Finite Group Databasebeta

← #11740532

Subgroups — #11740532

Order 896.

582 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 141 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×C7×#126C211yesmaximal, characteristic, complemented
2C2×C4×D28C211yesmaximal, characteristic, complemented
3#89763C211yesmaximal, characteristic, complemented
4#89762C212yesmaximal, complemented
5#89761C212yesmaximal, complemented
6#89755C211yesmaximal, characteristic, complemented
7#89732C211yesmaximal, characteristic
8#89729C212yesmaximal, complemented
9#89728C212yesmaximal, complemented
10#89725C211yesmaximal, characteristic, complemented
11#89704C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2212yes—
13C22×D28C2211yescharacteristic
14C22×C4×D7C2212yes—
15C7×#126—24no—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C42×C7C2211yescharacteristic
23C2×C22⋊Dic7C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2212yes—
31C4×D28—24no—
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2603—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—24no—
47C2×C22⋊D7—24no—
48C2×C22⋊D7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2312yes—
52C2×D28—24no—
53C2×D28—22no—
54C2×C4×D7—24no—
55C2×C4×D7—42no—
56C2×C4×D7—42no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—22no—
59C7×C4⋊C4—22no—
60C7×C22⋊C4—22no—
61C7×C22⋊C4—22no—
62C7×C22⋊C4—22no—
63C42×C7D414yescomplemented
64C22⋊Dic7—42no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—42no—
69C4⋊Dic7—22no—
70Dic7⋊C4—42no—
71C4×Dic7—24no—
72C2×#126D711yescharacteristic, complemented
73C2×C4×D4—71no—
74#398—71no—
75#397—71no—
76#394—71no—
77#392—71no—
78#390—72no—
79#388—72no—
80#386—72no—
81#385—72no—
82#376—71no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7C2411yescharacteristic
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7#4114yes—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7C2×D412yes—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C22⋊D7—48no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7#4114yes—
120D28—28no—
121C4×D7—48no—
122C22×D4—72no—
123C22×D4—71no—
124C23×C4—72no—
125#126—24no—
126C4×D4—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—72no—
129C2×C4⋊C4C2×D711yescharacteristic
130C2×C4⋊C4C2×D711yescharacteristic
131C2×C4⋊C4C2×D711yescharacteristic
132C2×C22⋊C4—72no—
133C2×C22⋊C4—72no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4C2×D711yescharacteristic
138C2×C22⋊C4C2×D711yescharacteristic
139C2×C22⋊C4C2×D711yescharacteristic
140C2×C42—72no—
141C2×C42C2×D711yescharacteristic
142#95—72no—
143#95—71no—
144#95—71no—
145#95—72no—
146#95—72no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7C22×D411yescharacteristic
153C22×C7C2×#4112yes—
154C22×C7C2×#4112yes—
155C22×C7H(5,𝔽2)11yescharacteristic
156C22×C7H(5,𝔽2)11yescharacteristic
157C2×D7—24no—
158C2×D7—24no—
159C2×D7—24no—
160C2×D7—42no—
161C2×D7—42no—
162C2×D7—42no—
163C2×D7—42no—
164C4×C7—22no—
165C4×C7—22no—
166C4×C7—22no—
167C4×C7—41no—
168C4×C7—41no—
169C4×C7—41no—
170Dic7—24no—
171Dic7—42no—
172Dic7—42no—
173C24—72no—
174C24C22×D711yescharacteristic
175C2×D4—144no—
176C2×D4—144no—
177C2×D4—144no—
178C2×D4—142no—
179C2×D4—282no—
180C22×C4—72no—
181C22×C4—72no—
182C22×C4—72no—
183C22×C4—144no—
184C22×C4—282no—
185C22×C4—282no—
186C22×C4C22×D711yescharacteristic
187C22×C4C22×D711yescharacteristic
188C22×C4C22×D711yescharacteristic
189C22×C4C22×D711yescharacteristic
190C22×C4C22×D711yescharacteristic
191C22×C4C22×D711yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—142no—
196C4⋊C4—282no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—144no—
201C22⋊C4—144no—
202C22⋊C4—144no—
203C22⋊C4—282no—
204C22⋊C4—282no—
205C42—144no—
206C42C22⋊D714yescomplemented
207C2×C7—81no—
208C2×C7#52511yescharacteristic
209C2×C7#53211yescharacteristic
210C2×C7#53511yescharacteristic
211C2×C7#55112yes—
212C2×C7#55312yes—
213D7—44no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—74no—
220C23—142no—
221C23—142no—
222C23—142no—
223C23—142no—
224C23—142no—
225C23—142no—
226C23C23×D711yescharacteristic
227D4—148no—
228D4—288no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—74no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—288no—
250C2×C4C2×C22⋊D712yes—
251C2×C4C2×C22⋊D712yes—
252C2×C4C2×C22⋊D712yes—
253C7#260311yescharacteristic, complemented, Sylow
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—144no—
260C22—144no—
261C22—144no—
262C22—282no—
263C22—282no—
264C22—282no—
265C22—282no—
266C22D7×#4112yes—
267C22D7×#4112yes—
268C22#655311yescharacteristic
269C22#655311yescharacteristic
270C22C22×C22⋊D711yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—144no—
278C4—282no—
279C4—282no—
280C2—81no—
281C2—284no—
282C2#9039612yes—
283C2#9040012yes—
284C2#9051911yescharacteristic
285C2#9052111yescharacteristic
286C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#126C2
prime radical#89732C2
hypercenterC2×#126D7
hypocenterC7#2603
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C7×#126C2semidirect
2. C2×C4×D28C2semidirect
3. #89763C2semidirect
4. #89762C2semidirect
5. #89761C2semidirect
6. #89755C2semidirect
8. #89729C2semidirect
9. #89728C2semidirect
10. #89725C2semidirect
11. #89704C2semidirect
63. C42×C7D4semidirect
72. C2×#126D7semidirect
206. C42C22⋊D7semidirect
253. C7#2603semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1262
C2×C4×D282
#897632
#897622
#897612
#897552
#897322
#897292
#897282
#897252
#897042
#26037