Finite Group Databasebeta

← #11734631

Subgroups — #11734631

Order 896.

556 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 181 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×#6507C212yesmaximal
2C2×#6451C212yesmaximal
3C7×#395C211yesmaximal, characteristic
4#89751C211yesmaximal, characteristic
5#89748C211yesmaximal, characteristic
6#89744C212yesmaximal
7#89428C212yesmaximal
8#89421C212yesmaximal
9#89419C212yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C22×Dic14C2212yes—
13C2×C7×C4⋊C4C2212yes—
14C2×C7×C4⋊C4C2211yescharacteristic
15#6507—28no—
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×C4×Dic7C2212yes—
25C2×C4×Dic7C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27#6451—28no—
28C7×#95C2211yescharacteristic
29C7×#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2211yescharacteristic
33#2551—71nomaximal, Sylow
34C2×C7×Q8—24no—
35C2×C7×Q8—41no—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C2×Dic14—24no—
46C2×Dic14—24no—
47C2×Dic14—24no—
48C2×Dic14—41no—
49C2×Dic14—42no—
50C7×C4⋊C4—24no—
51C7×C4⋊C4—41no—
52C4⋊Dic7—24no—
53C4⋊Dic7—41no—
54Dic7⋊C4—24no—
55Dic7⋊C4—24no—
56Dic7⋊C4—24no—
57Dic7⋊C4—24no—
58Dic7⋊C4D418yes—
59Dic7⋊C4Q818yescomplemented
60C4×Dic7—24no—
61C4×Dic7—24no—
62C4×Dic7Q814yescomplemented
63C2×C4⋊Q8—72no—
64C2×C4⋊Q8—72no—
65#395—71no—
66#395—72no—
67#395D711yescharacteristic
68#389—72no—
69#389—72no—
70#384—71no—
71C4×C4⋊C4—72no—
72C23×C7C2411yescharacteristic
73C7×Q8—44no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—21no—
77C2×C4×C7—22no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7C2×D414yes—
85C2×C4×C7C2×Q812yes—
86C2×C4×C7C2×Q814yes—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7C2×D414yes—
95C2×Dic7C2×D414yes—
96C2×Dic7C2×Q814yes—
97C2×Dic7C2×Q814yes—
98C2×Dic7C2×Q814yes—
99Dic14—44no—
100Dic14—48no—
101C22×Q8—71no—
102C22×Q8—72no—
103C22×Q8C2×D711yescharacteristic
104C4⋊Q8—148no—
105C4⋊Q8—148no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4C2×D712yes—
115C2×C4⋊C4C2×D711yescharacteristic
116C2×C42—71no—
117C2×C42—72no—
118C2×C42—72no—
119#95—72no—
120#95—71no—
121#95—72no—
122#95C2×D712yes—
123#95C2×D711yescharacteristic
124C22×C7C22×D412yes—
125C22×C7C22×Q811yescharacteristic
126C22×C7C22×Q812yes—
127C22×C7H(5,𝔽2)11yescharacteristic
128C22×C72−1+411yescharacteristic
129C4×C7—22no—
130C4×C7—24no—
131C4×C7—24no—
132C4×C7—41no—
133C4×C7—41no—
134Dic7—24no—
135Dic7—24no—
136Dic7—42no—
137Dic7C4⋊Q818yescomplemented
138C2×Q8—24no—
139C2×Q8—41no—
140C2×Q8—144no—
141C2×Q8—144no—
142C2×Q8—144no—
143C2×Q8—282no—
144C2×Q8—281no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4—72no—
149C22×C4C22×D711yescharacteristic
150C22×C4C22×D711yescharacteristic
151C22×C4C22×D712yes—
152C22×C4C22×D711yescharacteristic
153C22×C4C22×D712yes—
154C4⋊C4—24no—
155C4⋊C4—41no—
156C4⋊C4—78no—
157C4⋊C4—78no—
158C4⋊C4—144no—
159C4⋊C4—144no—
160C4⋊C4—144no—
161C4⋊C4—144no—
162C4⋊C4—144no—
163C4⋊C4—281no—
164C42—74no—
165C42—144no—
166C42—144no—
167C2×C7C2×C4⋊Q812yes—
168C2×C7#54411yescharacteristic
169C2×C7D4211yescharacteristic
170C2×C7D4×Q812yes—
171C2×C7Q8211yescharacteristic
172C23C23×D711yescharacteristic
173Q8—44no—
174Q8—288no—
175Q8—284no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—74no—
187C2×C4—74no—
188C2×C4—74no—
189C2×C4—74no—
190C2×C4—74no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4D4×D714yes—
199C2×C4Q8×D712yes—
200C2×C4Q8×D714yes—
201C7#255111yescharacteristic, complemented, Sylow
202C22C2×D4×D712yes—
203C22#654811yescharacteristic
204C22C2×Q8×D712yes—
205C22C2×Q8×D711yescharacteristic
206C22#655411yescharacteristic
207C4—24no—
208C4—22no—
209C4—24no—
210C4—41no—
211C4—41no—
212C4—78no—
213C4—144no—
214C4—144no—
215C4—282no—
216C2#9028111yescharacteristic
217C2#9031411yescharacteristic
218C2#9032312yes—
219C2#9041311yescharacteristic
220C2D7×C4⋊Q812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2551
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
59. Dic7⋊C4Q8semidirect
62. C4×Dic7Q8semidirect
137. Dic7C4⋊Q8semidirect
201. C7#2551semidirect

Maximal subgroups

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

subgroupindex
C2×#65072
C2×#64512
C7×#3952
#897512
#897482
#897442
#894282
#894212
#894192
#25517