Finite Group Databasebeta

← #13256788

Subgroups — #13256788

Order 1792.

640 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 109 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×#2646C211yesmaximal, characteristic
2#11727279C212yesmaximal
3#11727178C211yesmaximal, characteristic
4#11725430C212yesmaximal
5#11725419C211yesmaximal, characteristic
6C23×Dic14C2211yescharacteristic
7C7×#400—22no—
8C7×#398—22no—
9C7×#385—22no—
10C7×#380C2212yes—
11C2×C7×#95C2211yescharacteristic
12C2×C7⋊#95—22no—
13C2×C7⋊#95C2211yescharacteristic
14C2×C22⋊(C7⋊C8)C2212yes—
15C2×#6384—22no—
16C2×#6384—22no—
17#10345—71nomaximal, Sylow
18C23×C4×C7D412yes—
19C23×C4×C7C2311yescharacteristic
20C23×Dic7—21no—
21C23×Dic7—21no—
22C22×Dic14—22no—
23C22×Dic14—22no—
24C22×Dic14—21no—
25C22×Dic14—44no—
26C22×Dic14Q812yes—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C22⋊C4—22no—
32C22×C7⋊C8—22no—
33C7×#95—22no—
34C7×#95—22no—
35C7×#95—22no—
36C7×#95—22no—
37C7⋊#95—22no—
38C7⋊#95—44no—
39C22⋊(C7⋊C8)—22no—
40C22⋊(C7⋊C8)—22no—
41#6384—44no—
42#6384—44no—
43#2646D711yescharacteristic
44#1787—72no—
45#1446—71no—
46#1294—72no—
47#1280—71no—
48C24×C7C2×D411yescharacteristic
49C22×C4×C7—22no—
50C22×C4×C7—22no—
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—42no—
55C22×C4×C7—42no—
56C22×C4×C7—42no—
57C22×C4×C7C2×D412yes—
58C22×C4×C7C2×Q811yescharacteristic
59C22×C4×C7#4112yes—
60C22×C4×C7#4111yescharacteristic
61C22×Dic7—24no—
62C22×Dic7—22no—
63C22×Dic7—22no—
64C22×Dic7—42no—
65C22×Dic7—41no—
66C22×Dic7—41no—
67C22×Dic7—42no—
68C22×Dic7—41no—
69C22×Dic7—41no—
70C22×Dic7—42no—
71C2×Dic14—24no—
72C2×Dic14—24no—
73C2×Dic14—24no—
74C2×Dic14—44no—
75C2×Dic14—48no—
76C2×Dic14—44no—
77C2×Dic14—42no—
78C2×Dic14—42no—
79C2×Dic14—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—82no—
83C7×C4⋊C4—82no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C2×C7⋊C8—42no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C23×Q8—71no—
90C2×#103—142no—
91C2×#103—142no—
92C2×C22⋊C8—72no—
93#400—22no—
94#398—22no—
95#385—22no—
96#380C2×D712yes—
97C2×#95—71no—
98C2×#95—142no—
99C2×#95C2×D711yescharacteristic
100C23×C7—21no—
101C23×C7—21no—
102C23×C7—22no—
103C23×C7#12011yescharacteristic
104C23×C7C22⋊Q811yescharacteristic
105C23×C7#12411yescharacteristic
106C23×C7#13512yes—
107C23×C7#13712yes—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—41no—
111C2×C4×C7—41no—
112C2×C4×C7—42no—
113C2×C4×C7—44no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—82no—
124C2×C4×C7C22⋊Q812yes—
125C2×C4×C7#12412yes—
126C2×Dic7—24no—
127C2×Dic7—42no—
128C2×Dic7—44no—
129C2×Dic7—42no—
130C2×Dic7—44no—
131C2×Dic7—44no—
132C2×Dic7—42no—
133C2×Dic7—82no—
134C2×Dic7—81no—
135C2×Dic7—81no—
136Dic14—28no—
137Dic14—44no—
138Dic14—44no—
139Dic14—44no—
140C7⋊C8—82no—
141C22×Q8—72no—
142C22×Q8—141no—
143C22×Q8—142no—
144C22×Q8—142no—
145C22×Q8—284no—
146C23×C4—141no—
147C23×C4—141no—
148C23×C4C22⋊D712yes—
