Finite Group Databasebeta

← #13263985

Subgroups — #13263985

Order 1792.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2621C211yesmaximal, characteristic
2#11729346C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725455C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C7×#117C2211yescharacteristic
8C2×#6489—21no—
9C7×#398—21no—
10C7×#398C412yescomplemented
11C7×#397—21no—
12C7×#397C2212yes—
13C7×#391—21no—
14C7×#382C411yescharacteristic, complemented
15#89697—21no—
16#89323—21no—
17#89321—21no—
18#89321C2212yes—
19#89320C2212yes—
20#11694—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C23×C4×C7D411yescharacteristic
23C7×C4⋊Q8—41no—
24C7×C4⋊Q8—41no—
25C7×C22⋊Q8—42no—
26C7×C22⋊Q8—42no—
27C7×#117—41no—
28C7×#117—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C7×C22⋊C4C2311yescharacteristic
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×C22⋊Dic7—22no—
41#6489—24no—
42#6489—42no—
43C2×C4⋊Dic7—41no—
44C2×C4×Dic7—41no—
45C7×#95—21no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95C2×C412yes—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—41no—
51C4×C7⋊C8—41no—
52#2621D711yescharacteristic
53#1740—71no—
54#1458—71no—
55#1344—74no—
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—41no—
58C2×C7×Q8—41no—
59C2×C7×Q8—42no—
60C2×C7×Q8—42no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7D812yes—
71C22×C4×C7SD1612yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—42no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—44no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—42no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C22⋊Dic7—24no—
90C22⋊Dic7—42no—
91C4⋊Dic7—44no—
92C4×Dic7—44no—
93C2×C7⋊C8—42no—
94C2×C4⋊Q8Dic711yescharacteristic, complemented
95C2×C22⋊Q8—21no—
96C2×#117—141no—
97C2×#117C2×D711yescharacteristic
98#431—141no—
99#398—21no—
100#398Dic712yescomplemented
101#397—21no—
102#397C2×D712yes—
103#391—21no—
104#382Dic711yescharacteristic, complemented
105#342—72no—
106#342—141no—
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
116C7×Q8—82no—
117C7×Q8—82no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7#9911yescharacteristic
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7#10212yes—
141C2×C4×C7#10212yes—
142C2×C4×C7#10412yes—
143C2×Dic7—44no—
144C2×Dic7—82no—
145C2×Dic7—82no—
146C7⋊C8—82no—
147C22×Q8—21no—
148C23×C4C22⋊D711yescharacteristic
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153#117—41no—
154#117—41no—
155#117—144no—
156#117—282no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—281no—
161C2×C4⋊C4C2×Dic712yes—
162C2×C4⋊C4C2×Dic711yescharacteristic
163C2×C4⋊C4C22⋊D711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—142no—
166C2×C22⋊C4C22⋊D711yescharacteristic
167C2×C22⋊C4C22×D711yescharacteristic
168C2×C42—281no—
169C2×C42C2×Dic711yescharacteristic
170C2×C42C22⋊D711yescharacteristic
171C4⋊C8—281no—
172C22⋊C8—142no—
173C4×C8—281no—
174#95—21no—
175#95—22no—
176#95—22no—
177#95C2×Dic712yes—
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—82no—
196C4×C7—82no—
197C4×C7#32712yes—
198C4×C7#32712yes—
199Dic7—84no—
200C24C2×C22⋊D711yescharacteristic
201C2×Q8—42no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—282no—
211C22×C4#114112yes—
212C22×C4#114212yes—
213C22×C4C22⋊Dic711yescharacteristic
214C22×C4C22⋊Dic711yescharacteristic
215C22×C4C22⋊Dic711yescharacteristic
216C22×C4C22⋊Dic711yescharacteristic
217C22×C4C22×Dic711yescharacteristic
218C22×C4C2×C22⋊D711yescharacteristic
219C2×C8—282no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—284no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—144no—
232C22⋊C4—282no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—284no—
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×#114111yescharacteristic
250C23C2×#114211yescharacteristic
251C23C2×C22⋊Dic711yescharacteristic
252Q8—82no—
253Q8—82no—
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—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—284no—
275C2×C4—562no—
276C2×C4—562no—
277C2×C4#640612yes—
278C2×C4#640612yes—
279C2×C4#640811yescharacteristic
280C2×C4#640811yescharacteristic
281C2×C4#641112yes—
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×#640611yescharacteristic
294C22C2×#640811yescharacteristic
295C22C2×#641111yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—82no—
303C4—564no—
304C4#8933212yes—
305C4#8933212yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#8933211yescharacteristic
311C2#1172962411yescharacteristic
312C2#1172962511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2621C2
socleC22×C7C2×#99
prime radicalC24×C7C2×D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2621D7
hypocenterC7#11694
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
5. C2×C7×C4⋊Q8C4semidirect
10. C7×#398C4semidirect
14. C7×#382C4semidirect
94. C2×C4⋊Q8Dic7semidirect
100. #398Dic7semidirect
104. #382Dic7semidirect
283. C7#11694semidirect

Maximal subgroups

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

subgroupindex
C7×#26212
#117293462
#117293252
#117254552
#116947