Finite Group Databasebeta

← #13251581

Subgroups — #13251581

Order 1792.

912 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 123 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#11739747C211yesmaximal, characteristic, complemented
3#11729882C212yesmaximal, complemented
4#11729373C211yesmaximal, characteristic, complemented
5#11725431C212yesmaximal, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#121—21no—
8C2×C7×#120—21no—
9C2×C4×C7×D4C2211yescharacteristic
10C7×#394C412yescomplemented
11C7×#392C412yescomplemented
12C7×#390—21no—
13C7×#380C2211yescharacteristic
14C7×#379—21no—
15C2×#6408—21no—
16C2×#6406—24no—
17#89984C2212yes—
18#89979—21no—
19C2×C22⋊(C7⋊C8)C2212yes—
20#89783—24no—
21#89698—21no—
22#89321—21no—
23#9472—71nomaximal, Sylow
24C25×C7—21no—
25C22×C7×D4—22no—
26C22×C7×D4—22no—
27C22×C7×D4—21no—
28C22×C7×D4—22no—
29C22×C7×D4—44no—
30C22×C7×D4C2×C412yes—
31C22×C7×D4D411yescharacteristic
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⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4C2×C411yescharacteristic
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×C22⋊Dic7—22no—
48C2×C4⋊Dic7—22no—
49C2×#1136—22no—
50C22×C7⋊C8D414yescomplemented
51C7×#95C2×C412yes—
52#6408—42no—
53#6406—48no—
54C7⋊#95—22no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)—22no—
58C4⋊(C7⋊C8)—22no—
59#6377—22no—
60#2973—71no—
61#2408D711yescharacteristic
62#1799—71no—
63#1786—72no—
64#1289—72no—
65C24×C7—22no—
66C24×C7—44no—
67C24×C7—42no—
68C24×C7—42no—
69C24×C7—41no—
70C24×C7—41no—
71C24×C7—82no—
72C24×C7C22⋊C411yescharacteristic
73C24×C7C2×D411yescharacteristic
74C2×C7×D4—28no—
75C2×C7×D4—24no—
76C2×C7×D4—22no—
77C2×C7×D4—21no—
78C2×C7×D4—22no—
79C2×C7×D4—21no—
80C2×C7×D4—24no—
81C2×C7×D4—44no—
82C2×C7×D4—42no—
83C2×C7×D4—44no—
84C2×C7×D4—48no—
85C2×C7×D4—44no—
86C2×C7×D4—44no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—82no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C2×D412yes—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4112yes—
102C22×Dic7—42no—
103C7×C4⋊C4—22no—
104C7×C4⋊C4—42no—
105C7×C22⋊C4—22no—
106C7×C22⋊C4—22no—
107C7×C22⋊C4—44no—
108C7×C22⋊C4—44no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—82no—
111C42×C7—21no—
112C42×C7—21no—
113C22⋊Dic7—42no—
114C22⋊Dic7—42no—
115C4⋊Dic7—44no—
116#1136—82no—
117C2×C7⋊C8—24no—
118C2×C7⋊C8—44no—
119C23×D4C2×D711yescharacteristic
120C2×#121—21no—
121C2×#120—21no—
122C2×C4×D4C2×D711yescharacteristic
123#435—144no—
124#432—141no—
125C2×#102—144no—
126C2×#99—141no—
127#407—141no—
128C2×C22⋊C8—72no—
129#394Dic712yescomplemented
130#392Dic712yescomplemented
131#390—21no—
132#380—72no—
133#380C2×D711yescharacteristic
134#379—21no—
135#342—141no—
136C23×C7—24no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—42no—
145C23×C7—48no—
146C23×C7—42no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—42no—
150C23×C7—42no—
151C23×C7—84no—
152C23×C7—88no—
153C23×C7—84no—
154C23×C7—84no—
155C23×C7—82no—
156C23×C7—82no—
157C23×C7C2×C22⋊C411yescharacteristic
158C23×C7#12112yes—
159C23×C7AGL(1,ℤ/8)12yes—
160C23×C7AGL(1,ℤ/8)11yescharacteristic
161C23×C7#13711yescharacteristic
162C7×D4—28no—
163C7×D4—44no—
164C7×D4—44no—
165C7×D4—42no—
166C7×D4—42no—
167C7×D4—44no—
168C7×D4—88no—
169C2×C4×C7—24no—
170C2×C4×C7—22no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—81no—
185C2×C4×C7C4×D412yes—
186C2×C4×C7#12011yescharacteristic
187C2×C4×C7#12311yescharacteristic
188C2×Dic7—82no—
189C2×Dic7—82no—
190C2×Dic7—82no—
191C7⋊C8—44no—
192C25—21no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—44no—
198C22×D4C2×Dic712yes—
199C22×D4C22⋊D711yescharacteristic
200C23×C4C22⋊D711yescharacteristic
201C23×C4C22×D711yescharacteristic
202C2×M16—142no—
203C22×C8—74no—
204#121—44no—
205#120—44no—
206C4×D4—22no—
