Finite Group Databasebeta

← #11734959

Subgroups — #11734959

Order 896.

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

normal subgroups only

This group has 604 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×#6451C211yesmaximal, characteristic, complemented
2C2×#6448C212yesmaximal
3C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
4C7×#397C211yesmaximal, characteristic, complemented
5#89765C211yesmaximal, characteristic, complemented
6#89725C211yesmaximal, characteristic, complemented
7#89714C212yesmaximal, complemented
8#89446C211yesmaximal, characteristic, complemented
9#89439C212yesmaximal, complemented
10#89421C211yesmaximal, characteristic, complemented
11#89418C212yesmaximal, complemented
12C22×C22⋊D7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2212yes—
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×C4⋊Dic7C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6451—24no—
30#6448—28no—
31(C22⋊D7)⋊C4—24no—
32C7×#95C2212yes—
33C7×#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38#2547—71nomaximal, Sylow
39C24×C7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—22no—
46C2×C22⋊D7—22no—
47C2×C22⋊D7D414yes—
48C22×Dic7—22no—
49C22×Dic7—24no—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2312yes—
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C2×C4×D7—22no—
56C2×C4×D7—24no—
57C2×Dic14—24no—
58C2×Dic14—24no—
59C2×Dic14—24no—
60C7×C4⋊C4—22no—
61C7×C22⋊C4—24no—
62C7×C22⋊C4—22no—
63C22⋊Dic7—24no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—22no—
67C4⋊Dic7—22no—
68Dic7⋊C4—22no—
69Dic7⋊C4—24no—
70Dic7⋊C4—24no—
71C4×Dic7—24no—
72C4×Dic7D414yes—
73C2×C4⋊Q8—71no—
74C2×#124—72no—
75C2×C4×D4—71no—
76#397—71no—
77#397—71no—
78#397—71no—
79#397D711yescharacteristic, complemented
80#391—72no—
81#380—72no—
82#380—72no—
83C4×C4⋊C4—71no—
84C23×C7—21no—
85C23×C7—22no—
86C23×C7—21no—
87C23×C7—22no—
88C23×C7C2×D412yes—
89C23×C7C2411yescharacteristic
90C22×D7—21no—
91C22×D7—22no—
92C22×D7—21no—
93C22×D7—22no—
94C22×D7C2×D412yes—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7C2×D412yes—
105C22⋊D7—28no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—44no—
115C2×Dic7C2×D412yes—
116C2×Dic7C2×D412yes—
117C2×Dic7C2×D412yes—
118C2×Dic7#4112yes—
119C2×Dic7#4114yes—
120C4×D7—44no—
121Dic14—48no—
122C22×Q8—72no—
123C22×D4—71no—
124C23×C4—71no—
125C23×C4—71no—
126C4⋊Q8—144no—
127#124—148no—
128C4×D4—144no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C22⋊C4—72no—
135C2×C22⋊C4—71no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4C2×D711yescharacteristic
139C2×C22⋊C4C2×D712yes—
140C2×C42—71no—
141C2×C42—72no—
142#95—71no—
143#95—72no—
144#95—72no—
145#95—72no—
146#95C2×D711yescharacteristic
147#95C2×D712yes—
148C22×C7—22no—
149C22×C7—24no—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7—42no—
153C22×C7C22×D411yescharacteristic
154C22×C7C22×D411yescharacteristic
155C22×C7C2×#4111yescharacteristic
156C22×C7C2×#4112yes—
157C22×C72−1+412yes—
158C2×D7—22no—
159C2×D7—24no—
160C2×D7—41no—
161C2×D7—41no—
162C2×D7—42no—
163C4×C7—22no—
164C4×C7—42no—
165C4×C7—41no—
166C4×C7—42no—
167Dic7—42no—
168Dic7—41no—
169Dic7—42no—
170Dic7#12114yes—
171Dic7#12414yescomplemented
172C24—71no—
173C24C22×D711yescharacteristic
174C2×Q8—144no—
175C2×Q8—144no—
176C2×Q8—144no—
177C2×D4—74no—
178C2×D4—142no—
179C2×D4—142no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4—71no—
183C22×C4—72no—
184C22×C4—72no—
185C22×C4—142no—
186C22×C4—144no—
187C22×C4—142no—
188C22×C4—144no—
189C22×C4C22×D711yescharacteristic
190C22×C4C22×D712yes—
191C22×C4C22×D712yes—
192C22×C4C22×D711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—144no—
195C4⋊C4—142no—
196C4⋊C4—144no—
197C4⋊C4—142no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—144no—
201C22⋊C4—144no—
202C22⋊C4—144no—
203C22⋊C4—142no—
204C42—74no—
205C42—144no—
206C2×C7—42no—
207C2×C7C2×#12111yescharacteristic
208C2×C7C2×#12411yescharacteristic
209C2×C7#53611yescharacteristic
210C2×C7#54712yes—
211C2×C7#55612yes—
212D7—42no—
213C23—21no—
214C23—22no—
215C23—21no—
216C23—22no—
217C23—72no—
218C23—141no—
219C23—142no—
220C23—141no—
221C23—142no—
222C23D4×D712yes—
223C23C23×D711yescharacteristic
224Q8—288no—
225D4—148no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—72no—
236C2×C4—72no—
237C2×C4—72no—
238C2×C4—74no—
239C2×C4—72no—
240C2×C4—142no—
241C2×C4—142no—
242C2×C4—141no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—284no—
249C2×C4—284no—
250C2×C4D4×D712yes—
251C7#254711yescharacteristic, complemented, Sylow
252C22—22no—
253C22—24no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—142no—
258C22—144no—
259C22—282no—
260C22—281no—
261C22—281no—
262C22C2×D4×D711yescharacteristic
263C22C2×D4×D711yescharacteristic
264C22D7×#4111yescharacteristic
265C22D7×#4112yes—
266C22#655412yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—74no—
272C4—74no—
273C4—282no—
274C4—282no—
275C4—281no—
276C2—42no—
277C2—282no—
278C2D7×#12111yescharacteristic
279C2#9033911yescharacteristic
280C2#9034612yes—
281C2D7×#12411yescharacteristic
282C2#9039312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC22×C22⋊D7C22
hypercenter#397D7
hypocenterC7#2547
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6451C2semidirect
4. C7×#397C2semidirect
5. #89765C2semidirect
6. #89725C2semidirect
7. #89714C2semidirect
8. #89446C2semidirect
9. #89439C2semidirect
10. #89421C2semidirect
11. #89418C2semidirect
79. #397D7semidirect
171. Dic7#124semidirect
251. C7#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64482
C2×(C22⋊D7)⋊C42
C7×#3972
#897652
#897252
#897142
#894462
#894392
#894212
#894182
#25477