Finite Group Databasebeta

← #11734391

Subgroups — #11734391

Order 896.

676 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 151 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×#6503C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic, complemented
3C2×#6448C212yesmaximal, complemented
4C7×#394C211yesmaximal, characteristic, complemented
5#89994C211yesmaximal, characteristic, complemented
6#89763C211yesmaximal, characteristic, complemented
7#89729C212yesmaximal, complemented
8#89729C212yesmaximal, complemented
9#89729C212yesmaximal, complemented
10#89444C211yesmaximal, characteristic, complemented
11#89421C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2212yes—
15C22×D28C2211yescharacteristic
16C22×Dic14C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22#6503—24no—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2212yes—
31C2×C4×Dic7C2211yescharacteristic
32C2×C4×Dic7C2211yescharacteristic
33C2×C4×Dic7C2211yescharacteristic
34#6450—24no—
35#6448—28no—
36C7×#95C2211yescharacteristic
37C7⋊#95C2211yescharacteristic
38#2548—71nomaximal, Sylow
39C24×C7C2312yes—
40C23×D7C2312yes—
41C2×C7×D4—22no—
42C2×C7×D4—42no—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—24no—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—24no—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—24no—
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2311yescharacteristic
58C2×D28—22no—
59C2×D28—42no—
60C2×Dic14—22no—
61C2×Dic14—22no—
62C2×Dic14D414yescomplemented
63C7×C4⋊C4—22no—
64C7×C22⋊C4—24no—
65C7×C22⋊C4—42no—
66C22⋊Dic7—24no—
67C22⋊Dic7—42no—
68(C2×D7)⋊C4—24no—
69(C2×D7)⋊C4—24no—
70(C2×D7)⋊C4—42no—
71(C2×D7)⋊C4—42no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74Dic7⋊C4—24no—
75C4×Dic7—24no—
76C4×Dic7—22no—
77C4×Dic7—22no—
78C4×Dic7D414yes—
79C2×#127—71no—
80C2×#124—72no—
81C2×C4×Q8—71no—
82#394—71no—
83#394—71no—
84#394—71no—
85#394D711yescharacteristic, complemented
86#390—72no—
87#390—72no—
88#390—72no—
89C4×C4⋊C4—71no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7C2411yescharacteristic
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7—22no—
104C22×D7—22no—
105C7×D4—44no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7C2×D412yes—
115C2×C4×C7C2×D412yes—
116C22⋊D7—48no—
117C22⋊D7—48no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7C2×D412yes—
126C2×Dic7C2×D412yes—
127C2×Dic7C2×D412yes—
128C2×Dic7C2×D412yes—
129C2×Dic7#4112yes—
130C2×Dic7#4114yes—
131D28—44no—
132Dic14—28no—
133C22×Q8—71no—
134C22×D4—72no—
135C22×D4—72no—
136C22×D4—71no—
137C22×D4C2×D711yescharacteristic
138#127—144no—
139#124—148no—
140C4×Q8—144no—
141C2×C4⋊C4—71no—
142C2×C4⋊C4—71no—
143C2×C4⋊C4—72no—
144C2×C4⋊C4C2×D711yescharacteristic
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4—72no—
149C2×C22⋊C4—72no—
150C2×C22⋊C4—72no—
151C2×C22⋊C4C2×D712yes—
152C2×C22⋊C4C2×D712yes—
153C2×C42—71no—
154C2×C42—71no—
155C2×C42—71no—
156C2×C42—72no—
157#95—71no—
158#95C2×D711yescharacteristic
159C22×C7—42no—
160C22×C7—42no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—42no—
164C22×C7—42no—
165C22×C7—42no—
166C22×C7C22×D411yescharacteristic
167C22×C7C22×D411yescharacteristic
168C22×C7C2×#4111yescharacteristic
169C22×C7C2×#4112yes—
170C22×C7H(5,𝔽2)12yes—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C2×D7—42no—
177C2×D7—42no—
178C4×C7—22no—
179C4×C7—22no—
180C4×C7—41no—
181C4×C7—42no—
182Dic7—22no—
183Dic7—41no—
184Dic7—42no—
185Dic7#12114yes—
186Dic7#12414yescomplemented
187C24—72no—
188C24C22×D712yes—
189C2×Q8—74no—
190C2×Q8—142no—
191C2×Q8—142no—
192C2×D4—22no—
193C2×D4—42no—
194C2×D4—144no—
195C2×D4—144no—
196C2×D4—144no—
197C2×D4—144no—
198C2×D4—144no—
199C2×D4—144no—
200C2×D4—142no—
201C2×D4—282no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—71no—
205C22×C4—72no—
206C22×C4—71no—
207C22×C4C22×D712yes—
208C22×C4C22×D711yescharacteristic
209C22×C4C22×D711yescharacteristic
210C22×C4C22×D711yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—144no—
213C4⋊C4—142no—
214C4⋊C4—142no—
215C22⋊C4—24no—
216C22⋊C4—42no—
217C22⋊C4—144no—
218C22⋊C4—144no—
219C22⋊C4—144no—
220C22⋊C4—282no—
221C22⋊C4—282no—
222C22⋊C4—282no—
223C42—74no—
224C42—142no—
225C42—142no—
226C42—144no—
227C2×C7—82no—
228C2×C7C2×#12111yescharacteristic
229C2×C7C2×#12411yescharacteristic
230C2×C7#53511yescharacteristic
231C2×C7#55012yes—
232C2×C7#55912yes—
233D7—82no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—142no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—142no—
246C23—142no—
247C23—142no—
248C23C23×D711yescharacteristic
249Q8—148no—
250D4—44no—
251D4—284no—
252D4—288no—
253D4—288no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—72no—
263C2×C4—72no—
264C2×C4—72no—
265C2×C4—74no—
266C2×C4—72no—
267C2×C4—72no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—142no—
271C2×C4—141no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4—142no—
275C2×C4D4×D712yes—
276C2×C4D4×D712yes—
277C7#254811yescharacteristic, complemented, Sylow
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—282no—
286C22—282no—
287C22—282no—
288C22—282no—
289C22—282no—
290C22—282no—
291C22—282no—
292C22C2×D4×D711yescharacteristic
293C22C2×D4×D711yescharacteristic
294C22#654812yes—
295C22D7×#4112yes—
296C22D7×#4111yescharacteristic
297C4—22no—
298C4—22no—
299C4—41no—
300C4—42no—
301C4—74no—
302C4—74no—
303C4—142no—
304C4—281no—
305C4—282no—
306C2—82no—
307C2—562no—
308C2#9028311yescharacteristic
309C2#9029012yes—
310C2D7×#12111yescharacteristic
311C2#9035512yes—
312C2D7×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6503C2
hypercenter#394D7
hypocenterC7#2548
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6450C2semidirect
3. C2×#6448C2semidirect
4. C7×#394C2semidirect
5. #89994C2semidirect
6. #89763C2semidirect
7. #89729C2semidirect
8. #89729C2semidirect
9. #89729C2semidirect
10. #89444C2semidirect
11. #89421C2semidirect
62. C2×Dic14D4semidirect
85. #394D7semidirect
186. Dic7#124semidirect
277. C7#2548semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
C2×#64502
C2×#64482
C7×#3942
#899942
#897632
#897292
#894442
#894212
#25487