Finite Group Databasebeta

← #11734317

Subgroups — #11734317

Order 896.

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

normal subgroups only

This group has 576 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×#6509C211yesmaximal, characteristic, complemented
2C2×#6447C211yesmaximal, characteristic
3C7×#393C211yesmaximal, characteristic, complemented
4#90003C211yesmaximal, characteristic, complemented
5#89731C212yesmaximal, complemented
6#89728C212yesmaximal, complemented
7#89725C211yesmaximal, characteristic, complemented
8#89446C211yesmaximal, characteristic, complemented
9#89437C212yesmaximal, complemented
10D7×#95C211yesmaximal, characteristic, complemented
11#89430C212yesmaximal, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C22×D28C2211yescharacteristic
16C22×C4×D7C2212yes—
17C22×C4×D7C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2212yes—
21#6509—24no—
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29#6447—24no—
30C7×#95C2212yes—
31C7×#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2639—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C2×C7×Q8—22no—
42C2×C7×Q8—42no—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2312yes—
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—22no—
48C2×C22⋊D7—24no—
49C2×C22⋊D7—22no—
50C2×C22⋊D7—42no—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2312yes—
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C2×D28—22no—
56C2×D28—24no—
57C2×C4×D7—24no—
58C2×C4×D7—42no—
59C2×C4×D7—42no—
60C2×C4×D7—42no—
61C2×C4×D7D414yes—
62C7×C22⋊C4—22no—
63C7×C22⋊C4—42no—
64C22⋊Dic7—42no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—42no—
68(C2×D7)⋊C4—42no—
69C4⋊Dic7—22no—
70Dic7⋊C4—22no—
71Dic7⋊C4—24no—
72C2×C22⋊Q8—71no—
73C2×#121—71no—
74#400—72no—
75#397—71no—
76#397—71no—
77#394—72no—
78#393—71no—
79#393D711yescharacteristic, complemented
80#386—72no—
81#386—72no—
82C2×#95—71no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7—21no—
88C23×C7C2411yescharacteristic
89C22×D7—22no—
90C22×D7—21no—
91C22×D7—21no—
92C22×D7—21no—
93C22×D7—22no—
94C22×D7—21no—
95C22×D7—22no—
96C22×D7—21no—
97C22×D7C2×D412yes—
98C22×D7#4114yes—
99C7×Q8—44no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7C2×D412yes—
110C22⋊D7—44no—
111C22⋊D7—44no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7C2×D412yes—
122D28—44no—
123C4×D7—44no—
124C4×D7—48no—
125C22×Q8C2×D711yescharacteristic
126C22×D4—71no—
127C22×D4—71no—
128C22×D4—71no—
129C23×C4—71no—
130C23×C4—72no—
131C22⋊Q8—144no—
132#121—144no—
133C2×C4⋊C4—71no—
134C2×C4⋊C4—72no—
135C2×C4⋊C4—71no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4C2×D711yescharacteristic
142C2×C22⋊C4C2×D712yes—
143#95—71no—
144#95—72no—
145#95—72no—
146#95—72no—
147#95—72no—
148#95C2×D711yescharacteristic
149#95C2×D712yes—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7C22×D411yescharacteristic
156C22×C7C2×#4112yes—
157C22×C7H(5,𝔽2)12yes—
158C22×C7H(5,𝔽2)11yescharacteristic
159C22×C72−1+411yescharacteristic
160C2×D7—22no—
161C2×D7—22no—
162C2×D7—24no—
163C2×D7—24no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7—42no—
168C2×D7—42no—
169C2×D7#12414yes—
170C4×C7—22no—
171C4×C7—41no—
172C4×C7—42no—
173C4×C7—42no—
174Dic7—22no—
175Dic7—41no—
176Dic7—42no—
177Dic7—42no—
178C24—71no—
179C24—71no—
180C24C22×D711yescharacteristic
181C2×Q8—22no—
182C2×Q8—42no—
183C2×D4—142no—
184C2×D4—144no—
185C2×D4—142no—
186C2×D4—142no—
187C2×D4—144no—
188C2×D4—282no—
189C22×C4—71no—
190C22×C4—74no—
191C22×C4—72no—
192C22×C4—71no—
193C22×C4—72no—
194C22×C4—144no—
195C22×C4—282no—
196C22×C4—282no—
197C22×C4—282no—
198C22×C4C22×D712yes—
199C22×C4C22×D711yescharacteristic
200C22×C4C22×D712yes—
201C22×C4C22×D711yescharacteristic
202C4⋊C4—142no—
203C4⋊C4—144no—
204C4⋊C4—142no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—144no—
208C22⋊C4—142no—
209C22⋊C4—282no—
210C22⋊C4—282no—
211C22⋊C4—282no—
212C2×C7—81no—
213C2×C7C2×#12411yescharacteristic
214C2×C7#53411yescharacteristic
215C2×C7#53711yescharacteristic
216C2×C7#53812yes—
217C2×C7#53912yes—
218D7—24no—
219D7—81no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—72no—
226C23—74no—
227C23—142no—
228C23—141no—
229C23—141no—
230C23—141no—
231C23—142no—
232C23—141no—
233C23—142no—
234C23—141no—
235C23C23×D711yescharacteristic
236Q8—44no—
237D4—284no—
238D4—284no—
239D4—284no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—72no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—141no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—284no—
260C2×C4—288no—
261C2×C4D4×D712yes—
262C7#263911yescharacteristic, complemented, Sylow
263C22—42no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—74no—
269C22—144no—
270C22—142no—
271C22—142no—
272C22—144no—
273C22—282no—
274C22—281no—
275C22—281no—
276C22—282no—
277C22—281no—
278C22C2×D4×D711yescharacteristic
279C22#654812yes—
280C22#655111yescharacteristic
281C22D7×#4112yes—
282C22#655311yescharacteristic
283C4—22no—
284C4—42no—
285C4—42no—
286C4—41no—
287C4—142no—
288C4—282no—
289C4—282no—
290C4—281no—
291C2—81no—
292C2—144no—
293C2—561no—
294C2#9028011yescharacteristic
295C2#9032011yescharacteristic
296C2#9032912yes—
297C2D7×#12411yescharacteristic
298C2#9036912yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC2×#6447C2
hypercenter#393D7
hypocenterC7#2639
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#6509C2semidirect
3. C7×#393C2semidirect
4. #90003C2semidirect
5. #89731C2semidirect
6. #89728C2semidirect
7. #89725C2semidirect
8. #89446C2semidirect
9. #89437C2semidirect
10. D7×#95C2semidirect
11. #89430C2semidirect
79. #393D7semidirect
262. C7#2639semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×#64472
C7×#3932
#900032
#897312
#897282
#897252
#894462
#894372
D7×#952
#894302
#26397