Finite Group Databasebeta

← #11731752

Subgroups — #11731752

Order 896.

604 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×D28C211yesmaximal, characteristic, complemented
2C2×#6435C211yesmaximal, characteristic, complemented
3C7×#383C211yesmaximal, characteristic, complemented
4#89449C211yesmaximal, characteristic, complemented
5#89444C211yesmaximal, characteristic, complemented
6#89443C211yesmaximal, characteristic
7#89442C212yesmaximal, complemented
8#89440C212yesmaximal, complemented
9#89437C212yesmaximal, complemented
10#89437C212yesmaximal, complemented
11#89431C211yesmaximal, characteristic, complemented
12C22×D28C2211yescharacteristic
13C22×D28C2212yes—
14C22×C4×D7C2211yescharacteristic
15C22×C4×D7C2212yes—
16C2×C42×C7C2211yescharacteristic
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2212yes—
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C4×D28—24no—
30#6435—24no—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7×#95C2212yes—
34C7×#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2598—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×D28—24no—
50C2×D28—22no—
51C2×D28—24no—
52C2×D28—24no—
53C2×D28—42no—
54C2×C4×D7—22no—
55C2×C4×D7—22no—
56C2×C4×D7—22no—
57C2×C4×D7—24no—
58C2×C4×D7—42no—
59C2×C4×D7—42no—
60C42×C7D414yescomplemented
61(C2×D7)⋊C4—24no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—42no—
67(C2×D7)⋊C4—42no—
68C4⋊Dic7—22no—
69C4⋊Dic7—42no—
70Dic7⋊C4—22no—
71Dic7⋊C4—22no—
72C4×Dic7—22no—
73C2×C4×D4—71no—
74C2×#117—71no—
75#400—71no—
76#399—71no—
77#394—71no—
78#392—71no—
79#388—72no—
80#386—72no—
81#386—72no—
82#385—72no—
83#383D711yescharacteristic, complemented
84C23×C7C2411yescharacteristic
85C22×D7—21no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—21no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7#4114yes—
96C22×D7#4112yes—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7C2×D412yes—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7#4112yes—
120D28—28no—
121D28—48no—
122C4×D7—28no—
123C4×D7—48no—
124C22×D4—71no—
125C22×D4—72no—
126C23×C4—71no—
127C23×C4—72no—
128C4×D4—144no—
129#117—144no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—71no—
133C2×C4⋊C4—72no—
134C2×C22⋊C4—71no—
135C2×C22⋊C4—71no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C42—71no—
142C2×C42C2×D711yescharacteristic
143#95—72no—
144#95—72no—
145#95C2×D712yes—
146#95C2×D712yes—
147#95C2×D711yescharacteristic
148#95C2×D711yescharacteristic
149C22×C7C22×D411yescharacteristic
150C22×C7C2×#4111yescharacteristic
151C22×C7C2×#4112yes—
152C22×C7C2×#4111yescharacteristic
153C22×C7H(5,𝔽2)12yes—
154C2×D7—22no—
155C2×D7—24no—
156C2×D7—24no—
157C2×D7—24no—
158C2×D7—24no—
159C2×D7—42no—
160C2×D7—41no—
161C2×D7—42no—
162C2×D7—41no—
163C2×D7—42no—
164C2×D7—42no—
165C2×D7—42no—
166C4×C7—22no—
167C4×C7—22no—
168C4×C7—22no—
169C4×C7—42no—
170C4×C7—42no—
171Dic7—22no—
172Dic7—41no—
173Dic7—42no—
174Dic7—41no—
175C24—72no—
176C24—71no—
177C2×D4—144no—
178C2×D4—144no—
179C2×D4—144no—
180C2×D4—142no—
181C2×D4—282no—
182C22×C4—71no—
183C22×C4—71no—
184C22×C4—72no—
185C22×C4—71no—
186C22×C4—142no—
187C22×C4—142no—
188C22×C4—142no—
189C22×C4—144no—
190C22×C4—282no—
191C22×C4—282no—
192C22×C4C22×D712yes—
193C22×C4C22×D712yes—
194C22×C4C22×D711yescharacteristic
195C22×C4C22×D711yescharacteristic
196C22×C4C22×D711yescharacteristic
197C4⋊C4—142no—
198C4⋊C4—142no—
199C4⋊C4—142no—
200C4⋊C4—282no—
201C22⋊C4—144no—
202C22⋊C4—144no—
203C22⋊C4—142no—
204C22⋊C4—142no—
205C22⋊C4—144no—
206C22⋊C4—282no—
207C22⋊C4—282no—
208C42—142no—
209C42D2814yescomplemented
210C2×C7#52511yescharacteristic
211C2×C7#52511yescharacteristic
212C2×C7#53511yescharacteristic
213C2×C7#55112yes—
214C2×C7#55312yes—
215D7—42no—
216D7—44no—
217C23—74no—
218C23—72no—
219C23—141no—
220C23—142no—
221C23—142no—
222C23—141no—
223C23—142no—
224C23—142no—
225C23—142no—
226C23—142no—
227C23—142no—
228C23—142no—
229C23C23×D711yescharacteristic
230D4—148no—
231D4—288no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—72no—
243C2×C4—148no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—288no—
254C2×C4C2×D2812yes—
255C2×C4C2×D2812yes—
256C2×C4C2×D2812yes—
257C7#259811yescharacteristic, complemented, Sylow
258C22—144no—
259C22—144no—
260C22—144no—
261C22—142no—
262C22—144no—
263C22—282no—
264C22—282no—
265C22—282no—
266C22—282no—
267C22—281no—
268C22—282no—
269C22—281no—
270C22C22×D2811yescharacteristic
271C22D7×#4111yescharacteristic
272C22D7×#4112yes—
273C22D7×#4111yescharacteristic
274C22#655312yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—42no—
280C4—142no—
281C4—281no—
282C4—282no—
283C4—281no—
284C2—282no—
285C2—284no—
286C2#9021511yescharacteristic
287C2#9021511yescharacteristic
288C2#9021611yescharacteristic
289C2#9039612yes—
290C2#9040012yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#383C2
prime radical#89443C2
hypercenter#383D7
hypocenterC7#2598
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×C4×D28C2semidirect
2. C2×#6435C2semidirect
3. C7×#383C2semidirect
4. #89449C2semidirect
5. #89444C2semidirect
7. #89442C2semidirect
8. #89440C2semidirect
9. #89437C2semidirect
10. #89437C2semidirect
11. #89431C2semidirect
60. C42×C7D4semidirect
83. #383D7semidirect
209. C42D28semidirect
257. C7#2598semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D282
C2×#64352
C7×#3832
#894492
#894442
#894432
#894422
#894402
#894372
#894312
#25987