Finite Group Databasebeta

← #13329901

Subgroups — #13329901

Order 1792.

1630 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 195 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
1C2×C7×D42C211yesmaximal, characteristic
2#11740604C211yesmaximal, characteristic, complemented
3#11739760C212yesmaximal, complemented
4#11729662C218yesmaximal, complemented
5#11729514C211yesmaximal, characteristic, complemented
6#11729353C212yesmaximal, complemented
7C23×C7×D4—22no—
8C7×D42—22no—
9C7×D42—24no—
10C7×D42C2214yes—
11C2×C7×#127C2211yescharacteristic
12C2×C7×#121—22no—
13C2×C7×#120—22no—
14C2×C4×C7×D4C2212yes—
15#90161C2211yescharacteristic, complemented
16#89990—22no—
17#89987—22no—
18#89843—22no—
19#89842C2214yescomplemented
20#89791C2214yescomplemented
21#89790—24no—
22#89779C2214yescomplemented
23#89742—22no—
24#89698C2212yescomplemented
25C4×#1136C2211yescharacteristic, complemented
26#89339C2214yescomplemented
27D4⋊(C7⋊C8)C2218yescomplemented
28#18529—71nomaximal, Sylow
29C25×C7—22no—
30C22×C7×D4—24no—
31C22×C7×D4—28no—
32C22×C7×D4—28no—
33C22×C7×D4—22no—
34C22×C7×D4—416no—
35C22×C7×D4D412yes—
36C22×C7×D4D412yes—
37C23×C4×C7—22no—
38C7×#127—22no—
39C7×#127D412yes—
40C7×#127C2312yes—
41C7×#121—28no—
42C7×#121—28no—
43C7×#120—28no—
44C7×#120—28no—
45C4×C7×D4—22no—
46C4×C7×D4—22no—
47C4×C7×D4D414yes—
48C4×C7×D4C2314yes—
49C2×C7×C4⋊C4D412yes—
50C2×C7×C22⋊C4—22no—
51C2×C7×C22⋊C4—22no—
52C2×C42×C7C2311yescharacteristic
53C2×#1142—24no—
54#6489—21no—
55#6487—22no—
56C2×#1136—21no—
57#6433—21no—
58#6432C2312yescomplemented
59C4×Dic14—22no—
60#6406—28no—
61#6406—24no—
62C22⋊(C7⋊C8)—22no—
63#6384—24no—
64#6381—24no—
65C4⋊(C7⋊C8)C2314yescomplemented
66#6377—21no—
67#6377—22no—
68C4×C7⋊C8C2312yescomplemented
69C2×D42D711yescharacteristic
70#3138—71no—
71#3110—72no—
72#1639—78no—
73#1555—71no—
74#1490—72no—
75C24×C7—22no—
76C24×C7—28no—
77C24×C7—22no—
78C24×C7—416no—
79C24×C7—84no—
80C2×C7×D4—28no—
81C2×C7×D4—216no—
82C2×C7×D4—22no—
83C2×C7×D4—24no—
84C2×C7×D4—22no—
85C2×C7×D4—24no—
86C2×C7×D4—48no—
87C2×C7×D4—44no—
88C2×C7×D4—432no—
89C2×C7×D4—416no—
90C2×C7×D4—416no—
91C2×C7×D4—416no—
92C2×C7×D4C2×D414yes—
93C2×C7×D4C2×D414yes—
94C22×C4×C7—22no—
95C22×C4×C7—24no—
96C22×C4×C7—24no—
97C22×C4×C7—48no—
98C22×C4×C7C2×D412yes—
99C22×C4×C7C2×D411yescharacteristic
100C2×Dic14—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4C2×D414yes—
103C7×C22⋊C4—28no—
104C7×C22⋊C4—24no—
105C7×C22⋊C4—24no—
106C42×C7C2×D412yes—
107C42×C7C2×D411yescharacteristic
108C42×C7C2411yescharacteristic
109C22⋊Dic7—41no—
110#1142—48no—
111C4⋊Dic7—22no—
112C4⋊Dic7—22no—
113Dic7⋊C4—42no—
114C4×Dic7—41no—
115#1136—42no—
116C2×C7⋊C8—24no—
117C2×C7⋊C8—22no—
118C23×D4—22no—
119D42—22no—
120D42—24no—
121D42C2×D714yes—
122#533—71no—
123C2×#127C2×D711yescharacteristic
124C2×#121—22no—
125C2×#120—22no—
126C2×C4×D4C2×D712yes—
127#490—142no—
128#486—74no—
129#484—142no—
130#474—74no—
131#461—74no—
132#450—142no—
133#440—144no—
134#432—72no—
135#418—142no—
136C4×M16—71no—
137#331—74no—
138D4⋊C8—78no—
139C23×C7—28no—
140C23×C7—24no—
141C23×C7—24no—
142C23×C7—44no—
143C23×C7—48no—
144C23×C7—432no—
145C23×C7—44no—
146C23×C7—44no—
147C23×C7—88no—
148C23×C7—832no—
149C23×C7#12011yescharacteristic
150C7×D4—28no—
151C7×D4—24no—
152C7×D4—44no—
153C7×D4—48no—
154C7×D4—432no—
155C7×D4—48no—
156C7×D4—82no—
157C7×D4AGL(1,ℤ/8)18yes—
158C2×C4×C7—22no—
159C2×C4×C7—24no—
160C2×C4×C7—24no—
161C2×C4×C7—48no—
162C2×C4×C7—44no—
163C2×C4×C7—48no—
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12012yes—