207C4×D4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—142no—
210C2×C4⋊C4C2×Dic711yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—142no—
216C2×C22⋊C4C2×Dic711yescharacteristic
217C2×C22⋊C4C22⋊D711yescharacteristic
218C2×C42C22⋊D711yescharacteristic
219C4⋊C8—142no—
220#102—288no—
221#99—282no—
222C22⋊C8—142no—
223C22⋊C8—142no—
224C22⋊C8—142no—
225#97—142no—
226#95—142no—
227#95C2×Dic712yes—
228C22×C7—22no—
229C22×C7—21no—
230C22×C7—22no—
231C22×C7—21no—
232C22×C7—42no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—42no—
236C22×C7—44no—
237C22×C7—44no—
238C22×C7—41no—
239C22×C7—44no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—81no—
243C22×C7—82no—
244C22×C7—82no—
245C22×C7—88no—
246C22×C7—88no—
247C22×C7—82no—
248C22×C7—82no—
249C22×C7#38611yescharacteristic
250C22×C7#41711yescharacteristic
251C22×C7#41811yescharacteristic
252C22×C7#46812yes—
253C22×C7#46912yes—
254C4×C7—21no—
255C4×C7—22no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7—82no—
260C4×C7B(4,𝔽2)12yes—
261Dic7—162no—
262C24—22no—
263C24—44no—
264C24—42no—
265C24—42no—
266C24—41no—
267C24—41no—
268C24—82no—
269C24C22⋊Dic711yescharacteristic
270C24C2×C22⋊D711yescharacteristic
271C2×D4—28no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—24no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—82no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—282no—
293C22×C4C22⋊Dic711yescharacteristic
294C22×C4C22⋊Dic711yescharacteristic
295C22×C4C22⋊Dic711yescharacteristic
296C22×C4D4×D712yes—
297C22×C4#115912yes—
298C22×C4C22×Dic711yescharacteristic
299C22×C4C2×C22⋊D711yescharacteristic
300M16—562no—
301C2×C8—144no—
302C2×C8—284no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—284no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—82no—
312C22⋊C4—282no—
313C22⋊C4—282no—
314C42—21no—
315C42—21no—
316C2×C7—22no—
317C2×C7—21no—
318C2×C7—21no—
319C2×C7—22no—
320C2×C7—81no—
321C2×C7—84no—
322C2×C7#188911yescharacteristic
323C2×C7#189511yescharacteristic
324C2×C7#191011yescharacteristic
325C23—24no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—48no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—84no—
341C23—88no—
342C23—84no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#649512yes—
347C23#650212yes—
348C23#651211yescharacteristic
349C23#651411yescharacteristic
350C23C2×C22⋊Dic711yescharacteristic
351D4—28no—
352D4—44no—
353D4—44no—
354D4—42no—
355D4—42no—
356D4—44no—
357D4—88no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—562no—
375C2×C4—562no—
376C2×C4—562no—
377C2×C4D4×Dic712yes—
378C2×C4#649811yescharacteristic
379C2×C4#650011yescharacteristic
380C8—284no—
381C7#947211yescharacteristic, complemented, Sylow
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—44no—
391C22—44no—
392C22—41no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—88no—
400C22—88no—
401C22—82no—
402C22—82no—
403C22#8981112yes—
404C22#8981412yes—
405C22#8998711yescharacteristic
406C22#8999211yescharacteristic
407C22#9000411yescharacteristic
408C4—21no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—82no—
414C4—1122no—
415C4#8980412yes—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—84no—
422C2#1172942311yescharacteristic
423C2#1172943611yescharacteristic
424C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC7×#2408C2
socleC22×C7#386
cosocleC23×C4×C7C23
hypercenter#2408D7
hypocenterC7#9472
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. #11739747C2semidirect
3. #11729882C2semidirect
4. #11729373C2semidirect
5. #11725431C2semidirect
10. C7×#394C4semidirect
11. C7×#392C4semidirect
50. C22×C7⋊C8D4semidirect
129. #394Dic7semidirect
130. #392Dic7semidirect
381. C7#9472semidirect

Maximal subgroups

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

subgroupindex
C7×#24082
#117397472
#117298822
#117293732
#117254312
#94727