Finite Group Databasebeta

← #13251191

Subgroups — #13251191

Order 1792.

514 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2975C211yesmaximal, characteristic
2#11739829C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11728533C212yesmaximal, complemented
5#11725552C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×#436—24no—
9C7×#431C2212yes—
10C7×#407—22no—
11#89993—21no—
12C2×#6407—21no—
13#89984—22no—
14#89749—21no—
15#89719C414yes—
16C4×C4⋊Dic7C2211yescharacteristic
17#89696C2212yes—
18#89325—21no—
19#89276—24no—
20#9399—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7D412yes—
23C22×Dic14C2×C412yes—
24C2×C7×M16—24no—
25C4×C7×D4—22no—
26C4×C7×D4—22no—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C22⋊C4C2×C412yes—
29C2×C42×C7C2311yescharacteristic
30C2×C22⋊Dic7—24no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4×Dic7—21no—
35C2×#1136—22no—
36#6432—22no—
37#6432—22no—
38C7×C4⋊C8D414yes—
39C7×C22⋊C8—24no—
40C7×C22⋊C8—22no—
41C4×C7×C8—22no—
42#6407—42no—
43C7⋊#95—24no—
44C7⋊#95C2×C412yes—
45C4⋊(C7⋊C8)—22no—
46C4⋊(C7⋊C8)—22no—
47#2975D711yescharacteristic
48#2563—71no—
49#1800—71no—
50#1548—72no—
51#1303—72no—
52C24×C7C22⋊C412yes—
53C2×C7×D4—22no—
54C2×C7×D4—21no—
55C2×C7×D4—22no—
56C2×C7×D4—21no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—44no—
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×Dic7—22no—
65C22×Dic7—22no—
66C2×Dic14—24no—
67C2×Dic14—82no—
68C7×M16—84no—
69C2×C7×C8—24no—
70C7×C4⋊C4—22no—
71C7×C22⋊C4—24no—
72C42×C7C2×D412yes—
73C42×C7#4112yes—
74C22⋊Dic7—84no—
75C4⋊Dic7—42no—
76C4⋊Dic7—41no—
77C4⋊Dic7—42no—
78C4⋊Dic7—41no—
79C4×Dic7—41no—
80C4×Dic7—41no—
81#1136—42no—
82#1136—42no—
83C2×C7⋊C8—42no—
84C2×C4⋊Q8—71no—
85C2×C4×D4C2×D711yescharacteristic
86#436—24no—
87#431C2×D712yes—
88#423—72no—
89C2×#100—141no—
90#407—22no—
91#393—74no—
92#390—141no—
93#382—141no—
94#380—142no—
95C4×C4⋊C4—71no—
96#343—141no—
97#333—144no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—22no—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7C2×C22⋊C411yescharacteristic
104C7×D4—42no—
105C7×D4—42no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—41no—
109C2×C4×C7—44no—
110C2×C4×C7—42no—
111C2×C4×C7—82no—
112C2×C4×C7C4×D412yes—
113C2×C4×C7#12011yescharacteristic
114C2×C4×C7#12112yes—
115C2×C4×C7#12311yescharacteristic
116C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C7#13712yes—
118C2×C4×C7#13711yescharacteristic
119C2×Dic7—42no—
120C2×Dic7—42no—
121C2×Dic7—44no—
122C2×Dic7—42no—
123C2×Dic7—42no—
124Dic14—84no—
125C7×C8—44no—
126C7⋊C8—82no—
127C22×Q8—72no—
128C22×D4D2811yescharacteristic
129C23×C4C22⋊D712yes—
130C2×M16—24no—
131C2×M16—142no—
132C4⋊Q8—142no—
133C4⋊Q8—142no—
134C4×D4—22no—
135C4×D4—22no—
136C2×C4⋊C4—141no—
137C2×C4⋊C4—141no—
138C2×C4⋊C4—141no—
139C2×C4⋊C4D2811yescharacteristic
140C2×C22⋊C4—144no—
141C2×C22⋊C4C4×D712yes—
142C2×C42—141no—
143C2×C42C22×D711yescharacteristic
144C4⋊C8—142no—
145C4⋊C8—142no—
146C4⋊C8C22⋊D714yes—
147#100—282no—
148C22⋊C8—24no—
149C22⋊C8—22no—
150C4×C8—22no—
151#95—72no—
152#95—144no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7—44no—
156C22×C7—42no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—82no—
160C22×C7#38611yescharacteristic
161C22×C7#41711yescharacteristic
162C22×C7#41911yescharacteristic
163C4×C7—21no—
164C4×C7—22no—
165C4×C7—42no—
166C4×C7#45512yes—
167C4×C7#46812yes—
168C4×C7#47012yes—
169Dic7—82no—
170Dic7—82no—
171C24(C2×D7)⋊C412yes—
172C2×Q8—144no—
173C2×Q8—562no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—21no—
177C2×D4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—44no—
181C22×C4—142no—
182C22×C4—142no—
183C22×C4(C2×D7)⋊C412yes—
184C22×C4C2×C4×D711yescharacteristic
185C22×C4C2×D2811yescharacteristic
186C22×C4C2×C22⋊D711yescharacteristic
187M16—84no—
188M16—282no—
189M16—282no—
190C2×C8—24no—
191C2×C8—282no—
192C4⋊C4—22no—
193C4⋊C4—282no—
194C4⋊C4—281no—
195C4⋊C4—282no—
196C4⋊C4—281no—
197C22⋊C4—24no—
198C22⋊C4—564no—
199C42—281no—
200C42—281no—
201C42#115712yes—
202C42C2×C22⋊D712yes—
203C2×C7—21no—
204C2×C7—21no—
205C2×C7—82no—
206C2×C7#188511yescharacteristic
207C2×C7#189411yescharacteristic
208C2×C7#191411yescharacteristic
209C23—22no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23C2×(C2×D7)⋊C411yescharacteristic
215Q8—564no—
216D4—42no—
217D4—42no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—82no—
224C2×C4—282no—
225C2×C4—284no—
226C2×C4—282no—
227C2×C4—282no—
228C2×C4—282no—
229C2×C4#647111yescharacteristic
230C2×C4#647212yes—
231C2×C4#647211yescharacteristic
232C2×C4C4×C22⋊D712yes—
233C2×C4#649211yescharacteristic
234C2×C4#649312yes—
235C2×C4#651611yescharacteristic
236C8—44no—
237C8—562no—
238C7#939911yescharacteristic, complemented, Sylow
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#8989211yescharacteristic
247C22#8990511yescharacteristic
248C22#8998511yescharacteristic
249C4—22no—
250C4—21no—
251C4—42no—
252C4—562no—
253C4—562no—
254C4#8990612yes—
255C4#8990912yes—
256C4#9001412yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#1172940311yescharacteristic
261C2#1172974311yescharacteristic
262C2#1172978411yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2975C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2975D7
hypocenterC7#9399
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
4. #11728533C2semidirect
5. #11725552C2semidirect
238. C7#9399semidirect

Maximal subgroups

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

subgroupindex
C7×#29752
#117398292
#117293712
#117285332
#117255522
#93997