Finite Group Databasebeta

← #13251575

Subgroups — #13251575

Order 1792.

716 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 119 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#11739794C211yesmaximal, characteristic
3#11729373C211yesmaximal, characteristic
4#11727717C212yesmaximal, complemented
5#11725482C212yesmaximal, complemented
6C23×Dic14C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C2×#6487—21no—
9C7×#435—24no—
10C7×#432—21no—
11C7×#407—21no—
12C2×C7×C22⋊C8C2212yes—
13#89992—21no—
14C2×#6408—21no—
15#89984C2212yes—
16#89984C2211yescharacteristic
17C2×#6390—24no—
18#89751C412yescomplemented
19#89719C412yescomplemented
20#89718—21no—
21#89701—21no—
22#89321—21no—
23#9471—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7—21no—
28C22×Dic14—22no—
29C22×Dic14—21no—
30C22×Dic14—22no—
31C22×Dic14—44no—
32C22×Dic14C2×C412yes—
33C2×C7×M16—22no—
34C22×C7×C8D414yes—
35C4×C7×D4—22no—
36C4×C7×D4—22no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C22⋊Dic7—22no—
42C2×C22⋊Dic7—21no—
43C2×C22⋊Dic7—22no—
44C2×C4⋊Dic7—21no—
45C2×C4⋊Dic7—22no—
46#6487—44no—
47C2×Dic7⋊C4—22no—
48C7×C4⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#97—22no—
53#6408—42no—
54C7⋊#95—22no—
55C7⋊#95—22no—
56C7⋊#95—22no—
57C7⋊#95C2×C412yes—
58#6390—48no—
59#2973D711yescharacteristic
60#2409—71no—
61#1799—71no—
62#1787—72no—
63#1289—72no—
64C24×C7C22⋊C411yescharacteristic
65C24×C7C2×D411yescharacteristic
66C2×C7×D4—22no—
67C2×C7×D4—21no—
68C2×C7×D4—21no—
69C2×C7×D4—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D412yes—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7#4112yes—
81C22×Dic7—22no—
82C22×Dic7—22no—
83C22×Dic7—44no—
84C22×Dic7—42no—
85C22×Dic7—42no—
86C2×Dic14—28no—
87C2×Dic14—24no—
88C2×Dic14—24no—
89C2×Dic14—44no—
90C2×Dic14—48no—
91C2×Dic14—44no—
92C2×Dic14—42no—
93C2×Dic14—41no—
94C2×Dic14—41no—
95C7×M16—82no—
96C2×C7×C8—24no—
97C2×C7×C8—44no—
98C7×C4⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—22no—
101C42×C7—21no—
102C42×C7—21no—
103C22⋊Dic7—44no—
104C22⋊Dic7—41no—
105C22⋊Dic7—41no—
106C22⋊Dic7—42no—
107C22⋊Dic7—42no—
108C4⋊Dic7—42no—
109C4⋊Dic7—44no—
110Dic7⋊C4—44no—
111C23×Q8—71no—
112C2×C22⋊Q8—141no—
113C2×C4×D4C2×D711yescharacteristic
114#435—24no—
115#432—21no—
116C2×#103—144no—
117C2×#99—141no—
118#407—21no—
119C2×C22⋊C8C2×D712yes—
120#395—72no—
121#393—72no—
122#391—141no—
123#386—141no—
124#385—141no—
125#380—71no—
126#380—72no—
127#342—141no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7C2×C22⋊C411yescharacteristic
136C23×C7#12112yes—
137C23×C7AGL(1,ℤ/8)11yescharacteristic
138C23×C7#13711yescharacteristic
139C23×C7#13712yes—
140C7×D4—42no—
141C7×D4—42no—
142C2×C4×C7—24no—
143C2×C4×C7—22no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—81no—
155C2×C4×C7C4×D412yes—
156C2×C4×C7#12011yescharacteristic
157C2×C4×C7#12311yescharacteristic
158C2×Dic7—24no—
159C2×Dic7—48no—
160C2×Dic7—42no—
161C2×Dic7—42no—
162C2×Dic7—42no—
163C2×Dic7—42no—
164C2×Dic7—42no—
165C2×Dic7—82no—
166C2×Dic7—82no—
167C2×Dic7—82no—
168Dic14—28no—
169Dic14—44no—
170Dic14—44no—
171Dic14—44no—
172C7×C8—44no—
173C22×Q8—72no—
174C22×Q8—141no—
175C22×Q8—142no—
176C22×Q8—142no—
177C22×Q8—284no—
178C22×D4D2811yescharacteristic
179C23×C4—141no—
180C23×C4C22⋊D711yescharacteristic
181C23×C4C22×D711yescharacteristic
182C2×M16—22no—
183C22×C8C22⋊D714yes—
