Finite Group Databasebeta

← #13321550

Subgroups — #13321550

Order 1792.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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
1#11742871C211yesmaximal, characteristic
2C7×#2982C211yesmaximal, characteristic
3#11739905C211yesmaximal, characteristic
4#11728547C216yesmaximal
5#11728546C216yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7#90503—21no—
8C2×Q8×Dic7—21no—
9#90261C414yescomplemented
10#90260—24no—
11Q8×Dic14C414yescomplemented
12#90191—23no—
13#90161C2216yes—
14C7×C8×Q8—24no—
15C7×#431C2216yes—
16#89757—23no—
17#89708C2213yes—
18C4×C4⋊Dic7C2213yes—
19#89262C2214yes—
20#89262C22112yes—
21#17619—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C4×C7×Q8C2×C414yes—
24C4×C7×Q8D414yes—
25C2×C7×C4⋊C4D413yes—
26C2×C42×C7C2313yes—
27Q8×Dic7—24no—
28Q8×Dic7—24no—
29Q8×Dic7—44no—
30#6489—26no—
31C2×C4⋊Dic7—23no—
32C2×C4⋊Dic7—23no—
33#6487—26no—
34#6487C2314yes—
35C2×C4×Dic7—23no—
36#6453—212no—
37#6451—212no—
38#6433—26no—
39#6432C2×C416yes—
40#6432C2×C416yes—
41C4×Dic14—26no—
42C4×Dic14C2×C4112yes—
43C7×C4⋊C8—26no—
44C7×C22⋊C8C2314yes—
45C4×C7×C8—26no—
46C7⋊#95—23no—
47C7⋊#95C2311yescharacteristic
48C7⋊#95C2313yes—
49#3502—71no—
50#2982D711yescharacteristic
51#2351—71no—
52#1518—76no—
53#1503—76no—
54C2×C7×Q8C22⋊C414yes—
55C2×C7×Q8SD1614yes—
56C2×C7×Q8Q1614yes—
57C22×C4×C7C2×D413yes—
58C22×C4×C7C2×D413yes—
59C22×C4×C7C2411yescharacteristic
60C22×Dic7—21no—
61C22×Dic7—23no—
62C2×Dic14—26no—
63C2×Dic14C22×C414yes—
64C2×C7×C8—24no—
65C7×C4⋊C4C22⋊C4112yes—
66C42×C7C22×C416yes—
67C42×C7C2×D416yes—
68C22⋊Dic7—24no—
69C4⋊Dic7—26no—
70C4⋊Dic7—26no—
71C4⋊Dic7—26no—
72C4⋊Dic7—46no—
73C4⋊Dic7—46no—
74C4⋊Dic7C22×C414yes—
75Dic7⋊C4—212no—
76Dic7⋊C4—24no—
77C4×Dic7—212no—
78C4×Dic7—46no—
79Q82—74no—
80#557—74no—
81#557—144no—
82#544—143no—
83#533—76no—
84#519—141no—
85C2×C4×Q8—141no—
86C2×C4×Q8C2×D711yescharacteristic
87C8×Q8—24no—
88#431C2×D716yes—
89#389—143no—
90#382—73no—
91C4×C4⋊C4—73no—
92#328—74no—
93#328—712no—
94C23×C7C22×D411yescharacteristic
95C7×Q8#103116yes—
96C2×C4×C7—43no—
97C2×C4×C7—41no—
98C2×C4×C7C2×C22⋊C416yes—
99C2×C4×C7C2×C22⋊C416yes—
100C2×C4×C7C2×SD1616yes—
101C2×C4×C7C2×Q1616yes—
102C2×C4×C7C23×C411yescharacteristic
103C2×C4×C7C22×D411yescharacteristic
104C2×Dic7—22no—
105C2×Dic7—24no—
106C2×Dic7—24no—
107C2×Dic7—41no—
108C2×Dic7—41no—
109C2×Dic7—43no—
110C2×Dic7—43no—
111C2×Dic7—43no—
112Dic14—44no—
113Dic14—86no—
114C7×C8—44no—
115C22×Q8D2811yescharacteristic
116C4⋊Q8—76no—
117C4⋊Q8—76no—
