Finite Group Databasebeta

← #13263413

Subgroups — #13263413

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×#2486C211yesmaximal, characteristic
2#11729356C211yesmaximal, characteristic
3#11729355C211yesmaximal, characteristic
4#11725433C214yesmaximal, complemented
5C2×C7×#123C412yes—
6C2×C4×C7×D4C2211yescharacteristic
7C22×C7×C4⋊C4C411yescharacteristic
8C7×#398—21no—
9C7×#397—21no—
10C7×#389—21no—
11C7×#381C2212yes—
12C7×#380—21no—
13C7×#380C411yescharacteristic
14#89979—21no—
15#89697—21no—
16#89323—21no—
17#89320—21no—
18#89320C2212yes—
19#89320C2212yes—
20#11562—71nomaximal, Sylow
21C22×C7×D4C2×C411yescharacteristic
22C23×C4×C7C2×C412yes—
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×#123—44no—
26C4×C7×D4—41no—
27C4×C7×D4—41no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—41no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4C2×C412yes—
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C7×C22⋊C4C2311yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×#1136—41no—
41C7×#95—21no—
42C7×#95—22no—
43C7×#95—22no—
44C22⋊(C7⋊C8)—22no—
45C22⋊(C7⋊C8)—22no—
46C22⋊(C7⋊C8)—41no—
47C4⋊(C7⋊C8)—41no—
48C4×C7⋊C8—41no—
49#2486D711yescharacteristic
50#1465—71no—
51#1460—71no—
52#1317—74no—
53C24×C7C22⋊C411yescharacteristic
54C24×C7C22×C411yescharacteristic
55C2×C7×D4—22no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—24no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
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×C4⋊C4—44no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—42no—
84C42×C7—21no—
85C42×C7—21no—
86#1136—82no—
87C2×C7⋊C8—42no—
88C2×C7⋊C8—42no—
89C2×#123Dic712yes—
90C2×C4×D4C2×D711yescharacteristic
91C22×C4⋊C4Dic711yescharacteristic
92#431—141no—
93#407—141no—
94#398—21no—
95#397—21no—
96#389—21no—
97#381C2×D712yes—
98#380—21no—
99#380Dic711yescharacteristic
100#324—141no—
101#323—72no—
102#323—72no—
103#323—141no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7C2×C22⋊C411yescharacteristic
114C23×C7#13112yes—
115C7×D4—44no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—81no—
135C2×C4×C7—84no—
136C2×C4×C7#9911yescharacteristic
137C2×C4×C7#9911yescharacteristic
138C2×C4×C7#10012yes—
139C7⋊C8—82no—
140C7⋊C8—82no—
141C22×D4C2×Dic711yescharacteristic
142C23×C4C2×Dic712yes—
143C23×C4C2×Dic711yescharacteristic
144C23×C4C22⋊D711yescharacteristic
145C2×M16—281no—
146#123—44no—
147C4×D4—41no—
148C4×D4—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C22⋊D711yescharacteristic
156C2×C22⋊C4—22no—
157C2×C22⋊C4C2×Dic712yes—
158C2×C22⋊C4C22⋊D711yescharacteristic
159C2×C22⋊C4C22×D711yescharacteristic
160C2×C42C22⋊D711yescharacteristic
161C4⋊C8—281no—
162C22⋊C8—142no—
163C22⋊C8—142no—
164C22⋊C8—281no—
165C4×C8—281no—
166#95—21no—
167#95—22no—
168#95—22no—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—22no—
172C22×C7—42no—
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#40811yescharacteristic
182C22×C7C2×#9911yescharacteristic
183C22×C7C2×#10011yescharacteristic
184C4×C7—21no—
185C4×C7—22no—
186C4×C7—41no—
187C4×C7—41no—
188C4×C7—41no—
189C4×C7—82no—
190C4×C7—82no—
191C4×C7#35212yes—
192C24C22⋊Dic711yescharacteristic
193C24C22×Dic711yescharacteristic
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—41no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4C22⋊Dic711yescharacteristic
209C22×C4C22⋊Dic711yescharacteristic
210C22×C4C22⋊Dic711yescharacteristic
211C22×C4C2×C22⋊D711yescharacteristic
212C22×C4C2×C22⋊D711yescharacteristic
213M16—562no—
214C2×C8—282no—
215C2×C8—282no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C42—21no—
227C42—21no—
228C2×C7—21no—
229C2×C7—21no—
230C2×C7—42no—
231C2×C7—81no—
232C2×C7#147111yescharacteristic
233C2×C7C2×#35211yescharacteristic
234C2×C7#212311yescharacteristic
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#651112yes—
245C23C2×C22⋊Dic711yescharacteristic
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—81no—
266C2×C4—84no—
267C2×C4#640712yes—
268C2×C4#640811yescharacteristic
269C2×C4#640811yescharacteristic
270C8—562no—
271C8—562no—
272C7#1156211yescharacteristic, complemented, Sylow
273C22—21no—
274C22—21no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22C2×#640711yescharacteristic
286C22C2×#640811yescharacteristic
287C22#9000611yescharacteristic
288C4—22no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#8933412yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—81no—
300C2#1172943011yescharacteristic
301C2C2×#8933411yescharacteristic
302C2#1172962411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2486C2
socleC22×C7C2×#99
prime radicalC22×C7×D4C2×C4
cosocleC2×C7×C22⋊C4C23
hypercenter#2486D7
hypocenterC7#11562
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #11725433C2semidirect
272. C7#11562semidirect

Maximal subgroups

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

subgroupindex
C7×#24862
#117293562
#117293552
#117254332
#115627