Finite Group Databasebeta

← #11740418

Subgroups — #11740418

Order 896.

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

normal subgroups only

This group has 604 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×C7×#124C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×#6499C211yesmaximal, characteristic
4C2×D4×Dic7C211yesmaximal, characteristic
5#90002C211yesmaximal, characteristic, complemented
6#89724C214yesmaximal, complemented
7#89714C214yesmaximal, complemented
8C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
9#89701C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12C23×Dic7C2212yes—
13C22×Dic14C2211yescharacteristic
14C7×#124—24no—
15C2×C7×C22⋊C4C2214yes—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2214yes—
18C2×C22⋊Dic7C2212yes—
19#6507—24no—
20#6499—24no—
21D4×Dic7—24no—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28C7⋊#95C2214yes—
29C7⋊#95C2212yes—
30C7⋊#95C2214yes—
31#2547—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×Q8—22no—
36C2×C7×D4—22no—
37C2×C7×D4—22no—
38C2×C7×D4D414yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×Dic7—24no—
43C22×Dic7—28no—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C7×C22⋊C4—28no—
53C42×C7—22no—
54C22⋊Dic7—28no—
55C22⋊Dic7—24no—
56C4⋊Dic7—22no—
57Dic7⋊C4—24no—
58Dic7⋊C4—24no—
59Dic7⋊C4—24no—
60C4×Dic7—22no—
61C4×Dic7D414yes—
62C2×C4⋊Q8—71no—
63C2×#124—71no—
64C2×#124D711yescharacteristic
65C2×C4×D4—71no—
66#397—74no—
67#391—71no—
68#391—71no—
69#380—74no—
70C4×C4⋊C4—71no—
71C23×C7—22no—
72C23×C7—24no—
73C23×C7—22no—
74C23×C7—24no—
75C23×C7C2×D414yes—
76C23×C7C2411yescharacteristic
77C7×Q8—44no—
78C7×D4—28no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7C2×D412yes—
86C2×C4×C7C2×D412yes—
87C2×C4×C7#4112yes—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—24no—
96C2×Dic7—22no—
97C2×Dic7—48no—
98C2×Dic7C2×D412yes—
99C2×Dic7C2×D412yes—
100C2×Dic7#4114yes—
101Dic14—44no—
102C22×Q8—71no—
103C22×Q8C2×D711yescharacteristic
104C22×D4C2×D711yescharacteristic
105C23×C4—72no—
106C4⋊Q8—144no—
107#124—24no—
108#124—144no—
109C4×D4—144no—
110C2×C4⋊C4—71no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—74no—
116C2×C22⋊C4C2×D714yes—
117C2×C42—71no—
118C2×C42—71no—
119C2×C42C2×D711yescharacteristic
120#95—72no—
121#95—74no—
122#95—74no—
123C22×C7—24no—
124C22×C7—28no—
125C22×C7—42no—
126C22×C7—42no—
127C22×C7—44no—
128C22×C7C22×D411yescharacteristic
129C22×C7C22×D411yescharacteristic
130C22×C7C2×#4111yescharacteristic
131C22×C7C2×#4112yes—
132C22×C72−1+412yes—
133C4×C7—42no—
134C4×C7—42no—
135C4×C7#12114yes—
136Dic7—22no—
137Dic7—42no—
138Dic7—42no—
139Dic7—42no—
140Dic7#12414yescomplemented
141C24C22×D712yes—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—142no—
146C2×Q8—142no—
147C2×Q8—142no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4C22⋊D714yes—
151C22×C4—71no—
152C22×C4—71no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4—72no—
156C22×C4—144no—
157C22×C4—148no—
158C22×C4C22×D712yes—
159C22×C4C22×D712yes—
160C22×C4C22×D711yescharacteristic
161C4⋊C4—144no—
162C4⋊C4—144no—
163C4⋊C4—144no—
164C4⋊C4—142no—
165C22⋊C4—28no—
166C22⋊C4—144no—
167C22⋊C4—148no—
168C42—22no—
169C42—74no—
170C42—142no—
171C2×C7—44no—
172C2×C7C2×#12111yescharacteristic
173C2×C7C2×#12411yescharacteristic
174C2×C7#53611yescharacteristic
175C2×C7#54712yes—
176C2×C7#55612yes—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—24no—
181C23C2×C22⋊D714yes—
182C23C23×D711yescharacteristic
183Q8—44no—
184Q8—284no—
185D4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—72no—
193C2×C4—74no—
194C2×C4—72no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—144no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—288no—
205C2×C4D4×D712yes—
206C2×C4#115912yes—
207C2×C4C2×C22⋊D712yes—
208C7#254711yescharacteristic, complemented, Sylow
209C22—24no—
210C22—28no—
211C22—44no—
212C22—42no—
213C22—42no—
214C22C2×D4×D711yescharacteristic
215C22C2×#115911yescharacteristic
216C22D7×#4112yes—
217C22#655412yes—
218C22C22×C22⋊D711yescharacteristic
219C4—42no—
220C4—42no—
221C4—74no—
222C4—142no—
223C4—282no—
224C4—282no—
225C4—282no—
226C4#650214yes—
227C2—44no—
228C2#9036212yes—
229C2D7×#12411yescharacteristic
230C2#9036611yescharacteristic
231C2C2×#650211yescharacteristic
232C2#9051512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radicalC22×C7×D4C22
hypercenterC2×#124D7
hypocenterC7#2547
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#6507C2semidirect
5. #90002C2semidirect
6. #89724C2semidirect
7. #89714C2semidirect
8. C4×Dic7⋊C4C2semidirect
9. #89701C2semidirect
140. Dic7#124semidirect
208. C7#2547semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#65072
C2×#64992
C2×D4×Dic72
#900022
#897242
#897142
C4×Dic7⋊C42
#897012
#25477