Finite Group Databasebeta

← #11740435

Subgroups — #11740435

Order 896.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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×#124C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3#90002C211yesmaximal, characteristic, complemented
4#89994C212yesmaximal
5#89720C212yesmaximal, complemented
6#89719C212yesmaximal, complemented
7#89717C212yesmaximal, complemented
8#89716C212yesmaximal, complemented
9#89705C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C7×#124—24no—
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2212yes—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21#6507—24no—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33#2731—71nomaximal, Sylow
34C24×C7C2312yes—
35C2×C7×Q8—41no—
36C2×C7×Q8—41no—
37C2×C7×Q8D414yes—
38C2×C7×D4—22no—
39C2×C7×D4—41no—
40C2×C7×D4—41no—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C2×Dic14—22no—
50C2×Dic14—41no—
51C2×Dic14—41no—
52C7×C22⋊C4—24no—
53C7×C22⋊C4—24no—
54C42×C7—22no—
55C22⋊Dic7—24no—
56C22⋊Dic7—24no—
57C22⋊Dic7—42no—
58C22⋊Dic7—42no—
59C4⋊Dic7—41no—
60Dic7⋊C4—24no—
61Dic7⋊C4—24no—
62Dic7⋊C4—24no—
63C4×Dic7—22no—
64C4×Dic7—41no—
65C2×C4⋊Q8—71no—
66C2×#124D711yescharacteristic
67#396—72no—
68#394—72no—
69#393—72no—
70#391—71no—
71#388—72no—
72#388—72no—
73#387—72no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7—22no—
81C23×C7C2411yescharacteristic
82C7×Q8—44no—
83C7×D4—44no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×D412yes—
95C2×C4×C7C2×D412yes—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7#4114yes—
108Dic14—44no—
109C22×Q8—71no—
110C22×Q8C2×D711yescharacteristic
111C22×D4C2×D711yescharacteristic
112C4⋊Q8—144no—
113#124—24no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—71no—
118C2×C22⋊C4—72no—
119C2×C22⋊C4—72no—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—72no—
122C2×C22⋊C4C2×D712yes—
123C2×C22⋊C4C2×D712yes—
124C2×C42—71no—
125C2×C42—71no—
126C2×C42C2×D711yescharacteristic
127#95—72no—
128#95—72no—
129#95—72no—
130#95—72no—
131#95—72no—
132C22×C7—42no—
133C22×C7—42no—
134C22×C7—42no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7—42no—
138C22×C7—42no—
139C22×C7C22×D411yescharacteristic
140C22×C7C2×#4112yes—
141C22×C7H(5,𝔽2)11yescharacteristic
142C22×C7H(5,𝔽2)11yescharacteristic
143C22×C72−1+412yes—
144C4×C7—22no—
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—42no—
148Dic7—24no—
149Dic7—42no—
150Dic7—42no—
151Dic7—42no—
152C24C22×D712yes—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—142no—
156C2×Q8—281no—
157C2×Q8—281no—
158C2×Q8C22⋊D714yes—
159C2×D4—22no—
160C2×D4—41no—
161C2×D4—41no—
162C22×C4—72no—
163C22×C4—72no—
164C22×C4—72no—
165C22×C4—72no—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D711yescharacteristic
170C4⋊C4—144no—
171C4⋊C4—144no—
172C4⋊C4—144no—
173C4⋊C4—281no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—144no—
177C22⋊C4—144no—
178C22⋊C4—282no—
179C22⋊C4—282no—
180C42—22no—
181C42—142no—
182C42—281no—
183C2×C7—82no—
184C2×C7#53411yescharacteristic
185C2×C7#54212yes—
186C2×C7#54812yes—
187C2×C7#55511yescharacteristic
188C2×C7#56311yescharacteristic
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23C23×D711yescharacteristic
197Q8—44no—
198Q8—284no—
199D4—44no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—74no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4C2×C22⋊D712yes—
222C2×C4C2×C22⋊D712yes—
223C2×C4C2×C22⋊D712yes—
224C7#273111yescharacteristic, complemented, Sylow
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22#654811yescharacteristic
233C22#654811yescharacteristic
234C22D7×#4112yes—
235C22#655412yes—
236C22C22×C22⋊D711yescharacteristic
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—144no—
242C4—282no—
243C4—282no—
244C4—282no—
245C2—82no—
246C2#9036012yes—
247C2#9036111yescharacteristic
248C2#9036311yescharacteristic
249C2#9049511yescharacteristic
250C2#9052212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radicalC22×C7×D4C22
hypercenterC2×#124D7
hypocenterC7#2731
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#6507C2semidirect
3. #90002C2semidirect
5. #89720C2semidirect
6. #89719C2semidirect
7. #89717C2semidirect
8. #89716C2semidirect
9. #89705C2semidirect
224. C7#2731semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#65072
#900022
#899942
#897202
#897192
#897172
#897162
#897052
#27317