Finite Group Databasebeta

← #11734530

Subgroups — #11734530

Order 896.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6459C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89994C211yesmaximal, characteristic, complemented
5#89755C211yesmaximal, characteristic, complemented
6#89733C212yesmaximal, complemented
7#89731C212yesmaximal, complemented
8#89721C212yesmaximal, complemented
9#89719C212yesmaximal, complemented
10#89436C211yesmaximal, characteristic, complemented
11#89433C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22#6501—24no—
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2212yes—
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2212yes—
31#6459—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2816—71nomaximal, Sylow
38C24×C7C2312yes—
39C23×D7C2311yescharacteristic
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—42no—
48C2×C22⋊D7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C2×C4×D7—42no—
55C2×C4×D7D414yes—
56C2×Dic14—22no—
57C2×Dic14—42no—
58C7×C4⋊C4—22no—
59C7×C22⋊C4—42no—
60C7×C22⋊C4—42no—
61C22⋊Dic7—24no—
62C22⋊Dic7—42no—
63C22⋊Dic7—42no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—42no—
67C4⋊Dic7—22no—
68C4⋊Dic7—42no—
69Dic7⋊C4—22no—
70Dic7⋊C4—22no—
71Dic7⋊C4—42no—
72C2×C22⋊Q8—71no—
73C2×#121—71no—
74#399—72no—
75#398—71no—
76#398—71no—
77#394—71no—
78#394—72no—
79#394—72no—
80#394D711yescharacteristic, complemented
81#393—72no—
82#381—71no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7C2411yescharacteristic
91C22×D7—22no—
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—22no—
95C22×D7C2×D412yes—
96C7×D4—44no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7C2×D412yes—
107C22⋊D7—48no—
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—21no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7C2×D412yes—
121C4×D7—44no—
122Dic14—44no—
123C22×Q8—71no—
124C22×D4—72no—
125C22×D4C2×D711yescharacteristic
126C23×C4—71no—
127C22⋊Q8—144no—
128#121—144no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4C2×D711yescharacteristic
135C2×C22⋊C4—72no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—71no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4C2×D712yes—
142C2×C22⋊C4C2×D712yes—
143#95—72no—
144#95—71no—
145#95—72no—
146#95—72no—
147#95C2×D711yescharacteristic
148C22×C7—42no—
149C22×C7—42no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7C22×D411yescharacteristic
156C22×C7H(5,𝔽2)12yes—
157C22×C7H(5,𝔽2)11yescharacteristic
158C22×C7H(5,𝔽2)11yescharacteristic
159C22×C72−1+412yes—
160C2×D7—22no—
161C2×D7—22no—
162C2×D7—22no—
163C2×D7—42no—
164C2×D7—42no—
165C4×C7—22no—
166C4×C7—41no—
167C4×C7—41no—
168C4×C7—42no—
169Dic7—22no—
170Dic7—41no—
171Dic7—41no—
172Dic7—42no—
173Dic7—42no—
174C24—71no—
175C24C22×D712yes—
176C2×Q8—142no—
177C2×Q8—282no—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—144no—
181C2×D4—282no—
182C2×D4—282no—
183C22×C4—71no—
184C22×C4—74no—
185C22×C4—72no—
186C22×C4—71no—
187C22×C4—72no—
188C22×C4—71no—
189C22×C4—282no—
190C22×C4C22×D712yes—
191C22×C4C22×D711yescharacteristic
192C22×C4C22×D711yescharacteristic
193C22×C4C22×D711yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—142no—
196C4⋊C4—142no—
197C4⋊C4—142no—
198C4⋊C4—282no—
199C4⋊C4—282no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—142no—
203C22⋊C4—142no—
204C22⋊C4—144no—
205C22⋊C4—282no—
206C22⋊C4—282no—
207C22⋊C4—282no—
208C2×C7—82no—
209C2×C7#53411yescharacteristic
210C2×C7#53511yescharacteristic
211C2×C7#53611yescharacteristic
212C2×C7#56212yes—
213C2×C7#56212yes—
214D7—42no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—72no—
223C23—142no—
224C23—141no—
225C23—141no—
226C23—142no—
227C23C23×D711yescharacteristic
228Q8—284no—
229D4—44no—
230D4—288no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—72no—
241C2×C4—142no—
242C2×C4—141no—
243C2×C4—142no—
244C2×C4—141no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—141no—
248C2×C4—142no—
249C2×C4—141no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—284no—
254C2×C4D4×D712yes—
255C7#281611yescharacteristic, complemented, Sylow
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—142no—
264C22—142no—
265C22—142no—
266C22—282no—
267C22—282no—
268C22C2×D4×D711yescharacteristic
269C22#654812yes—
270C22#654811yescharacteristic
271C22#655311yescharacteristic
272C22#655412yes—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—42no—
277C4—142no—
278C4—281no—
279C4—282no—
280C4—281no—
281C4—282no—
282C2—82no—
283C2—282no—
284C2#9028011yescharacteristic
285C2#9029811yescharacteristic
286C2#9031212yes—
287C2#9035812yes—
288C2#9036611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6501C2
hypercenter#394D7
hypocenterC7#2816
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6459C2semidirect
3. C7×#394C2semidirect
4. #89994C2semidirect
5. #89755C2semidirect
6. #89733C2semidirect
7. #89731C2semidirect
8. #89721C2semidirect
9. #89719C2semidirect
10. #89436C2semidirect
11. #89433C2semidirect
80. #394D7semidirect
255. C7#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×#64592
C7×#3942
#899942
#897552
#897332
#897312
#897212
#897192
#894362
#894332
#28167