Finite Group Databasebeta

← #13371503

Subgroups — #13371503

Order 1792.

810 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2345C211yesmaximal, characteristic
2C22×#6381C211yesmaximal, characteristic
3#11730902C211yesmaximal, characteristic
4#11729308C211yesmaximal, characteristic
5#11727374C212yesmaximal, complemented
6#11727354C212yesmaximal, complemented
7#11727265C211yesmaximal, characteristic
8#11727256C211yesmaximal, characteristic
9#11727235C212yesmaximal, complemented
10#11727212C212yesmaximal, complemented
11#11727201C211yesmaximal, characteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C23×C7⋊C8C2211yescharacteristic
15C7×#389C2212yes—
16C7×#387—22no—
17C7×#380—21no—
18C7×#380C2211yescharacteristic
19C4×C7×C4⋊C4C2212yes—
20C4×C7×C22⋊C4—21no—
21C4×C7×C22⋊C4C2211yescharacteristic
22#89984C2211yescharacteristic
23C2×C22⋊(C7⋊C8)C2211yescharacteristic
24C2×#6382C2212yes—
25C2×#6381C418yescomplemented
26C2×#6381C2211yescharacteristic
27C2×#6381C2211yescharacteristic
28C2×#6381C2211yescharacteristic
29C2×#6381C2211yescharacteristic
30#89714—21no—
31C22⋊C4×Dic7—21no—
32C4×C4⋊Dic7C2212yes—
33#89700C2212yes—
34C2×#6377C2212yes—
35#89426—22no—
36#89324C2212yes—
37#89324C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89319C2212yes—
42#22618—71nomaximal, Sylow
43C23×C4×C7—21no—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4C2×C414yes—
50C2×C7×C4⋊C4C2312yes—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4D411yescharacteristic
55C2×C7×C22⋊C4D411yescharacteristic
56C2×C42×C7—22no—
57C2×C42×C7C2312yes—
58C2×C22⋊Dic7—21no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—22no—
61C2×C4⋊Dic7C2×C414yes—
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4⋊Dic7C2311yescharacteristic
65C2×Dic7⋊C4—22no—
66C2×C4×Dic7—22no—
67C22×C7⋊C8—24no—
68C22×C7⋊C8C2×C414yes—
69C22×C7⋊C8C2312yes—
70C22×C7⋊C8C2311yescharacteristic
71C22×C7⋊C8C2311yescharacteristic
72C7×#95—22no—
73C7×#95—22no—
74C7×#95—21no—
75C7×#95D412yes—
76C7⋊#95—21no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C22⋊(C7⋊C8)—22no—
80#6382—24no—
81#6381C2×C414yes—
82#6381C2×C414yes—
83#6381C2×C414yes—
84#6381C2×C414yes—
85#6377—24no—
86C22×#107—71no—
87#2345—71no—
88#2345D711yescharacteristic
89#1943—72no—
90#1918—72no—
91#1918—72no—
92#1824—71no—
93#1824—71no—
94#1807—71no—
95#1776—72no—
96#1752—71no—
97C24×C7C2×D411yescharacteristic
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—24no—
103C22×C4×C7—22no—
104C22×C4×C7—82no—
105C22×C4×C7C22×C412yes—
106C22×C4×C7C2×D412yes—
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4112yes—
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—21no—
112C22×Dic7—21no—
113C22×Dic7—24no—
114C22×Dic7—22no—
115C22×Dic7—82no—
116C7×C4⋊C4—24no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—82no—
122C7×C4⋊C4C22×C412yes—
123C7×C4⋊C4C22×C412yes—
124C7×C22⋊C4—22no—
125C7×C22⋊C4—22no—
126C7×C22⋊C4—42no—
127C42×C7—22no—
128C42×C7—22no—
129C42×C7—82no—
130C22⋊Dic7—42no—
131C4⋊Dic7—24no—
132C4⋊Dic7—21no—
133C4⋊Dic7—21no—
134C4⋊Dic7—22no—
135C4⋊Dic7—22no—
136C4⋊Dic7C22×C412yes—
137C4⋊Dic7C22×C412yes—
138Dic7⋊C4—82no—
139C4×Dic7—82no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—28no—
142C2×C7⋊C8—22no—
143C2×C7⋊C8—22no—
144C2×C7⋊C8—22no—
145C2×C7⋊C8C22×C412yes—
146C2×C7⋊C8C22×C412yes—
147C2×C7⋊C8#4114yes—
148C23×C8—71no—
149C22×C4⋊C4—71no—
150C22×C4⋊C4C2×D711yescharacteristic
151C2×#107—78no—
152C2×#107—71no—
153C2×#107—71no—
154C2×#107—71no—
155C2×#107—71no—
156C2×#106—72no—
157C2×C22⋊C8—71no—
158C2×#97—72no—
159#389—72no—
160#389C2×D712yes—
161#387—22no—
162#387—142no—
163#380—21no—
164#380—71no—
165#380—141no—
166#380C2×D711yescharacteristic
167C4×C4⋊C4—72no—
168C4×C4⋊C4C2×D712yes—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—141no—
171C4×C22⋊C4C2×D711yescharacteristic
172#340—72no—
173#340—72no—
174#340—72no—
175#340—72no—
176#340—72no—
177#336—72no—
178C23×C7—22no—
179C23×C7—22no—
180C23×C7C4×D412yes—
181C23×C7#13712yes—
182C23×C7#13712yes—
183C23×C7C22×D411yescharacteristic
184C2×C4×C7—22no—
185C2×C4×C7—22no—
186C2×C4×C7—22no—
187C2×C4×C7—24no—
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—84no—
202C2×C4×C7C4×D412yes—
203C2×C4×C7C23×C411yescharacteristic
