Finite Group Databasebeta

← #13390075

Subgroups — #13390075

Order 1792.

1014 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742834C211yesmaximal, characteristic
2C7×#2326C211yesmaximal, characteristic
3#11739876C212yesmaximal, complemented
4#11729477C212yesmaximal
5#11729371C211yesmaximal, characteristic
6#11727427C212yesmaximal, complemented
7#11727396C212yesmaximal
8#11727327C212yesmaximal, complemented
9#11727195C212yesmaximal, complemented
10C2×C7×#124C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13#90489C2212yes—
14C2×#6487—22no—
15#90234—22no—
16#90230—24no—
17D4×Dic14—22no—
18#90191C2212yes—
19#90174—22no—
20#90161C2212yes—
21C7×#391C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#388—22no—
24C7×#385—22no—
25C7×#384C2211yescharacteristic
26C7×#376C2211yescharacteristic
27#89997C2214yes—
28C2×#6407C2212yes—
29#89794—24no—
30#89793—22no—
31#89792—22no—
32#89736C2214yes—
33#89696C2212yes—
34#89325C2212yes—
35#89318C2214yes—
36#89318C2214yes—
37#24318—71nomaximal, Sylow
38C22×C7×Q8C2311yescharacteristic
39C22×C7×D4C2311yescharacteristic
40C23×C4×C7D412yes—
41C22×Dic14—22no—
42C7×#124—22no—
43C7×#124Q814yescomplemented
44C4×C7×Q8—22no—
45C4×C7×Q8—22no—
46C4×C7×D4—22no—
47C4×C7×D4—22no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4C2311yescharacteristic
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4D412yes—
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57C2×C42×C7C2311yescharacteristic
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—24no—
60D4×Dic7—24no—
61D4×Dic7—24no—
62#6489C2314yes—
63C2×C4⋊Dic7—22no—
64#6487—24no—
65#6487—24no—
66#6487—22no—
67#6487—22no—
68#6487—24no—
69C2×Dic7⋊C4—24no—
70C2×#1136C2314yes—
71#6453—24no—
72#6451—24no—
73#6441—28no—
74#6441—28no—
75#6433D414yes—
76#6432D412yes—
77#6432D412yes—
78C4×Dic14—22no—
79C4×Dic14—22no—
80C7×#95—21no—
81C7×#95—21no—
82C7×#95—22no—
83#6409—24no—
84#6409—24no—
85#6407—24no—
86#6382—24no—
87#6381—24no—
88C4⋊(C7⋊C8)—22no—
89C4⋊(C7⋊C8)—22no—
90#3501—71no—
91#3078—72no—
92#2326D711yescharacteristic
93#2060—72no—
94#2028—72no—
95#1995—72no—
96#1885—72no—
97#1856—72no—
98#1800—71no—
99C24×C7C2×D412yes—
100C2×C7×Q8—21no—
101C2×C7×Q8—22no—
102C2×C7×Q8—22no—
103C2×C7×Q8C2×Q812yescomplemented
104C2×C7×D4—22no—
105C2×C7×D4—22no—
106C2×C7×D4C2×Q812yes—
107C2×C7×D4#4112yes—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—82no—
115C22×C4×C7C2×D412yes—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2411yescharacteristic
119C22×Dic7—24no—
120C22×Dic7—24no—
121C2×Dic14—22no—
122C2×Dic14—24no—
123C2×Dic14—22no—
124C2×Dic14—84no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—82no—
130C7×C22⋊C4—28no—
131C7×C22⋊C4—24no—
132C7×C22⋊C4—82no—
133C42×C7—21no—
134C42×C7—22no—
135C42×C7—42no—
136C42×C7C2×D412yes—
137C42×C7C2×D412yes—
138C42×C7C2×Q812yescomplemented
139C22⋊Dic7—28no—
140C22⋊Dic7—24no—
141C4⋊Dic7—24no—
142C4⋊Dic7—22no—
143C4⋊Dic7—22no—
144C4⋊Dic7C2×D414yes—
145C4⋊Dic7C2×D414yes—
146Dic7⋊C4—28no—
147Dic7⋊C4—24no—
148Dic7⋊C4—24no—
149C4×Dic7—24no—
150#1136—24no—
151#1136—24no—
152C2×C7⋊C8—24no—
153#554—144no—
154#554—142no—
155D4×Q8—142no—
156#544—72no—
157#543—142no—
158#533—72no—
159C2×#124C2×D711yescharacteristic
160C2×C22⋊Q8—142no—
161#518—72no—
162C2×C4×Q8C2×D711yescharacteristic
163C2×C4×D4C2×D711yescharacteristic
164#479—144no—
165#477—142no—
166#475—142no—
167#428—74no—
168#423—72no—
169#419—74no—
170C2×#100—72no—
171#391C2×D711yescharacteristic
172#390C2×D711yescharacteristic
173#388—22no—
174#385—22no—
175#384C2×D711yescharacteristic
176#376C2×D711yescharacteristic
177#343—72no—
178#339—74no—
179#339—74no—
180C23×C7—22no—
181C23×C7—22no—
182C23×C7—42no—
183C23×C7—42no—
184C23×C7C22⋊Q814yes—
185C23×C7C22×D411yescharacteristic
186C7×Q8—24no—
187C7×Q8—42no—
188C7×D4—24no—
189C7×D4—24no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—44no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—42no—
201C2×C4×C7—41no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—82no—
207C2×C4×C7#12011yescharacteristic
208C2×C4×C7#12012yes—
209C2×C4×C7#12011yescharacteristic
210C2×C4×C7C22×D412yes—
211C2×C4×C7C22×Q811yescharacteristic
212C2×C4×C7C2×#4111yescharacteristic
213C2×C4×C72−1+412yes—
214C2×Dic7—24no—
215C2×Dic7—24no—
216C2×Dic7—84no—
217C2×Dic7—84no—
