Finite Group Databasebeta

← #11733868

Subgroups — #11733868

Order 896.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C7×#390C211yesmaximal, characteristic
2#89994C212yesmaximal
3#89755C211yesmaximal, characteristic, complemented
4#89751C211yesmaximal, characteristic, complemented
5#89724C212yesmaximal, complemented
6#89724C212yesmaximal, complemented
7#89720C212yesmaximal, complemented
8#89719C212yesmaximal, complemented
9#89705C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2212yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2211yescharacteristic
31#2830—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×D4—22no—
34C2×C7×D4—41no—
35C2×C7×D4—41no—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C2×Dic14—42no—
45C2×Dic14—41no—
46C7×C4⋊C4—41no—
47C7×C22⋊C4—42no—
48C7×C22⋊C4—42no—
49C42×C7—41no—
50C22⋊Dic7—42no—
51C22⋊Dic7—42no—
52C22⋊Dic7—42no—
53C22⋊Dic7—42no—
54C4⋊Dic7—41no—
55Dic7⋊C4—24no—
56Dic7⋊C4—42no—
57Dic7⋊C4—42no—
58#398—71no—
59#397—72no—
60#397—72no—
61#396—72no—
62#395—71no—
63#394—72no—
64#393—72no—
65#390D711yescharacteristic
66#387—72no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—22no—
74C23×C7C2411yescharacteristic
75C7×D4—44no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—21no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7#4112yes—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98Dic14—44no—
99C22×Q8—71no—
100C22×D4C2×D711yescharacteristic
101C2×C4⋊C4—72no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—71no—
105C2×C4⋊C4C2×D711yescharacteristic
106C2×C22⋊C4—72no—
107C2×C22⋊C4—72no—
108C2×C22⋊C4—72no—
109C2×C22⋊C4—72no—
110C2×C22⋊C4C2×D712yes—
111C2×C22⋊C4C2×D712yes—
112C2×C42C2×D711yescharacteristic
113#95—72no—
114#95—72no—
115#95—72no—
116#95—72no—
117#95—72no—
118#95—71no—
119#95—71no—
120C22×C7—42no—
121C22×C7—42no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7—42no—
126C22×C7—42no—
127C22×C7C2×#4111yescharacteristic
128C22×C7H(5,𝔽2)11yescharacteristic
129C22×C7H(5,𝔽2)11yescharacteristic
130C22×C7H(5,𝔽2)12yes—
131C22×C72−1+412yes—
132C4×C7—22no—
133C4×C7—41no—
134C4×C7—41no—
135C4×C7—42no—
136Dic7—42no—
137Dic7—42no—
138Dic7—42no—
139Dic7—42no—
140C24C22×D712yes—
141C2×Q8—282no—
142C2×Q8—281no—
143C2×D4—22no—
144C2×D4—41no—
145C2×D4—41no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4—72no—
149C22×C4—72no—
150C22×C4C22×D712yes—
151C22×C4C22×D711yescharacteristic
152C22×C4C22×D711yescharacteristic
153C22×C4C22×D711yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—144no—
156C4⋊C4—282no—
157C4⋊C4—282no—
158C4⋊C4—281no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—282no—
162C22⋊C4—282no—
163C22⋊C4—282no—
164C22⋊C4—282no—
165C42—41no—
166C2×C7—82no—
167C2×C7#53811yescharacteristic
168C2×C7#54212yes—
169C2×C7#56011yescharacteristic
170C2×C7#56212yes—
171C2×C7#56311yescharacteristic
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23C23×D711yescharacteristic
180Q8—284no—
181D4—44no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4#115712yes—
204C7#283011yescharacteristic, complemented, Sylow
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22C2×#115711yescharacteristic
213C22#654811yescharacteristic
214C22#654812yes—
215C22#654811yescharacteristic
216C22#655412yes—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—42no—
221C4—282no—
222C4—282no—
223C4—282no—
224C4—282no—
225C2—82no—
226C2#9018511yescharacteristic
227C2#9025212yes—
228C2#9031212yes—
229C2#9036311yescharacteristic
230C2#9041111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2830
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #89755C2semidirect
4. #89751C2semidirect
5. #89724C2semidirect
6. #89724C2semidirect
7. #89720C2semidirect
8. #89719C2semidirect
9. #89705C2semidirect
204. C7#2830semidirect

Maximal subgroups

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

subgroupindex
C7×#3902
#899942
#897552
#897512
#897242
#897202
#897192
#897052
#28307