Finite Group Databasebeta

← #11731520

Subgroups — #11731520

Order 896.

624 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6492C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
3C2×#6449C211yesmaximal, characteristic
4C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
5C7×#381C211yesmaximal, characteristic, complemented
6#89715C211yesmaximal, characteristic, complemented
7#89714C211yesmaximal, characteristic, complemented
8#89447C212yesmaximal, complemented
9#89439C212yesmaximal, complemented
10#89429C212yesmaximal, complemented
11#89419C212yesmaximal, complemented
12C23×C4×C7C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C23×Dic7C2211yescharacteristic
16C22×C4×D7C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C22⋊Dic7C2211yescharacteristic
20#6492—24no—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4—24no—
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×C4×Dic7C2211yescharacteristic
32#6449—24no—
33(C22⋊D7)⋊C4—24no—
34C7×#95C2212yes—
35C7×#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2486—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C22×C4×C7—22no—
43C22×C4×C7—22no—
44C22×C4×C7—22no—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2312yes—
48C22×C4×C7C2312yes—
49C2×C22⋊D7—22no—
50C2×C22⋊D7—22no—
51C2×C22⋊D7D414yes—
52C22×Dic7—24no—
53C22×Dic7—22no—
54C22×Dic7—22no—
55C22×Dic7—22no—
56C22×Dic7D414yes—
57C22×Dic7C2312yes—
58C22×Dic7C2312yes—
59C22×Dic7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C2×C4×D7—22no—
63C2×C4×D7—24no—
64C7×C22⋊C4—22no—
65C7×C22⋊C4—22no—
66C22⋊Dic7—22no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—24no—
69(C2×D7)⋊C4—22no—
70C4⋊Dic7—24no—
71Dic7⋊C4—22no—
72Dic7⋊C4—24no—
73Dic7⋊C4—24no—
74Dic7⋊C4—24no—
75Dic7⋊C4—24no—
76C4×Dic7—22no—
77C2×#123—71no—
78C2×#123—71no—
79C2×C4×D4—71no—
80C22×C4⋊C4—71no—
81#398—72no—
82#397—72no—
83#389—72no—
84#381—71no—
85#381D711yescharacteristic, complemented
86#380—71no—
87#380—72no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7C2×D412yes—
91C23×C7C2×D412yes—
92C23×C7#4112yes—
93C23×C7C2411yescharacteristic
94C22×D7—21no—
95C22×D7—22no—
96C22×D7—21no—
97C22×D7—22no—
98C22×D7C2×D412yes—
99C2×C4×C7—21no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—44no—
110C22⋊D7—28no—
111C2×Dic7—21no—
112C2×Dic7—22no—
113C2×Dic7—21no—
114C2×Dic7—28no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—44no—
122C2×Dic7C2×D412yes—
123C2×Dic7C2×D412yes—
124C2×Dic7C2×D412yes—
125C2×Dic7#4114yes—
126C4×D7—44no—
127C22×D4—71no—
128C23×C4—71no—
129C23×C4—71no—
130C23×C4—71no—
131C23×C4C2×D711yescharacteristic
132#123—144no—
133#123—144no—
134C4×D4—144no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—72no—
141C2×C4⋊C4—144no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4C2×D711yescharacteristic
147C2×C22⋊C4C2×D711yescharacteristic
148C2×C42—71no—
149#95—72no—
150#95—72no—
151#95—72no—
152#95C2×D712yes—
153#95C2×D712yes—
154C22×C7—22no—
155C22×C7—22no—
156C22×C7—24no—
157C22×C7—24no—
158C22×C7#12114yes—
159C22×C7C22×D411yescharacteristic
160C22×C7C22×D411yescharacteristic
161C22×C7C2×#4111yescharacteristic
162C22×C7C2×#4112yes—
163C22×C72−1+412yes—
164C2×D7—22no—
165C2×D7—24no—
166C2×D7—41no—
167C2×D7—41no—
168C2×D7—42no—
169C4×C7—42no—
170C4×C7—41no—
171C4×C7—41no—
172C4×C7—42no—
173Dic7—22no—
174Dic7—22no—
175Dic7—42no—
176Dic7—42no—
177Dic7#12314yescomplemented
178C24—71no—
179C24C22×D711yescharacteristic
180C2×D4—74no—
181C2×D4—142no—
182C2×D4—142no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—71no—
187C22×C4—74no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—72no—
191C22×C4—72no—
192C22×C4—142no—
193C22×C4—144no—
194C22×C4—144no—
195C22×C4—142no—
196C22×C4—142no—
197C22×C4—142no—
198C22×C4C22×D711yescharacteristic
199C22×C4C22×D711yescharacteristic
200C22×C4C22×D712yes—
201C22×C4C22×D712yes—
202C4⋊C4—144no—
203C4⋊C4—144no—
204C4⋊C4—144no—
205C4⋊C4—142no—
206C4⋊C4—144no—
207C4⋊C4—144no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—142no—
211C22⋊C4—144no—
212C22⋊C4—144no—
213C22⋊C4—142no—
214C42—142no—
215C2×C7—24no—
216C2×C7C2×#12111yescharacteristic
217C2×C7C2×#12311yescharacteristic
218C2×C7#53611yescharacteristic
219C2×C7#54712yes—
220C2×C7#55212yes—
221D7—42no—
222C23—22no—
223C23—22no—
224C23—72no—
225C23—141no—
226C23—141no—
227C23—142no—
228C23—142no—
229C23#115712yes—
230C23D4×D712yes—
231C23D4×D712yes—
232C23C23×D711yescharacteristic
233D4—148no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—72no—
246C2×C4—74no—
247C2×C4—72no—
248C2×C4—72no—
249C2×C4—148no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—284no—
260C2×C4—284no—
261C7#248611yescharacteristic, complemented, Sylow
262C22—24no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—142no—
267C22—144no—
268C22—282no—
269C22—281no—
270C22—281no—
271C22#644714yes—
272C22C2×#115711yescharacteristic
273C22C2×D4×D711yescharacteristic
274C22C2×D4×D711yescharacteristic
275C22D7×#4112yes—
276C22#655412yes—
277C4—42no—
278C4—41no—
279C4—41no—
280C4—42no—
281C4—74no—
282C4—142no—
283C4—142no—
284C4—282no—
285C4—282no—
286C2—24no—
287C2—282no—
288C2C2×#644711yescharacteristic
289C2#9022212yes—
290C2#9033911yescharacteristic
291C2D7×#12311yescharacteristic
292C2#9034612yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC22×C22⋊D7C22
hypercenter#381D7
hypocenterC7#2486
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C4C2semidirect
5. C7×#381C2semidirect
6. #89715C2semidirect
7. #89714C2semidirect
8. #89447C2semidirect
9. #89439C2semidirect
10. #89429C2semidirect
11. #89419C2semidirect
85. #381D7semidirect
177. Dic7#123semidirect
261. C7#2486semidirect

Maximal subgroups

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

subgroupindex
C2×#64922
C22×Dic7⋊C42
C2×#64492
C2×(C22⋊D7)⋊C42
C7×#3812
#897152
#897142
#894472
#894392
#894292
#894192
#24867