Finite Group Databasebeta

← #11733705

Subgroups — #11733705

Order 896.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6458C212yesmaximal, complemented
2C2×#6457C212yesmaximal, complemented
3C7×#389C211yesmaximal, characteristic, complemented
4#89766C212yesmaximal, complemented
5#89763C212yesmaximal, complemented
6#89711C211yesmaximal, characteristic, complemented
7#89711C211yesmaximal, characteristic, complemented
8#89443C212yesmaximal, complemented
9#89440C212yesmaximal, complemented
10C22×D28C2212yes—
11C22×D28C2212yes—
12C22×D28C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C22×C4×D7C2212yes—
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C42×C7C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29#6458—28no—
30#6457—28no—
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33#2682—71nomaximal, Sylow
34C23×D7C2312yes—
35C23×D7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×D28—24no—
44C2×D28—24no—
45C2×D28—24no—
46C2×D28—22no—
47C2×D28—41no—
48C2×D28—41no—
49C2×D28—42no—
50C2×D28—41no—
51C2×D28—41no—
52C2×D28—42no—
53C2×D28—42no—
54C2×D28D414yescomplemented
55C2×C4×D7—24no—
56C2×C4×D7—24no—
57C2×C4×D7—42no—
58C7×C4⋊C4—24no—
59C7×C4⋊C4—41no—
60C7×C4⋊C4D414yescomplemented
61C42×C7—41no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—24no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—42no—
69C4⋊Dic7—41no—
70C4⋊Dic7—41no—
71Dic7⋊C4—24no—
72C2×#123—72no—
73C2×#121—72no—
74#396—72no—
75#394—72no—
76#392—72no—
77#390—71no—
78#390—71no—
79#389D711yescharacteristic, complemented
80#385—72no—
81C23×C7C2411yescharacteristic
82C22×D7—22no—
83C22×D7—22no—
84C22×D7—22no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7C2×D414yes—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7C2×D414yes—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C2×C4×C7#4112yes—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117D28—44no—
118D28—48no—
119D28—44no—
120D28—48no—
121C4×D7—48no—
122C22×D4—71no—
123C22×D4—72no—
124C22×D4—72no—
125C22×D4—71no—
126C23×C4—72no—
127#123—148no—
128#121—148no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4C2×D712yes—
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C4⋊C4C2×D711yescharacteristic
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—72no—
142C2×C42C2×D711yescharacteristic
143#95—72no—
144#95C2×D712yes—
145C22×C7C22×D411yescharacteristic
146C22×C7C22×D411yescharacteristic
147C22×C7C2×#4111yescharacteristic
148C22×C7H(5,𝔽2)11yescharacteristic
149C22×C7H(5,𝔽2)11yescharacteristic
150C22×C7H(5,𝔽2)11yescharacteristic
151C22×C72−1+411yescharacteristic
152C2×D7—24no—
153C2×D7—24no—
154C2×D7—24no—
155C2×D7—42no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—42no—
160C2×D7—42no—
161C2×D7—42no—
162C2×D7—42no—
163C2×D7—42no—
164C2×D7—42no—
165C2×D7—42no—
166C4×C7—22no—
167C4×C7—22no—
168C4×C7—22no—
169C4×C7—24no—
170C4×C7—42no—
171Dic7—42no—
172Dic7—42no—
173C24—72no—
174C24—72no—
175C2×D4—74no—
176C2×D4—144no—
177C2×D4—144no—
178C2×D4—144no—
179C2×D4—142no—
180C2×D4—281no—
181C2×D4—281no—
182C2×D4—281no—
183C2×D4—281no—
184C2×D4—282no—
185C2×D4—282no—
186C2×D4—282no—
187C22×C4—72no—
188C22×C4—72no—
189C22×C4—144no—
190C22×C4—144no—
191C22×C4—282no—
192C22×C4C22×D712yes—
193C22×C4C22×D712yes—
194C22×C4C22×D711yescharacteristic
195C22×C4C22×D711yescharacteristic
196C22×C4C22×D711yescharacteristic
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—144no—
200C4⋊C4—281no—
201C4⋊C4—281no—
202C4⋊C4D2814yescomplemented
203C22⋊C4—144no—
204C22⋊C4—144no—
205C22⋊C4—144no—
206C22⋊C4—144no—
207C22⋊C4—144no—
208C22⋊C4—144no—
209C22⋊C4—282no—
210C42—41no—
211C2×C7#53411yescharacteristic
212C2×C7#53711yescharacteristic
213C2×C7#53811yescharacteristic
214C2×C7D4211yescharacteristic
215C2×C7#54711yescharacteristic
216C2×C7#55011yescharacteristic
217C2×C7#56211yescharacteristic
218D7—44no—
219D7—82no—
220C23—74no—
221C23—142no—
222C23—142no—
223C23—142no—
224C23—142no—
225C23—142no—
226C23—142no—
227C23—142no—
228C23—142no—
229C23—142no—
230C23—142no—
231C23—142no—
232C23—142no—
233C23—142no—
234C23C23×D711yescharacteristic
235D4—284no—
236D4—288no—
237D4—284no—
238D4—288no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—288no—
257C2×C4C2×D2812yes—
258C2×C4C2×D2814yes—
259C2×C4D4×D712yes—
260C2×C4#116112yes—
261C7#268211yescharacteristic, complemented, Sylow
262C22—144no—
263C22—144no—
264C22—144no—
265C22—282no—
266C22—282no—
267C22—282no—
268C22—282no—
269C22—282no—
270C22—282no—
271C22—282no—
272C22—282no—
273C22—282no—
274C22—282no—
275C22—282no—
276C22C22×D2811yescharacteristic
277C22C2×D4×D711yescharacteristic
278C22#654811yescharacteristic
279C22C2×#116111yescharacteristic
280C22#655111yescharacteristic
281C22#655311yescharacteristic
282C22#655311yescharacteristic
283C4—22no—
284C4—24no—
285C4—22no—
286C4—22no—
287C4—42no—
288C4—282no—
289C4—282no—
290C2—284no—
291C2—562no—
292C2#9020111yescharacteristic
293C2D4×D2811yescharacteristic
294C2#9026811yescharacteristic
295C2#9029611yescharacteristic
296C2#9033511yescharacteristic
297C2#9039111yescharacteristic
298C2#9041711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#389C2
hypercenter#389D7
hypocenterC7#2682
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#6458C2semidirect
2. C2×#6457C2semidirect
3. C7×#389C2semidirect
4. #89766C2semidirect
5. #89763C2semidirect
6. #89711C2semidirect
7. #89711C2semidirect
8. #89443C2semidirect
9. #89440C2semidirect
54. C2×D28D4semidirect
60. C7×C4⋊C4D4semidirect
79. #389D7semidirect
202. C4⋊C4D28semidirect
261. C7#2682semidirect

Maximal subgroups

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

subgroupindex
C2×#64582
C2×#64572
C7×#3892
#897662
#897632
#897112
#894432
#894402
#26827