Finite Group Databasebeta

← #11735229

Subgroups — #11735229

Order 896.

604 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 147 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×#6459C212yesmaximal, complemented
2C2×#6456C211yesmaximal, characteristic
3C2×#6450C211yesmaximal, characteristic, complemented
4C7×#398C211yesmaximal, characteristic, complemented
5#89763C211yesmaximal, characteristic
6#89762C212yesmaximal, complemented
7#89752C211yesmaximal, characteristic, complemented
8#89444C211yesmaximal, characteristic
9#89441C212yesmaximal, complemented
10#89439C212yesmaximal, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30#6459—28no—
31#6456—24no—
32#6450—24no—
33C7×#95C2211yescharacteristic
34C7×#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37C7⋊#95C2212yes—
38#2536—71nomaximal, Sylow
39C23×D7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C2×D28—24no—
51C2×D28—22no—
52C2×C4×D7—24no—
53C2×C4×D7—24no—
54C2×C4×D7—24no—
55C2×Dic14—22no—
56C2×Dic14—22no—
57C2×Dic14D414yescomplemented
58C7×C4⋊C4—22no—
59C7×C4⋊C4—22no—
60C7×C4⋊C4—24no—
61(C2×D7)⋊C4—24no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—24no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66Dic7⋊C4—22no—
67Dic7⋊C4—22no—
68Dic7⋊C4—24no—
69Dic7⋊C4—24no—
70C4×Dic7—22no—
71C4×Dic7—22no—
72C4×Dic7D414yes—
73C2×C22⋊Q8—72no—
74C2×C4×Q8—71no—
75C2×C4×D4—71no—
76#398—71no—
77#398D711yescharacteristic, complemented
78#394—71no—
79#394—71no—
80#388—72no—
81#388—72no—
82#382—71no—
83#380—72no—
84C23×C7C2411yescharacteristic
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7#4114yes—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7C2×D412yes—
104C2×C4×C7C2×D412yes—
105C2×Dic7—21no—
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7C2×D412yes—
114C2×Dic7C2×D412yes—
115C2×Dic7C2×D412yes—
116C2×Dic7C2×D412yes—
117C2×Dic7#4112yes—
118D28—28no—
119C4×D7—48no—
120Dic14—28no—
121C22×Q8—71no—
122C22×D4—71no—
123C23×C4—72no—
124C22⋊Q8—148no—
125C4×Q8—144no—
126C4×D4—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4C2×D711yescharacteristic
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C4⋊C4C2×D712yes—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C42—71no—
140C2×C42—71no—
141C2×C42—71no—
142#95—72no—
143#95—72no—
144#95—71no—
145#95C2×D712yes—
146#95C2×D711yescharacteristic
147C22×C7C22×D411yescharacteristic
148C22×C7C22×D411yescharacteristic
149C22×C7C2×#4111yescharacteristic
150C22×C7C2×#4112yes—
151C22×C72−1+412yes—
152C2×D7—24no—
153C2×D7—24no—
154C2×D7—24no—
155C2×D7—42no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C4×C7—22no—
160C4×C7—22no—
161C4×C7—42no—
162C4×C7—41no—
163C4×C7—42no—
164Dic7—22no—
165Dic7—22no—
166Dic7—41no—
167Dic7—42no—
168Dic7#12114yes—
169C24—72no—
170C2×Q8—74no—
171C2×Q8—142no—
172C2×Q8—142no—
173C2×D4—144no—
174C2×D4—142no—
175C22×C4—71no—
176C22×C4—71no—
177C22×C4—71no—
178C22×C4—72no—
179C22×C4—71no—
180C22×C4—144no—
181C22×C4—144no—
182C22×C4—144no—
183C22×C4C22×D712yes—
184C22×C4C22×D712yes—
185C22×C4C22×D711yescharacteristic
186C22×C4C22×D711yescharacteristic
187C22×C4C22×D711yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—24no—
191C4⋊C4—142no—
192C4⋊C4—144no—
193C4⋊C4—144no—
194C4⋊C4—142no—
195C22⋊C4—144no—
196C22⋊C4—144no—
197C22⋊C4—144no—
198C22⋊C4—144no—
199C22⋊C4—144no—
200C42—74no—
201C42—142no—
202C42—142no—
203C2×C7C2×#12111yescharacteristic
204C2×C7#52511yescharacteristic
205C2×C7#53611yescharacteristic
206C2×C7#54812yes—
207C2×C7#55212yes—
208D7—44no—
209C23—74no—
210C23—142no—
211C23—142no—
212C23—142no—
213C23—142no—
214C23—142no—
215C23—142no—
216C23C23×D711yescharacteristic
217Q8—148no—
218D4—148no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—72no—
231C2×C4—72no—
232C2×C4—72no—
233C2×C4—72no—
234C2×C4—72no—
235C2×C4—141no—
236C2×C4—142no—
237C2×C4—141no—
238C2×C4—141no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—288no—
244C2×C4D4×D712yes—
245C2×C4D4×D712yes—
246C7#253611yescharacteristic, complemented, Sylow
247C22—144no—
248C22—144no—
249C22—144no—
250C22—282no—
251C22—282no—
252C22—282no—
253C22—282no—
254C22C2×D4×D711yescharacteristic
255C22C2×D4×D711yescharacteristic
256C22#655112yes—
257C22D7×#4111yescharacteristic
258C22D7×#4112yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—74no—
265C4—142no—
266C4—142no—
267C4—282no—
268C4—281no—
269C2—284no—
270C2D7×#12111yescharacteristic
271C2#9032412yes—
272C2#9034711yescharacteristic
273C2#9038412yes—
274C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC22×D28C22
hypercenter#398D7
hypocenterC7#2536
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#6459C2semidirect
3. C2×#6450C2semidirect
4. C7×#398C2semidirect
6. #89762C2semidirect
7. #89752C2semidirect
9. #89441C2semidirect
10. #89439C2semidirect
11. #89422C2semidirect
57. C2×Dic14D4semidirect
77. #398D7semidirect
246. C7#2536semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64562
C2×#64502
C7×#3982
#897632
#897622
#897522
#894442
#894412
#894392
#894222
#25367