Finite Group Databasebeta

← #13263984

Subgroups — #13263984

Order 1792.

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

normal subgroups only

This group has 468 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
1#11739707C211yesmaximal, characteristic
2C7×#1740C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725297C214yesmaximal
5C2×C7×#117—21no—
6C2×C7×#117C2211yescharacteristic
7C2×#6487—21no—
8C2×#6432C411yescharacteristic, complemented
9#89755—21no—
10#89755C412yescomplemented
11#89725—21no—
12#89725C2212yes—
13#89708C411yescharacteristic, complemented
14#89701—21no—
15#89697—21no—
16C7×#342—21no—
17C7×#342C2212yes—
18#89323—21no—
19#89320C2212yes—
20#11694—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C22×Dic14—21no—
23C7×#117—24no—
24C7×#117—41no—
25C7×#117—41no—
26C7×#117—42no—
27C2×C7×C4⋊C4—41no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C7×C22⋊C4C2311yescharacteristic
33C2×C42×C7—41no—
34C2×C42×C7C2×C411yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C22⋊Dic7—22no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7C2×C412yes—
40#6487—42no—
41#6487—42no—
42C2×Dic7⋊C4—22no—
43#6432—41no—
44#6432—41no—
45C7⋊#95—22no—
46C7⋊#95—21no—
47C7⋊#95—22no—
48C7⋊#95C2×C412yes—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—41no—
51C4×C7⋊C8—41no—
52#2621—71no—
53#1740D711yescharacteristic
54#1458—71no—
55#1344—74no—
56C24×C7C2×D411yescharacteristic
57C22×C4×C7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—22no—
60C22×C4×C7—42no—
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7D812yes—
66C22×C4×C7SD1612yes—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×Dic7—22no—
70C22×Dic7—22no—
71C2×Dic14—41no—
72C2×Dic14—41no—
73C2×Dic14—42no—
74C2×Dic14—42no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—44no—
78C7×C22⋊C4—24no—
79C7×C22⋊C4—41no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—42no—
82C42×C7—21no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—44no—
87C22⋊Dic7—42no—
88C22⋊Dic7—42no—
89C4⋊Dic7—42no—
90C4⋊Dic7—42no—
91C4⋊Dic7—42no—
92Dic7⋊C4—44no—
93C2×C7⋊C8—42no—
94C2×C4⋊Q8—71no—
95C2×C22⋊Q8—141no—
96C2×#117—21no—
97C2×#117C2×D711yescharacteristic
98#431—141no—
99#398—72no—
100#398—141no—
101#397—72no—
102#397—141no—
103#391—141no—
104#382—71no—
105#342—21no—
106#342C2×D712yes—
107#324—141no—
108#323—72no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7C2×C22⋊C411yescharacteristic
114C23×C7C2×D811yescharacteristic
115C23×C7C2×SD1611yescharacteristic
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—44no—
131C2×C4×C7—82no—
132C2×C4×C7—82no—
133C2×C4×C7#9911yescharacteristic
134C2×C4×C7#9911yescharacteristic
135C2×C4×C7#10212yes—
136C2×C4×C7#10212yes—
137C2×C4×C7#10412yes—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C2×Dic7—42no—
141C2×Dic7—42no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144Dic14—82no—
145Dic14—82no—
146C7⋊C8—82no—
147C22×Q8—141no—
148C23×C4C22⋊D711yescharacteristic
149C4⋊Q8—281no—
150C4⋊Q8—281no—
151C22⋊Q8—282no—
152C22⋊Q8—282no—
153#117—24no—
154#117—41no—
155#117—41no—
156#117—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—72no—
159C2×C4⋊C4—142no—
160C2×C4⋊C4—141no—
161C2×C4⋊C4—141no—
162C2×C4⋊C4C4×D711yescharacteristic
163C2×C4⋊C4D2811yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—142no—
166C2×C22⋊C4C22⋊D711yescharacteristic
167C2×C22⋊C4C22×D711yescharacteristic
168C2×C42—41no—
169C2×C42C4×D711yescharacteristic
170C2×C42D2811yescharacteristic
171C4⋊C8—281no—
172C22⋊C8—142no—
173C4×C8—281no—
174#95—72no—
175#95—141no—
176#95—142no—
177#95—142no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—81no—
187C22×C7C2×#9911yescharacteristic
188C22×C7C2×#10211yescharacteristic
189C22×C7C2×#10411yescharacteristic
190C4×C7—22no—
191C4×C7—41no—
192C4×C7—41no—
193C4×C7—41no—
194C4×C7—41no—
195C4×C7—84no—
196C4×C7#32712yes—
197C4×C7#32712yes—
198Dic7—82no—
199Dic7—82no—
200C24C2×C22⋊D711yescharacteristic
201C2×Q8—282no—
202C2×Q8—281no—
203C2×Q8—281no—
204C2×Q8—282no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—142no—
210C22×C4—142no—
211C22×C4(C2×D7)⋊C411yescharacteristic
212C22×C4(C2×D7)⋊C411yescharacteristic
213C22×C4(C2×D7)⋊C411yescharacteristic
214C22×C4(C2×D7)⋊C411yescharacteristic
215C22×C4#114112yes—
216C22×C4#114312yes—
217C22×C4C2×C4×D711yescharacteristic
218C22×C4C2×D2811yescharacteristic
219C2×C8—282no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—44no—
223C4⋊C4—282no—
224C4⋊C4—282no—
225C4⋊C4—284no—
226C4⋊C4—282no—
227C22⋊C4—24no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—282no—
232C22⋊C4—282no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—44no—
238C2×C7—21no—
239C2×C7—21no—
240C2×C7—41no—
241C2×C7—41no—
242C2×C7C2×#32711yescharacteristic
243C2×C7#212311yescharacteristic
244C2×C7#212911yescharacteristic
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23C2×(C2×D7)⋊C411yescharacteristic
250C23C2×#114111yescharacteristic
251C23C2×#114311yescharacteristic
252Q8—562no—
253Q8—562no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—282no—
272C2×C4—282no—
273C2×C4—282no—
274C2×C4—282no—
275C2×C4—282no—
276C2×C4—282no—
277C2×C4#637912yes—
278C2×C4#638011yescharacteristic
279C2×C4#638011yescharacteristic
280C2×C4#638312yes—
281C2×C4#638312yes—
282C8—562no—
283C7#1169411yescharacteristic, complemented, Sylow
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#637911yescharacteristic
294C22C2×#638011yescharacteristic
295C22C2×#638311yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—84no—
302C4—562no—
303C4—562no—
304C4#8924612yes—
305C4#8924612yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#8924611yescharacteristic
311C2#1172781211yescharacteristic
312C2#1172781311yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#1740C2
socleC22×C7C2×#99
prime radicalC24×C7C2×D4
cosocleC2×C7×C22⋊C4C23
hypercenter#1740D7
hypocenterC7#11694
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
8. C2×#6432C4semidirect
10. #89755C4semidirect
13. #89708C4semidirect
283. C7#11694semidirect

Maximal subgroups

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

subgroupindex
#117397072
C7×#17402
#117293252
#117252972
#116947