Finite Group Databasebeta

← #13251212

Subgroups — #13251212

Order 1792.

570 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2542C211yesmaximal, characteristic
2#11739749C211yesmaximal, characteristic
3#11729529C212yesmaximal, complemented
4#11729371C211yesmaximal, characteristic
5#11725426C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×C22⋊Q8—22no—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#399C414yes—
10C7×#390—21no—
11C7×#384—21no—
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6407—21no—
14#89979—22no—
15#89784—24no—
16#89697C2212yes—
17#89696C2212yes—
18#89342—24no—
19#89325—21no—
20#9403—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D412yes—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—22no—
26C7×C22⋊Q8—48no—
27C4×C7×D4—22no—
28C4×C7×D4—22no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—24no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C22⋊C4—24no—
35C2×C7×C22⋊C4C2×C412yes—
36C2×C42×C7—21no—
37C2×C42×C7C2311yescharacteristic
38C2×#1136—24no—
39C2×#1136—22no—
40C7×#95C2×C412yes—
41#6407—42no—
42C22⋊(C7⋊C8)—24no—
43C22⋊(C7⋊C8)—22no—
44C4⋊(C7⋊C8)—22no—
45C4⋊(C7⋊C8)—22no—
46C4⋊(C7⋊C8)D414yes—
47C4×C7⋊C8—22no—
48#2975—71no—
49#2542D711yescharacteristic
50#1800—71no—
51#1549—72no—
52#1303—72no—
53C24×C7C22⋊C412yes—
54C2×C7×Q8—44no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×D4—22no—
58C2×C7×D4—21no—
59C2×C7×D4—22no—
60C2×C7×D4—21no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—44no—
66C22×C4×C7C22⋊C412yes—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C7×C4⋊C4—24no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—24no—
73C7×C4⋊C4—48no—
74C7×C4⋊C4—42no—
75C7×C22⋊C4—24no—
76C7×C22⋊C4—48no—
77C42×C7—41no—
78C42×C7—41no—
79C42×C7C2×D412yes—
80C42×C7#4112yes—
81#1136—42no—
82#1136—42no—
83#1136—84no—
84C2×C7⋊C8—24no—
85C2×C7⋊C8—42no—
86C2×C4⋊Q8C2×D711yescharacteristic
87C2×C22⋊Q8—22no—
88C2×C4×D4C2×D711yescharacteristic
89#436—144no—
90#431—72no—
91#423—72no—
92C2×#100—141no—
93#407—142no—
94#399Dic714yes—
95#390—21no—
96#384—21no—
97C4×C4⋊C4C2×D711yescharacteristic
98#343—141no—
99#332—144no—
100C23×C7—22no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7C2×C22⋊C411yescharacteristic
106C7×Q8—82no—
107C7×Q8—82no—
108C7×D4—42no—
109C7×D4—42no—
110C2×C4×C7—22no—
111C2×C4×C7—24no—
112C2×C4×C7—22no—
113C2×C4×C7—41no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—44no—
117C2×C4×C7—42no—
118C2×C4×C7—44no—
119C2×C4×C7—42no—
120C2×C4×C7—82no—
121C2×C4×C7C4×D412yes—
122C2×C4×C7#12011yescharacteristic
123C2×C4×C7#12112yes—
124C2×C4×C7#12311yescharacteristic
125C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C7#13712yes—
127C2×C4×C7#13711yescharacteristic
128C7⋊C8—44no—
129C7⋊C8—82no—
130C22×Q8—21no—
131C22×D4C22⋊D711yescharacteristic
132C23×C4C22⋊D712yes—
133C2×M16—144no—
134C2×M16—142no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C22⋊Q8—48no—
138C4×D4—22no—
139C4×D4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4C2×Dic712yes—
144C2×C4⋊C4C22⋊D711yescharacteristic
145C2×C22⋊C4—24no—
146C2×C22⋊C4C2×Dic712yes—
147C2×C42—21no—
148C2×C42C22×D711yescharacteristic
149C4⋊C8—74no—
150C4⋊C8—142no—
151C4⋊C8—142no—
152#100—282no—
153C22⋊C8—144no—
154C22⋊C8—142no—
155C4×C8—142no—
156#95C2×Dic712yes—
157C22×C7—21no—
158C22×C7—21no—
159C22×C7—44no—
160C22×C7—42no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—82no—
164C22×C7#38611yescharacteristic
165C22×C7#41711yescharacteristic
166C22×C7#41911yescharacteristic
167C4×C7—21no—
168C4×C7—22no—
169C4×C7—44no—
170C4×C7—42no—
171C4×C7—82no—
172C4×C7#45312yes—
173C4×C7#46812yes—
174C4×C7#47012yes—
175C24C22⋊Dic712yes—
176C2×Q8—44no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—21no—
182C2×D4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4C22⋊Dic712yes—
189C22×C4C22×Dic711yescharacteristic
190C22×C4C2×C22⋊D711yescharacteristic
191C22×C4C2×C22⋊D711yescharacteristic
192M16—282no—
193M16—282no—
194M16—564no—
195C2×C8—144no—
196C2×C8—282no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—48no—
201C4⋊C4—42no—
202C22⋊C4—24no—
203C22⋊C4—48no—
204C42—41no—
205C42—41no—
206C42D4×D712yes—
207C42#115912yes—
208C2×C7—21no—
209C2×C7—21no—
210C2×C7—82no—
211C2×C7#188711yescharacteristic
212C2×C7#189411yescharacteristic
213C2×C7#191511yescharacteristic
214C23—22no—
215C23—22no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23C2×C22⋊Dic711yescharacteristic
220Q8—82no—
221Q8—82no—
222D4—42no—
223D4—42no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4D4×Dic712yes—
236C2×C4#649811yescharacteristic
237C2×C4#650011yescharacteristic
238C2×C4#650212yes—
239C2×C4#650512yes—
240C2×C4#651211yescharacteristic
241C2×C4#651411yescharacteristic
242C8—284no—
243C8—562no—
244C7#940311yescharacteristic, complemented, Sylow
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#8999211yescharacteristic
253C22#8999711yescharacteristic
254C22#9000411yescharacteristic
255C4—21no—
256C4—22no—
257C4—44no—
258C4—42no—
259C4—82no—
260C4#8982112yes—
261C4#8982412yes—
262C4#8984912yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1172945111yescharacteristic
267C2#1172951711yescharacteristic
268C2#1172962011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2542C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2542D7
hypocenterC7#9403
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #11729529C2semidirect
5. #11725426C2semidirect
244. C7#9403semidirect

Maximal subgroups

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

subgroupindex
C7×#25422
#117397492
#117295292
#117293712
#117254262
#94037