Finite Group Databasebeta

← #13251586

Subgroups — #13251586

Order 1792.

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

normal subgroups only

This group has 556 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×#2487C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3#11729373C211yesmaximal, characteristic
4#11729308C212yesmaximal, complemented
5#11725431C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#400C412yescomplemented
9C7×#399C412yescomplemented
10C7×#389—21no—
11C7×#386—21no—
12C7×#385—21no—
13C7×#380—21no—
14C7×#380C2211yescharacteristic
15C2×#6408—21no—
16#89984C2212yes—
17#89979—21no—
18C2×C22⋊(C7⋊C8)C2212yes—
19#89783—24no—
20#89698—21no—
21#89324—24no—
22#89321—21no—
23#9473—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7D411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C2×C22⋊Dic7—22no—
43C2×C4⋊Dic7—22no—
44C2×#1136—22no—
45C22×C7⋊C8D414yes—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95C2×C412yes—
49#6408—42no—
50C7⋊#95—22no—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53C22⋊(C7⋊C8)—22no—
54C4⋊(C7⋊C8)—22no—
55#6377—22no—
56#2973—71no—
57#2487D711yescharacteristic
58#1824—72no—
59#1799—71no—
60#1289—72no—
61C24×C7C22⋊C411yescharacteristic
62C24×C7C2×D411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—21no—
65C2×C7×D4—21no—
66C2×C7×D4—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7—82no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D412yes—
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7#4112yes—
82C22×Dic7—42no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—24no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—82no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—82no—
94C42×C7—21no—
95C42×C7—21no—
96C22⋊Dic7—42no—
97C22⋊Dic7—42no—
98C4⋊Dic7—44no—
99#1136—82no—
100C2×C7⋊C8—24no—
101C2×C7⋊C8—44no—
102C2×C4×D4C2×D711yescharacteristic
103C22×C4⋊C4C2×D711yescharacteristic
104#435—144no—
105#432—141no—
106C2×#99—141no—
107#407—141no—
108C2×C22⋊C8—72no—
109#400Dic712yescomplemented
110#399Dic712yescomplemented
111#389—21no—
112#386—21no—
113#385—21no—
114#380—21no—
115#380—72no—
116#380C2×D711yescharacteristic
117#342—141no—
118#340—144no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7C2×C22⋊C411yescharacteristic
127C23×C7#12112yes—
128C23×C7AGL(1,ℤ/8)11yescharacteristic
129C23×C7#13712yes—
130C23×C7#13711yescharacteristic
131C7×D4—42no—
132C7×D4—42no—
133C2×C4×C7—24no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—41no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—81no—
152C2×C4×C7—84no—
153C2×C4×C7C4×D412yes—
154C2×C4×C7#12011yescharacteristic
155C2×C4×C7#12311yescharacteristic
156C2×Dic7—82no—
157C2×Dic7—82no—
158C2×Dic7—82no—
159C7⋊C8—44no—
160C22×D4C22⋊D711yescharacteristic
161C23×C4—21no—
162C23×C4C22⋊D711yescharacteristic
163C23×C4C22×D711yescharacteristic
164C2×M16—142no—
165C22×C8—74no—
166C4×D4—22no—
167C4×D4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—142no—
174C2×C4⋊C4C2×Dic712yes—
175C2×C4⋊C4C2×Dic711yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—142no—
179C2×C22⋊C4C2×Dic711yescharacteristic
180C2×C22⋊C4C22⋊D711yescharacteristic
181C2×C42C22⋊D711yescharacteristic
182C4⋊C8—142no—
183#99—282no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186C22⋊C8—142no—
187#97—142no—
188#95—22no—
189#95—22no—
190#95—142no—
191#95C2×Dic712yes—
192C22×C7—22no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—21no—
196C22×C7—41no—
197C22×C7—42no—
198C22×C7—41no—
199C22×C7—42no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—81no—
204C22×C7#38611yescharacteristic
205C22×C7#41711yescharacteristic
206C22×C7#41911yescharacteristic
207C22×C7#46812yes—
208C22×C7#47012yes—
209C4×C7—21no—
210C4×C7—22no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—82no—
215C4×C7—82no—
216C4×C7#45812yes—
217Dic7—162no—
218C24C22⋊Dic711yescharacteristic
219C24C2×C22⋊D711yescharacteristic
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—82no—
232C22×C4—282no—
233C22×C4C22⋊Dic711yescharacteristic
234C22×C4C22⋊Dic711yescharacteristic
235C22×C4C22⋊Dic711yescharacteristic
236C22×C4D4×D712yes—
237C22×C4#115912yes—
238C22×C4C22×Dic711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240M16—562no—
241C2×C8—144no—
242C2×C8—284no—
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—82no—
249C4⋊C4—284no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—82no—
255C22⋊C4—282no—
256C22⋊C4—282no—
257C42—21no—
258C42—21no—
259C2×C7—22no—
260C2×C7—21no—
261C2×C7—21no—
262C2×C7—22no—
263C2×C7—81no—
264C2×C7#189011yescharacteristic
265C2×C7#189411yescharacteristic
266C2×C7#191011yescharacteristic
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#650212yes—
275C23#650512yes—
276C23#651211yescharacteristic
277C23#651411yescharacteristic
278C23C2×C22⋊Dic711yescharacteristic
279D4—42no—
280D4—42no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—84no—
301C2×C4—562no—
302C2×C4—562no—
303C2×C4—562no—
304C2×C4D4×Dic712yes—
305C2×C4#649811yescharacteristic
306C2×C4#650011yescharacteristic
307C8—284no—
308C7#947311yescharacteristic, complemented, Sylow
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#8982112yes—
322C22#8982412yes—
323C22#8999211yescharacteristic
324C22#8999711yescharacteristic
325C22#9000411yescharacteristic
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—1122no—
334C4#8982512yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#1172945111yescharacteristic
341C2#1172945811yescharacteristic
342C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2487C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2487D7
hypocenterC7#9473
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #11729308C2semidirect
5. #11725431C2semidirect
8. C7×#400C4semidirect
9. C7×#399C4semidirect
109. #400Dic7semidirect
110. #399Dic7semidirect
308. C7#9473semidirect

Maximal subgroups

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

subgroupindex
C7×#24872
#117397472
#117293732
#117293082
#117254312
#94737