204C2×C4×C7C2×#4111yescharacteristic
205C2×C4×C7C2×#4112yes—
206C2×C4×C72−1+412yes—
207C2×Dic7—22no—
208C2×Dic7—22no—
209C2×Dic7—42no—
210C2×Dic7—42no—
211C2×Dic7—42no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7—84no—
215C7⋊C8—22no—
216C7⋊C8—42no—
217C7⋊C8#11714yes—
218C23×C4—21no—
219C23×C4—141no—
220C23×C4C22×D711yescharacteristic
221C22×C8—71no—
222C22×C8—74no—
223C22×C8—71no—
224C22×C8—72no—
225C22×C8—144no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—74no—
230C2×C4⋊C4—71no—
231C2×C4⋊C4—71no—
232C2×C4⋊C4—72no—
233C2×C4⋊C4—142no—
234C2×C4⋊C4—141no—
235C2×C4⋊C4—142no—
236C2×C4⋊C4C4×D714yes—
237C2×C4⋊C4C22×D712yes—
238C2×C4⋊C4C22×D711yescharacteristic
239C2×C4⋊C4C22×D711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—141no—
242C2×C22⋊C4C22⋊D711yescharacteristic
243C2×C22⋊C4C22⋊D711yescharacteristic
244C2×C42—22no—
245C2×C42—142no—
246C2×C42C22×D712yes—
247#107—74no—
248#107—74no—
249#107—74no—
250#107—74no—
251#106—144no—
252C22⋊C8—142no—
253#97—144no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—142no—
258#95—142no—
259#95—141no—
260#95C22⋊D712yes—
261C22×C7—24no—
262C22×C7—24no—
263C22×C7—22no—
264C22×C7—22no—
265C22×C7#44114yes—
266C22×C7C2×C4×D411yescharacteristic
267C22×C7#54712yes—
268C22×C7C2×#13711yescharacteristic
269C22×C7C2×#13711yescharacteristic
270C22×C7#57612yes—
271C4×C7—22no—
272C4×C7—42no—
273C4×C7—42no—
274C4×C7—42no—
275C4×C7—42no—
276C4×C7—82no—
277C4×C7C2×#11711yescharacteristic
278C4×C7#51911yescharacteristic
279C4×C7#55212yes—
280Dic7—42no—
281Dic7—42no—
282Dic7—82no—
283C24C2×C22⋊D711yescharacteristic
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—82no—
291C22×C4—141no—
292C22×C4—141no—
293C22×C4—144no—
294C22×C4—142no—
295C22×C4—562no—
296C22×C4C2×C4×D712yes—
297C22×C4#115712yes—
298C22×C4C2×C22⋊D712yes—
299C22×C4C2×C22⋊D712yes—
300C22×C4C2×C22⋊D711yescharacteristic
301C22×C4C23×D711yescharacteristic
302C2×C8—72no—
303C2×C8—74no—
304C2×C8—72no—
305C2×C8—148no—
306C2×C8—142no—
307C2×C8—142no—
308C2×C8—142no—
309C2×C8—142no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—72no—
316C4⋊C4—72no—
317C4⋊C4—82no—
318C4⋊C4—144no—
319C4⋊C4—141no—
320C4⋊C4—141no—
321C4⋊C4—142no—
322C4⋊C4—142no—
323C4⋊C4—562no—
324C4⋊C4C2×C4×D712yes—
325C4⋊C4C2×C4×D712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—42no—
329C22⋊C4—282no—
330C42—22no—
331C42—22no—
332C42—82no—
333C42—562no—
334C2×C7—24no—
335C2×C7#234411yescharacteristic
336C2×C7C2×#44111yescharacteristic
337C2×C7#294811yescharacteristic
338C2×C7#334612yes—
339C2×C7#334612yes—
340C23—22no—
341C23—22no—
342C23C4×C22⋊D712yes—
343C23#650512yes—
344C23#651412yes—
345C23C22×C22⋊D711yescharacteristic
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—84no—
364C2×C4—142no—
365C2×C4—142no—
366C2×C4—282no—
367C2×C4—282no—
368C2×C4—282no—
369C2×C4—282no—
370C2×C4—282no—
371C2×C4—564no—
372C2×C4C4×C22⋊D712yes—
373C2×C4C22×C4×D711yescharacteristic
374C2×C4C2×#115711yescharacteristic
375C2×C4D7×#4112yes—
376C2×C4#655412yes—
377C8—74no—
378C8—142no—
379C8—282no—
380C7#2261811yescharacteristic, complemented, Sylow
381C22—22no—
382C22—22no—
383C22—24no—
384C22—24no—
385C22#8980214yes—
386C22C2×C4×C22⋊D711yescharacteristic
387C22C2×#650511yescharacteristic
388C22C2×#651411yescharacteristic
389C22#9051512yes—
390C22#9052612yes—
391C4—22no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—82no—
397C4—282no—
398C4—282no—
399C4—562no—
400C4#9020711yescharacteristic
401C4D7×#11711yescharacteristic
402C4#9022212yes—
403C2—24no—
404C2#1173968311yescharacteristic
405C2C2×#8980211yescharacteristic
406C2#1173998311yescharacteristic
407C2#1174105412yes—
408C2#1174107012yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2345C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#2345D7
hypocenterC7#22618
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
5. #11727374C2semidirect
6. #11727354C2semidirect
9. #11727235C2semidirect
10. #11727212C2semidirect
25. C2×#6381C4semidirect
380. C7#22618semidirect

Maximal subgroups

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

subgroupindex
C7×#23452
C22×#63812
#117309022
#117293082
#117273742
#117273542
#117272652
#117272562
#117272352
#117272122
#117272012
#226187