149C23×C4C22×D711yescharacteristic
150C22×C8—142no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D712yes—
154C2×C4⋊C4C22⋊D712yes—
155C2×C22⋊C4—22no—
156#103—284no—
157#103—284no—
158C22⋊C8—142no—
159C22⋊C8—142no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—142no—
165#95—284no—
166C22×C7—22no—
167C22×C7—22no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7#39311yescharacteristic
175C22×C7#45212yes—
176C22×C7#45511yescharacteristic
177C22×C7#45611yescharacteristic
178C22×C7B(4,𝔽2)12yes—
179C22×C7#47912yes—
180C22×C7#48811yescharacteristic
181C22×C7#49111yescharacteristic
182C4×C7—22no—
183C4×C7—22no—
184C4×C7—82no—
185C4×C7—82no—
186C4×C7—82no—
187Dic7—44no—
188Dic7—82no—
189C24C2×C22⋊D711yescharacteristic
190C2×Q8—144no—
191C2×Q8—144no—
192C2×Q8—144no—
193C2×Q8—282no—
194C2×Q8—282no—
195C2×Q8—284no—
196C2×Q8—284no—
197C2×Q8—288no—
198C2×Q8—282no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—144no—
208C22×C4—142no—
209C22×C4—142no—
210C22×C4—282no—
211C22×C4—281no—
212C22×C4—281no—
213C22×C4—282no—
214C22×C4—281no—
215C22×C4—281no—
216C22×C4—282no—
217C22×C4Q8×D711yescharacteristic
218C22×C4#116112yes—
219C22×C4#116111yescharacteristic
220C22×C4C2×C22⋊D712yes—
221C2×C8—282no—
222C2×C8—282no—
223C2×C8—282no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C2×C7—21no—
231C2×C7—22no—
232C2×C7—21no—
233C2×C7#160312yes—
234C2×C7#160612yes—
235C2×C7#202611yescharacteristic
236C2×C7#202811yescharacteristic
237C2×C7#202911yescharacteristic
238C23—21no—
239C23—22no—
240C23—21no—
241C23#650911yescharacteristic
242C23#651011yescharacteristic
243C23#651312yes—
244C23#651412yes—
245C23#651611yescharacteristic
246Q8—148no—
247Q8—284no—
248Q8—284no—
249Q8—284no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—144no—
267C2×C4—282no—
268C2×C4—284no—
269C2×C4—282no—
270C2×C4—284no—
271C2×C4—284no—
272C2×C4—282no—
273C2×C4—562no—
274C2×C4—561no—
275C2×C4—561no—
276C2×C4#650912yes—
277C2×C4#651012yes—
278C8—562no—
279C7#1034511yescharacteristic, complemented, Sylow
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22#8983512yes—
289C22#8984011yescharacteristic
290C22#8984111yescharacteristic
291C22#9000311yescharacteristic
292C22#9001112yes—
293C22#9001411yescharacteristic
294C22#9001612yes—
295C22#9001811yescharacteristic
296C4—22no—
297C4—22no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—284no—
302C4—562no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2#1172949811yescharacteristic
307C2#1172963912yes—
308C2#1172964312yes—
309C2#1172965111yescharacteristic
310C2#1172965911yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#2646C2
socleC23×C7#120
prime radicalC24×C7C2×D4
hypercenter#2646D7
hypocenterC7#10345
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
279. C7#10345semidirect

Maximal subgroups

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

subgroupindex
C7×#26462
#117272792
#117271782
#117254302
#117254192
#103457