Finite Group Databasebeta

← #11731900

Subgroups — #11731900

Order 896.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×(C2×D7)⋊C4C211yesmaximal, characteristic
2C2×D7×C22⋊C4C211yesmaximal, characteristic
3C7×#385C211yesmaximal, characteristic, complemented
4#89982C211yesmaximal, characteristic, complemented
5#89765C212yesmaximal, complemented
6#89761C212yesmaximal, complemented
7#89727C211yesmaximal, characteristic
8#89715C211yesmaximal, characteristic, complemented
9#89715C211yesmaximal, characteristic, complemented
10#89447C212yesmaximal, complemented
11#89436C212yesmaximal, complemented
12C24×D7C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C22×C4×D7C2211yescharacteristic
17C2×C7×C4⋊C4C2212yescomplemented
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊Dic7C2211yescharacteristic
22C2×(C2×D7)⋊C4—24no—
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×Dic7⋊C4C2212yescomplemented
30D7×C22⋊C4—24no—
31C7×#95C2212yescomplemented
32C7⋊#95C2212yescomplemented
33C7⋊#95C2212yescomplemented
34C7⋊#95C2212yescomplemented
35C7⋊#95C2212yescomplemented
36C7⋊#95C2212yescomplemented
37#2483—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7—22no—
40C23×D7—22no—
41C23×D7—24no—
42C23×D7—42no—
43C23×D7D414yes—
44C23×D7C2311yescharacteristic
45C23×D7C2311yescharacteristic
46C22×C4×C7—22no—
47C22×C4×C7—42no—
48C22×C4×C7C2311yescharacteristic
49C22×C4×C7C2312yes—
50C22×C4×C7C2311yescharacteristic
51C22×C4×C7C2312yes—
52C22×Dic7—22no—
53C22×Dic7—22no—
54C22×Dic7—22no—
55C22×Dic7C2312yes—
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2312yes—
58C22×Dic7C2311yescharacteristic
59C2×C4×D7—22no—
60C2×C4×D7—22no—
61C2×C4×D7—24no—
62C2×C4×D7—42no—
63C7×C4⋊C4—42no—
64C7×C22⋊C4—22no—
65C7×C22⋊C4—22no—
66C22⋊Dic7—22no—
67C22⋊Dic7—22no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—24no—
71(C2×D7)⋊C4—24no—
72(C2×D7)⋊C4—24no—
73(C2×D7)⋊C4—42no—
74Dic7⋊C4—42no—
75C22×C22⋊C4—71no—
76C22×C22⋊C4—71no—
77#397—72no—
78#397—72no—
79#385—71no—
80#385—72no—
81#385D711yescharacteristic, complemented
82#381—71no—
83#381—71no—
84#381—72no—
85#379—71no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7C2×D412yes—
89C23×C7#4112yes—
90C23×C7#4112yes—
91C23×C7C2411yescharacteristic
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—22no—
95C22×D7—28no—
96C22×D7—21no—
97C22×D7—21no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—28no—
101C22×D7—44no—
102C22×D7—48no—
103C22×D7—48no—
104C22×D7C2×D412yes—
105C22×D7C2×D412yes—
106C22×D7#4114yes—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—44no—
118C2×Dic7—22no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—22no—
128C2×Dic7—44no—
129C4×D7—44no—
130C4×D7—44no—
131C25—71no—
132C23×C4—71no—
133C23×C4—71no—
134C23×C4—71no—
135C23×C4C2×D711yescharacteristic
136C2×C4⋊C4—72no—
137C2×C4⋊C4C2×D712yescomplemented
138C2×C22⋊C4—71no—
139C2×C22⋊C4—71no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—71no—
142C2×C22⋊C4—72no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—144no—
147C2×C22⋊C4—144no—
148C2×C22⋊C4C2×D711yescharacteristic
149C2×C22⋊C4C2×D711yescharacteristic
150#95—72no—
151#95—72no—
152#95—72no—
153#95—72no—
154#95—72no—
155#95C2×D712yescomplemented
156C22×C7—22no—
157C22×C7—22no—
158C22×C7—24no—
