Finite Group Databasebeta

← #11740437

Subgroups — #11740437

Order 896.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6433C211yesmaximal, characteristic, complemented
3#90001C212yesmaximal, complemented
4#89994C212yesmaximal
5#89725C212yesmaximal, complemented
6#89721C212yesmaximal, complemented
7#89720C214yesmaximal, complemented
8#89708C211yesmaximal, characteristic, complemented
9C22×C7×Q8C2211yescharacteristic
10C22×C7×D4C2211yescharacteristic
11C7×#124—24no—
12C2×C7×C22⋊C4C2212yes—
13C2×C7×C22⋊C4C2212yes—
14C2×C42×C7C2211yescharacteristic
15C2×C22⋊Dic7C2214yes—
16C2×C22⋊Dic7C2214yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2214yes—
23#6433—24no—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2214yes—
27#2820—71nomaximal, Sylow
28C24×C7C2312yes—
29C2×C7×Q8—22no—
30C2×C7×Q8—42no—
31C2×C7×D4—22no—
32C2×C7×D4—42no—
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2314yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C7×C22⋊C4—24no—
41C7×C22⋊C4—24no—
42C42×C7D414yes—
43C22⋊Dic7—44no—
44C22⋊Dic7—44no—
45C4⋊Dic7—22no—
46C4⋊Dic7—22no—
47C4⋊Dic7—42no—
48Dic7⋊C4—24no—
49Dic7⋊C4—24no—
50Dic7⋊C4—44no—
51C2×#125—71no—
52C2×#124D711yescharacteristic
53#399—72no—
54#397—72no—
55#396—74no—
56#395—72no—
57#394—72no—
58#382—71no—
59C23×C7—22no—
60C23×C7—22no—
61C23×C7—24no—
62C23×C7—24no—
63C23×C7—22no—
64C23×C7C2411yescharacteristic
65C7×Q8—44no—
66C7×D4—44no—
67C2×C4×C7—21no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—21no—
71C2×C4×C7—22no—
72C2×C4×C7—24no—
73C2×C4×C7C2×D412yes—
74C2×C4×C7C2×D412yes—
75C2×C4×C7C2×D412yes—
76C2×Dic7—22no—
77C2×Dic7—22no—
78C2×Dic7—24no—
79C2×Dic7—24no—
80C2×Dic7—24no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—24no—
84C22×Q8C2×D711yescharacteristic
85C22×D4C2×D711yescharacteristic
86#125—144no—
87#124—24no—
88C2×C4⋊C4—72no—
89C2×C4⋊C4—72no—
90C2×C4⋊C4—71no—
91C2×C4⋊C4—71no—
92C2×C4⋊C4—74no—
93C2×C4⋊C4—72no—
94C2×C22⋊C4—74no—
95C2×C22⋊C4—74no—
96C2×C22⋊C4C2×D712yes—
97C2×C22⋊C4C2×D712yes—
98C2×C42C2×D711yescharacteristic
99#95—72no—
100#95—72no—
101#95—74no—
102C22×C7—42no—
103C22×C7—42no—
104C22×C7—42no—
105C22×C7—44no—
106C22×C7—44no—
107C22×C7C22×D411yescharacteristic
108C22×C7H(5,𝔽2)12yes—
109C22×C7H(5,𝔽2)11yescharacteristic
110C22×C72−1+412yes—
111C22×C72−1+411yescharacteristic
112C4×C7—22no—
113C4×C7—22no—
114C4×C7—22no—
115C4×C7—42no—
116Dic7—42no—
117Dic7—42no—
118Dic7—44no—
119C24C22×D712yes—
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—22no—
123C2×D4—42no—
124C22×C4—74no—
125C22×C4—72no—
126C22×C4—72no—
127C22×C4C22×D712yes—
128C22×C4C22×D711yescharacteristic
129C22×C4C22×D711yescharacteristic
130C22×C4C22×D711yescharacteristic
131C4⋊C4—144no—
132C4⋊C4—144no—
133C4⋊C4—142no—
134C4⋊C4—142no—
135C4⋊C4—284no—
136C4⋊C4—282no—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—284no—
140C22⋊C4—284no—
141C42C22⋊D714yes—
142C2×C7—82no—
143C2×C7#53511yescharacteristic
144C2×C7#53611yescharacteristic
145C2×C7#53711yescharacteristic
146C2×C7#56212yes—
147C2×C7#56312yes—
148C23—24no—
149C23—22no—
150C23—24no—
151C23—22no—
152C23—22no—
153C23C23×D711yescharacteristic
154Q8—44no—
155D4—44no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—144no—
163C2×C4—144no—
164C2×C4—142no—
165C2×C4—144no—
166C2×C4—144no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4C2×C22⋊D712yes—
171C2×C4C2×C22⋊D712yes—
172C2×C4C2×C22⋊D712yes—
173C7#282011yescharacteristic, complemented, Sylow
174C22—44no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—44no—
179C22#654812yes—
180C22#655311yescharacteristic
181C22#655412yes—
182C22#655411yescharacteristic
183C22C22×C22⋊D711yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—284no—
189C4—282no—
190C4—282no—
191C2—82no—
192C2#9036312yes—
193C2#9037512yes—
194C2#9049611yescharacteristic
195C2#9051611yescharacteristic
196C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radicalC22×C7×D4C22
hypercenterC2×#124D7
hypocenterC7#2820
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#6433C2semidirect
3. #90001C2semidirect
5. #89725C2semidirect
6. #89721C2semidirect
7. #89720C2semidirect
8. #89708C2semidirect
173. C7#2820semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#64332
#900012
#899942
#897252
#897212
#897202
#897082
#28207