Finite Group Databasebeta

← #13321548

Subgroups — #13321548

Order 1792.

938 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 395 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
1C7×#3502C211yesmaximal, characteristic
2#11739905C211yesmaximal, characteristic
3#11739880C211yesmaximal, characteristic
4#11729592C216yesmaximal
5#11729589C216yesmaximal
6C7×Q82C414yescomplemented
7C7×#557—24no—
8C7×#557C414yescomplemented
9C7×#544—23no—
10C7×#533C2216yes—
11C7×#519—21no—
12C2×C4×C7×Q8C2211yescharacteristic
13C2×Q8×Dic7—21no—
14Q8×C7⋊C8—24no—
15#89757—23no—
16#89708C2213yes—
17C4×C4⋊Dic7C2213yes—
18#89697C2216yes—
19#89333C2214yes—
20#89333C22112yes—
21#17619—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C7×C4⋊Q8—212no—
24C7×C4⋊Q8C2×C416yes—
25C7×C4⋊Q8C2×C416yes—
26C7×#125—212no—
27C7×#125—26no—
28C7×C22⋊Q8—26no—
29C7×C22⋊Q8C2314yes—
30C4×C7×Q8—24no—
31C4×C7×Q8—24no—
32C4×C7×Q8—26no—
33C4×C7×Q8C2×C4112yes—
34C4×C7×Q8C2×C414yes—
35C4×C7×Q8D414yes—
36C7×#117—26no—
37C2×C7×C4⋊C4D413yes—
38C2×C42×C7C2313yes—
39Q8×Dic7—44no—
40C2×C4⋊Dic7—23no—
41C2×C4⋊Dic7—23no—
42C2×C4×Dic7—23no—
43C7⋊#95—23no—
44C7⋊#95C2311yescharacteristic
45C7⋊#95C2313yes—
46C22⋊(C7⋊C8)C2314yes—
47C4⋊(C7⋊C8)—26no—
48C4×C7⋊C8—26no—
49#3502D711yescharacteristic
50#2982—71no—
51#2351—71no—
52#1518—76no—
53#1503—76no—
54C2×C7×Q8—26no—
55C2×C7×Q8C22⋊C414yes—
56C2×C7×Q8SD1614yes—
57C2×C7×Q8Q1614yes—
58C2×C7×Q8C22×C414yes—
59C22×C4×C7C2×D413yes—
60C22×C4×C7C2×D413yes—
61C22×C4×C7C2411yescharacteristic
62C22×Dic7—21no—
63C22×Dic7—23no—
64C7×C4⋊C4—26no—
65C7×C4⋊C4—26no—
66C7×C4⋊C4—26no—
67C7×C4⋊C4—212no—
68C7×C4⋊C4—24no—
69C7×C4⋊C4C22⋊C4112yes—
70C7×C4⋊C4C22×C414yes—
71C7×C22⋊C4—24no—
72C42×C7—212no—
73C42×C7C22×C416yes—
74C42×C7C2×D416yes—
75C4⋊Dic7—46no—
76C4⋊Dic7—46no—
77C4×Dic7—46no—
78C2×C7⋊C8—24no—
79Q82Dic714yescomplemented
80#557—24no—
81#557Dic714yescomplemented
82#544—23no—
83#533C2×D716yes—
84#519—21no—
85C2×C4×Q8—141no—
86C2×C4×Q8C2×D711yescharacteristic
87C8×Q8—144no—
88#431—76no—
89#389—143no—
90#382—73no—
91C4×C4⋊C4—73no—
92#328—74no—
93#328—712no—
94C23×C7C22×D411yescharacteristic
95C7×Q8—44no—
96C7×Q8—86no—
97C7×Q8#103116yes—
98C2×C4×C7—24no—
99C2×C4×C7—24no—
100C2×C4×C7—43no—
101C2×C4×C7—41no—
102C2×C4×C7C2×C22⋊C416yes—
103C2×C4×C7C2×C22⋊C416yes—
104C2×C4×C7C2×SD1616yes—
105C2×C4×C7C2×Q1616yes—
106C2×C4×C7C23×C411yescharacteristic
107C2×C4×C7C22×D411yescharacteristic
108C2×Dic7—22no—
109C2×Dic7—41no—
110C2×Dic7—41no—
111C2×Dic7—43no—
112C2×Dic7—43no—
113C2×Dic7—43no—
114C7⋊C8—44no—
