Finite Group Databasebeta

← #13248843

Subgroups — #13248843

Order 1792.

768 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#89725C211yesmaximal, characteristic
2#11739539C211yesmaximal, characteristic
3C2×C7×#323C211yesmaximal, characteristic
4#11727678C212yesmaximal, complemented
5#11725481C212yesmaximal, complemented
6C22×C7×C22⋊C4C2211yescharacteristic
7C22×C22⋊Dic7—21no—
8C22×C22⋊Dic7—21no—
9C22×C4⋊Dic7C2211yescharacteristic
10C2×C7×C22⋊C8C2212yes—
11#89984—21no—
12#89984C2212yes—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2211yescharacteristic
15#89725—22no—
16#89725C414yescomplemented
17#89722—24no—
18C7×#323—22no—
19#89346—24no—
20#9025—71nomaximal, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C23×Dic7—22no—
27C22×C7×C8—22no—
28C2×C7×C22⋊C4—22no—
29C2×C7×C22⋊C4—22no—
30C2×C7×C22⋊C4C2×C412yes—
31C2×C7×C22⋊C4D412yes—
32C2×C22⋊Dic7—22no—
33C2×C22⋊Dic7—24no—
34C2×C22⋊Dic7—22no—
35C2×C22⋊Dic7—42no—
36C2×C22⋊Dic7—42no—
37C2×C22⋊Dic7—42no—
38C2×C22⋊Dic7—42no—
39C2×C4⋊Dic7—22no—
40C2×C4⋊Dic7—22no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7—22no—
43C2×C4⋊Dic7C2×C412yescomplemented
44C2×C4⋊Dic7D414yes—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7⋊#95—22no—
48C7⋊#95—24no—
49C7⋊#95—22no—
50C7⋊#95—21no—
51C7⋊#95C2×C412yescomplemented
52C2×#397—71no—
53#2315—71no—
54#1823—72no—
55C2×#323D711yescharacteristic
56#1294—72no—
57C24×C7—21no—
58C24×C7—21no—
59C24×C7—21no—
60C24×C7—21no—
61C24×C7—42no—
62C24×C7—42no—
63C24×C7C22⋊C412yes—
64C24×C7D812yes—
65C24×C7SD1612yes—
66C24×C7C2×D411yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D412yes—
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7#4112yes—
76C22×Dic7—22no—
77C22×Dic7—22no—
78C22×Dic7—24no—
79C22×Dic7—44no—
80C22×Dic7—42no—
81C22×Dic7—42no—
82C22×Dic7—41no—
83C22×Dic7—41no—
84C22×Dic7—41no—
85C22×Dic7—41no—
86C22×Dic7—41no—
87C22×Dic7—41no—
88C22×Dic7—41no—
89C22×Dic7—41no—
90C22×Dic7—82no—
91C22×Dic7—82no—
92C2×C7×C8—44no—
93C2×C7×C8—42no—
94C7×C22⋊C4—24no—
95C7×C22⋊C4—24no—
96C22⋊Dic7—44no—
97C22⋊Dic7—48no—
98C22⋊Dic7—42no—
99C22⋊Dic7—42no—
100C4⋊Dic7—24no—
101C4⋊Dic7—42no—
102C4⋊Dic7—44no—
103C4⋊Dic7—82no—
104C22×C4⋊C4—71no—
105C22×C22⋊C4—141no—
106C22×C22⋊C4—141no—
107C22×C22⋊C4C2×D711yescharacteristic
108C2×C22⋊C8C2×D712yes—
109#397—74no—
110#397—142no—
111#385—144no—
112#380—72no—
113#380—141no—
114C2×#95—71no—
115C2×#95—141no—
116#340—144no—
117#323—22no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—22no—
121C23×C7—22no—
122C23×C7—22no—
123C23×C7—22no—
124C23×C7—22no—
125C23×C7—22no—
126C23×C7—42no—
127C23×C7—42no—
128C23×C7—44no—
129C23×C7—44no—
130C23×C7—42no—
131C23×C7—42no—
132C23×C7—82no—
133C23×C7—82no—
134C23×C7—81no—
135C23×C7—81no—
136C23×C7#10214yes—
137C23×C7C2×C22⋊C411yescharacteristic
138C23×C7#12011yescharacteristic
139C23×C7#12311yescharacteristic
140C23×C7C2×D811yescharacteristic
141C23×C7C2×SD1611yescharacteristic
142C23×C7AGL(1,ℤ/8)11yescharacteristic
143C23×C7#13711yescharacteristic
144C2×C4×C7—24no—
145C2×C4×C7—22no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—81no—
152C2×C4×C7C4×D412yes—
153C2×C4×C7#12112yes—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159C2×Dic7—42no—
160C2×Dic7—44no—
161C2×Dic7—44no—
162C2×Dic7—44no—
163C2×Dic7—82no—
164C2×Dic7—82no—
165C2×Dic7—88no—
166C2×Dic7—81no—
167C2×Dic7—81no—
168C2×Dic7—81no—
169C2×Dic7—81no—
170C7×C8—82no—
171C25D2811yescharacteristic
172C23×C4—141no—
173C23×C4—141no—
174C23×C4—142no—
175C23×C4D2811yescharacteristic
176C23×C4C22×D711yescharacteristic
177C22×C8—22no—
178C2×C4⋊C4—72no—
179C2×C4⋊C4—74no—
180C2×C4⋊C4—142no—
181C2×C4⋊C4—142no—
182C2×C4⋊C4—141no—
183C2×C4⋊C4—142no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—142no—
187C2×C22⋊C4—144no—
