Finite Group Databasebeta

← #11734429

Subgroups — #11734429

Order 896.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6503C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89994C211yesmaximal, characteristic, complemented
5#89763C211yesmaximal, characteristic, complemented
6#89733C212yesmaximal, complemented
7#89731C212yesmaximal, complemented
8#89729C212yesmaximal, complemented
9#89729C212yesmaximal, complemented
10#89444C211yesmaximal, characteristic, complemented
11#89420C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2212yes—
15C22×D28C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21#6503—24no—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×C4×Dic7C2212yes—
32C2×C4×Dic7C2211yescharacteristic
33#6452—24no—
34C7×#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2641—71nomaximal, Sylow
38C24×C7C2312yes—
39C23×D7C2312yes—
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—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—42no—
51C2×C22⋊D7—42no—
52C22×Dic7C2312yes—
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C2×D28—22no—
58C2×D28—42no—
59C7×C4⋊C4—22no—
60C7×C22⋊C4—42no—
61C7×C22⋊C4—42no—
62C22⋊Dic7—42no—
63C22⋊Dic7—42no—
64(C2×D7)⋊C4—42no—
65(C2×D7)⋊C4—42no—
66(C2×D7)⋊C4—42no—
67(C2×D7)⋊C4—42no—
68C4⋊Dic7—22no—
69Dic7⋊C4—22no—
70Dic7⋊C4—22no—
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73C4×Dic7—24no—
74C4×Dic7D414yes—
75C2×#127—71no—
76C2×#125—71no—
77#394—71no—
78#394—71no—
79#394—72no—
80#394—72no—
81#394—71no—
82#394D711yescharacteristic, complemented
83#390—72no—
84#390—72no—
85#376—71no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7C2411yescharacteristic
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C7×D4—44no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—21no—
111C2×C4×C7C2×D412yes—
112C22⋊D7—48no—
113C22⋊D7—48no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7—22no—
119C2×Dic7—21no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—21no—
124C2×Dic7C2×D412yes—
125C2×Dic7C2×D412yes—
126C2×Dic7#4114yes—
127D28—44no—
128C22×D4—72no—
129C22×D4—72no—
130C22×D4—71no—
131C22×D4C2×D711yescharacteristic
132#127—144no—
133#125—144no—
134C2×C4⋊C4—71no—
135C2×C4⋊C4—71no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4C2×D711yescharacteristic
140C2×C22⋊C4—72no—
141C2×C22⋊C4—72no—
142C2×C22⋊C4—72no—
143C2×C22⋊C4—72no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4C2×D712yes—
147C2×C22⋊C4C2×D712yes—
148C2×C42—71no—
149C2×C42—72no—
150#95—72no—
151#95—71no—
152#95C2×D711yescharacteristic
153C22×C7—42no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7C22×D411yescharacteristic
161C22×C7C2×#4112yes—
162C22×C7H(5,𝔽2)12yes—
163C22×C7H(5,𝔽2)11yescharacteristic
164C22×C7H(5,𝔽2)11yescharacteristic
165C2×D7—42no—
166C2×D7—42no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—42no—
172C4×C7—22no—
173C4×C7—41no—
174C4×C7—41no—
175C4×C7—42no—
176Dic7—22no—
177Dic7—41no—
178Dic7—41no—
179Dic7—42no—
180Dic7#12414yes—
181C24—72no—
182C24C22×D712yes—
183C2×D4—22no—
184C2×D4—42no—
185C2×D4—144no—
186C2×D4—144no—
187C2×D4—144no—
188C2×D4—144no—
189C2×D4—142no—
190C2×D4—282no—
191C2×D4—282no—
192C2×D4—282no—
193C22×C4—71no—
194C22×C4—71no—
195C22×C4—72no—
196C22×C4—71no—
197C22×C4—71no—
198C22×C4C22×D712yes—
199C22×C4C22×D711yescharacteristic
200C22×C4C22×D711yescharacteristic
201C22×C4C22×D711yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—142no—
204C4⋊C4—142no—
205C4⋊C4—142no—
206C4⋊C4—142no—
207C4⋊C4—142no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—282no—
211C22⋊C4—282no—
212C22⋊C4—282no—
213C22⋊C4—282no—
214C22⋊C4—282no—
215C22⋊C4—282no—
216C42—74no—
217C42—144no—
218C2×C7—82no—
219C2×C7C2×#12411yescharacteristic
220C2×C7#53511yescharacteristic
221C2×C7#53511yescharacteristic
222C2×C7#54012yes—
223C2×C7#54012yes—
224D7—82no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—142no—
233C23—142no—
234C23—142no—
235C23—142no—
236C23—142no—
237C23—142no—
238C23—142no—
239C23C23×D711yescharacteristic
240D4—44no—
241D4—284no—
242D4—288no—
243D4—288no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—72no—
254C2×C4—74no—
255C2×C4—72no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4—141no—
260C2×C4—142no—
261C2×C4—141no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4—141no—
265C2×C4—142no—
266C2×C4D4×D712yes—
267C7#264111yescharacteristic, complemented, Sylow
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—282no—
276C22—282no—
277C22—282no—
278C22—282no—
279C22—282no—
280C22—282no—
281C22—282no—
282C22C2×D4×D711yescharacteristic
283C22#654812yes—
284C22#654811yescharacteristic
285C22D7×#4112yes—
286C22#655311yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—74no—
292C4—142no—
293C4—281no—
294C4—281no—
295C4—282no—
296C2—82no—
297C2—562no—
298C2#9028311yescharacteristic
299C2#9029811yescharacteristic
300C2#9030612yes—
301C2#9035412yes—
302C2D7×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6503C2
hypercenter#394D7
hypocenterC7#2641
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6452C2semidirect
3. C7×#394C2semidirect
4. #89994C2semidirect
5. #89763C2semidirect
6. #89733C2semidirect
7. #89731C2semidirect
8. #89729C2semidirect
9. #89729C2semidirect
10. #89444C2semidirect
11. #89420C2semidirect
82. #394D7semidirect
267. C7#2641semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
C2×#64522
C7×#3942
#899942
#897632
#897332
#897312
#897292
#894442
#894202
#26417