Finite Group Databasebeta

← #13247398

Subgroups — #13247398

Order 1792.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#11739783C211yesmaximal, characteristic
2C7×#1460C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11725552C212yesmaximal, complemented
5#11725544C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C7×#431C2212yes—
8#89993—21no—
9C2×#6407—21no—
10#89983C412yes—
11C4×C22⋊Dic7C412yes—
12C4⋊C4×Dic7—21no—
13#89723—22no—
14#89708C2211yescharacteristic
15C4×C4⋊Dic7C2211yescharacteristic
16#89696C2212yes—
17C7×#323—22no—
18#89325—21no—
19#8771—71nomaximal, Sylow
20C22×C7×D4D411yescharacteristic
21C23×C4×C7C2×C412yes—
22C4×C7×D4—22no—
23C4×C7×D4—22no—
24C2×C7×C4⋊C4D411yescharacteristic
25C2×C7×C22⋊C4D412yes—
26C2×C42×C7C2311yescharacteristic
27C2×C22⋊Dic7—22no—
28C2×C22⋊Dic7—22no—
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4×Dic7—21no—
32C2×C4×Dic7—22no—
33C2×#1136—22no—
34C7×C4⋊C8—22no—
35C7×C22⋊C8—22no—
36C4×C7×C8—22no—
37#6407—42no—
38C7⋊#95—22no—
39C7⋊#95—21no—
40C7⋊#95C2×C411yescharacteristic
41C7⋊#95C2×C411yescharacteristic
42C7⋊#95C2×C412yes—
43C4⋊(C7⋊C8)—22no—
44C4⋊(C7⋊C8)—22no—
45#2357—71no—
46#1800—71no—
47#1460D711yescharacteristic
48#1303—72no—
49#1286—72no—
50C24×C7C22⋊C412yes—
51C2×C7×D4—22no—
52C2×C7×D4—22no—
53C2×C7×D4—21no—
54C2×C7×D4—21no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—82no—
58C22×C4×C7C22⋊C412yes—
59C22×C4×C7C22×C411yescharacteristic
60C22×C4×C7C2×D411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×Dic7—21no—
63C22×Dic7—22no—
64C22×Dic7—21no—
65C2×C7×C8—42no—
66C7×C4⋊C4—22no—
67C7×C22⋊C4—24no—
68C42×C7#4112yes—
69C42×C7#4112yes—
70C22⋊Dic7—44no—
71C22⋊Dic7—82no—
72C4⋊Dic7—42no—
73C4⋊Dic7—41no—
74C4⋊Dic7—41no—
75C4×Dic7—42no—
76C4×Dic7—82no—
77#1136—42no—
78#1136—42no—
79C2×C7⋊C8—42no—
80C2×C4×D4C2×D711yescharacteristic
81#431C2×D712yes—
82#423—72no—
83C2×#100—141no—
84#390—141no—
85#388—142no—
86#382—71no—
87#381—72no—
88C4×C4⋊C4—71no—
89C4×C4⋊C4—141no—
90C4×C22⋊C4—72no—
91#343—141no—
92#323—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—42no—
97C23×C7—42no—
98C23×C7C2×C22⋊C411yescharacteristic
99C7×D4—42no—
100C7×D4—42no—
101C2×C4×C7—22no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7—82no—
106C2×C4×C7—82no—
107C2×C4×C7#11711yescharacteristic
108C2×C4×C7#11711yescharacteristic
109C2×C4×C7#12312yes—
110C2×C4×C7#12312yes—
111C2×C4×C7#13512yes—
112C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
113C2×C4×C7#13711yescharacteristic
114C2×Dic7—22no—
115C2×Dic7—41no—
116C2×Dic7—41no—
117C2×Dic7—42no—
118C2×Dic7—41no—
119C2×Dic7—42no—
120C2×Dic7—41no—
121C2×Dic7—41no—
122C2×Dic7—42no—
123C7×C8—82no—
124C7⋊C8—82no—
125C22×D4D2811yescharacteristic
126C23×C4C4×D712yes—
127C2×M16—142no—
128C4×D4—22no—
129C4×D4—22no—
130C2×C4⋊C4—141no—
131C2×C4⋊C4—141no—
132C2×C4⋊C4D2811yescharacteristic
133C2×C22⋊C4—142no—
134C2×C22⋊C4—142no—
135C2×C22⋊C4C22⋊D712yes—
136C2×C42—141no—
137C2×C42—142no—
138C2×C42C22×D711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—142no—
141C4⋊C8—142no—
142#100—282no—
143C22⋊C8—22no—
144C4×C8—22no—
145#95—71no—
146#95—72no—
147#95—71no—
148#95—142no—
149#95—141no—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7—82no—
156C22×C7—82no—
157C22×C7#38111yescharacteristic
158C22×C7#41611yescharacteristic
159C22×C7#41711yescharacteristic
160C4×C7—22no—
161C4×C7—21no—
162C4×C7—21no—
163C4×C7—42no—
164C4×C7#48212yes—
165C4×C7#48512yes—
166Dic7—42no—
167Dic7—81no—
168Dic7—82no—
169C24(C2×D7)⋊C412yes—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—21no—
173C2×D4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—82no—
177C22×C4—141no—
178C22×C4—142no—
179C22×C4—141no—
180C22×C4(C2×D7)⋊C412yes—
181C22×C4C2×C4×D711yescharacteristic
182C22×C4C2×D2811yescharacteristic
183C22×C4C2×C22⋊D711yescharacteristic
184M16—282no—
185M16—282no—
186C2×C8—42no—
187C2×C8—282no—
188C4⋊C4—22no—
189C4⋊C4—282no—
190C4⋊C4—281no—
191C4⋊C4—281no—
192C22⋊C4—24no—
193C22⋊C4—284no—
194C22⋊C4—562no—
195C42—282no—
196C42—562no—
197C42#115912yes—
198C42#115912yes—
199C2×C7—21no—
200C2×C7—21no—
201C2×C7—82no—
202C2×C7#182511yescharacteristic
203C2×C7#182711yescharacteristic
204C2×C7#182911yescharacteristic
205C23—22no—
206C23—22no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23C2×(C2×D7)⋊C411yescharacteristic
211D4—42no—
212D4—42no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—142no—
220C2×C4—281no—
221C2×C4—281no—
222C2×C4—282no—
223C2×C4—282no—
224C2×C4—281no—
225C2×C4—281no—
226C2×C4—281no—
227C2×C4—282no—
228C2×C4#644011yescharacteristic
229C2×C4#644011yescharacteristic
230C2×C4#644912yes—
231C2×C4#646712yes—
232C2×C4#647111yescharacteristic
233C2×C4#647211yescharacteristic
234C2×C4#649812yes—
235C8—82no—
236C8—562no—
237C7#877111yescharacteristic, complemented, Sylow
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#8971511yescharacteristic
246C22#8988511yescharacteristic
247C22#8990511yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—282no—
253C4—561no—
254C4—562no—
255C4#8949512yes—
256C4#8950912yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#1172768011yescharacteristic
261C2#1172779311yescharacteristic
262C2#1172782111yescharacteristic
subgroupG/H
centerC22#89715
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1460C2
socleC22×C7#381
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#1460D7
hypocenterC7#8771
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
4. #11725552C2semidirect
5. #11725544C2semidirect
237. C7#8771semidirect

Maximal subgroups

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

subgroupindex
#117397832
C7×#14602
#117293712
#117255522
#117255442
#87717