Finite Group Databasebeta

← #13251121

Subgroups — #13251121

Order 1792.

548 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2973C211yesmaximal, characteristic
2#11739830C211yesmaximal, characteristic
3#11729373C211yesmaximal, characteristic
4#11727700C212yesmaximal, complemented
5#11725481C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487C2211yescharacteristic
8C7×#435—24no—
9C7×#432—21no—
10C7×#407—21no—
11C2×C7×C22⋊C8C2212yes—
12#89992—21no—
13C2×#6408—21no—
14#89984C2212yes—
15#89982—21no—
16C2×C7⋊#95C2211yescharacteristic
17#89758C411yescharacteristic, complemented
18#89751C411yescharacteristic, complemented
19#89721C411yescharacteristic, complemented
20#89719C411yescharacteristic, complemented
21#89714—21no—
22#89705—21no—
23#89321—21no—
24#89262—24no—
25#9385—71nomaximal, Sylow
26C22×C7×D4D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7C2311yescharacteristic
29C23×Dic7—21no—
30C22×Dic14C2×C411yescharacteristic
31C2×C7×M16—22no—
32C22×C7×C8—22no—
33C4×C7×D4—22no—
34C4×C7×D4—22no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C22⋊Dic7—21no—
40C2×C22⋊Dic7—21no—
41C2×C22⋊Dic7—22no—
42C2×C22⋊Dic7—21no—
43C2×C4⋊Dic7—22no—
44C2×C4⋊Dic7C2×C411yescharacteristic
45#6487—22no—
46#6487—22no—
47C2×Dic7⋊C4—21no—
48C2×Dic7⋊C4—21no—
49C2×Dic7⋊C4—21no—
50C7×C4⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×C22⋊C8—22no—
53C7×C22⋊C8D414yes—
54C7×#97—22no—
55#6408—42no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—22no—
62C7⋊#95C2×C411yescharacteristic
63C7⋊#95C2×C411yescharacteristic
64#2973D711yescharacteristic
65#2564—71no—
66#1799—71no—
67#1520—72no—
68#1294—72no—
69C24×C7C22⋊C411yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×D4—22no—
72C2×C7×D4—21no—
73C2×C7×D4—21no—
74C2×C7×D4—22no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—81no—
78C22×C4×C7—81no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D412yes—
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7#4112yes—
86C22×Dic7—21no—
87C22×Dic7—21no—
88C22×Dic7—21no—
89C22×Dic7—21no—
90C22×Dic7—42no—
91C22×Dic7—41no—
92C22×Dic7—41no—
93C22×Dic7—42no—
94C22×Dic7—81no—
95C2×Dic14—22no—
96C2×Dic14—81no—
97C7×M16—82no—
98C2×C7×C8—24no—
99C2×C7×C8—82no—
100C7×C4⋊C4—22no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C42×C7—21no—
104C42×C7—21no—
105C22⋊Dic7—42no—
106C22⋊Dic7—42no—
107C22⋊Dic7—42no—
108C22⋊Dic7—42no—
109C22⋊Dic7—81no—
110C4⋊Dic7—22no—
111C4⋊Dic7—82no—
112Dic7⋊C4—42no—
113Dic7⋊C4—41no—
114Dic7⋊C4—41no—
115Dic7⋊C4—81no—
116C2×C22⋊Q8—71no—
117C2×C4×D4C2×D711yescharacteristic
118#435—24no—
119#432—21no—
120C2×#99—141no—
121#407—21no—
122C2×C22⋊C8C2×D712yes—
123#400—71no—
124#399—71no—
125#395—71no—
126#393—71no—
127#387—141no—
128#386—141no—
129#385—141no—
130#380—72no—
131#380—141no—
132C2×#95—71no—
133#342—141no—
134#328—144no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7C2×C22⋊C411yescharacteristic
143C23×C7#12112yes—
144C23×C7AGL(1,ℤ/8)11yescharacteristic
145C23×C7#13711yescharacteristic
146C23×C7#13712yes—
147C7×D4—42no—
148C7×D4—42no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—81no—
162C2×C4×C7C4×D412yes—
163C2×C4×C7#12011yescharacteristic
164C2×C4×C7#12311yescharacteristic
165C2×Dic7—22no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—82no—
178C2×Dic7—82no—
179C2×Dic7—82no—
180C2×Dic7—82no—
181C2×Dic7—82no—
182Dic14—82no—
183C7×C8—44no—
184C22×Q8—71no—
185C22×D4D2811yescharacteristic
186C23×C4—141no—
187C23×C4C22⋊D711yescharacteristic
188C23×C4C22×D711yescharacteristic
189C2×M16—22no—
190C22×C8—22no—
191C22⋊Q8—142no—
192C22⋊Q8—142no—
193C4×D4—22no—