218Dic14—44no—
219Dic14—82no—
220Dic14—84no—
221C7⋊C8—44no—
222C22×Q8—142no—
223C22×Q8C22×D711yescharacteristic
224C22×D4C22×D711yescharacteristic
225C23×C4C22⋊D712yes—
226C2×M16—74no—
227C4⋊Q8—72no—
228C4⋊Q8—72no—
229C4⋊Q8—144no—
230#125—74no—
231#125—144no—
232#124—22no—
233#124Dic1414yescomplemented
234C22⋊Q8—148no—
235C22⋊Q8—148no—
236C22⋊Q8—144no—
237C22⋊Q8—144no—
238C22⋊Q8—142no—
239C22⋊Q8—144no—
240C22⋊Q8—142no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—142no—
244C4×Q8—142no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—144no—
248C4×D4—144no—
249#117—74no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—144no—
253C2×C4⋊C4—142no—
254C2×C4⋊C4C22×D711yescharacteristic
255C2×C4⋊C4C22×D711yescharacteristic
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—144no—
259C2×C22⋊C4C22⋊D712yes—
260C2×C42—21no—
261C2×C42C22×D711yescharacteristic
262C2×C42C22×D711yescharacteristic
263C2×C42C22×D711yescharacteristic
264#107—144no—
265#106—144no—
266C4⋊C8—142no—
267C4⋊C8—142no—
268#103—144no—
269#103—144no—
270#100—144no—
271#95—21no—
272#95—21no—
273#95—22no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—24no—
277C22×C7—44no—
278C22×C7—42no—
279C22×C7—42no—
280C22×C7—82no—
281C22×C7C2×#12011yescharacteristic
282C22×C7C2×C22⋊Q811yescharacteristic
283C22×C7#53611yescharacteristic
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—22no—
287C4×C7—42no—
288C4×C7—42no—
289C4×C7—42no—
290C4×C7—81no—
291C4×C7—81no—
292C4×C7#54812yes—
293C4×C7D4×Q812yes—
294Dic7—44no—
295Dic7—44no—
296C24C2×C22⋊D712yes—
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—144no—
301C2×Q8—142no—
302C2×Q8—142no—
303C2×Q8—564no—
304C2×Q8C2×Dic1412yescomplemented
305C2×D4—22no—
306C2×D4—22no—
307C2×D4C2×Dic1412yes—
308C2×D4#115712yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—82no—
316C22×C4—144no—
317C22×C4—144no—
318C22×C4C2×C22⋊D712yes—
319C22×C4C2×C22⋊D711yescharacteristic
320C22×C4C2×C22⋊D711yescharacteristic
321C22×C4C23×D711yescharacteristic
322M16—144no—
323M16—144no—
324C2×C8—144no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—74no—
330C4⋊C4—74no—
331C4⋊C4—82no—
332C4⋊C4—148no—
333C4⋊C4—144no—
334C4⋊C4—144no—
335C4⋊C4—144no—
336C4⋊C4—142no—
337C4⋊C4—142no—
338C22⋊C4—28no—
339C22⋊C4—24no—
340C22⋊C4—82no—
341C22⋊C4—148no—
342C22⋊C4—144no—
343C42—21no—
344C42—22no—
345C42—42no—
346C42—144no—
347C42C2×Dic1412yescomplemented
348C42D4×D712yes—
349C42D4×D712yes—
350C2×C7—21no—
351C2×C7—21no—
352C2×C7—44no—
353C2×C7#240911yescharacteristic
354C2×C7#301911yescharacteristic
355C2×C7#302211yescharacteristic
356C23—22no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23#648714yes—
361C23C22×C22⋊D711yescharacteristic
362Q8—24no—
363Q8—42no—
364Q8—284no—
365Q8—562no—
366Q8—564no—
367D4—24no—
368D4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—144no—
387C2×C4—144no—
388C2×C4—564no—
389C2×C4—564no—
390C2×C4#650012yes—
391C2×C4#650011yescharacteristic
392C2×C4#650011yescharacteristic
393C2×C4C22×Dic1411yescharacteristic
394C2×C4C2×#115711yescharacteristic
395C2×C4C2×D4×D712yes—
396C2×C4#655412yes—
397C8—284no—
398C7#2431811yescharacteristic, complemented, Sylow
399C22—21no—
400C22—21no—
401C22—24no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22C2×#648711yescharacteristic
407C22C2×#650011yescharacteristic
408C22#9051611yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4—284no—
418C4—284no—
419C4D4×Dic1412yes—
420C4#9024312yes—
421C2—21no—
422C2—21no—
423C2—44no—
424C2#1173979411yescharacteristic
425C2#1174083911yescharacteristic
426C2#1174086111yescharacteristic
subgroupG/H
centerC22C2×#6500
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2326C2
socleC22×C7C2×#120
prime radicalC22×C7×D4C23
hypercenter#2326D7
hypocenterC7#24318
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
3. #11739876C2semidirect
6. #11727427C2semidirect
8. #11727327C2semidirect
9. #11727195C2semidirect
43. C7×#124Q8semidirect
103. C2×C7×Q8C2×Q8semidirect
138. C42×C7C2×Q8semidirect
233. #124Dic14semidirect
304. C2×Q8C2×Dic14semidirect
347. C42C2×Dic14semidirect
398. C7#24318semidirect

Maximal subgroups

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

subgroupindex
#117428342
C7×#23262
#117398762
#117294772
#117293712
#117274272
#117273962
#117273272
#117271952
#243187