Finite Group Databasebeta

← #13256791

Subgroups — #13256791

Order 1792.

640 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#11730805C211yesmaximal, characteristic
2C7×#1446C211yesmaximal, characteristic
3#11729886C212yesmaximal
4#11725419C211yesmaximal, characteristic
5#11725418C212yesmaximal
6C23×C7×Q8C2211yescharacteristic
7C2×C7×#95—22no—
8C2×C7×#95C2211yescharacteristic
9C2×C7×#95C2211yescharacteristic
10C2×#6409—22no—
11C2×#6409—22no—
12#89984C2212yes—
13#89982—22no—
14C2×C22⋊(C7⋊C8)C2212yes—
15#89432—22no—
16#89429—22no—
17#10345—71nomaximal, Sylow
18C22×C7×Q8—21no—
19C22×C7×Q8—22no—
20C22×C7×Q8—22no—
21C22×C7×Q8—44no—
22C22×C7×Q8Q812yes—
23C23×C4×C7—21no—
24C23×C4×C7—21no—
25C23×C4×C7D412yes—
26C23×C4×C7C2311yescharacteristic
27C2×C22⋊Dic7—22no—
28C2×C4⋊Dic7D412yes—
29C2×C4⋊Dic7D412yes—
30C2×Dic7⋊C4—22no—
31C2×Dic7⋊C4—22no—
32C22×C7⋊C8—22no—
33C7×#95—22no—
34C7×#95—22no—
35C7×#95—44no—
36#6409—44no—
37#6409—44no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40C7⋊#95—22no—
41C22⋊(C7⋊C8)—22no—
42C22⋊(C7⋊C8)—22no—
43#2646—71no—
44#1787—72no—
45#1446D711yescharacteristic
46#1294—72no—
47#1280—71no—
48C24×C7C2×D411yescharacteristic
49C2×C7×Q8—24no—
50C2×C7×Q8—24no—
51C2×C7×Q8—24no—
52C2×C7×Q8—44no—
53C2×C7×Q8—42no—
54C2×C7×Q8—42no—
55C2×C7×Q8—44no—
56C2×C7×Q8—48no—
57C2×C7×Q8—42no—
58C22×C4×C7—24no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—42no—
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×Q811yescharacteristic
76C22×C4×C7#4112yes—
77C22×C4×C7#4111yescharacteristic
78C22×Dic7—22no—
79C22×Dic7—22no—
80C22⋊Dic7—42no—
81C22⋊Dic7—42no—
82C4⋊Dic7—42no—
83C4⋊Dic7—42no—
84Dic7⋊C4—82no—
85Dic7⋊C4—82no—
86C2×C7⋊C8—42no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C23×Q8C2×D711yescharacteristic
90C2×#103—142no—
91C2×#103—142no—
92C2×C22⋊C8—72no—
93#400—142no—
94#398—142no—
95#385—142no—
96#380—72no—
97C2×#95—22no—
98C2×#95C2×D711yescharacteristic
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—
108C7×Q8—28no—
109C7×Q8—44no—
110C7×Q8—44no—
111C7×Q8—44no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—24no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—44no—
120C2×C4×C7—42no—
121C2×C4×C7—44no—
122C2×C4×C7—44no—
123C2×C4×C7—42no—
124C2×C4×C7—44no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—82no—
129C2×C4×C7—81no—
130C2×C4×C7—81no—
131C2×C4×C7—82no—
132C2×C4×C7C22⋊Q812yes—
133C2×C4×C7#12412yes—
134C2×Dic7—42no—
135C2×Dic7—42no—
136C2×Dic7—42no—
137C2×Dic7—42no—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C7⋊C8—82no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—44no—
145C22×Q8Dic1412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22⋊D712yes—
149C23×C4C22×D711yescharacteristic
150C22×C8—142no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—72no—
153C2×C4⋊C4—142no—
154C2×C4⋊C4—142no—
155C2×C22⋊C4—142no—
156#103—284no—
157#103—284no—
158C22⋊C8—142no—
159C22⋊C8—142no—
160#95—22no—
161#95—22no—
162#95—44no—
163#95—142no—
164#95—142no—
165#95—142no—
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—44no—
185C4×C7—82no—
186C4×C7—82no—
187Dic7—82no—
188Dic7—82no—
189C24C2×C22⋊D711yescharacteristic
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—48no—
198C2×Q8—42no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—142no—
216C22×C4—142no—
217C22×C4C2×Dic1411yescharacteristic
218C22×C4#115711yescharacteristic
219C22×C4D4×D712yes—
220C22×C4#115912yes—
221C2×C8—282no—
222C2×C8—282no—
223C2×C8—282no—
224C4⋊C4—282no—
225C4⋊C4—282no—
226C4⋊C4—562no—
227C4⋊C4—562no—
228C22⋊C4—282no—
229C22⋊C4—282no—
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#648711yescharacteristic
242C23#649911yescharacteristic
243C23#650011yescharacteristic
244C23#650512yes—
245C23#651312yes—
246Q8—28no—
247Q8—44no—
248Q8—44no—
249Q8—44no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—282no—
271C2×C4—282no—
272C2×C4—282no—
273C2×C4—282no—
274C2×C4—282no—
275C2×C4—282no—
276C2×C4#644112yes—
277C2×C4#644812yes—
278C8—562no—
279C7#1034511yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—21no—
287C22—21no—
288C22#8971911yescharacteristic
289C22#8979412yes—
290C22#8980412yes—
291C22#8981912yes—
292C22#8982811yescharacteristic
293C22#8983311yescharacteristic
294C22#8985411yescharacteristic
295C22#8986711yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—562no—
302C4—562no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2#1172725111yescharacteristic
307C2#1172742311yescharacteristic
308C2#1172742711yescharacteristic
309C2#1172745412yes—
310C2#1172745912yes—
subgroupG/H
centerC23#6500
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#1446C2
socleC23×C7#120
prime radicalC24×C7C2×D4
hypercenter#1446D7
hypocenterC7#10345
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

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
#117308052
C7×#14462
#117298862
#117254192
#117254182
#103457