Finite Group Databasebeta

← #13264384

Subgroups — #13264384

Order 1792.

464 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 87 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#11739818C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic
3C4×#6407C211yesmaximal, characteristic
4#11725264C214yesmaximal, complemented
5C2×C4×C7×D4C2211yescharacteristic
6C2×#6498—21no—
7C22×C4⋊Dic7C411yescharacteristic
8C7×#407—21no—
9C2×#6407—21no—
10#89984—21no—
11#89984C411yescharacteristic
12#89755C2212yes—
13#89725C412yes—
14#89715—21no—
15#89700—21no—
16C4×#1136—21no—
17C7×#324C2212yes—
18C7×#323—21no—
19#89325C2212yes—
20#11780—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×Dic7—21no—
25C2×C7×M16—41no—
26C4×C7×D4—41no—
27C4×C7×D4—41no—
28C2×C7×C4⋊C4C2311yescharacteristic
29C2×C7×C22⋊C4C2×C411yescharacteristic
30C2×C7×C22⋊C4D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C22⋊Dic7—21no—
33C2×C22⋊Dic7—22no—
34#6498—44no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—41no—
38C2×C4⋊Dic7—41no—
39C2×C4⋊Dic7C2×C412yes—
40C2×Dic7⋊C4—22no—
41C2×#1136—22no—
42C7×C22⋊C8—22no—
43C7×C22⋊C8—41no—
44#6407—24no—
45C7⋊#95—22no—
46C7⋊#95—22no—
47C7⋊#95C2×C412yes—
48#6377—41no—
49C4×C7⋊C8—41no—
50#2486—71no—
51C4×#100—71no—
52#1465D711yescharacteristic
53#1346—74no—
54C24×C7C22⋊C411yescharacteristic
55C24×C7C22⋊C411yescharacteristic
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4—41no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—42no—
64C22×C4×C7—42no—
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—42no—
73C22×Dic7—42no—
74C22×Dic7—42no—
75C7×M16—82no—
76C2×C7×C8—42no—
77C7×C4⋊C4—41no—
78C7×C22⋊C4—41no—
79C7×C22⋊C4—41no—
80C42×C7—21no—
81C42×C7—21no—
82C22⋊Dic7—42no—
83C22⋊Dic7—44no—
84C4⋊Dic7—42no—
85C4⋊Dic7—41no—
86C4⋊Dic7—41no—
87C4⋊Dic7—42no—
88Dic7⋊C4—44no—
89#1136—24no—
90#1136—42no—
91C2×C7⋊C8—42no—
92C2×#123—141no—
93C2×C4×D4C2×D711yescharacteristic
94C22×C4⋊C4—71no—
95C2×#100—141no—
96#407—21no—
97C4×M16—141no—
98#398—72no—
99#397—72no—
100#389—141no—
101#381—141no—
102#380—71no—
103#380—141no—
104#343—72no—
105#324C2×D712yes—
106#323—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7#10412yes—
117C23×C7C2×C22⋊C411yescharacteristic
118C7×D4—44no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—81no—
132C2×C4×C7#9911yescharacteristic
133C2×C4×C7#9911yescharacteristic
134C2×C4×C7AGL(1,ℤ/8)12yes—
135C2×Dic7—42no—
136C2×Dic7—42no—
137C2×Dic7—42no—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C2×Dic7—42no—
141C2×Dic7—84no—
142C7×C8—82no—
143C7⋊C8—44no—
144C22×D4D2811yescharacteristic
145C23×C4—141no—
146C23×C4C4×D711yescharacteristic
147C23×C4C22⋊D711yescharacteristic
148C2×M16—41no—
149C2×M16—142no—
150#123—284no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—72no—
154C2×C4⋊C4—141no—
155C2×C4⋊C4—141no—
156C2×C4⋊C4—142no—
157C2×C4⋊C4—281no—
158C2×C4⋊C4—281no—
159C2×C4⋊C4C22×D711yescharacteristic
160C2×C22⋊C4—141no—
161C2×C22⋊C4—142no—
162C2×C22⋊C4C4×D711yescharacteristic
163C2×C22⋊C4C22⋊D711yescharacteristic
164C2×C42D2811yescharacteristic
165#100—144no—
166C22⋊C8—22no—
167C22⋊C8—41no—
168#97—281no—
169C4×C8—281no—
170#95—72no—
171#95—142no—
172#95—142no—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—22no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—42no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—81no—
185C22×C7C2×#9911yescharacteristic
186C22×C7#41811yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C4×C7—21no—
189C4×C7—22no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7—41no—
193C4×C7#35212yes—
194Dic7—82no—
195Dic7—82no—
196C24(C2×D7)⋊C411yescharacteristic
197C24(C2×D7)⋊C411yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—142no—
208C22×C4—142no—
209C22×C4—282no—
210C22×C4—282no—
211C22×C4—282no—
212C22×C4(C2×D7)⋊C411yescharacteristic
213C22×C4(C2×D7)⋊C411yescharacteristic
214C22×C4C2×C4×D711yescharacteristic
215C22×C4C2×D2811yescharacteristic
216C22×C4C2×C22⋊D711yescharacteristic
217M16—82no—
218M16—144no—
219M16—282no—
220C2×C8—42no—
221C2×C8—282no—
222C4⋊C4—41no—
223C4⋊C4—282no—
224C4⋊C4—282no—
225C4⋊C4—284no—
226C4⋊C4—281no—
227C4⋊C4—281no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—282no—
231C22⋊C4—284no—
232C42—21no—
233C42—21no—
234C2×C7—21no—
235C2×C7—21no—
236C2×C7—42no—
237C2×C7—81no—
238C2×C7#150711yescharacteristic
239C2×C7C2×#35211yescharacteristic
240C2×C7#213511yescharacteristic
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#639912yes—
251C23C2×(C2×D7)⋊C411yescharacteristic
252D4—44no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—282no—
267C2×C4—282no—
268C2×C4—282no—
269C2×C4—282no—
270C2×C4—282no—
271C2×C4—282no—
272C2×C4—564no—
273C2×C4#638011yescharacteristic
274C2×C4#638011yescharacteristic
275C2×C4#647112yes—
276C8—82no—
277C8—284no—
278C7#1178011yescharacteristic, complemented, Sylow
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22C2×#638011yescharacteristic
292C22#8990311yescharacteristic
293C22C2×#639911yescharacteristic
294C4—22no—
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—562no—
300C4—562no—
301C4#8926712yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#1172773111yescharacteristic
307C2C2×#8926711yescharacteristic
308C2#1172790411yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1465C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C4⋊C4C23
hypercenter#1465D7
hypocenterC7#11780
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
4. #11725264C2semidirect
278. C7#11780semidirect

Maximal subgroups

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

subgroupindex
#117398182
C7×#14652
C4×#64072
#117252642
#117807