Finite Group Databasebeta

← #13248783

Subgroups — #13248783

Order 1792.

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

normal subgroups only

This group has 908 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×#89719C211yesmaximal, characteristic
2#11739539C211yesmaximal, characteristic
3C2×C7×#323C211yesmaximal, characteristic
4#11727717C212yesmaximal, complemented
5#11725481C212yesmaximal, complemented
6C23×Dic14C2211yescharacteristic
7C22×C7×C22⋊C4C2211yescharacteristic
8C22×C22⋊Dic7—21no—
9C22×C22⋊Dic7—21no—
10C2×C7×C22⋊C8C2212yes—
11#89984—21no—
12#89984C2212yes—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2211yescharacteristic
15C2×#6390—24no—
16#89722—24no—
17#89719—22no—
18#89719C414yescomplemented
19C7×#323—22no—
20#9013—71nomaximal, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C23×Dic7—22no—
27C22×Dic14—22no—
28C22×Dic14—22no—
29C22×Dic14—21no—
30C22×Dic14—44no—
31C22×Dic14C2×C412yescomplemented
32C22×C7×C8—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4C2×C412yes—
36C2×C7×C22⋊C4D412yes—
37C2×C22⋊Dic7—22no—
38C2×C22⋊Dic7—24no—
39C2×C22⋊Dic7—22no—
40C2×C22⋊Dic7—44no—
41C2×C22⋊Dic7—42no—
42C2×C22⋊Dic7—42no—
43C2×C4⋊Dic7—22no—
44C2×C4⋊Dic7D414yes—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7⋊#95—22no—
48C7⋊#95—24no—
49C7⋊#95—21no—
50C7⋊#95—22no—
51C7⋊#95C2×C412yescomplemented
52#6390—48no—
53C2×#393—71no—
54#2315—71no—
55#1787—72no—
56C2×#323D711yescharacteristic
57#1294—72no—
58C24×C7—21no—
59C24×C7—21no—
60C24×C7—21no—
61C24×C7—21no—
62C24×C7—42no—
63C24×C7—42no—
64C24×C7C22⋊C412yes—
65C24×C7SD1612yes—
66C24×C7Q1612yes—
67C24×C7C2×D411yescharacteristic
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7#4112yes—
77C22×Dic7—24no—
78C22×Dic7—22no—
79C22×Dic7—22no—
80C22×Dic7—24no—
81C22×Dic7—42no—
82C22×Dic7—41no—
83C22×Dic7—41no—
84C22×Dic7—42no—
85C22×Dic7—44no—
86C22×Dic7—42no—
87C22×Dic7—41no—
88C22×Dic7—41no—
89C22×Dic7—82no—
90C22×Dic7—82no—
91C2×Dic14—28no—
92C2×Dic14—24no—
93C2×Dic14—44no—
94C2×Dic14—48no—
95C2×Dic14—44no—
96C2×Dic14—44no—
97C2×Dic14—42no—
98C2×C7×C8—44no—
99C2×C7×C8—42no—
100C7×C22⋊C4—24no—
101C7×C22⋊C4—24no—
102C22⋊Dic7—48no—
103C22⋊Dic7—44no—
104C22⋊Dic7—42no—
105C22⋊Dic7—42no—
106C4⋊Dic7—44no—
107C4⋊Dic7—82no—
108C23×Q8—71no—
109C22×C22⋊C4—141no—
110C22×C22⋊C4—141no—
111C22×C22⋊C4C2×D711yescharacteristic
112C2×#103—144no—
113C2×C22⋊C8C2×D712yes—
114#393—74no—
115#393—142no—
116#385—144no—
117#380—72no—
118#380—141no—
119C2×#95—71no—
120C2×#95—141no—
121#323—22no—
122C23×C7—22no—
123C23×C7—22no—
124C23×C7—22no—
125C23×C7—22no—
126C23×C7—22no—
127C23×C7—22no—
128C23×C7—22no—
129C23×C7—22no—
130C23×C7—42no—
131C23×C7—42no—
132C23×C7—44no—
133C23×C7—44no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—82no—
137C23×C7—82no—
138C23×C7—81no—
139C23×C7—81no—
140C23×C7#10314yescomplemented
141C23×C7C2×C22⋊C411yescharacteristic
142C23×C7#12011yescharacteristic
143C23×C7#12311yescharacteristic
144C23×C7C2×SD1611yescharacteristic
145C23×C7C2×Q1611yescharacteristic
146C23×C7AGL(1,ℤ/8)11yescharacteristic
147C23×C7#13711yescharacteristic
148C2×C4×C7—24no—
149C2×C4×C7—22no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—81no—
156C2×C4×C7C4×D412yes—
157C2×C4×C7#12112yes—
158C2×Dic7—24no—
159C2×Dic7—42no—
160C2×Dic7—44no—
161C2×Dic7—42no—
162C2×Dic7—44no—
163C2×Dic7—42no—
164C2×Dic7—42no—
165C2×Dic7—42no—
166C2×Dic7—44no—
167C2×Dic7—44no—
168C2×Dic7—44no—
169C2×Dic7—82no—
170C2×Dic7—82no—
171C2×Dic7—88no—
172Dic14—28no—
173Dic14—48no—
174Dic14—44no—
175C7×C8—82no—
176C25D2811yescharacteristic
177C22×Q8—72no—
178C22×Q8—141no—
179C22×Q8—142no—
180C22×Q8—142no—
181C22×Q8—284no—
182C23×C4—141no—
183C23×C4—141no—
184C23×C4—142no—
185C23×C4D2811yescharacteristic
186C23×C4C22×D711yescharacteristic
187C22×C8—22no—
188C2×C4⋊C4—74no—
189C2×C4⋊C4—142no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—142no—