118C4⋊Q8—1412no—
119#125—1412no—
120#125—146no—
121C22⋊Q8—74no—
122C22⋊Q8—146no—
123C4×Q8—712no—
124C4×Q8—144no—
125C4×Q8—144no—
126C4×Q8—146no—
127C4×Q8—284no—
128C4×Q8C4×D714yes—
129C4×Q8C22⋊D714yes—
130#117—146no—
131C2×C4⋊C4—143no—
132C2×C4⋊C4—143no—
133C2×C4⋊C4D2813yes—
134C2×C42—143no—
135C2×C42C22×D713yes—
136C4⋊C8—26no—
137C22⋊C8C22×D714yes—
138C4×C8—26no—
139#95—71no—
140#95—73no—
141#95—143no—
142C22×C7—21no—
143C22×C7—21no—
144C22×C7C22×C22⋊C411yescharacteristic
145C22×C7C22×SD1611yescharacteristic
146C22×C7C22×Q1611yescharacteristic
147C4×C7—21no—
148C4×C7—21no—
149C4×C7—26no—
150C4×C7C2×#103112yes—
151Dic7—42no—
152Dic7—44no—
153Dic7—44no—
154Dic7—83no—
155C2×Q8—74no—
156C2×Q8—146no—
157C2×Q8#113214yes—
158C2×Q8Dic2814yes—
159C2×Q8(C2×D7)⋊C414yes—
160C22×C4—141no—
161C22×C4—143no—
162C22×C4C2×D2813yes—
163C22×C4C2×D2813yes—
164C22×C4C23×D711yescharacteristic
165C2×C8—24no—
166C4⋊C4—74no—
167C4⋊C4—1412no—
168C4⋊C4—146no—
169C4⋊C4—146no—
170C4⋊C4—146no—
171C4⋊C4—144no—
172C4⋊C4—286no—
173C4⋊C4—286no—
174C4⋊C4(C2×D7)⋊C4112yes—
175C22⋊C4—144no—
176C42—1412no—
177C42—286no—
178C42C2×C4×D716yes—
179C42C2×C22⋊D716yes—
180C2×C7—21no—
181C2×C7—21no—
182C2×C7#288411yescharacteristic
183C2×C7C22×#10311yescharacteristic
184C2×C7#290111yescharacteristic
185C23C22×D2811yescharacteristic
186Q8—284no—
187Q8—566no—
188Q8#6390116yes—
189C2×C4—43no—
190C2×C4—41no—
191C2×C4—142no—
192C2×C4—144no—
193C2×C4—144no—
194C2×C4—281no—
195C2×C4—281no—
196C2×C4—283no—
197C2×C4—283no—
198C2×C4—283no—
199C2×C4C2×#113216yes—
200C2×C4C2×Dic2816yes—
201C2×C4C2×(C2×D7)⋊C416yes—
202C2×C4C2×(C2×D7)⋊C416yes—
203C2×C4C22×C4×D711yescharacteristic
204C2×C4C22×C22⋊D711yescharacteristic
205C8—44no—
206C7#1761911yescharacteristic, complemented, Sylow
207C22—21no—
208C22—21no—
209C22C22×#113211yescharacteristic
210C22C22×Dic2811yescharacteristic
211C22C22×(C2×D7)⋊C411yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—282no—
216C4—284no—
217C4—284no—
218C4—563no—
219C4C2×#6390112yes—
220C2—21no—
221C2—21no—
222C2#1174000111yescharacteristic
223C2C22×#639011yescharacteristic
224C2#1174126211yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2982C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#2982D7
hypocenterC7#17619
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
9. #90261C4semidirect
11. Q8×Dic14C4semidirect
206. C7#17619semidirect

Maximal subgroups

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

subgroupindex
#117428712
C7×#29822
#117399052
#117285472
#117285462
#176197