Finite Group Databasebeta

← #11733808

Subgroups — #11733808

Order 896.

548 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 161 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×#6453C211yesmaximal, characteristic, complemented
3C7×#390C211yesmaximal, characteristic
4#89993C211yesmaximal, characteristic
5#89748C211yesmaximal, characteristic, complemented
6#89720C214yesmaximal, complemented
7#89718C214yesmaximal, complemented
8#89708C211yesmaximal, characteristic, complemented
9C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C23×Dic7C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2214yes—
14C2×C42×C7C2211yescharacteristic
15C2×C22⋊Dic7C2214yes—
16C2×C22⋊Dic7C2212yes—
17D4×Dic7—24no—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2214yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2211yescharacteristic
26#6453—24no—
27C7⋊#95C2212yes—
28C7⋊#95C2214yes—
29#2566—71nomaximal, Sylow
30C24×C7C2312yes—
31C2×C7×D4—22no—
32C2×C7×D4—22no—
33C2×C7×D4Q814yes—
34C22×C4×C7C2312yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7—24no—
38C22×Dic7—44no—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2311yescharacteristic
42C22×Dic7C2311yescharacteristic
43C22×Dic7C2312yes—
44C7×C4⋊C4—22no—
45C7×C22⋊C4—44no—
46C42×C7—22no—
47C22⋊Dic7—24no—
48C22⋊Dic7—28no—
49C4⋊Dic7—22no—
50C4⋊Dic7—22no—
51C4⋊Dic7—24no—
52Dic7⋊C4—24no—
53Dic7⋊C4—24no—
54Dic7⋊C4—44no—
55C4×Dic7—22no—
56C4×Dic7D414yes—
57C2×#125—71no—
58C2×C4×D4—71no—
59#396—74no—
60#390—71no—
61#390D711yescharacteristic
62#385—74no—
63#384—71no—
64#382—71no—
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—24no—
77C2×C4×C7—22no—
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—22no—
87C2×Dic7—24no—
88C2×Dic7—24no—
89C2×Dic7—48no—
90C2×Dic7C2×D412yes—
91C2×Dic7C2×D412yes—
92C2×Dic7#4114yes—
93C22×D4C2×D711yescharacteristic
94C23×C4—72no—
95#125—144no—
96C4×D4—144no—
97C2×C4⋊C4—71no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—71no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—74no—
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—
112C22×C7—24no—
113C22×C7—28no—
114C22×C7—42no—
115C22×C7—42no—
116C22×C7—44no—
117C22×C7C22×D411yescharacteristic
118C22×C7C22×Q811yescharacteristic
119C22×C7C2×#4111yescharacteristic
120C22×C7C2×#4112yes—
121C22×C7H(5,𝔽2)12yes—
122C4×C7—42no—
123C4×C7—42no—
124C4×C7C22⋊Q814yes—
125Dic7—22no—
126Dic7—22no—
127Dic7—24no—
128Dic7—42no—
129Dic7—42no—
130C24C22×D712yes—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4Dic1414yes—
134C22×C4—71no—
135C22×C4—72no—
136C22×C4—71no—
137C22×C4—72no—
138C22×C4—72no—
139C22×C4—144no—
140C22×C4—284no—
141C22×C4C22×D712yes—
142C22×C4C22×D712yes—
143C22×C4C22×D711yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—144no—
146C4⋊C4—144no—
147C4⋊C4—144no—
148C4⋊C4—142no—
149C4⋊C4—142no—
150C4⋊C4—284no—
151C22⋊C4—44no—
152C22⋊C4—144no—
153C22⋊C4—148no—
154C42—22no—
155C42—74no—
156C42—142no—
157C2×C7—44no—
158C2×C7C2×C22⋊Q811yescharacteristic
159C2×C7#53211yescharacteristic
160C2×C7#53511yescharacteristic
161C2×C7#55312yes—
162C2×C7#55412yes—
163C23—22no—
164C23—22no—
165C23—24no—
166C23—24no—
167C23C2×Dic1414yes—
168C23C23×D711yescharacteristic
169D4—28no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—72no—
176C2×C4—72no—
177C2×C4—74no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—144no—
185C2×C4—144no—
186C2×C4—288no—
187C2×C4C2×Dic1412yes—
188C2×C4D4×D712yes—
189C2×C4#115912yes—
190C7#256611yescharacteristic, complemented, Sylow
191C22—24no—
192C22—28no—
193C22—42no—
194C22—44no—
195C22—42no—
196C22C22×Dic1411yescharacteristic
197C22C2×D4×D711yescharacteristic
198C22C2×#115911yescharacteristic
199C22#654812yes—
200C22D7×#4112yes—
201C4—42no—
202C4—42no—
203C4—142no—
204C4—144no—
205C4—142no—
206C4—282no—
207C4—282no—
208C4#644114yes—
209C2—44no—
210C2C2×#644111yescharacteristic
211C2#9023012yes—
212C2#9030512yes—
213C2#9035911yescharacteristic
214C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2566
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#6453C2semidirect
5. #89748C2semidirect
6. #89720C2semidirect
7. #89718C2semidirect
8. #89708C2semidirect
9. C4×Dic7⋊C4C2semidirect
190. C7#2566semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C2×#64532
C7×#3902
#899932
#897482
#897202
#897182
#897082
C4×Dic7⋊C42
#25667