Finite Group Databasebeta

← #11730998

Subgroups — #11730998

Order 896.

444 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4×C7×C22⋊C4C211yesmaximal, characteristic
2#89983C211yesmaximal, characteristic
3C4×C22⋊Dic7C211yesmaximal, characteristic
4#89721C212yesmaximal
5#89719C212yesmaximal
6#89705C212yesmaximal, complemented
7#89701C212yesmaximal, complemented
8#89426C212yesmaximal, complemented
9#89418C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×Dic14C2212yes—
12C2×C7×C22⋊C4C2212yes—
13C2×C42×C7C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C4⋊Dic7C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2211yescharacteristic
22C2×C4×Dic7C2211yescharacteristic
23C7×#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2607—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7—41no—
34C22×C4×C7—41no—
35C22×C4×C7Q814yescomplemented
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C2×Dic14—24no—
45C2×Dic14—24no—
46C2×Dic14—42no—
47C7×C22⋊C4—24no—
48C42×C7—41no—
49C42×C7—41no—
50C22⋊Dic7—24no—
51C22⋊Dic7—41no—
52C22⋊Dic7—41no—
53C4⋊Dic7—24no—
54Dic7⋊C4—24no—
55Dic7⋊C4—42no—
56C4×Dic7—41no—
57C4×Dic7—41no—
58#399—72no—
59#393—72no—
60#391—72no—
61#391—72no—
62#387—72no—
63#387—72no—
64#381—71no—
65C4×C22⋊C4—71no—
66C4×C22⋊C4D711yescharacteristic
67C23×C7—22no—
68C23×C7—22no—
69C23×C7—21no—
70C23×C7—21no—
71C23×C7C2×Q812yes—
72C23×C7C2411yescharacteristic
73C2×C4×C7—22no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—22no—
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—44no—
83C2×C4×C7C2×Q812yes—
84C2×C4×C7C2×Q812yes—
85C2×C4×C7#4114yes—
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#4114yes—
98Dic14—48no—
99C22×Q8—72no—
100C23×C4C2×D711yescharacteristic
101C2×C4⋊C4—72no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4—72no—
104C2×C22⋊C4—72no—
105C2×C22⋊C4—71no—
106C2×C22⋊C4—71no—
107C2×C22⋊C4C2×D712yes—
108C2×C42—71no—
109C2×C42—71no—
110C2×C42C2×D711yescharacteristic
111C2×C42C2×D711yescharacteristic
112#95—72no—
113#95—72no—
114#95—72no—
115#95—72no—
116#95—72no—
117#95—72no—
118#95—72no—
119#95C2×D712yes—
120C22×C7—22no—
121C22×C7—22no—
122C22×C7—22no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7C22×Q811yescharacteristic
126C22×C7C2×#4112yes—
127C22×C7C2×#4112yes—
128C22×C7H(5,𝔽2)11yescharacteristic
129C22×C7H(5,𝔽2)11yescharacteristic
130C4×C7—22no—
131C4×C7—22no—
132C4×C7—24no—
133C4×C7—42no—
134Dic7—24no—
135Dic7—42no—
136Dic7—42no—
137Dic7—42no—
138C24C22×D711yescharacteristic
139C2×Q8—144no—
140C2×Q8—144no—
141C2×Q8—282no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—72no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4Dic1414yescomplemented
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C22×C4C22×D711yescharacteristic
152C22×C4C22×D712yes—
153C4⋊C4—144no—
154C4⋊C4—144no—
155C4⋊C4—282no—
156C22⋊C4—24no—
157C22⋊C4—144no—
158C22⋊C4—281no—
159C22⋊C4—281no—
160C42—41no—
161C42—41no—
162C42—281no—
163C42—281no—
164C2×C7—42no—
165C2×C7#53311yescharacteristic
166C2×C7#53311yescharacteristic
167C2×C7#54311yescharacteristic
168C2×C7#55112yes—
169C2×C7#55912yes—
170C23—22no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23C2×Dic1412yes—
175C23C23×D711yescharacteristic
176Q8—288no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—44no—
187C2×C4—74no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
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×C4C2×Dic1412yes—
200C2×C4C2×Dic1412yes—
201C2×C4#115714yes—
202C7#260711yescharacteristic, complemented, Sylow
203C22—22no—
204C22—22no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22C22×Dic1411yescharacteristic
209C22C2×#115712yes—
210C22#654811yescharacteristic
211C22#654811yescharacteristic
212C22D7×#4112yes—
213C4—22no—
214C4—22no—
215C4—24no—
216C4—42no—
217C4—144no—
218C4—282no—
219C4—282no—
220C4—282no—
221C2—42no—
222C2#9016111yescharacteristic
223C2#9017411yescharacteristic
224C2#9020811yescharacteristic
225C2#9024712yes—
226C2#9025012yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC4×C7×C22⋊C4C2
prime radicalC24×C7C23
hypercenterC4×C22⋊C4D7
hypocenterC7#2607
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #89705C2semidirect
7. #89701C2semidirect
8. #89426C2semidirect
9. #89418C2semidirect
35. C22×C4×C7Q8semidirect
148. C22×C4Dic14semidirect
202. C7#2607semidirect

Maximal subgroups

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

subgroupindex
C4×C7×C22⋊C42
#899832
C4×C22⋊Dic72
#897212
#897192
#897052
#897012
#894262
#894182
#26077