Finite Group Databasebeta

← #13251741

Subgroups — #13251741

Order 1792.

878 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2408C211yesmaximal, characteristic
2#11739749C211yesmaximal, characteristic, complemented
3#11729883C212yesmaximal, complemented
4#11729356C211yesmaximal, characteristic, complemented
5#11725432C212yesmaximal, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#121—21no—
8C2×C7×#120C412yes—
9C2×C4×C7×D4C2211yescharacteristic
10C7×#394—21no—
11C7×#392—21no—
12C7×#390—21no—
13C7×#380C2211yescharacteristic
14C7×#379C412yes—
15C2×#6407—24no—
16#89979—21no—
17#89979C2212yes—
18#89979C2212yes—
19#89784—24no—
20#89697—21no—
21#89323—21no—
22#89320—21no—
23#9504—71nomaximal, Sylow
24C25×C7C2×C412yes—
25C22×C7×D4—22no—
26C22×C7×D4—22no—
27C22×C7×D4—21no—
28C22×C7×D4—21no—
29C22×C7×D4—22no—
30C22×C7×D4—44no—
31C22×C7×D4C2×C411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7C2311yescharacteristic
34C7×#121—44no—
35C7×#120—44no—
36C4×C7×D4—22no—
37C4×C7×D4—22no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4C2×C412yes—
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×#1136—22no—
48C2×#1136—22no—
49C2×#1136D414yescomplemented
50C7×#95—21no—
51#6407—48no—
52C22⋊(C7⋊C8)—22no—
53C22⋊(C7⋊C8)—22no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—22no—
58C4×C7⋊C8—22no—
59#2975—71no—
60#2408D711yescharacteristic
61#1783—72no—
62#1465—71no—
63#1305—72no—
64C24×C7—22no—
65C24×C7—24no—
66C24×C7—44no—
67C24×C7—42no—
68C24×C7—42no—
69C24×C7—82no—
70C24×C7C22⋊C411yescharacteristic
71C24×C7C22×C411yescharacteristic
72C2×C7×D4—28no—
73C2×C7×D4—24no—
74C2×C7×D4—22no—
75C2×C7×D4—22no—
76C2×C7×D4—41no—
77C2×C7×D4—44no—
78C2×C7×D4—42no—
79C2×C7×D4—44no—
80C2×C7×D4—48no—
81C2×C7×D4—44no—
82C2×C7×D4—44no—
83C2×C7×D4—41no—
84C2×C7×D4—42no—
85C2×C7×D4—41no—
86C2×C7×D4—82no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—22no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7#4112yes—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—42no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—44no—
104C7×C22⋊C4—44no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—82no—
107C42×C7—21no—
108C42×C7—21no—
109#1136—44no—
110#1136—44no—
111#1136—82no—
112C2×C7⋊C8—24no—
113C2×C7⋊C8—42no—
114C23×D4C2×D711yescharacteristic
115C2×#121—21no—
116C2×#120Dic712yes—
117C2×C4×D4C2×D711yescharacteristic
118#436—144no—
119#431—141no—
120C2×#100—144no—
121#407—72no—
122#407—72no—
123#407—141no—
124#394—21no—
125#392—21no—
126#390—21no—
127#380C2×D711yescharacteristic
128#379Dic712yes—
129#324—141no—
130#323—141no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—24no—
134C23×C7—28no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—42no—
143C23×C7—42no—
144C23×C7—44no—
145C23×C7—42no—
146C23×C7—42no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—84no—
150C23×C7—84no—
151C23×C7—88no—
152C23×C7—88no—
153C23×C7C2×C22⋊C411yescharacteristic
154C23×C7C4×D412yes—
155C7×D4—44no—
156C7×D4—48no—
157C7×D4—44no—
158C7×D4—44no—
159C7×D4—88no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—24no—
163C2×C4×C7—22no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—81no—
175C2×C4×C7#10012yescomplemented
176C2×C4×C7#12011yescharacteristic
177C2×C4×C7#12112yes—
178C2×C4×C7#12311yescharacteristic
179C7⋊C8—44no—
180C7⋊C8—82no—
181C25C2×Dic712yes—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—44no—
188C22×D4C2×Dic711yescharacteristic