166C2×C4×C7C22×D411yescharacteristic
167C2×C4×C7C22×D412yes—
168C2×Dic7—42no—
169Dic14—44no—
170C7⋊C8—42no—
171C7⋊C8—44no—
172C25—22no—
173C22×D4—24no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—416no—
178C22×D4C22⋊D712yes—
179C22×D4C22⋊D712yes—
180C23×C4—22no—
181C2×SD16—144no—
182C2×M16—141no—
183C4⋊Q8—72no—
184#127—22no—
185#127C22⋊D712yes—
186#127C22×D712yes—
187#125—141no—
188C22⋊Q8—142no—
189#121—28no—
190#121—28no—
191#120—28no—
192#120—28no—
193C4×Q8—142no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4C22⋊D714yes—
197C4×D4C22×D714yes—
198#117—141no—
199C2×C4⋊C4C22⋊D712yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42C22×D711yescharacteristic
203#107—144no—
204C4⋊C8—74no—
205#103—144no—
206#102—148no—
207#102—144no—
208C22⋊C8—142no—
209#97—141no—
210#97—142no—
211C4×C8—72no—
212C22×C7—22no—
213C22×C7—28no—
214C22×C7—416no—
215C22×C7—44no—
216C22×C7—44no—
217C22×C7—48no—
218C22×C7—44no—
219C22×C7—88no—
220C22×C7—832no—
221C22×C7#45512yes—
222C22×C7C2×#12011yescharacteristic
223C4×C7—22no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—44no—
227C4×C7C2×AGL(1,ℤ/8)14yes—
228Dic7—82no—
229C24—22no—
230C24—28no—
231C24—22no—
232C24—416no—
233C24—84no—
234C2×Q8—142no—
235C2×D4—28no—
236C2×D4—216no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—48no—
242C2×D4—44no—
243C2×D4—432no—
244C2×D4—416no—
245C2×D4—416no—
246C2×D4—416no—
247C2×D4C2×C22⋊D714yes—
248C2×D4C2×C22⋊D714yes—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—48no—
253C22×C4C2×C22⋊D712yes—
254C22×C4C2×C22⋊D711yescharacteristic
255SD16—288no—
256M16—282no—
257C2×C8—142no—
258C2×C8—144no—
259C4⋊C4—22no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—282no—
263C4⋊C4C2×C22⋊D714yes—
264C22⋊C4—28no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—281no—
268C42—281no—
269C42C2×C22⋊D711yescharacteristic
270C42C2×C22⋊D712yes—
271C42C23×D711yescharacteristic
272C2×C7—22no—
273C2×C7—48no—
274C2×C7—84no—
275C2×C7#300312yes—
276C2×C7C2×#45511yescharacteristic
277C23—28no—
278C23—24no—
279C23—24no—
280C23—44no—
281C23—48no—
282C23—432no—
283C23—44no—
284C23—44no—
285C23—88no—
286C23—832no—
287C23#651611yescharacteristic
288Q8—284no—
289D4—28no—
290D4—24no—
291D4—44no—
292D4—48no—
293D4—432no—
294D4—48no—
295D4—82no—
296D4#649518yes—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—282no—
304C2×C4#651611yescharacteristic
305C2×C4#651612yes—
306C2×C4C22×C22⋊D711yescharacteristic
307C2×C4C22×C22⋊D712yes—
308C8—282no—
309C8—284no—
310C7#1852911yescharacteristic, complemented, Sylow
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—88no—
319C22—832no—
320C22#9001412yes—
321C22C2×#651611yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—44no—
326C4—562no—
327C4C2×#649514yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2#1174148312yes—
332C2C2×#9001411yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×D42C2
socleC22×C7C2×#120
hypercenterC2×D42D7
hypocenterC7#18529
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
2. #11740604C2semidirect
3. #11739760C2semidirect
4. #11729662C2semidirect
5. #11729514C2semidirect
6. #11729353C2semidirect
15. #90161C22semidirect
19. #89842C22semidirect
20. #89791C22semidirect
22. #89779C22semidirect
24. #89698C22semidirect
25. C4×#1136C22semidirect
26. #89339C22semidirect
27. D4⋊(C7⋊C8)C22semidirect
58. #6432C23semidirect
65. C4⋊(C7⋊C8)C23semidirect
68. C4×C7⋊C8C23semidirect
310. C7#18529semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D422
#117406042
#117397602
#117296622
#117295142
#117293532
#185297