Finite Group Databasebeta

← #13251491

Subgroups — #13251491

Order 1792.

448 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2567C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic
3#11729530C212yesmaximal
4#11729379C211yesmaximal, characteristic
5#11725427C212yesmaximal
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#395—21no—
9C7×#395—21no—
10C7×#391C412yes—
11C7×#389—21no—
12C7×#389—21no—
13C7×#384C412yes—
14C4×C7×C4⋊C4C2211yescharacteristic
15#89796—24no—
16#89698—21no—
17#89698C2212yes—
18#89698C2212yes—
19#89697—21no—
20#89342—24no—
21#89323—21no—
22#89323—21no—
23#9454—71nomaximal, Sylow
24C22×C7×Q8—21no—
25C22×C7×Q8C2×C411yescharacteristic
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—22no—
28C4×C7×Q8—22no—
29C4×C7×Q8—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4C2×C412yes—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C42×C7C2×C412yes—
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C42×C7C2311yescharacteristic
42C7×#95—22no—
43C7×#95—22no—
44C7×#95—21no—
45C22⋊(C7⋊C8)—22no—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)D414yes—
51#6377—22no—
52#6377—22no—
53#6377—22no—
54C4×C7⋊C8—22no—
55#2983—71no—
56#2567D711yescharacteristic
57#1556—72no—
58#1467—71no—
59#1285—72no—
60C2×C7×Q8—22no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C2×C7×Q8—42no—
64C2×C7×Q8—82no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C7×C4⋊C4—24no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—24no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—82no—
82C7×C4⋊C4—82no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—42no—
88C42×C7C2×D412yes—
89C42×C7#4112yes—
90C2×C7⋊C8—24no—
91C2×C7⋊C8—42no—
92C2×C4⋊Q8C2×D711yescharacteristic
93C2×C4×Q8C2×D711yescharacteristic
94#446—144no—
95#432—72no—
96#432—72no—
97#432—141no—
98#431—141no—
99#395—21no—
100#395—21no—
101#391Dic712yes—
102#389—21no—
103#389—21no—
104#384Dic712yes—
105C4×C4⋊C4C2×D711yescharacteristic
106#332—144no—
107#324—141no—
108#324—141no—
109C23×C7C2×C22⋊C411yescharacteristic
110C7×Q8—44no—
111C7×Q8—84no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—24no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—41no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7#10112yes—
132C2×C4×C7C4×D412yes—
133C2×C4×C7#12011yescharacteristic
134C2×C4×C7#12112yes—
135C2×C4×C7#12311yescharacteristic
136C7⋊C8—44no—
137C7⋊C8—82no—
138C22×Q8—21no—
139C22×Q8C2×Dic711yescharacteristic
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C2×Dic712yes—
149C2×C4⋊C4C2×Dic711yescharacteristic
150C2×C4⋊C4C22⋊D711yescharacteristic
151C2×C4⋊C4C22⋊D711yescharacteristic
152C2×C42C2×Dic712yes—
153C2×C42C22⋊D711yescharacteristic
154C2×C42C22⋊D711yescharacteristic
155C2×C42C22×D711yescharacteristic
156C4⋊C8—74no—
157C4⋊C8—142no—
158C4⋊C8—142no—
159C4⋊C8—142no—
160C22⋊C8—142no—
161C22⋊C8—142no—
162#97—142no—
163#97—142no—
164#97—142no—
165C4×C8—142no—
166#95—22no—
167#95—22no—
168#95—21no—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7#38611yescharacteristic
172C22×C7C2×#10111yescharacteristic
173C22×C7#41311yescharacteristic
174C4×C7—21no—
175C4×C7—22no—
176C4×C7—22no—
177C4×C7—22no—
178C4×C7—44no—
179C4×C7—41no—
180C4×C7—41no—
181C4×C7—41no—
182C4×C7—82no—
183C4×C7#45612yes—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—82no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C22⋊Dic711yescharacteristic
192C22×C4C22⋊Dic711yescharacteristic
193C22×C4C22⋊Dic711yescharacteristic
194C22×C4C22⋊Dic711yescharacteristic
195C22×C4C22×Dic711yescharacteristic
196C22×C4C2×C22⋊D711yescharacteristic
197C22×C4C2×C22⋊D711yescharacteristic
198C2×C8—144no—
199C2×C8—282no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—24no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42D4×D712yes—
215C42#115912yes—
216C2×C7—21no—
217C2×C7—21no—
218C2×C7#188611yescharacteristic
219C2×C7#190111yescharacteristic
220C2×C7#191011yescharacteristic
221C23C2×C22⋊Dic711yescharacteristic
222Q8—44no—
223Q8—84no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4#641012yes—
244C2×C4D4×Dic712yes—
245C2×C4#649811yescharacteristic
246C2×C4#650011yescharacteristic
247C2×C4#650212yes—
248C8—284no—
249C8—562no—
250C7#945411yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22#8999211yescharacteristic
254C22C2×#641011yescharacteristic
255C22#9000911yescharacteristic
256C4—22no—
257C4—22no—
258C4—21no—
259C4—22no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4#8986712yes—
266C2—21no—
267C2—21no—
268C2#1172945211yescharacteristic
269C2#1172953811yescharacteristic
270C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2567C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2567D7
hypocenterC7#9454
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
250. C7#9454semidirect

Maximal subgroups

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

subgroupindex
C7×#25672
#117398812
#117295302
#117293792
#117254272
#94547