Finite Group Databasebeta

← #11740254

Subgroups — #11740254

Order 896.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic, complemented
3C2×#6493C211yesmaximal, characteristic
4C22×C4×Dic7C211yesmaximal, characteristic, complemented
5#90002C211yesmaximal, characteristic, complemented
6#89762C214yesmaximal, complemented
7#89760C211yesmaximal, characteristic, complemented
8#89749C211yesmaximal, characteristic, complemented
9#89730C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C22×C22⋊D7C2212yes—
13C23×Dic7C2212yes—
14C22×D28C2211yescharacteristic
15C7×C22⋊Q8—24no—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18C2×C7×C22⋊C4C2212yes—
19#6510—24no—
20#6493—24no—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2214yes—
23C2×(C2×D7)⋊C4C2214yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4×Dic7—24no—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29C7⋊#95C2214yes—
30C7⋊#95C2214yes—
31#2850—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C23×D7C2312yes—
34C2×C7×Q8—22no—
35C2×C7×Q8—22no—
36C2×C7×Q8—22no—
37C22×C4×C7—22no—
38C22×C4×C7—22no—
39C22×C4×C7D414yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C2×C22⋊D7—24no—
45C2×C22⋊D7—28no—
46C22×Dic7—24no—
47C22×Dic7—24no—
48C22×Dic7—24no—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2312yes—
52C2×D28—22no—
53C2×D28—22no—
54C2×D28—22no—
55C7×C4⋊C4—22no—
56C7×C4⋊C4—24no—
57C7×C22⋊C4—24no—
58(C2×D7)⋊C4—24no—
59(C2×D7)⋊C4—28no—
60(C2×D7)⋊C4—28no—
61C4⋊Dic7—22no—
62C4×Dic7—22no—
63C4×Dic7—24no—
64C4×Dic7D414yescomplemented
65C2×#124—71no—
66C2×C22⋊Q8D711yescharacteristic, complemented
67C2×#121—71no—
68C22×C42—71no—
69#391—71no—
70#390—71no—
71#388—74no—
72#386—74no—
73#382—71no—
74C23×C7—21no—
75C23×C7—22no—
76C23×C7—21no—
77C23×C7C2×D412yes—
78C23×C7#4114yes—
79C23×C7C2411yescharacteristic
80C22×D7—22no—
81C22×D7—22no—
82C22×D7—24no—
83C22×D7—22no—
84C22×D7—24no—
85C7×Q8—44no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—28no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—24no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D412yes—
95C2×C4×C7C2×D412yes—
96C2×C4×C7C2×D412yes—
97C2×C4×C7#4112yes—
98C22⋊D7—48no—
99C2×Dic7—216no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—24no—
106C2×Dic7C2×D414yes—
107C2×Dic7#4114yes—
108D28—44no—
109C22×Q8C2×D711yescharacteristic
110C22×D4—72no—
111C22×D4—71no—
112C23×C4—72no—
113C23×C4C2×D711yescharacteristic
114#124—144no—
115C22⋊Q8—24no—
116#121—144no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D712yes—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—74no—
122C2×C22⋊C4—74no—
123C2×C22⋊C4C2×D712yes—
124C2×C42—71no—
125C2×C42—72no—
126C2×C42—71no—
127C2×C42—144no—
128#95—74no—
129#95—74no—
130C22×C7—22no—
131C22×C7—24no—
132C22×C7—24no—
133C22×C7—22no—
134C22×C7#12414yescomplemented
135C22×C7C22×D411yescharacteristic
136C22×C7C22×D411yescharacteristic
137C22×C7C2×#4111yescharacteristic
138C22×C7C2×#4112yes—
139C22×C7C2×#4112yes—
140C2×D7—42no—
141C2×D7—44no—
142C2×D7—42no—
143C2×D7—44no—
144C2×D7—42no—
145C4×C7—22no—
146C4×C7—42no—
147C4×C7—42no—
148C4×C7#12114yes—
149Dic7—24no—
150Dic7—24no—
151Dic7—42no—
152C24—72no—
153C24C22×D711yescharacteristic
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×D4—148no—
158C2×D4—144no—
159C2×D4—142no—
160C2×D4—142no—
161C2×D4—142no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—72no—
165C22×C4—72no—
166C22×C4—72no—
167C22×C4—144no—
168C22×C4—144no—
169C22×C4—144no—
170C22×C4C22⋊D714yes—
171C22×C4C22×D712yes—
172C22×C4C22×D712yes—
173C22×C4C22×D711yescharacteristic
174C22×C4C22×D711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—142no—
178C22⋊C4—24no—
179C22⋊C4—148no—
180C22⋊C4—148no—
181C22⋊C4—144no—
182C42—74no—
183C42—144no—
184C42—142no—
185C2×C7—24no—
186C2×C7C2×#12111yescharacteristic
187C2×C7#52512yes—
188C2×C7C2×#12411yescharacteristic
189C2×C7#52912yes—
190C2×C7#53211yescharacteristic
191D7—82no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—142no—
196C23—142no—
197C23—144no—
198C23—144no—
199C23—142no—
200C23#116114yes—
201C23C2×C22⋊D712yes—
202C23C23×D711yescharacteristic
203Q8—44no—
204D4—288no—
205D4—284no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—28no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—74no—
215C2×C4—74no—
216C2×C4—1416no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—144no—
222C2×C4—142no—
223C2×C4D4×D712yes—
224C2×C4#115912yes—
225C2×C4C2×C22⋊D712yes—
226C2×C4C2×C22⋊D712yes—
227C7#285011yescharacteristic, complemented, Sylow
228C22—22no—
229C22—22no—
230C22—24no—
231C22—24no—
232C22—282no—
233C22—282no—
234C22—282no—
235C22—284no—
236C22—284no—
237C22#651014yescomplemented
238C22C2×D4×D711yescharacteristic
239C22C2×#115911yescharacteristic
240C22C2×#116112yes—
241C22D7×#4112yes—
242C22C22×C22⋊D711yescharacteristic
243C4—22no—
244C4—42no—
245C4—42no—
246C4—144no—
247C4—144no—
248C4—282no—
249C4#650214yes—
250C2—24no—
251C2—562no—
252C2#9031512yes—
253C2#9031812yes—
254C2C2×#650211yescharacteristic
255C2C2×#651011yescharacteristic
256C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC2×#6493C2
hypercenterC2×C22⋊Q8D7
hypocenterC7#2850
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6510C2semidirect
4. C22×C4×Dic7C2semidirect
5. #90002C2semidirect
6. #89762C2semidirect
7. #89760C2semidirect
8. #89749C2semidirect
9. #89730C2semidirect
64. C4×Dic7D4semidirect
66. C2×C22⋊Q8D7semidirect
134. C22×C7#124semidirect
227. C7#2850semidirect
237. C22#6510semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65102
C2×#64932
C22×C4×Dic72
#900022
#897622
#897602
#897492
#897302
#28507