Finite Group Databasebeta

← #11739729

Subgroups — #11739729

Order 896.

604 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 145 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×C7×#117C211yesmaximal, characteristic, complemented
2C2×C4×D28C211yesmaximal, characteristic, complemented
3#89985C212yesmaximal, complemented
4#89763C211yesmaximal, characteristic, complemented
5#89761C212yesmaximal, complemented
6#89758C211yesmaximal, characteristic, complemented
7#89732C211yesmaximal, characteristic
8#89728C212yesmaximal, complemented
9#89721C211yesmaximal, characteristic, complemented
10#89712C212yesmaximal, complemented
11#89709C211yesmaximal, characteristic, complemented
12C23×C4×C7C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×D28C2211yescharacteristic
15C22×C4×D7C2212yes—
16C7×#117—24no—
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C42×C7C2211yescharacteristic
22C2×C42×C7C2211yescharacteristic
23C2×C22⋊Dic7C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C4×D28—24no—
31C7⋊#95C2212yes—
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2598—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2312yes—
40C22×C4×C7—22no—
41C22×C4×C7—22no—
42C22×C4×C7—22no—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C22×C4×C7C2311yescharacteristic
49C2×C22⋊D7—24no—
50C2×C22⋊D7—24no—
51C2×C22⋊D7—42no—
52C22×Dic7C2312yes—
53C22×Dic7C2312yes—
54C22×Dic7C2312yes—
55C2×D28—24no—
56C2×D28—22no—
57C2×C4×D7—24no—
58C2×C4×D7—42no—
59C2×C4×D7—42no—
60C7×C4⋊C4—22no—
61C7×C4⋊C4—22no—
62C7×C22⋊C4—22no—
63C7×C22⋊C4—22no—
64C42×C7—22no—
65C42×C7D414yescomplemented
66C22⋊Dic7—42no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—24no—
69(C2×D7)⋊C4—24no—
70(C2×D7)⋊C4—42no—
71C4⋊Dic7—22no—
72Dic7⋊C4—42no—
73C2×C4×D4—71no—
74C2×#117D711yescharacteristic, complemented
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#383—71no—
84C23×C7—21no—
85C23×C7—22no—
86C23×C7—21no—
87C23×C7—22no—
88C23×C7#4112yes—
89C23×C7C2411yescharacteristic
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7#4114yes—
97C2×C4×C7—28no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2×D412yes—
111C2×C4×C7#4112yes—
112C22⋊D7—48no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122D28—28no—
123C4×D7—48no—
124C22×D4—72no—
125C22×D4—71no—
126C23×C4—72no—
127C23×C4C2×D711yescharacteristic
128C4×D4—144no—
129#117—24no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4C2×D711yescharacteristic
140C2×C22⋊C4C2×D711yescharacteristic
141C2×C42C2×D711yescharacteristic
142C2×C42C2×D711yescharacteristic
143#95—72no—
144#95—72no—
145#95—71no—
146#95—71no—
147#95—72no—
148#95—72no—
149C22×C7—22no—
150C22×C7—24no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—42no—
154C22×C7C22×D411yescharacteristic
155C22×C7C2×#4111yescharacteristic
156C22×C7C2×#4111yescharacteristic
157C22×C7C2×#4112yes—
158C22×C7H(5,𝔽2)12yes—
159C2×D7—24no—
160C2×D7—24no—
161C2×D7—24no—
162C2×D7—42no—
163C2×D7—42no—
164C2×D7—42no—
165C2×D7—42no—
166C4×C7—22no—
167C4×C7—22no—
168C4×C7—22no—
169C4×C7—22no—
170C4×C7—41no—
171C4×C7—41no—
172Dic7—42no—
173Dic7—42no—
174Dic7—42no—
175C24—72no—
176C24C22×D711yescharacteristic
177C2×D4—144no—
178C2×D4—144no—
179C2×D4—144no—
180C2×D4—142no—
181C2×D4—282no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—72no—
186C22×C4—72no—
187C22×C4—72no—
188C22×C4—144no—
189C22×C4—282no—
190C22×C4—282no—
191C22×C4C22×D711yescharacteristic
192C22×C4C22×D711yescharacteristic
193C22×C4C22×D711yescharacteristic
194C22×C4C22×D711yescharacteristic
195C22×C4C22×D711yescharacteristic
196C22×C4C22×D711yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—142no—
200C4⋊C4—282no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—144no—
204C22⋊C4—144no—
205C22⋊C4—144no—
206C22⋊C4—282no—
207C22⋊C4—282no—
208C42—22no—
209C42C22⋊D714yescomplemented
210C2×C7—42no—
211C2×C7#52511yescharacteristic
212C2×C7#52511yescharacteristic
213C2×C7#53511yescharacteristic
214C2×C7#55112yes—
215C2×C7#55312yes—
216D7—44no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—74no—
222C23—142no—
223C23—142no—
224C23—142no—
225C23—142no—
226C23—142no—
227C23—142no—
228C23#115712yes—
229C23C23×D711yescharacteristic
230D4—148no—
231D4—288no—
232C2×C4—28no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—288no—
253C2×C4#115712yes—
254C2×C4C2×C22⋊D712yes—
255C2×C4C2×C22⋊D712yes—
256C2×C4C2×C22⋊D712yes—
257C7#259811yescharacteristic, complemented, Sylow
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—144no—
264C22—144no—
265C22—144no—
266C22—282no—
267C22—282no—
268C22—282no—
269C22—282no—
270C22C2×#115711yescharacteristic
271C22C2×#115711yescharacteristic
272C22D7×#4112yes—
273C22#655312yes—
274C22C22×C22⋊D711yescharacteristic
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—282no—
282C4—282no—
283C4—282no—
284C2—42no—
285C2—284no—
286C2#9021812yes—
287C2#9022112yes—
288C2#9048211yescharacteristic
289C2#9051911yescharacteristic
290C2#9052111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#117C2
prime radical#89732C2
hypercenterC2×#117D7
hypocenterC7#2598
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C7×#117C2semidirect
2. C2×C4×D28C2semidirect
3. #89985C2semidirect
4. #89763C2semidirect
5. #89761C2semidirect
6. #89758C2semidirect
8. #89728C2semidirect
9. #89721C2semidirect
10. #89712C2semidirect
11. #89709C2semidirect
65. C42×C7D4semidirect
74. C2×#117D7semidirect
209. C42C22⋊D7semidirect
257. C7#2598semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1172
C2×C4×D282
#899852
#897632
#897612
#897582
#897322
#897282
#897212
#897122
#897092
#25987