Finite Group Databasebeta

← #11734414

Subgroups — #11734414

Order 896.

576 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 135 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×#6499C211yesmaximal, characteristic, complemented
2C2×#6455C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic
5#89751C211yesmaximal, characteristic, complemented
6#89729C212yesmaximal, complemented
7#89728C212yesmaximal, complemented
8#89723C212yesmaximal, complemented
9#89717C212yesmaximal, complemented
10#89445C211yesmaximal, characteristic, complemented
11#89420C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22#6499—24no—
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31#6455—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2610—71nomaximal, Sylow
38C24×C7C2312yes—
39C23×D7C2311yescharacteristic
40C2×C7×D4—22no—
41C2×C7×D4—42no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—24no—
47C2×C22⋊D7—24no—
48C2×C22⋊D7—42no—
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2312yes—
54C2×C4×D7—22no—
55C2×C4×D7—22no—
56C2×C4×D7—22no—
57C2×Dic14—22no—
58C2×Dic14—42no—
59C7×C4⋊C4—22no—
60C7×C22⋊C4—24no—
61C7×C22⋊C4—42no—
62C22⋊Dic7—24no—
63C22⋊Dic7—24no—
64C22⋊Dic7—24no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—42no—
68C4⋊Dic7—22no—
69Dic7⋊C4—42no—
70C4×Dic7—22no—
71C4×Dic7—24no—
72C4×Dic7D414yescomplemented
73C2×#124—71no—
74C2×#117—71no—
75#395—71no—
76#394D711yescharacteristic, complemented
77#393—71no—
78#392—71no—
79#390—72no—
80#388—72no—
81#388—72no—
82#386—72no—
83#376—71no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7C2411yescharacteristic
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7#4112yes—
97C7×D4—44no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
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×C7#4112yes—
108C22⋊D7—48no—
109C2×Dic7—22no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7C2×D412yes—
120C2×Dic7C2×D412yes—
121C2×Dic7#4114yes—
122C4×D7—28no—
123Dic14—44no—
124C22×Q8—71no—
125C22×D4—72no—
126C22×D4C2×D711yescharacteristic
127C23×C4—71no—
128#124—144no—
129#117—144no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C22⋊C4—72no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—71no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4C2×D712yes—
140C2×C22⋊C4C2×D712yes—
141C2×C42—71no—
142C2×C42—71no—
143C2×C42—72no—
144#95—72no—
145#95—71no—
146#95—72no—
147#95—72no—
148#95C2×D711yescharacteristic
149C22×C7—42no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7C22×D411yescharacteristic
157C22×C7C2×#4111yescharacteristic
158C22×C7C2×#4111yescharacteristic
159C22×C7C2×#4112yes—
160C22×C7H(5,𝔽2)12yes—
161C2×D7—22no—
162C2×D7—24no—
163C2×D7—41no—
164C2×D7—41no—
165C2×D7—42no—
166C4×C7—22no—
167C4×C7—22no—
168C4×C7—41no—
169C4×C7—42no—
170Dic7—22no—
171Dic7—22no—
172Dic7—24no—
173Dic7—41no—
174Dic7—42no—
175C24—71no—
176C24C22×D712yes—
177C2×Q8—142no—
178C2×Q8—282no—
179C2×D4—22no—
180C2×D4—42no—
181C2×D4—144no—
182C2×D4—144no—
183C2×D4—282no—
184C22×C4—72no—
185C22×C4—71no—
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—72no—
189C22×C4—142no—
190C22×C4—142no—
191C22×C4—142no—
192C22×C4C22×D712yes—
193C22×C4C22×D711yescharacteristic
194C22×C4C22×D711yescharacteristic
195C22×C4C22×D711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—142no—
198C4⋊C4—282no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—142no—
202C22⋊C4—144no—
203C22⋊C4—144no—
204C22⋊C4—144no—
205C22⋊C4—142no—
206C22⋊C4—282no—
207C42—74no—
208C42—144no—
209C42—142no—
210C2×C7—82no—
211C2×C7#52511yescharacteristic
212C2×C7#53211yescharacteristic
213C2×C7#53511yescharacteristic
214C2×C7#55112yes—
215C2×C7#55512yes—
216D7—42no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—72no—
225C23—141no—
226C23—141no—
227C23—142no—
228C23—142no—
229C23C23×D711yescharacteristic
230Q8—284no—
231D4—44no—
232D4—288no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—72no—
242C2×C4—72no—
243C2×C4—74no—
244C2×C4—148no—
245C2×C4—141no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—141no—
249C2×C4—142no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4D4×D712yes—
256C2×C4#115912yes—
257C7#261011yescharacteristic, complemented, Sylow
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—142no—
266C22—144no—
267C22—281no—
268C22—282no—
269C22—281no—
270C22C2×D4×D711yescharacteristic
271C22C2×#115911yescharacteristic
272C22#654812yes—
273C22D7×#4111yescharacteristic
274C22D7×#4112yes—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—142no—
280C4—144no—
281C4—142no—
282C4—282no—
283C4—281no—
284C2—82no—
285C2—282no—
286C2#9028311yescharacteristic
287C2#9029711yescharacteristic
288C2#9030712yes—
289C2#9035012yes—
290C2#9035911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radical#89995C2
hypercenter#394D7
hypocenterC7#2610
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6499C2semidirect
2. C2×#6455C2semidirect
3. C7×#394C2semidirect
5. #89751C2semidirect
6. #89729C2semidirect
7. #89728C2semidirect
8. #89723C2semidirect
9. #89717C2semidirect
10. #89445C2semidirect
11. #89420C2semidirect
72. C4×Dic7D4semidirect
76. #394D7semidirect
257. C7#2610semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
C2×#64552
C7×#3942
#899952
#897512
#897292
#897282
#897232
#897172
#894452
#894202
#26107