Finite Group Databasebeta

← #13264963

Subgroups — #13264963

Order 1792.

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

normal subgroups only

This group has 388 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
1C7×#2567C211yesmaximal, characteristic
2#11729391C211yesmaximal, characteristic
3#11729378C211yesmaximal, characteristic
4#11725454C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C4×C7×Q8C2211yescharacteristic
7C7×#395C412yes—
8C7×#395C2212yes—
9C7×#391—21no—
10C7×#389—21no—
11C7×#389—21no—
12C7×#384—21no—
13C4×C7×C4⋊C4C411yescharacteristic
14C2×#6410—21no—
15#89697—21no—
16#89696—21no—
17#89325C2212yes—
18#89323—21no—
19#89323C2212yes—
20#11908—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×Q8D411yescharacteristic
23C7×C4⋊Q8—41no—
24C7×C4⋊Q8—41no—
25C4×C7×Q8—41no—
26C4×C7×Q8—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C42×C7—21no—
36C2×C42×C7C2×C411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×#1136—22no—
40C7×#95—22no—
41C7×#95—22no—
42C7×#95C2×C412yes—
43#6410—42no—
44C22⋊(C7⋊C8)—22no—
45C4⋊(C7⋊C8)—41no—
46C4⋊(C7⋊C8)—41no—
47C4⋊(C7⋊C8)—41no—
48C4×C7⋊C8—41no—
49#2567D711yescharacteristic
50#1801—71no—
51#1461—71no—
52#1354—74no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C2×C7×Q8SD1612yes—
58C2×C7×Q8Q1612yes—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C7×C4⋊C4—24no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—82no—
76C7×C4⋊C4—82no—
77C42×C7—21no—
78C42×C7—21no—
79C42×C7—21no—
80C42×C7—21no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7—42no—
84#1136—42no—
85#1136—42no—
86C2×C7⋊C8—42no—
87C2×C7⋊C8—42no—
88C2×C4⋊Q8Dic711yescharacteristic
89C2×C4×Q8C2×D711yescharacteristic
90#431—141no—
91#423—141no—
92C2×#101—141no—
93#395Dic712yes—
94#395C2×D712yes—
95#391—21no—
96#389—21no—
97#389—21no—
98#384—21no—
99C4×C4⋊C4Dic711yescharacteristic
100#343—72no—
101#324—72no—
102#324—141no—
103C23×C7C2×C22⋊C411yescharacteristic
104C7×Q8—42no—
105C7×Q8—42no—
106C7×Q8—84no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—22no—
114C2×C4×C7—24no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7#9911yescharacteristic
127C2×C4×C7#9911yescharacteristic
128C2×C4×C7#10312yes—
129C2×C4×C7#10312yes—
130C2×C4×C7C2×SD1611yescharacteristic
131C2×C4×C7C2×Q1611yescharacteristic
132C7⋊C8—82no—
133C7⋊C8—82no—
134C22×Q8C2×Dic712yes—
135C22×Q8C22⋊D711yescharacteristic
136C2×M16—142no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C2×Dic711yescharacteristic
147C2×C4⋊C4C22⋊D711yescharacteristic
148C2×C4⋊C4C22×D711yescharacteristic
149C2×C42—21no—
150C2×C42C2×Dic711yescharacteristic
151C2×C42C22⋊D711yescharacteristic
152C2×C42C22⋊D711yescharacteristic
153C4⋊C8—281no—
154C4⋊C8—281no—
155C4⋊C8—281no—
156#101—282no—
157C22⋊C8—142no—
158C4×C8—281no—
159#95—22no—
160#95—22no—
161#95C2×Dic712yes—
162C22×C7—21no—
163C22×C7—21no—
164C22×C7C2×#9911yescharacteristic
165C22×C7C2×#10311yescharacteristic
166C22×C7#42111yescharacteristic
167C4×C7—21no—
168C4×C7—21no—
169C4×C7—22no—
170C4×C7—22no—
171C4×C7—21no—
172C4×C7—44no—
173C4×C7—41no—
174C4×C7—41no—
175C4×C7—41no—
176C4×C7—82no—
177C4×C7#35612yes—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—42no—
181C2×Q8—82no—
182C2×Q8#114312yes—
183C2×Q8#114412yes—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4C22⋊Dic711yescharacteristic
187C22×C4C22⋊Dic711yescharacteristic
188C22×C4C22⋊Dic711yescharacteristic
189C22×C4C22⋊Dic711yescharacteristic
190C22×C4C22×Dic711yescharacteristic
191C22×C4C2×C22⋊D711yescharacteristic
192C22×C4C2×C22⋊D711yescharacteristic
193M16—282no—
194M16—282no—
195C2×C8—282no—
196C2×C8—282no—
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—42no—
213C2×C7—21no—
214C2×C7—21no—
215C2×C7#152011yescharacteristic
216C2×C7#212211yescharacteristic
217C2×C7C2×#35611yescharacteristic
218C23C2×C22⋊Dic711yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—84no—
222C2×C4—21no—
223C2×C4—21no—
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—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4#640811yescharacteristic
242C2×C4#640811yescharacteristic
243C2×C4#640912yes—
244C2×C4#640912yes—
245C2×C4C2×#114311yescharacteristic
246C2×C4C2×#114411yescharacteristic
247C8—562no—
248C8—562no—
249C7#1190811yescharacteristic, complemented, Sylow
250C22—21no—
251C22—21no—
252C22C2×#640811yescharacteristic
253C22C2×#640911yescharacteristic
254C22#9000811yescharacteristic
255C4—21no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4#8934312yes—
266C2—21no—
267C2—21no—
268C2#1172944411yescharacteristic
269C2C2×#8934311yescharacteristic
270C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2567C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2567D7
hypocenterC7#11908
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
249. C7#11908semidirect

Maximal subgroups

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

subgroupindex
C7×#25672
#117293912
#117293782
#117254542
#119087