188C2×C22⋊C4—142no—
189C2×C22⋊C4—282no—
190C2×C22⋊C4—282no—
191C2×C22⋊C4—282no—
192C2×C22⋊C4—282no—
193C2×C22⋊C4C4×D712yes—
194C2×C22⋊C4C22⋊D712yes—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—72no—
198#95—142no—
199#95—144no—
200#95—142no—
201#95—141no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—24no—
211C22×C7—44no—
212C22×C7—44no—
213C22×C7—44no—
214C22×C7—42no—
215C22×C7—42no—
216C22×C7—82no—
217C22×C7—82no—
218C22×C7—82no—
219C22×C7#38611yescharacteristic
220C22×C7C2×#10211yescharacteristic
221C22×C7#41711yescharacteristic
222C22×C7#44711yescharacteristic
223C22×C7#44811yescharacteristic
224C22×C7#45512yes—
225C22×C7#45812yes—
226C22×C7#48011yescharacteristic
227C22×C7#48111yescharacteristic
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—42no—
231Dic7—82no—
232Dic7—84no—
233Dic7—82no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—42no—
239C24—42no—
240C24#113212yes—
241C24D5612yes—
242C24(C2×D7)⋊C412yes—
243C24C2×D2811yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—144no—
249C22×C4—142no—
250C22×C4—142no—
251C22×C4—284no—
252C22×C4—282no—
253C22×C4—282no—
254C22×C4—281no—
255C22×C4—281no—
256C22×C4—281no—
257C22×C4—281no—
258C22×C4—281no—
259C22×C4—281no—
260C22×C4—281no—
261C22×C4—281no—
262C22×C4—562no—
263C22×C4—562no—
264C22×C4(C2×D7)⋊C412yes—
265C22×C4C2×C4×D711yescharacteristic
266C22×C4D4×D712yes—
267C22×C4#115912yes—
268C22×C4C2×C22⋊D711yescharacteristic
269C2×C8—44no—
270C2×C8—42no—
271C4⋊C4—144no—
272C4⋊C4—284no—
273C4⋊C4—282no—
274C4⋊C4—562no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—284no—
278C22⋊C4—282no—
279C22⋊C4—282no—
280C22⋊C4—288no—
281C2×C7—21no—
282C2×C7—21no—
283C2×C7—21no—
284C2×C7—21no—
285C2×C7—44no—
286C2×C7#160712yes—
287C2×C7#160812yes—
288C2×C7#187411yescharacteristic
289C2×C7#188511yescharacteristic
290C2×C7#189011yescharacteristic
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—82no—
306C23—82no—
307C23—81no—
308C23—81no—
309C23#639514yes—
310C23#644511yescharacteristic
311C23#644911yescharacteristic
312C23C2×#113211yescharacteristic
313C23C2×D5611yescharacteristic
314C23#647111yescharacteristic
315C23#647211yescharacteristic
316C23C2×(C2×D7)⋊C411yescharacteristic
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4—282no—
331C2×C4—284no—
332C2×C4—284no—
333C2×C4—284no—
334C2×C4—562no—
335C2×C4—562no—
336C2×C4—568no—
337C2×C4—561no—
338C2×C4—561no—
339C2×C4—561no—
340C2×C4—561no—
341C2×C4(C22⋊D7)⋊C412yes—
342C2×C4#650212yes—
343C8—82no—
344C7#902511yescharacteristic, complemented, Sylow
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—24no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22#8950411yescharacteristic
363C22#8950711yescharacteristic
364C22#8950811yescharacteristic
365C22#8951011yescharacteristic
366C22#8951412yes—
367C22#8951812yes—
368C22#8973011yescharacteristic
369C22C2×#639511yescharacteristic
370C22#8990511yescharacteristic
371C4—22no—
372C4—22no—
373C4—42no—
374C4—562no—
375C4—564no—
376C4—562no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#1172586412yes—
383C2#1172587012yes—
384C2#1172768611yescharacteristic
385C2#1172777711yescharacteristic
386C2#1172782311yescharacteristic
subgroupG/H
centerC23#6445
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#323C2
socleC23×C7#120
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenterC2×#323D7
hypocenterC7#9025
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
4. #11727678C2semidirect
5. #11725481C2semidirect
16. #89725C4semidirect
43. C2×C4⋊Dic7C2×C4semidirect
51. C7⋊#95C2×C4semidirect
344. C7#9025semidirect

Maximal subgroups

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

subgroupindex
C2×#897252
#117395392
C2×C7×#3232
#117276782
#117254812
#90257