115C22×Q8C22⋊D711yescharacteristic
116C4⋊Q8—212no—
117C4⋊Q8C2×Dic716yes—
118C4⋊Q8C2×Dic716yes—
119#125—212no—
120#125—26no—
121C22⋊Q8—26no—
122C22⋊Q8C22×D714yes—
123C4×Q8—24no—
124C4×Q8—24no—
125C4×Q8—26no—
126C4×Q8—284no—
127C4×Q8C2×Dic7112yes—
128C4×Q8C2×Dic714yes—
129C4×Q8C22⋊D714yes—
130#117—26no—
131C2×C4⋊C4—143no—
132C2×C4⋊C4—143no—
133C2×C4⋊C4C22⋊D713yes—
134C2×C42—143no—
135C2×C42C22×D713yes—
136C4⋊C8—146no—
137C22⋊C8—74no—
138C4×C8—146no—
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×C7—44no—
151C4×C7—44no—
152C4×C7C2×#103112yes—
153Dic7—42no—
154Dic7—83no—
155C2×Q8—26no—
156C2×Q8#114314yes—
157C2×Q8#114414yes—
158C2×Q8C22⋊Dic714yes—
159C2×Q8C22×Dic714yes—
160C22×C4—141no—
161C22×C4—143no—
162C22×C4C2×C22⋊D713yes—
163C22×C4C2×C22⋊D713yes—
164C22×C4C23×D711yescharacteristic
165C2×C8—144no—
166C4⋊C4—26no—
167C4⋊C4—26no—
168C4⋊C4—26no—
169C4⋊C4—212no—
170C4⋊C4—24no—
171C4⋊C4—286no—
172C4⋊C4—286no—
173C4⋊C4C22⋊Dic7112yes—
174C4⋊C4C22×Dic714yes—
175C22⋊C4—24no—
176C42—212no—
177C42—286no—
178C42C22×Dic716yes—
179C42C2×C22⋊D716yes—
180C2×C7—21no—
181C2×C7—21no—
182C2×C7#288411yescharacteristic
183C2×C7C22×#10311yescharacteristic
184C2×C7#290111yescharacteristic
185C23C22×C22⋊D711yescharacteristic
186Q8—44no—
187Q8—86no—
188Q8#6409116yes—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—43no—
192C2×C4—41no—
193C2×C4—142no—
194C2×C4—281no—
195C2×C4—281no—
196C2×C4—283no—
197C2×C4—283no—
198C2×C4—283no—
199C2×C4C2×#114316yes—
200C2×C4C2×#114416yes—
201C2×C4C2×C22⋊Dic716yes—
202C2×C4C2×C22⋊Dic716yes—
203C2×C4C23×Dic711yescharacteristic
204C2×C4C22×C22⋊D711yescharacteristic
205C8—284no—
206C7#1761911yescharacteristic, complemented, Sylow
207C22—21no—
208C22—21no—
209C22C22×#114311yescharacteristic
210C22C22×#114411yescharacteristic
211C22C22×C22⋊Dic711yescharacteristic
212C4—21no—
213C4—21no—
214C4—26no—
215C4—44no—
216C4—44no—
217C4—282no—
218C4—563no—
219C4C2×#6409112yes—
220C2—21no—
221C2—21no—
222C2C22×#640911yescharacteristic
223C2#1174156411yescharacteristic
224C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#3502C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#3502D7
hypocenterC7#17619
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3502

Complements

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

NH ≅ G/Ntype
6. C7×Q82C4semidirect
8. C7×#557C4semidirect
79. Q82Dic7semidirect
81. #557Dic7semidirect
206. C7#17619semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117399052
#117398802
#117295922
#117295892
#176197