159C22×C7—24no—
160C22×C7#12314yes—
161C22×C7C22×D411yescharacteristic
162C22×C7C2×#4111yescharacteristic
163C22×C7C2×#4111yescharacteristic
164C22×C7C2×#4112yes—
165C22×C7H(5,𝔽2)12yes—
166C2×D7—22no—
167C2×D7—22no—
168C2×D7—24no—
169C2×D7—24no—
170C2×D7—22no—
171C2×D7—22no—
172C2×D7—22no—
173C2×D7—416no—
174C2×D7—42no—
175C2×D7—42no—
176C2×D7#12314yes—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7—42no—
180C4×C7—42no—
181Dic7—41no—
182Dic7—42no—
183Dic7—41no—
184Dic7—42no—
185C24—71no—
186C24—71no—
187C24—74no—
188C24—144no—
189C24—142no—
190C24—142no—
191C24—282no—
192C24C22×D711yescharacteristic
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—72no—
196C22×C4—71no—
197C22×C4—72no—
198C22×C4—71no—
199C22×C4—142no—
200C22×C4—142no—
201C22×C4—144no—
202C22×C4—142no—
203C22×C4—142no—
204C22×C4—142no—
205C22×C4—282no—
206C22×C4C22×D711yescharacteristic
207C22×C4C22×D712yes—
208C22×C4C22×D711yescharacteristic
209C22×C4C22×D712yes—
210C4⋊C4—42no—
211C4⋊C4—282no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—142no—
215C22⋊C4—142no—
216C22⋊C4—144no—
217C22⋊C4—144no—
218C22⋊C4—144no—
219C22⋊C4—142no—
220C22⋊C4—142no—
221C22⋊C4—282no—
222C2×C7—24no—
223C2×C7C2×#12311yescharacteristic
224C2×C7C2×#12311yescharacteristic
225C2×C7#53411yescharacteristic
226C2×C7#55112yes—
227C2×C7#55112yes—
228D7—24no—
229D7—42no—
230C23—22no—
231C23—22no—
232C23—72no—
233C23—74no—
234C23—72no—
235C23—148no—
236C23—141no—
237C23—142no—
238C23—141no—
239C23—148no—
240C23—141no—
241C23—142no—
242C23—141no—
243C23—142no—
244C23—284no—
245C23—288no—
246C23—288no—
247C23#115712yes—
248C23D4×D712yes—
249C23#116112yes—
250C23C23×D711yescharacteristic
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—44no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—284no—
273C2×C4—284no—
274C2×C4—284no—
275C7#248311yescharacteristic, complemented, Sylow
276C22—24no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—74no—
281C22—142no—
282C22—144no—
283C22—142no—
284C22—144no—
285C22—142no—
286C22—142no—
287C22—142no—
288C22—2816no—
289C22—282no—
290C22—282no—
291C22#645714yes—
292C22C2×#115711yescharacteristic
293C22C2×D4×D711yescharacteristic
294C22#654812yes—
295C22C2×#116111yescharacteristic
296C22D7×#4112yes—
297C4—42no—
298C4—42no—
299C4—41no—
300C4—41no—
301C4—282no—
302C4—281no—
303C4—282no—
304C4—281no—
305C2—24no—
306C2—144no—
307C2—282no—
308C2C2×#645711yescharacteristic
309C2#9024712yes—
310C2#9028011yescharacteristic
311C2#9032812yes—
312C2D7×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×D7C22
hypercenter#385D7
hypocenterC7#2483
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C7×#385C2semidirect
4. #89982C2semidirect
5. #89765C2semidirect
6. #89761C2semidirect
8. #89715C2semidirect
9. #89715C2semidirect
10. #89447C2semidirect
11. #89436C2semidirect
17. C2×C7×C4⋊C4C22semidirect
29. C2×Dic7⋊C4C22semidirect
31. C7×#95C22semidirect
32. C7⋊#95C22semidirect
33. C7⋊#95C22semidirect
34. C7⋊#95C22semidirect
35. C7⋊#95C22semidirect
36. C7⋊#95C22semidirect
81. #385D7semidirect
137. C2×C4⋊C4C2×D7semidirect
155. #95C2×D7semidirect
275. C7#2483semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D7)⋊C42
C2×D7×C22⋊C42
C7×#3852
#899822
#897652
#897612
#897272
#897152
#894472
#894362
#24837