Finite Group Databasebeta

← #11734984

Subgroups — #11734984

Order 896.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6455C211yesmaximal, characteristic, complemented
2C2×#6447C211yesmaximal, characteristic
3C7×#397C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89733C212yesmaximal, complemented
6#89729C212yesmaximal, complemented
7#89721C211yesmaximal, characteristic, complemented
8#89449C211yesmaximal, characteristic, complemented
9#89444C212yesmaximal, complemented
10#89439C211yesmaximal, characteristic, complemented
11#89419C212yesmaximal, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×D28C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2212yes—
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2212yes—
31#6455—24no—
32#6447—24no—
33C7×#95C2212yes—
34C7×#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2624—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—22no—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—24no—
48C2×C22⋊D7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C2×D28—22no—
54C2×D28—42no—
55C2×C4×D7—24no—
56C2×C4×D7D414yes—
57C7×C4⋊C4—22no—
58C7×C22⋊C4—22no—
59C7×C22⋊C4—42no—
60C22⋊Dic7—42no—
61(C2×D7)⋊C4—24no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—42no—
64(C2×D7)⋊C4—42no—
65(C2×D7)⋊C4—42no—
66C4⋊Dic7—22no—
67C4⋊Dic7—42no—
68Dic7⋊C4—22no—
69Dic7⋊C4—24no—
70Dic7⋊C4—42no—
71C4×Dic7—24no—
72C2×#121—71no—
73C2×#117—71no—
74#399—71no—
75#399—71no—
76#397—71no—
77#397D711yescharacteristic, complemented
78#394—72no—
79#394—72no—
80#390—72no—
81#389—72no—
82#380—71no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7C2411yescharacteristic
89C22×D7—22no—
90C22×D7—21no—
91C22×D7—22no—
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—22no—
95C22×D7—21no—
96C22×D7—22no—
97C22×D7—21no—
98C22×D7C2×D412yes—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7C2×D412yes—
109C22⋊D7—44no—
110C22⋊D7—44no—
111C2×Dic7—22no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7C2×D412yes—
120C2×Dic7#4114yes—
121D28—44no—
122C4×D7—28no—
123C22×D4—71no—
124C22×D4—71no—
125C22×D4—71no—
126C23×C4—71no—
127#121—144no—
128#117—144no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4C2×D711yescharacteristic
135C2×C22⋊C4—72no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4C2×D711yescharacteristic
142C2×C22⋊C4C2×D712yes—
143C2×C42—72no—
144#95—71no—
145#95—72no—
146#95C2×D711yescharacteristic
147#95C2×D712yes—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—41no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7C22×D411yescharacteristic
154C22×C7C2×#4112yes—
155C22×C7H(5,𝔽2)11yescharacteristic
156C22×C7H(5,𝔽2)12yes—
157C22×C72−1+411yescharacteristic
158C2×D7—22no—
159C2×D7—22no—
160C2×D7—22no—
161C2×D7—42no—
162C2×D7—42no—
163C2×D7—41no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—42no—
167C2×D7—42no—
168C4×C7—22no—
169C4×C7—41no—
170C4×C7—42no—
171C4×C7—42no—
172Dic7—42no—
173Dic7—41no—
174Dic7—42no—
175Dic7#12314yes—
176C24—71no—
177C24—71no—
178C24C22×D711yescharacteristic
179C2×D4—142no—
180C2×D4—142no—
181C2×D4—144no—
182C2×D4—142no—
183C2×D4—282no—
184C2×D4—282no—
185C22×C4—71no—
186C22×C4—74no—
187C22×C4—72no—
188C22×C4—71no—
189C22×C4—72no—
190C22×C4—144no—
191C22×C4C22×D712yes—
192C22×C4C22×D712yes—
193C22×C4C22×D711yescharacteristic
194C22×C4C22×D711yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—144no—
197C4⋊C4—142no—
198C4⋊C4—142no—
199C4⋊C4—282no—
200C4⋊C4—282no—
201C22⋊C4—22no—
202C22⋊C4—42no—
203C22⋊C4—144no—
204C22⋊C4—142no—
205C22⋊C4—282no—
206C22⋊C4—282no—
207C22⋊C4—282no—
208C22⋊C4—282no—
209C42—144no—
210C2×C7—81no—
211C2×C7C2×#12311yescharacteristic
212C2×C7#53511yescharacteristic
213C2×C7#53711yescharacteristic
214C2×C7#54012yes—
215C2×C7#54212yes—
216D7—42no—
217D7—81no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—72no—
224C23—141no—
225C23—142no—
226C23—142no—
227C23—141no—
228C23—141no—
229C23—141no—
230C23—142no—
231C23—142no—
232C23—141no—
233C23C23×D711yescharacteristic
234D4—284no—
235D4—284no—
236D4—284no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—72no—
247C2×C4—74no—
248C2×C4—148no—
249C2×C4—142no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4D4×D712yes—
258C7#262411yescharacteristic, complemented, Sylow
259C22—42no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—142no—
265C22—142no—
266C22—142no—
267C22—282no—
268C22—282no—
269C22—282no—
270C22—281no—
271C22—281no—
272C22—282no—
273C22—281no—
274C22C2×D4×D711yescharacteristic
275C22#654811yescharacteristic
276C22D7×#4112yes—
277C22#655312yes—
278C22#655411yescharacteristic
279C4—22no—
280C4—42no—
281C4—42no—
282C4—41no—
283C4—74no—
284C4—282no—
285C4—281no—
286C4—282no—
287C2—81no—
288C2—282no—
289C2—561no—
290C2#9029911yescharacteristic
291C2D7×#12311yescharacteristic
292C2#9035412yes—
293C2#9036511yescharacteristic
294C2#9040112yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC2×#6447C2
hypercenter#397D7
hypocenterC7#2624
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6455C2semidirect
3. C7×#397C2semidirect
4. #89765C2semidirect
5. #89733C2semidirect
6. #89729C2semidirect
7. #89721C2semidirect
8. #89449C2semidirect
9. #89444C2semidirect
10. #89439C2semidirect
11. #89419C2semidirect
77. #397D7semidirect
258. C7#2624semidirect

Maximal subgroups

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

subgroupindex
C2×#64552
C2×#64472
C7×#3972
#897652
#897332
#897292
#897212
#894492
#894442
#894392
#894192
#26247