Finite Group Databasebeta

← #11740548

Subgroups — #11740548

Order 896.

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

normal subgroups only

This group has 430 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×#126C211yesmaximal, characteristic
2C2×#6432C211yesmaximal, characteristic, complemented
3#89758C212yesmaximal, complemented
4#89755C212yesmaximal, complemented
5#89751C212yesmaximal, complemented
6#89725C212yesmaximal
7#89721C212yesmaximal
8#89719C212yesmaximal
9#89709C211yesmaximal, characteristic, complemented
10C22×Dic14C2212yes—
11C7×#126—24no—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C42×C7C2211yescharacteristic
19C2×C22⋊Dic7C2214yes—
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2212yes—
22C2×Dic7⋊C4C2214yes—
23#6432—24no—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2214yes—
28C7⋊#95C2214yes—
29#2821—71nomaximal, Sylow
30C24×C7C2311yescharacteristic
31C22×C4×C7C2311yescharacteristic
32C22×C4×C7C2311yescharacteristic
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2314yes—
38C22×Dic7C2314yes—
39C2×Dic14—24no—
40C2×Dic14—44no—
41C7×C4⋊C4—22no—
42C7×C4⋊C4—22no—
43C7×C4⋊C4—22no—
44C7×C22⋊C4—22no—
45C7×C22⋊C4—22no—
46C7×C22⋊C4—22no—
47C42×C7D414yes—
48C22⋊Dic7—44no—
49C4⋊Dic7—24no—
50C4⋊Dic7—24no—
51Dic7⋊C4—44no—
52C2×C4⋊Q8—71no—
53C2×#126D711yescharacteristic
54#400—72no—
55#399—72no—
56#398—72no—
57#397—72no—
58#395—72no—
59#393—72no—
60#383—71no—
61C23×C7—21no—
62C23×C7—22no—
63C23×C7—22no—
64C23×C7—22no—
65C23×C7C2411yescharacteristic
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—21no—
70C2×C4×C7—21no—
71C2×C4×C7—21no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7C2×D412yes—
76C2×C4×C7C2×D412yes—
77C2×C4×C7C2×D412yes—
78C2×Dic7—24no—
79C2×Dic7—24no—
80C2×Dic7—24no—
81C2×Dic7—24no—
82C2×Dic7—24no—
83C2×Dic7—24no—
84Dic14—48no—
85C22×Q8—72no—
86C4⋊Q8—144no—
87#126—24no—
88C2×C4⋊C4—72no—
89C2×C4⋊C4—72no—
90C2×C4⋊C4—74no—
91C2×C4⋊C4C2×D711yescharacteristic
92C2×C4⋊C4C2×D711yescharacteristic
93C2×C4⋊C4C2×D711yescharacteristic
94C2×C22⋊C4—74no—
95C2×C22⋊C4C2×D711yescharacteristic
96C2×C22⋊C4C2×D711yescharacteristic
97C2×C22⋊C4C2×D711yescharacteristic
98C2×C42C2×D711yescharacteristic
99#95—72no—
100#95—72no—
101#95—72no—
102#95—74no—
103#95—74no—
104C22×C7—41no—
105C22×C7—41no—
106C22×C7—41no—
107C22×C7—44no—
108C22×C7C22×D411yescharacteristic
109C22×C7H(5,𝔽2)12yes—
110C22×C72−1+412yes—
111C22×C72−1+412yes—
112C4×C7—22no—
113C4×C7—22no—
114C4×C7—22no—
115C4×C7—41no—
116C4×C7—41no—
117C4×C7—41no—
118Dic7—44no—
119Dic7—44no—
120C24C22×D711yescharacteristic
121C2×Q8—144no—
122C2×Q8—284no—
123C22×C4—74no—
124C22×C4—74no—
125C22×C4C22×D711yescharacteristic
126C22×C4C22×D711yescharacteristic
127C22×C4C22×D711yescharacteristic
128C22×C4C22×D711yescharacteristic
129C22×C4C22×D711yescharacteristic
130C22×C4C22×D711yescharacteristic
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—144no—
135C4⋊C4—144no—
136C4⋊C4—284no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—284no—
141C42C22⋊D714yes—
142C2×C7—81no—
143C2×C7#53511yescharacteristic
144C2×C7#53611yescharacteristic
145C2×C7#53611yescharacteristic
146C2×C7#56314yes—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—21no—
151C23C23×D711yescharacteristic
152Q8—288no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—144no—
163C2×C4—144no—
164C2×C4—144no—
165C2×C4—144no—
166C2×C4—144no—
167C2×C4—144no—
168C2×C4C2×C22⋊D712yes—
169C2×C4C2×C22⋊D712yes—
170C2×C4C2×C22⋊D712yes—
171C7#282111yescharacteristic, complemented, Sylow
172C22—44no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#655312yes—
177C22#655412yes—
178C22#655412yes—
179C22C22×C22⋊D711yescharacteristic
180C4—22no—
181C4—22no—
182C4—22no—
183C4—41no—
184C4—41no—
185C4—41no—
186C4—284no—
187C4—284no—
188C2—81no—
189C2#9040314yes—
190C2#9051611yescharacteristic
191C2#9051611yescharacteristic
192C2#9052111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#126C2
prime radicalC24×C7C23
hypercenterC2×#126D7
hypocenterC7#2821
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×#6432C2semidirect
3. #89758C2semidirect
4. #89755C2semidirect
5. #89751C2semidirect
9. #89709C2semidirect
171. C7#2821semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1262
C2×#64322
#897582
#897552
#897512
#897252
#897212
#897192
#897092
#28217