Finite Group Databasebeta

← #13263421

Subgroups — #13263421

Order 1792.

444 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2547C211yesmaximal, characteristic
2#11729356C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4#11725449C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C7×#124C412yes—
7C2×C4×C7×D4C2211yescharacteristic
8C7×#397—21no—
9C7×#397—21no—
10C7×#391C2212yes—
11C7×#380—21no—
12C7×#380—21no—
13C4×C7×C4⋊C4C411yescharacteristic
14#89979—21no—
15C2×C22⋊(C7⋊C8)—21no—
16#89323C2212yes—
17#89323C2212yes—
18#89320—21no—
19#89320—21no—
20#11564—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×D4C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—41no—
27C7×#124—44no—
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2311yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7C2×C412yes—
40C2×#1136—41no—
41C22×C7⋊C8—41no—
42C7×#95—21no—
43C7×#95—22no—
44C7×#95—22no—
45C22⋊(C7⋊C8)—22no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)—41no—
48C22⋊(C7⋊C8)—41no—
49#2547D711yescharacteristic
50#1465—71no—
51#1459—71no—
52#1319—74no—
53C24×C7C22⋊C411yescharacteristic
54C24×C7C22⋊C411yescharacteristic
55C2×C7×Q8—24no—
56C2×C7×Q8—42no—
57C2×C7×Q8—42no—
58C2×C7×D4—22no—
59C2×C7×D4—22no—
60C2×C7×D4—41no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C7×C4⋊C4—41no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C22⋊C4—41no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—44no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—42no—
86#1136—82no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C2×C7⋊C8—82no—
90C2×C4⋊Q8Dic711yescharacteristic
91C2×#124Dic712yes—
92C2×C4×D4C2×D711yescharacteristic
93#407—141no—
94C2×C22⋊C8—141no—
95#397—21no—
96#397—21no—
97#391C2×D712yes—
98#380—21no—
99#380—21no—
100C4×C4⋊C4Dic711yescharacteristic
101#324—72no—
102#324—72no—
103#323—141no—
104#323—141no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7#10012yes—
115C23×C7C2×C22⋊C411yescharacteristic
116C7×Q8—84no—
117C7×D4—44no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—22no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—81no—
136C2×C4×C7—81no—
137C2×C4×C7#9911yescharacteristic
138C2×C4×C7#9911yescharacteristic
139C2×C4×C7#13112yes—
140C7⋊C8—82no—
141C7⋊C8—82no—
142C22×Q8C2×Dic712yes—
143C22×D4C2×Dic711yescharacteristic
144C23×C4C22⋊D711yescharacteristic
145C23×C4C22⋊D711yescharacteristic
146C2×M16—281no—
147C22×C8—281no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150#124—44no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C22×D711yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C22⋊D711yescharacteristic
160C2×C22⋊C4C22⋊D711yescharacteristic
161C2×C42C2×Dic711yescharacteristic
162C2×C42C2×Dic712yes—
163C22⋊C8—142no—
164C22⋊C8—142no—
165C22⋊C8—281no—
166C22⋊C8—281no—
167#95—21no—
168#95—22no—
169#95—22no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—22no—
173C22×C7—42no—
174C22×C7—42no—
175C22×C7—41no—
176C22×C7—41no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—81no—
181C22×C7—81no—
182C22×C7#40811yescharacteristic
183C22×C7C2×#9911yescharacteristic
184C22×C7C2×#10011yescharacteristic
185C4×C7—21no—
186C4×C7—22no—
187C4×C7—41no—
188C4×C7—41no—
189C4×C7—41no—
190C4×C7—82no—
191C4×C7—82no—
192C4×C7#35412yes—
193C24C22⋊Dic711yescharacteristic
194C24C22⋊Dic711yescharacteristic
195C2×Q8—24no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4C22⋊Dic711yescharacteristic
210C22×C4C22⋊Dic711yescharacteristic
211C22×C4C22×Dic711yescharacteristic
212C22×C4C2×C22⋊D711yescharacteristic
213C22×C4C2×C22⋊D711yescharacteristic
214M16—562no—
215C2×C8—282no—
216C2×C8—282no—
217C2×C8—562no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C42—21no—
228C42—21no—
229C42—42no—
230C2×C7—21no—
231C2×C7—21no—
232C2×C7—42no—
233C2×C7—81no—
234C2×C7#147111yescharacteristic
235C2×C7C2×#35411yescharacteristic
236C2×C7#213511yescharacteristic
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23#640712yes—
247C23C2×C22⋊Dic711yescharacteristic
248Q8—84no—
249D4—44no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4#640811yescharacteristic
270C2×C4#640811yescharacteristic
271C2×C4#651112yes—
272C8—562no—
273C8—562no—
274C7#1156411yescharacteristic, complemented, Sylow
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22C2×#640711yescharacteristic
288C22C2×#640811yescharacteristic
289C22#9000611yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4#8933612yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#1172943011yescharacteristic
303C2C2×#8933611yescharacteristic
304C2#1172962711yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2547C2
socleC22×C7C2×#99
prime radicalC22×C7×D4C2×C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2547D7
hypocenterC7#11564
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
4. #11725449C2semidirect
274. C7#11564semidirect

Maximal subgroups

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

subgroupindex
C7×#25472
#117293562
#117293542
#117254492
#115647