194C4×D4—22no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—141no—
199C2×C4⋊C4—142no—
200C2×C4⋊C4C4×D711yescharacteristic
201C2×C22⋊C4—141no—
202C2×C22⋊C4—141no—
203C2×C22⋊C4—142no—
204C2×C22⋊C4—141no—
205C2×C22⋊C4C4×D711yescharacteristic
206C2×C22⋊C4D2811yescharacteristic
207C2×C42C22⋊D711yescharacteristic
208C4⋊C8—22no—
209#99—282no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8C22⋊D714yes—
213#97—22no—
214#95—71no—
215#95—71no—
216#95—141no—
217#95—141no—
218#95—141no—
219#95—141no—
220#95—141no—
221#95—142no—
222C22×C7—21no—
223C22×C7—22no—
224C22×C7—21no—
225C22×C7—22no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—81no—
234C22×C7#38611yescharacteristic
235C22×C7#41711yescharacteristic
236C22×C7#41911yescharacteristic
237C22×C7#46812yes—
238C22×C7#47012yes—
239C4×C7—21no—
240C4×C7—22no—
241C4×C7—21no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245Dic7—42no—
246Dic7—81no—
247Dic7—81no—
248Dic7—81no—
249Dic7—162no—
250C24(C2×D7)⋊C411yescharacteristic
251C24C2×D2811yescharacteristic
252C2×Q8—142no—
253C2×Q8—561no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—141no—
263C22×C4—141no—
264C22×C4—141no—
265C22×C4—141no—
266C22×C4—282no—
267C22×C4—281no—
268C22×C4—281no—
269C22×C4—282no—
270C22×C4—561no—
271C22×C4(C2×D7)⋊C411yescharacteristic
272C22×C4(C2×D7)⋊C411yescharacteristic
273C22×C4(C2×D7)⋊C411yescharacteristic
274C22×C4C2×C4×D711yescharacteristic
275C22×C4#115712yes—
276C22×C4C2×C22⋊D712yes—
277C22×C4C2×C22⋊D711yescharacteristic
278M16—82no—
279C2×C8—24no—
280C2×C8—82no—
281C4⋊C4—22no—
282C4⋊C4—142no—
283C4⋊C4—282no—
284C4⋊C4—281no—
285C4⋊C4—281no—
286C4⋊C4—561no—
287C4⋊C4—562no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—282no—
291C22⋊C4—282no—
292C22⋊C4—282no—
293C22⋊C4—282no—
294C22⋊C4—561no—
295C42—21no—
296C42—21no—
297C2×C7—21no—
298C2×C7—22no—
299C2×C7—21no—
300C2×C7—22no—
301C2×C7—81no—
302C2×C7#188411yescharacteristic
303C2×C7#189411yescharacteristic
304C2×C7#191311yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23#647111yescharacteristic
313C23#647212yes—
314C23#647211yescharacteristic
315C23C2×(C2×D7)⋊C411yescharacteristic
316C23#649312yes—
317Q8—562no—
318D4—42no—
319D4—42no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—142no—
334C2×C4—281no—
335C2×C4—281no—
336C2×C4—281no—
337C2×C4—281no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4—281no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4—281no—
345C2×C4—562no—
346C2×C4—562no—
347C2×C4—562no—
348C2×C4—562no—
349C2×C4—562no—
350C2×C4C4×C22⋊D712yes—
351C2×C4#649211yescharacteristic
352C2×C4#651611yescharacteristic
353C8—44no—
354C7#938511yescharacteristic, complemented, Sylow
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#8989211yescharacteristic
368C22#8990511yescharacteristic
369C22#8990612yes—
370C22#8990912yes—
371C22#8998511yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—282no—
379C4—561no—
380C4—561no—
381C4—561no—
382C4—1122no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#1172941011yescharacteristic
389C2#1172974311yescharacteristic
390C2#1172977511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2973C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2973D7
hypocenterC7#9385
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #11727700C2semidirect
5. #11725481C2semidirect
17. #89758C4semidirect
18. #89751C4semidirect
19. #89721C4semidirect
20. #89719C4semidirect
354. C7#9385semidirect

Maximal subgroups

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

subgroupindex
C7×#29732
#117398302
#117293732
#117277002
#117254812
#93857