Finite Group Databasebeta

← #11734569

Subgroups — #11734569

Order 896.

632 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 143 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×#6501C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#6446C212yesmaximal, complemented
4C7×#394C211yesmaximal, characteristic, complemented
5#89994C211yesmaximal, characteristic, complemented
6#89755C211yesmaximal, characteristic, complemented
7#89728C212yesmaximal, complemented
8#89720C212yesmaximal, complemented
9#89720C212yesmaximal, complemented
10#89439C211yesmaximal, characteristic, complemented
11#89433C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C4×D7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C22×C4×D7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23#6501—24no—
24C2×(C2×D7)⋊C4C2212yes—
25C2×C4⋊Dic7C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2212yes—
32D7×C4⋊C4—24no—
33#6446—28no—
34C7×#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37C7⋊#95C2212yes—
38#2618—71nomaximal, Sylow
39C24×C7C2312yes—
40C23×D7C2311yescharacteristic
41C2×C7×D4—22no—
42C2×C7×D4—42no—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—24no—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C22×Dic7C2312yes—
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C2×C4×D7—22no—
56C2×C4×D7—22no—
57C2×C4×D7—22no—
58C2×C4×D7—42no—
59C2×C4×D7—42no—
60C2×C4×D7D414yes—
61C2×C4×D7D414yescomplemented
62C7×C4⋊C4—22no—
63C7×C22⋊C4—24no—
64C7×C22⋊C4—42no—
65C22⋊Dic7—24no—
66C22⋊Dic7—24no—
67C22⋊Dic7—42no—
68(C2×D7)⋊C4—24no—
69C4⋊Dic7—22no—
70C4⋊Dic7—24no—
71C4⋊Dic7—22no—
72Dic7⋊C4—24no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75Dic7⋊C4—42no—
76C2×#123—72no—
77C2×#121—71no—
78C22×C4⋊C4—71no—
79#398—71no—
80#398—71no—
81#396—72no—
82#396—72no—
83#394—71no—
84#394D711yescharacteristic, complemented
85#386—72no—
86#380—71no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7C2411yescharacteristic
95C22×D7—22no—
96C22×D7—22no—
97C22×D7C2×D412yes—
98C22×D7C2×D412yes—
99C22×D7#4112yes—
100C7×D4—44no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D412yes—
110C2×C4×C7C2×D412yes—
111C22⋊D7—48no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—22no—
115C2×Dic7—21no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7C2×D412yes—
124C2×Dic7C2×D412yes—
125C4×D7—28no—
126C4×D7—44no—
127C4×D7—44no—
128C22×D4—72no—
129C22×D4C2×D711yescharacteristic
130C23×C4—71no—
131C23×C4—71no—
132C23×C4—71no—
133#123—148no—
134#121—144no—
135C2×C4⋊C4—71no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—72no—
141C2×C4⋊C4—72no—
142C2×C4⋊C4—144no—
143C2×C4⋊C4C2×D711yescharacteristic
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4C2×D712yes—
149C2×C22⋊C4C2×D712yes—
150#95—71no—
151#95—72no—
152#95—72no—
153#95C2×D711yescharacteristic
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7—42no—
161C22×C7C22×D411yescharacteristic
162C22×C7C22×D411yescharacteristic
163C22×C7C2×#4111yescharacteristic
164C22×C7H(5,𝔽2)12yes—
165C22×C72−1+412yes—
166C2×D7—22no—
167C2×D7—22no—
168C2×D7—24no—
169C2×D7—24no—
170C2×D7#12114yes—
171C4×C7—22no—
172C4×C7—22no—
173C4×C7—41no—
174C4×C7—42no—
175Dic7—22no—
176Dic7—22no—
177Dic7—42no—
178Dic7—41no—
179Dic7—42no—
180C24—71no—
181C24C22×D712yes—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—144no—
185C2×D4—144no—
186C2×D4—144no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—74no—
190C22×C4—74no—
191C22×C4—72no—
192C22×C4—71no—
193C22×C4—72no—
194C22×C4—142no—
195C22×C4—142no—
196C22×C4—142no—
197C22×C4—282no—
198C22×C4—282no—
199C22×C4C22×D711yescharacteristic
200C22×C4C22×D712yes—
201C22×C4C22×D711yescharacteristic
202C22×C4C22×D711yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—142no—
205C4⋊C4—142no—
206C4⋊C4—144no—
207C4⋊C4—142no—
208C4⋊C4—144no—
209C4⋊C4—142no—
210C4⋊C4—282no—
211C22⋊C4—24no—
212C22⋊C4—42no—
213C22⋊C4—144no—
214C22⋊C4—144no—
215C22⋊C4—144no—
216C22⋊C4—282no—
217C2×C7—82no—
218C2×C7C2×#12111yescharacteristic
219C2×C7#53411yescharacteristic
220C2×C7#53611yescharacteristic
221C2×C7#53712yes—
222C2×C7#53912yes—
223D7—24no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—72no—
232C23—72no—
233C23—72no—
234C23—142no—
235C23—142no—
236C23C23×D711yescharacteristic
237D4—44no—
238D4—288no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—72no—
248C2×C4—72no—
249C2×C4—148no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—141no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—141no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—284no—
262C2×C4—284no—
263C2×C4D4×D712yes—
264C2×C4D4×D712yes—
265C7#261811yescharacteristic, complemented, Sylow
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—74no—
274C22—142no—
275C22—144no—
276C22—144no—
277C22—142no—
278C22C2×D4×D711yescharacteristic
279C22C2×D4×D711yescharacteristic
280C22#654812yes—
281C22D7×#4111yescharacteristic
282C22#655412yes—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—42no—
287C4—142no—
288C4—142no—
289C4—281no—
290C4—282no—
291C4—282no—
292C2—82no—
293C2—144no—
294C2#9028011yescharacteristic
295C2D7×#12111yescharacteristic
296C2#9029912yes—
297C2#9035112yes—
298C2#9036611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6501C2
hypercenter#394D7
hypocenterC7#2618
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×D7×C4⋊C4C2semidirect
3. C2×#6446C2semidirect
4. C7×#394C2semidirect
5. #89994C2semidirect
6. #89755C2semidirect
7. #89728C2semidirect
8. #89720C2semidirect
9. #89720C2semidirect
10. #89439C2semidirect
11. #89433C2semidirect
61. C2×C4×D7D4semidirect
84. #394D7semidirect
265. C7#2618semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×D7×C4⋊C42
C2×#64462
C7×#3942
#899942
#897552
#897282
#897202
#894392
#894332
#26187