184C22⋊Q8—284no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—142no—
188C2×C4⋊C4—141no—
189C2×C4⋊C4—142no—
190C2×C4⋊C4C4×D711yescharacteristic
191C2×C22⋊C4—142no—
192C2×C22⋊C4—141no—
193C2×C22⋊C4—142no—
194C2×C22⋊C4C4×D711yescharacteristic
195C2×C22⋊C4D2811yescharacteristic
196C2×C42C22⋊D711yescharacteristic
197C4⋊C8—22no—
198#103—288no—
199#99—282no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203#97—22no—
204#95—72no—
205#95—142no—
206#95—142no—
207#95—142no—
208C22×C7—21no—
209C22×C7—22no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—42no—
215C22×C7—42no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—81no—
220C22×C7#38611yescharacteristic
221C22×C7#41711yescharacteristic
222C22×C7#41911yescharacteristic
223C22×C7#46812yes—
224C22×C7#47012yes—
225C4×C7—22no—
226C4×C7—21no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7B(4,𝔽2)12yes—
231Dic7—44no—
232Dic7—82no—
233Dic7—162no—
234C24(C2×D7)⋊C411yescharacteristic
235C24C2×D2811yescharacteristic
236C2×Q8—144no—
237C2×Q8—148no—
238C2×Q8—144no—
239C2×Q8—284no—
240C2×Q8—282no—
241C2×Q8—284no—
242C2×Q8—288no—
243C2×Q8—281no—
244C2×Q8—281no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—142no—
254C22×C4—142no—
255C22×C4—284no—
256C22×C4—282no—
257C22×C4—282no—
258C22×C4(C2×D7)⋊C411yescharacteristic
259C22×C4(C2×D7)⋊C411yescharacteristic
260C22×C4(C2×D7)⋊C411yescharacteristic
261C22×C4C2×C4×D711yescharacteristic
262C22×C4#115712yes—
263C22×C4C2×C22⋊D712yes—
264C22×C4C2×C22⋊D711yescharacteristic
265M16—82no—
266C2×C8—24no—
267C2×C8—44no—
268C4⋊C4—22no—
269C4⋊C4—282no—
270C4⋊C4—284no—
271C4⋊C4—284no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—284no—
275C22⋊C4—281no—
276C22⋊C4—281no—
277C22⋊C4—282no—
278C22⋊C4—282no—
279C42—21no—
280C42—21no—
281C2×C7—21no—
282C2×C7—22no—
283C2×C7—21no—
284C2×C7—22no—
285C2×C7—81no—
286C2×C7#188911yescharacteristic
287C2×C7#189411yescharacteristic
288C2×C7#190911yescharacteristic
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#647111yescharacteristic
297C23#647212yes—
298C23#647211yescharacteristic
299C23C2×(C2×D7)⋊C411yescharacteristic
300C23#649312yes—
301Q8—148no—
302Q8—284no—
303Q8—284no—
304Q8—284no—
305D4—42no—
306D4—42no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—144no—
321C2×C4—282no—
322C2×C4—288no—
323C2×C4—282no—
324C2×C4—282no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4—562no—
328C2×C4—562no—
329C2×C4—562no—
330C2×C4C4×C22⋊D712yes—
331C2×C4#649211yescharacteristic
332C2×C4#651611yescharacteristic
333C8—44no—
334C7#947111yescharacteristic, complemented, Sylow
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#8989211yescharacteristic
348C22#8990511yescharacteristic
349C22#8990612yes—
350C22#8990912yes—
351C22#8998511yescharacteristic
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—284no—
358C4—562no—
359C4—1122no—
360C4#9001612yes—
361C2—21no—
362C2—22no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#1172940811yescharacteristic
367C2#1172974311yescharacteristic
368C2#1172977811yescharacteristic
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#9471
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. #11727717C2semidirect
5. #11725482C2semidirect
18. #89751C4semidirect
19. #89719C4semidirect
334. C7#9471semidirect

Maximal subgroups

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

subgroupindex
C7×#29732
#117397942
#117293732
#117277172
#117254822
#94717