Finite Group Databasebeta

← #11733806

Subgroups — #11733806

Order 896.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×D4×Dic7C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic, complemented
3C7×#390C211yesmaximal, characteristic
4#89993C211yesmaximal, characteristic
5#89749C211yesmaximal, characteristic, complemented
6#89719C214yesmaximal, complemented
7#89714C214yesmaximal, complemented
8#89708C211yesmaximal, characteristic, complemented
9C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C23×Dic7C2212yes—
12C22×Dic14C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2214yes—
18D4×Dic7—24no—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24C2×C4×Dic7C2211yescharacteristic
25#6451—24no—
26C7⋊#95C2212yes—
27C7⋊#95C2214yes—
28C7⋊#95C2214yes—
29#2563—71nomaximal, Sylow
30C24×C7C2312yes—
31C2×C7×D4—22no—
32C2×C7×D4—22no—
33C2×C7×D4Q814yescomplemented
34C22×C4×C7C2312yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7—24no—
38C22×Dic7—28no—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2311yescharacteristic
43C22×Dic7C2311yescharacteristic
44C2×Dic14—24no—
45C2×Dic14—44no—
46C7×C4⋊C4—22no—
47C7×C22⋊C4—44no—
48C42×C7—22no—
49C22⋊Dic7—24no—
50C22⋊Dic7—44no—
51C4⋊Dic7—22no—
52C4⋊Dic7—22no—
53Dic7⋊C4—24no—
54Dic7⋊C4—24no—
55C4×Dic7—22no—
56C4×Dic7D414yes—
57C2×C4⋊Q8—71no—
58C2×C4×D4—71no—
59#393—74no—
60#390—71no—
61#390D711yescharacteristic
62#382—71no—
63#382—71no—
64#380—74no—
65C4×C4⋊C4—71no—
66C23×C7—22no—
67C23×C7—24no—
68C23×C7—22no—
69C23×C7—24no—
70C23×C7C2×Q814yes—
71C23×C7C2411yescharacteristic
72C7×D4—28no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—24no—
78C2×C4×C7C2×D412yes—
79C2×C4×C7C2×Q812yes—
80C2×C4×C7#4112yes—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—24no—
87C2×Dic7—22no—
88C2×Dic7—24no—
89C2×Dic7—48no—
90C2×Dic7C2×D412yes—
91C2×Dic7C2×D412yes—
92C2×Dic7#4114yes—
93Dic14—48no—
94C22×Q8—72no—
95C22×D4C2×D711yescharacteristic
96C23×C4—72no—
97C4⋊Q8—144no—
98C4×D4—144no—
99C2×C4⋊C4—71no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—71no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4C2×D711yescharacteristic
104C2×C22⋊C4—72no—
105C2×C22⋊C4—74no—
106C2×C22⋊C4C2×D714yes—
107C2×C42—71no—
108C2×C42—71no—
109C2×C42C2×D711yescharacteristic
110#95—72no—
111#95—74no—
112#95—74no—
113C22×C7—24no—
114C22×C7—28no—
115C22×C7—42no—
116C22×C7—42no—
117C22×C7—44no—
118C22×C7C22×D411yescharacteristic
119C22×C7C22×Q811yescharacteristic
120C22×C7C2×#4111yescharacteristic
121C22×C7C2×#4112yes—
122C22×C7H(5,𝔽2)12yes—
123C4×C7—42no—
124C4×C7—42no—
125C4×C7C22⋊Q814yes—
126Dic7—22no—
127Dic7—42no—
128Dic7—42no—
129Dic7—42no—
130Dic7#12414yes—
131C24C22×D712yes—
132C2×Q8—144no—
133C2×Q8—284no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4Dic1414yescomplemented
137C22×C4—71no—
138C22×C4—71no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4—144no—
143C22×C4—148no—
144C22×C4C22×D712yes—
145C22×C4C22×D712yes—
146C22×C4C22×D711yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—144no—
149C4⋊C4—144no—
150C4⋊C4—142no—
151C4⋊C4—142no—
152C22⋊C4—44no—
153C22⋊C4—144no—
154C22⋊C4—284no—
155C42—22no—
156C42—74no—
157C42—142no—
158C2×C7—44no—
159C2×C7C2×C22⋊Q811yescharacteristic
160C2×C7C2×#12411yescharacteristic
161C2×C7#53511yescharacteristic
162C2×C7#55412yes—
163C2×C7#55512yes—
164C23—22no—
165C23—22no—
166C23—24no—
167C23—24no—
168C23C2×Dic1414yes—
169C23C23×D711yescharacteristic
170Q8—288no—
171D4—28no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—72no—
178C2×C4—74no—
179C2×C4—72no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—144no—
186C2×C4—142no—
187C2×C4—144no—
188C2×C4—288no—
189C2×C4C2×Dic1412yes—
190C2×C4D4×D712yes—
191C2×C4#115912yes—
192C7#256311yescharacteristic, complemented, Sylow
193C22—24no—
194C22—28no—
195C22—42no—
196C22—44no—
197C22—42no—
198C22C22×Dic1411yescharacteristic
199C22C2×D4×D711yescharacteristic
200C22C2×#115911yescharacteristic
201C22#654812yes—
202C22D7×#4112yes—
203C4—42no—
204C4—42no—
205C4—74no—
206C4—142no—
207C4—282no—
208C4—282no—
209C4—282no—
210C4#644114yes—
211C2—44no—
212C2C2×#644111yescharacteristic
213C2#9023012yes—
214C2#9030712yes—
215C2D7×#12411yescharacteristic
216C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2563
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#6451C2semidirect
5. #89749C2semidirect
6. #89719C2semidirect
7. #89714C2semidirect
8. #89708C2semidirect
9. C4×Dic7⋊C4C2semidirect
33. C2×C7×D4Q8semidirect
136. C2×D4Dic14semidirect
192. C7#2563semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C2×#64512
C7×#3902
#899932
#897492
#897192
#897142
#897082
C4×Dic7⋊C42
#25637