Finite Group Databasebeta

← #11740334

Subgroups — #11740334

Order 896.

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

normal subgroups only

This group has 478 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×C7×#123C211yesmaximal, characteristic
2C2×#6487C211yesmaximal, characteristic, complemented
3#89994C212yesmaximal
4#89984C211yesmaximal, characteristic, complemented
5#89756C212yesmaximal, complemented
6#89755C211yesmaximal, characteristic, complemented
7#89751C211yesmaximal, characteristic, complemented
8#89725C211yesmaximal, characteristic, complemented
9#89724C212yesmaximal, complemented
10#89720C212yesmaximal, complemented
11#89719C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C7×#123—24no—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24C2×C22⋊Dic7C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2212yes—
27#6487—24no—
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2814—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C24×C7C2311yescharacteristic
40C2×C7×D4—22no—
41C2×C7×D4—42no—
42C22×C4×C7—42no—
43C22×C4×C7D414yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C22×C4×C7C2311yescharacteristic
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2312yes—
52C22×Dic7C2312yes—
53C2×Dic14—22no—
54C2×Dic14—42no—
55C7×C4⋊C4—22no—
56C7×C4⋊C4—22no—
57C7×C22⋊C4—22no—
58C7×C22⋊C4—22no—
59C7×C22⋊C4—22no—
60C22⋊Dic7—24no—
61C22⋊Dic7—42no—
62C22⋊Dic7—42no—
63C22⋊Dic7—42no—
64C4⋊Dic7—22no—
65C4⋊Dic7—42no—
66Dic7⋊C4—24no—
67Dic7⋊C4—42no—
68Dic7⋊C4—42no—
69C2×#123D711yescharacteristic
70C2×C22⋊Q8—71no—
71#398—71no—
72#398—72no—
73#397—72no—
74#397—71no—
75#396—72no—
76#395—71no—
77#394—72no—
78#393—71no—
79#380—71no—
80C23×C7—21no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—21no—
89C23×C7C2×D412yes—
90C23×C7C2411yescharacteristic
91C7×D4—44no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—44no—
102C2×C4×C7C2×D412yes—
103C2×C4×C7C2×D412yes—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116Dic14—44no—
117C22×Q8—71no—
118C22×D4C2×D711yescharacteristic
119C23×C4C2×D711yescharacteristic
120#123—24no—
121C22⋊Q8—144no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C4⋊C4C2×D711yescharacteristic
129C2×C22⋊C4—72no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4C2×D711yescharacteristic
134C2×C22⋊C4C2×D711yescharacteristic
135C2×C22⋊C4C2×D711yescharacteristic
136#95—71no—
137#95—71no—
138#95—72no—
139#95—72no—
140#95—72no—
141#95—72no—
142C22×C7—22no—
143C22×C7—22no—
144C22×C7—22no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7C22×D411yescharacteristic
153C22×C7H(5,𝔽2)12yes—
154C22×C7H(5,𝔽2)11yescharacteristic
155C22×C72−1+412yes—
156C22×C72−1+411yescharacteristic
157C4×C7—22no—
158C4×C7—22no—
159C4×C7—41no—
160C4×C7—41no—
161C4×C7—41no—
162Dic7—42no—
163Dic7—42no—
164Dic7—42no—
165Dic7—42no—
166C24C22×D711yescharacteristic
167C24C22×D711yescharacteristic
168C2×Q8—142no—
169C2×Q8—282no—
170C2×D4—22no—
171C2×D4—42no—
172C22×C4—42no—
173C22×C4—72no—
174C22×C4—72no—
175C22×C4—72no—
176C22×C4—72no—
177C22×C4C22⋊D714yes—
178C22×C4C22×D711yescharacteristic
179C22×C4C22×D711yescharacteristic
180C22×C4C22×D711yescharacteristic
181C22×C4C22×D711yescharacteristic
182C22×C4C22×D711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—144no—
186C4⋊C4—142no—
187C4⋊C4—282no—
188C4⋊C4—282no—
189C4⋊C4—282no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—144no—
194C22⋊C4—282no—
195C22⋊C4—282no—
196C22⋊C4—282no—
197C2×C7—42no—
198C2×C7—81no—
199C2×C7#53411yescharacteristic
200C2×C7#53611yescharacteristic
201C2×C7#53711yescharacteristic
202C2×C7#56212yes—
203C2×C7#56312yes—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23C2×C22⋊D712yes—
214C23C23×D711yescharacteristic
215Q8—284no—
216D4—44no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—44no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4C2×C22⋊D712yes—
240C2×C4C2×C22⋊D712yes—
241C7#281411yescharacteristic, complemented, Sylow
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22#654812yes—
253C22#655311yescharacteristic
254C22#655412yes—
255C22#655411yescharacteristic
256C22C22×C22⋊D711yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—282no—
263C4—282no—
264C4—282no—
265C4—282no—
266C2—42no—
267C2—81no—
268C2#9034212yes—
269C2#9035812yes—
270C2#9049511yescharacteristic
271C2#9051611yescharacteristic
272C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radicalC22×C7×D4C22
hypercenterC2×#123D7
hypocenterC7#2814
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#6487C2semidirect
4. #89984C2semidirect
5. #89756C2semidirect
6. #89755C2semidirect
7. #89751C2semidirect
8. #89725C2semidirect
9. #89724C2semidirect
10. #89720C2semidirect
11. #89719C2semidirect
241. C7#2814semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1232
C2×#64872
#899942
#899842
#897562
#897552
#897512
#897252
#897242
#897202
#897192
#28147