Finite Group Databasebeta

← #11734388

Subgroups — #11734388

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×D4×Dic7C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic, complemented
3C2×#6441C212yesmaximal, complemented
4C7×#394C211yesmaximal, characteristic
5#89994C211yesmaximal, characteristic
6#89755C211yesmaximal, characteristic, complemented
7#89723C212yesmaximal, complemented
8#89717C212yesmaximal, complemented
9#89714C212yesmaximal, complemented
10#89433C211yesmaximal, characteristic, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C23×Dic7C2212yes—
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21D4×Dic7—24no—
22C2×C4⋊Dic7C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31#6450—24no—
32#6441—28no—
33C7×#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38#2536—71nomaximal, Sylow
39C24×C7C2312yes—
40C2×C7×D4—24no—
41C2×C7×D4—22no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×Dic7—24no—
47C22×Dic7—24no—
48C22×Dic7—24no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C2×Dic14—22no—
56C2×Dic14—22no—
57C2×Dic14D414yescomplemented
58C7×C4⋊C4—22no—
59C7×C22⋊C4—24no—
60C7×C22⋊C4—24no—
61C22⋊Dic7—24no—
62C22⋊Dic7—24no—
63C22⋊Dic7—24no—
64C4⋊Dic7—22no—
65C4⋊Dic7—24no—
66Dic7⋊C4—24no—
67Dic7⋊C4—22no—
68Dic7⋊C4—24no—
69Dic7⋊C4—22no—
70C4×Dic7—22no—
71C4×Dic7—22no—
72C4×Dic7D414yes—
73C2×C22⋊Q8—72no—
74C2×C4×Q8—71no—
75C2×C4×D4—71no—
76#398—71no—
77#398—71no—
78#394—71no—
79#394D711yescharacteristic
80#388—72no—
81#388—72no—
82#382—71no—
83#380—72no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7#4114yes—
91C23×C7C2411yescharacteristic
92C7×D4—28no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7C2×D412yes—
102C2×C4×C7C2×D412yes—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—48no—
115C2×Dic7C2×D412yes—
116C2×Dic7C2×D412yes—
117C2×Dic7C2×D412yes—
118C2×Dic7C2×D412yes—
119C2×Dic7#4112yes—
120Dic14—28no—
121C22×Q8—71no—
122C22×D4C2×D711yescharacteristic
123C23×C4—72no—
124C22⋊Q8—148no—
125C4×Q8—144no—
126C4×D4—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4C2×D712yes—
138C2×C22⋊C4C2×D712yes—
139C2×C42—71no—
140C2×C42—71no—
141C2×C42—71no—
142#95—72no—
143#95—72no—
144#95—72no—
145#95—71no—
146#95C2×D711yescharacteristic
147C22×C7—24no—
148C22×C7—24no—
149C22×C7—24no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7C22×D411yescharacteristic
155C22×C7C22×D411yescharacteristic
156C22×C7C2×#4112yes—
157C22×C7C2×#4111yescharacteristic
158C22×C72−1+412yes—
159C4×C7—22no—
160C4×C7—22no—
161C4×C7—41no—
162C4×C7—42no—
163Dic7—22no—
164Dic7—22no—
165Dic7—42no—
166Dic7—41no—
167Dic7—42no—
168Dic7#12114yes—
169C24C22×D712yes—
170C2×Q8—74no—
171C2×Q8—142no—
172C2×Q8—142no—
173C2×D4—24no—
174C2×D4—22no—
175C22×C4—71no—
176C22×C4—71no—
177C22×C4—71no—
178C22×C4—72no—
179C22×C4—71no—
180C22×C4—72no—
181C22×C4—144no—
182C22×C4—144no—
183C22×C4—144no—
184C22×C4C22×D712yes—
185C22×C4C22×D711yescharacteristic
186C22×C4C22×D711yescharacteristic
187C22×C4C22×D711yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—142no—
190C4⋊C4—144no—
191C4⋊C4—144no—
192C4⋊C4—142no—
193C4⋊C4—144no—
194C4⋊C4—142no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—144no—
198C22⋊C4—144no—
199C22⋊C4—144no—
200C42—74no—
201C42—142no—
202C42—142no—
203C2×C7—44no—
204C2×C7C2×#12111yescharacteristic
205C2×C7#52511yescharacteristic
206C2×C7#53611yescharacteristic
207C2×C7#54812yes—
208C2×C7#55212yes—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23#115914yes—
216C23C23×D711yescharacteristic
217Q8—148no—
218D4—28no—
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—72no—
228C2×C4—72no—
229C2×C4—72no—
230C2×C4—72no—
231C2×C4—72no—
232C2×C4—141no—
233C2×C4—141no—
234C2×C4—142no—
235C2×C4—141no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
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—24no—
248C22—24no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×D4×D711yescharacteristic
255C22C2×D4×D711yescharacteristic
256C22C2×#115912yes—
257C22D7×#4111yescharacteristic
258C22#655412yes—
259C4—22no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—74no—
264C4—142no—
265C4—142no—
266C4—281no—
267C4—282no—
268C4—282no—
269C2—44no—
270C2#9027711yescharacteristic
271C2#9028912yes—
272C2D7×#12111yescharacteristic
273C2#9034112yes—
274C2#9036611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2536
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×#6441C2semidirect
6. #89755C2semidirect
7. #89723C2semidirect
8. #89717C2semidirect
9. #89714C2semidirect
10. #89433C2semidirect
11. #89422C2semidirect
57. C2×Dic14D4semidirect
246. C7#2536semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C2×#64502
C2×#64412
C7×#3942
#899942
#897552
#897232
#897172
#897142
#894332
#894222
#25367