C4×Dic7—22no—
139C4×Dic7—22no—
140C4×Dic7—44no—
141C4×Dic7—82no—
142C2×C7⋊C8—22no—
143C2×C7⋊C8—22no—
144C2×C7⋊C8—22no—
145C23×C8C2×D711yescharacteristic
146C2×#117—71no—
147C2×#117—71no—
148#427—78no—
149C2×#107—71no—
150C2×#107—71no—
151C2×#106—71no—
152C2×#106—71no—
153C2×#106C2×D712yes—
154C2×C22⋊C8—71no—
155C2×C4×C8—72no—
156#389—72no—
157#389—72no—
158#388—142no—
159#388—142no—
160#383—141no—
161#382—141no—
162#380C2×D711yescharacteristic
163#380C2×D711yescharacteristic
164C4×C4⋊C4—72no—
165C4×C4⋊C4—72no—
166C4×C4⋊C4—141no—
167C4×C22⋊C4—71no—
168#376—141no—
169#340—72no—
170#340—72no—
171#340—72no—
172#340—72no—
173#340C2×D712yes—
174#336—72no—
175C23×C7—22no—
176C23×C7—22no—
177C23×C7—21no—
178C23×C7—21no—
179C23×C7C4×D412yes—
180C23×C7C22×D411yescharacteristic
181C2×C4×C7—22no—
182C2×C4×C7—22no—
183C2×C4×C7—22no—
184C2×C4×C7—24no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7C4×D412yes—
192C2×C4×C7C23×C411yescharacteristic
193C2×C4×C7C2×#4112yes—
194C2×C4×C7C2×#4111yescharacteristic
195C2×C4×C72−1+412yes—
196C2×Dic7—22no—
197C2×Dic7—22no—
198C2×Dic7—22no—
199C2×Dic7—22no—
200C2×Dic7—22no—
201C2×Dic7—22no—
202C2×Dic7—22no—
203C2×Dic7—22no—
204C2×Dic7—22no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7—41no—
208C2×Dic7—41no—
209C2×Dic7—41no—
210C2×Dic7—41no—
211C2×Dic7#13514yes—
212C7×C8—22no—
213C7×C8#11714yes—
214C7⋊C8—42no—
215C23×C4C22×D711yescharacteristic
216C22×C8—24no—
217C22×C8—72no—
218C22×C8C4×D714yes—
219C22×C8C22×D711yescharacteristic
220C22×C8C22×D711yescharacteristic
221#117—74no—
222#117—74no—
223#117—142no—
224#117—142no—
225C2×C4⋊C4—71no—
226C2×C4⋊C4—71no—
227C2×C4⋊C4—71no—
228C2×C4⋊C4—71no—
229C2×C4⋊C4—142no—
230C2×C4⋊C4—142no—
231C2×C4⋊C4C22×D712yes—
232C2×C4⋊C4C22×D712yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—72no—
236C2×C22⋊C4—141no—
237C2×C22⋊C4—141no—
238C2×C42—72no—
239C2×C42—141no—
240C2×C42—141no—
241C2×C42—142no—
242C2×C42—142no—
243#107—74no—
244#107—74no—
245#106—24no—
246#106—74no—
247#106—74no—
248C22⋊C8—142no—
249C4×C8—144no—
250#95—21no—
251#95—21no—
252#95—71no—
253#95—71no—
254#95—142no—
255#95—142no—
256#95—142no—
257#95—142no—
258C22×C7—22no—
259C22×C7—22no—
260C22×C7—22no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7C2×C4×D411yescharacteristic
264C22×C7#54712yes—
265C22×C7C2×#13512yes—
266C22×C7#57711yescharacteristic
267C22×C7#57711yescharacteristic
268C4×C7—22no—
269C4×C7—82no—
270C4×C7—82no—
271C4×C7C2×#11711yescharacteristic
272C4×C7#51911yescharacteristic
273C4×C7#55212yes—
274Dic7—24no—
275Dic7—42no—
276Dic7—42no—
277Dic7—42no—
278Dic7—42no—
279Dic7—42no—
280C24D4×D711yescharacteristic
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—72no—
286C22×C4—72no—
287C22×C4—141no—
288C22×C4—141no—
289C22×C4—141no—
290C22×C4—141no—
291C22×C4C2×C4×D712yes—
292C22×C4D4×D711yescharacteristic
293C22×C4#116112yes—
294C22×C4C23×D711yescharacteristic
295C2×C8—22no—
296C2×C8—28no—
297C2×C8—142no—
298C2×C8—142no—
299C2×C8—142no—
300C2×C8C2×C4×D712yes—
301C2×C8C2×C4×D712yes—
302C2×C8#115914yes—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—72no—
308C4⋊C4—72no—
309C4⋊C4—72no—
310C4⋊C4—72no—
311C4⋊C4—141no—
312C4⋊C4—141no—
313C4⋊C4—141no—
314C4⋊C4—141no—
315C4⋊C4—284no—
316C4⋊C4—562no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C22⋊C4—144no—
320C22⋊C4—144no—
321C22⋊C4—144no—
322C42—144no—
323C42—144no—
324C42—142no—
325C42—142no—
326C42—284no—
327C42—562no—
328C2×C7—42no—
329C2×C7#234411yescharacteristic
330C2×C7C4×#13511yescharacteristic
331C2×C7#295211yescharacteristic
332C2×C7#333712yes—
333C2×C7#334012yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23#645612yes—
339C23C2×D4×D711yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—74no—
351C2×C4—142no—
352C2×C4—142no—
353C2×C4—142no—
354C2×C4—142no—
355C2×C4—142no—
356C2×C4—142no—
357C2×C4—142no—
358C2×C4—142no—
359C2×C4—142no—
360C2×C4—281no—
361C2×C4—281no—
362C2×C4—281no—
363C2×C4—281no—
364C2×C4—282no—
365C2×C4—282no—
366C2×C4#645612yes—
367C2×C4C22×C4×D711yescharacteristic
368C2×C4C2×#115912yes—
369C2×C4C2×#116111yescharacteristic
370C2×C4#655412yes—
371C8—22no—
372C8—282no—
373C8#644014yes—
374C7#2261011yescharacteristic, complemented, Sylow
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22C2×#645611yescharacteristic
381C22#9030212yes—
382C22#9045311yescharacteristic
383C22#9045311yescharacteristic
384C22D7×#13512yes—
385C4—22no—
386C4—82no—
387C4—82no—
388C4—144no—
389C4—282no—
390C4—282no—
391C4—282no—
392C4—282no—
393C4—282no—
394C4C2×#644011yescharacteristic
395C4#9020711yescharacteristic
396C4#9033312yes—
397C2—42no—
398C2#1173132311yescharacteristic
399C2#1173655111yescharacteristic
400C2#1173661011yescharacteristic
401C2#1173809512yes—
402C2#1173811812yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1807C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1807D7
hypocenterC7#22610
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
6. #11728313C2semidirect
7. Dic7×#106C2semidirect
10. #11726574C2semidirect
11. #11726568C2semidirect
20. #89878C4semidirect
374. C7#22610semidirect

Maximal subgroups

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

subgroupindex
C2×#898782
#117313272
#117313262
C7×#18072
C8×C22⋊Dic72
#117283132
Dic7×#1062
#117272572
#117265742
#117265682
#226107