193C2×C22⋊C4—144no—
194C2×C22⋊C4—142no—
195C2×C22⋊C4—284no—
196C2×C22⋊C4—282no—
197C2×C22⋊C4—282no—
198C2×C22⋊C4C4×D712yes—
199C2×C22⋊C4C22⋊D712yes—
200#103—288no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203#95—72no—
204#95—142no—
205#95—144no—
206#95—142no—
207#95—141no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—22no—
211C22×C7—22no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—24no—
217C22×C7—44no—
218C22×C7—44no—
219C22×C7—44no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—82no—
223C22×C7—82no—
224C22×C7—82no—
225C22×C7#38611yescharacteristic
226C22×C7C2×#10311yescharacteristic
227C22×C7#41711yescharacteristic
228C22×C7#45011yescharacteristic
229C22×C7#45111yescharacteristic
230C22×C7#45512yes—
231C22×C7B(4,𝔽2)12yes—
232C22×C7#48311yescharacteristic
233C22×C7#48411yescharacteristic
234C4×C7—22no—
235C4×C7—22no—
236C4×C7—42no—
237Dic7—44no—
238Dic7—82no—
239Dic7—84no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—42no—
245C24—42no—
246C24#113212yes—
247C24Dic2812yes—
248C24(C2×D7)⋊C412yes—
249C24C2×D2811yescharacteristic
250C2×Q8—148no—
251C2×Q8—144no—
252C2×Q8—284no—
253C2×Q8—284no—
254C2×Q8—284no—
255C2×Q8—288no—
256C2×Q8—282no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—144no—
262C22×C4—142no—
263C22×C4—144no—
264C22×C4—142no—
265C22×C4—282no—
266C22×C4—281no—
267C22×C4—281no—
268C22×C4—282no—
269C22×C4—284no—
270C22×C4—282no—
271C22×C4—281no—
272C22×C4—281no—
273C22×C4—562no—
274C22×C4—562no—
275C22×C4(C2×D7)⋊C412yes—
276C22×C4C2×C4×D711yescharacteristic
277C22×C4D4×D712yes—
278C22×C4#115912yes—
279C22×C4C2×C22⋊D711yescharacteristic
280C2×C8—44no—
281C2×C8—42no—
282C4⋊C4—284no—
283C4⋊C4—562no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—288no—
287C22⋊C4—284no—
288C22⋊C4—282no—
289C22⋊C4—282no—
290C2×C7—21no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—21no—
294C2×C7—44no—
295C2×C7#160112yes—
296C2×C7#160212yes—
297C2×C7#187211yescharacteristic
298C2×C7#188511yescharacteristic
299C2×C7#188911yescharacteristic
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—82no—
315C23—82no—
316C23—81no—
317C23—81no—
318C23#639014yescomplemented
319C23#644511yescharacteristic
320C23#644911yescharacteristic
321C23C2×#113211yescharacteristic
322C23C2×Dic2811yescharacteristic
323C23#647111yescharacteristic
324C23#647211yescharacteristic
325C23C2×(C2×D7)⋊C411yescharacteristic
326Q8—148no—
327Q8—288no—
328Q8—284no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—144no—
338C2×C4—282no—
339C2×C4—284no—
340C2×C4—282no—
341C2×C4—284no—
342C2×C4—282no—
343C2×C4—282no—
344C2×C4—282no—
345C2×C4—284no—
346C2×C4—284no—
347C2×C4—284no—
348C2×C4—562no—
349C2×C4—562no—
350C2×C4—568no—
351C2×C4(C22⋊D7)⋊C412yes—
352C2×C4#650212yes—
353C8—82no—
354C7#901311yescharacteristic, complemented, Sylow
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22#8949311yescharacteristic
373C22#8949411yescharacteristic
374C22#8950311yescharacteristic
375C22#8951111yescharacteristic
376C22#8951312yes—
377C22#8951812yes—
378C22#8973011yescharacteristic
379C22C2×#639011yescharacteristic
380C22#8990511yescharacteristic
381C4—22no—
382C4—22no—
383C4—42no—
384C4—284no—
385C4—562no—
386C4—564no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2#1172586212yes—
393C2#1172586912yes—
394C2#1172768411yescharacteristic
395C2#1172777611yescharacteristic
396C2#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#9013
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. #11727717C2semidirect
5. #11725481C2semidirect
18. #89719C4semidirect
31. C22×Dic14C2×C4semidirect
51. C7⋊#95C2×C4semidirect
140. C23×C7#103semidirect
318. C23#6390semidirect
354. C7#9013semidirect

Maximal subgroups

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

subgroupindex
C2×#897192
#117395392
C2×C7×#3232
#117277172
#117254812
#90137