189C23×C4C22⋊D711yescharacteristic
190C23×C4C22×D711yescharacteristic
191C2×M16—74no—
192C2×M16—142no—
193C2×M16—142no—
194#121—44no—
195#120—44no—
196C4×D4—22no—
197C4×D4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22⋊D711yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C2×Dic711yescharacteristic
204C2×C22⋊C4C2×Dic712yes—
205C2×C22⋊C4C22⋊D711yescharacteristic
206C2×C42C22⋊D711yescharacteristic
207C4⋊C8—142no—
208#100—288no—
209C22⋊C8—142no—
210C22⋊C8—142no—
211C22⋊C8—142no—
212C22⋊C8—142no—
213C22⋊C8—142no—
214C4×C8—142no—
215#95—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—42no—
222C22×C7—42no—
223C22×C7—42no—
224C22×C7—44no—
225C22×C7—44no—
226C22×C7—41no—
227C22×C7—44no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—81no—
231C22×C7—82no—
232C22×C7—82no—
233C22×C7—88no—
234C22×C7—88no—
235C22×C7—82no—
236C22×C7—82no—
237C22×C7#38611yescharacteristic
238C22×C7C2×#10011yescharacteristic
239C22×C7#41311yescharacteristic
240C4×C7—22no—
241C4×C7—21no—
242C4×C7—22no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—82no—
246C4×C7B(4,𝔽2)12yes—
247C24—22no—
248C24—24no—
249C24—44no—
250C24—42no—
251C24—42no—
252C24—82no—
253C24C22⋊Dic711yescharacteristic
254C24C22×Dic711yescharacteristic
255C2×D4—28no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C22⋊Dic711yescharacteristic
276C22×C4C22⋊Dic711yescharacteristic
277C22×C4C22⋊Dic711yescharacteristic
278C22×C4D4×D712yes—
279C22×C4#115912yes—
280C22×C4C2×C22⋊D711yescharacteristic
281C22×C4C2×C22⋊D711yescharacteristic
282M16—284no—
283M16—284no—
284M16—562no—
285C2×C8—144no—
286C2×C8—282no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C22⋊C4—82no—
295C42—21no—
296C42—21no—
297C2×C7—21no—
298C2×C7—21no—
299C2×C7—42no—
300C2×C7—81no—
301C2×C7—84no—
302C2×C7#188911yescharacteristic
303C2×C7#190011yescharacteristic
304C2×C7#191411yescharacteristic
305C23—21no—
306C23—21no—
307C23—24no—
308C23—28no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—84no—
324C23—84no—
325C23—88no—
326C23—88no—
327C23D4×Dic712yes—
328C23C2×C22⋊Dic711yescharacteristic
329D4—44no—
330D4—48no—
331D4—44no—
332D4—44no—
333D4—88no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4#640712yescomplemented
350C2×C4#649811yescharacteristic
351C2×C4#650011yescharacteristic
352C2×C4#650212yes—
353C8—284no—
354C8—562no—
355C7#950411yescharacteristic, complemented, Sylow
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—44no—
365C22—44no—
366C22—41no—
367C22—44no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—88no—
374C22—88no—
375C22—82no—
376C22—82no—
377C22C2×#640711yescharacteristic
378C22#8999211yescharacteristic
379C22#9000911yescharacteristic
380C4—22no—
381C4—21no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4#8980412yes—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—81no—
391C2—84no—
392C2#1172942311yescharacteristic
393C2#1172943711yescharacteristic
394C2#1172960911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC7×#2408C2
socleC22×C7#386
cosocleC23×C4×C7C23
hypercenter#2408D7
hypocenterC7#9504
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. #11739749C2semidirect
3. #11729883C2semidirect
4. #11729356C2semidirect
5. #11725432C2semidirect
49. C2×#1136D4semidirect
175. C2×C4×C7#100semidirect
349. C2×C4#6407semidirect
355. C7#9504semidirect

Maximal subgroups

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

subgroupindex
C7×#24082
#117397492
#117298832
#117293562
#117254322
#95047