Finite Group Databasebeta

← #11735209

Subgroups — #11735209

Order 896.

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

normal subgroups only

This group has 624 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×#6457C212yesmaximal
2C2×#6456C211yesmaximal, characteristic
3C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
4C7×#398C211yesmaximal, characteristic, complemented
5#89765C211yesmaximal, characteristic, complemented
6#89755C211yesmaximal, characteristic, complemented
7#89744C212yesmaximal, complemented
8#89446C211yesmaximal, characteristic, complemented
9#89439C212yesmaximal, complemented
10#89439C211yesmaximal, characteristic, complemented
11#89436C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C22×C4×D7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6457—28no—
30#6456—24no—
31D7×C4⋊C4—24no—
32C7×#95C2212yes—
33C7×#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38#2486—71nomaximal, Sylow
39C23×D7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2311yescharacteristic
50C2×D28—22no—
51C2×D28—22no—
52C2×D28D414yes—
53C2×C4×D7—24no—
54C2×C4×D7—24no—
55C2×C4×D7—24no—
56C2×C4×D7—24no—
57C2×C4×D7—22no—
58C2×C4×D7—24no—
59C2×C4×D7D414yes—
60C7×C4⋊C4—22no—
61C7×C4⋊C4—22no—
62C7×C4⋊C4—24no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—22no—
68C4⋊Dic7—22no—
69Dic7⋊C4—24no—
70Dic7⋊C4—24no—
71Dic7⋊C4—24no—
72C4×Dic7—22no—
73C2×#123—72no—
74C2×C4×D4—71no—
75C22×C4⋊C4—71no—
76#398—71no—
77#398D711yescharacteristic, complemented
78#397—71no—
79#397—71no—
80#389—72no—
81#381—72no—
82#380—72no—
83#380—71no—
84C23×C7C2411yescharacteristic
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—21no—
88C22×D7—22no—
89C22×D7—21no—
90C22×D7—22no—
91C22×D7C2×D412yes—
92C22×D7C2×D412yes—
93C22×D7C2×D412yes—
94C22×D7#4112yes—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
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×C7C2×D412yes—
107C2×C4×C7C2×D412yes—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7C2×D412yes—
117C2×Dic7#4114yes—
118D28—28no—
119C4×D7—28no—
120C4×D7—48no—
121C4×D7—44no—
122C22×D4—71no—
123C23×C4—71no—
124C23×C4—72no—
125C23×C4—71no—
126#123—148no—
127C4×D4—144no—
128C2×C4⋊C4—72no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4—144no—
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C4⋊C4C2×D711yescharacteristic
135C2×C4⋊C4C2×D712yes—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—71no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C42—71no—
142#95—72no—
143#95—71no—
144#95—72no—
145#95—72no—
146#95C2×D712yes—
147#95C2×D711yescharacteristic
148C22×C7C22×D411yescharacteristic
149C22×C7C22×D411yescharacteristic
150C22×C7C2×#4111yescharacteristic
151C22×C7C2×#4112yes—
152C22×C72−1+412yes—
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—24no—
156C2×D7—24no—
157C2×D7—22no—
158C2×D7—24no—
159C2×D7—41no—
160C2×D7—41no—
161C2×D7—42no—
162C2×D7#12114yes—
163C4×C7—22no—
164C4×C7—22no—
165C4×C7—42no—
166C4×C7—41no—
167C4×C7—42no—
168Dic7—42no—
169Dic7—41no—
170Dic7—42no—
171Dic7#12314yescomplemented
172C24—71no—
173C24—71no—
174C2×D4—74no—
175C2×D4—142no—
176C2×D4—142no—
177C22×C4—71no—
178C22×C4—74no—
179C22×C4—71no—
180C22×C4—72no—
181C22×C4—72no—
182C22×C4—144no—
183C22×C4—142no—
184C22×C4—144no—
185C22×C4—144no—
186C22×C4—144no—
187C22×C4—144no—
188C22×C4C22×D711yescharacteristic
189C22×C4C22×D712yes—
190C22×C4C22×D712yes—
191C22×C4C22×D711yescharacteristic
192C22×C4C22×D711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—144no—
197C4⋊C4—144no—
198C4⋊C4—144no—
199C4⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—144no—
202C22⋊C4—144no—
203C22⋊C4—144no—
204C22⋊C4—142no—
205C42—142no—
206C2×C7C2×#12111yescharacteristic
207C2×C7C2×#12311yescharacteristic
208C2×C7#53611yescharacteristic
209C2×C7#54712yes—
210C2×C7#55212yes—
211D7—24no—
212D7—42no—
213C23—72no—
214C23—72no—
215C23—72no—
216C23—72no—
217C23—142no—
218C23—142no—
219C23—141no—
220C23—142no—
221C23—142no—
222C23—141no—
223C23C23×D711yescharacteristic
224D4—148no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—72no—
237C2×C4—74no—
238C2×C4—148no—
239C2×C4—141no—
240C2×C4—142no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—284no—
248C2×C4—288no—
249C2×C4D4×D712yes—
250C2×C4D4×D712yes—
251C7#248611yescharacteristic, complemented, Sylow
252C22—74no—
253C22—144no—
254C22—142no—
255C22—144no—
256C22—142no—
257C22—142no—
258C22—144no—
259C22—282no—
260C22—281no—
261C22—281no—
262C22C2×D4×D711yescharacteristic
263C22C2×D4×D711yescharacteristic
264C22#655112yes—
265C22D7×#4111yescharacteristic
266C22D7×#4112yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—74no—
273C4—281no—
274C4—282no—
275C4—282no—
276C2—144no—
277C2—282no—
278C2D7×#12111yescharacteristic
279C2#9032212yes—
280C2D7×#12311yescharacteristic
281C2#9038412yes—
282C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC22×D28C22
hypercenter#398D7
hypocenterC7#2486
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×D7×C4⋊C4C2semidirect
4. C7×#398C2semidirect
5. #89765C2semidirect
6. #89755C2semidirect
7. #89744C2semidirect
8. #89446C2semidirect
9. #89439C2semidirect
10. #89439C2semidirect
11. #89436C2semidirect
77. #398D7semidirect
171. Dic7#123semidirect
251. C7#2486semidirect

Maximal subgroups

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

subgroupindex
C2×#64572
C2×#64562
C2×D7×C4⋊C42
C7×#3982
#897652
#897552
#897442
#894462
#894392
#894362
#24867