Finite Group Databasebeta

← #11735009

Subgroups — #11735009

Order 896.

598 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 129 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×#6458C211yesmaximal, characteristic, complemented
2C2×#6446C211yesmaximal, characteristic, complemented
3C7×#397C211yesmaximal, characteristic, complemented
4#89752C211yesmaximal, characteristic, complemented
5#89732C211yesmaximal, characteristic, complemented
6#89732C212yesmaximal, complemented
7#89729C212yesmaximal, complemented
8#89444C211yesmaximal, characteristic, complemented
9#89444C212yesmaximal, complemented
10#89442C212yesmaximal, complemented
11#89438C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×D28C2212yes—
15C22×D28C2211yescharacteristic
16C22×C4×D7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2212yes—
31#6458—24no—
32#6446—24no—
33C7×#95C2211yescharacteristic
34C7×#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2661—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2312yes—
40C23×D7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—22no—
46C2×C22⋊D7—24no—
47C2×C22⋊D7—42no—
48C2×C22⋊D7—42no—
49C2×C22⋊D7—42no—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2312yes—
54C2×D28—24no—
55C2×D28—22no—
56C2×D28—42no—
57C2×D28—42no—
58C2×D28—42no—
59C2×C4×D7—42no—
60C2×C4×D7D414yescomplemented
61C7×C4⋊C4—22no—
62C7×C22⋊C4—22no—
63C7×C22⋊C4—42no—
64C22⋊Dic7—22no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—24no—
69(C2×D7)⋊C4—42no—
70(C2×D7)⋊C4—42no—
71C4⋊Dic7—22no—
72Dic7⋊C4—22no—
73Dic7⋊C4—42no—
74C4×Dic7—42no—
75C2×#123—71no—
76C2×#121—71no—
77#398—71no—
78#397D711yescharacteristic, complemented
79#394—71no—
80#394—72no—
81#392—71no—
82#392—72no—
83#390—72no—
84#388—72no—
85C4×C22⋊C4—71no—
86C23×C7—21no—
87C23×C7—22no—
88C23×C7—21no—
89C23×C7—22no—
90C23×C7—21no—
91C23×C7C2411yescharacteristic
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—21no—
95C22×D7—21no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7C2×D412yes—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7C2×D412yes—
114C22⋊D7—48no—
115C22⋊D7—44no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123C2×Dic7—22no—
124C2×Dic7C2×D412yes—
125C2×Dic7#4114yes—
126D28—44no—
127D28—48no—
128C4×D7—44no—
129C22×D4—71no—
130C22×D4—72no—
131C22×D4—72no—
132C22×D4—71no—
133C23×C4—71no—
134#123—144no—
135#121—144no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4C2×D711yescharacteristic
140C2×C22⋊C4—71no—
141C2×C22⋊C4—72no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—72no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4C2×D711yescharacteristic
148C2×C22⋊C4C2×D712yes—
149C2×C42—72no—
150#95—71no—
151#95—72no—
152#95C2×D711yescharacteristic
153#95C2×D712yes—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7C22×D411yescharacteristic
160C22×C7C2×#4112yes—
161C22×C7H(5,𝔽2)11yescharacteristic
162C22×C7H(5,𝔽2)12yes—
163C22×C7H(5,𝔽2)11yescharacteristic
164C2×D7—22no—
165C2×D7—22no—
166C2×D7—22no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C4×C7—22no—
177C4×C7—41no—
178C4×C7—42no—
179C4×C7—42no—
180Dic7—22no—
181Dic7—24no—
182Dic7—41no—
183Dic7—41no—
184C24—71no—
185C24—72no—
186C24C22×D711yescharacteristic
187C2×D4—144no—
188C2×D4—144no—
189C2×D4—142no—
190C2×D4—142no—
191C2×D4—282no—
192C2×D4—282no—
193C2×D4—282no—
194C2×D4—282no—
195C2×D4—282no—
196C2×D4—282no—
197C22×C4—72no—
198C22×C4—71no—
199C22×C4—74no—
200C22×C4—71no—
201C22×C4—71no—
202C22×C4—282no—
203C22×C4C22×D712yes—
204C22×C4C22×D712yes—
205C22×C4C22×D711yescharacteristic
206C22×C4C22×D711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—142no—
209C4⋊C4—142no—
210C4⋊C4—282no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—144no—
214C22⋊C4—144no—
215C22⋊C4—144no—
216C22⋊C4—142no—
217C22⋊C4—142no—
218C22⋊C4—282no—
219C22⋊C4—282no—
220C42—282no—
221C2×C7—81no—
222C2×C7#53211yescharacteristic
223C2×C7#53411yescharacteristic
224C2×C7#53511yescharacteristic
225C2×C7#53812yes—
226C2×C7#54012yes—
227D7—42no—
228D7—82no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—72no—
235C23—141no—
236C23—142no—
237C23—142no—
238C23—141no—
239C23—142no—
240C23—142no—
241C23—142no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—142no—
246C23C23×D711yescharacteristic
247D4—284no—
248D4—288no—
249D4—288no—
250D4—284no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—72no—
261C2×C4—74no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—141no—
266C2×C4—141no—
267C2×C4—141no—
268C2×C4—142no—
269C2×C4—141no—
270C2×C4—284no—
271C2×C4D4×D712yes—
272C7#266111yescharacteristic, complemented, Sylow
273C22—42no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—41no—
278C22—142no—
279C22—142no—
280C22—142no—
281C22—282no—
282C22—282no—
283C22—282no—
284C22—282no—
285C22—282no—
286C22—282no—
287C22—282no—
288C22—282no—
289C22—282no—
290C22C2×D4×D711yescharacteristic
291C22#654811yescharacteristic
292C22D7×#4112yes—
293C22#655311yescharacteristic
294C22#655312yes—
295C4—22no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—144no—
300C4—142no—
301C4—281no—
302C4—281no—
303C2—81no—
304C2—282no—
305C2—562no—
306C2#9029811yescharacteristic
307C2#9034411yescharacteristic
308C2#9035412yes—
309C2#9035911yescharacteristic
310C2#9039512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
hypercenter#397D7
hypocenterC7#2661
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6458C2semidirect
2. C2×#6446C2semidirect
3. C7×#397C2semidirect
4. #89752C2semidirect
5. #89732C2semidirect
6. #89732C2semidirect
7. #89729C2semidirect
8. #89444C2semidirect
9. #89444C2semidirect
10. #89442C2semidirect
11. #89438C2semidirect
60. C2×C4×D7D4semidirect
78. #397D7semidirect
272. C7#2661semidirect

Maximal subgroups

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

subgroupindex
C2×#64582
C2×#64462
C7×#3972
#897522
#897322
#897292
#894442
#894422
#894382
#26617