Finite Group Databasebeta

← #13263290

Subgroups — #13263290

Order 1792.

376 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 83 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#11739901C211yesmaximal, characteristic
2C7×#1467C211yesmaximal, characteristic
3#11729379C211yesmaximal, characteristic
4#11725261C212yesmaximal
5#11725258C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6507C411yescharacteristic
8C2×C4×Dic14C411yescharacteristic
9C7×#432—21no—
10#90002C411yescharacteristic
11#89750—21no—
12#89750—21no—
13#89749C2211yescharacteristic
14#89746C2211yescharacteristic
15#89705—21no—
16#89705—21no—
17#89704C411yescharacteristic
18#89698—21no—
19C7×#324—21no—
20C7×#324C2212yes—
21#89323—21no—
22#89323C2212yes—
23#11536—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×Dic14C2×C411yescharacteristic
26C4×C7×Q8—41no—
27C4×C7×Q8—41no—
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C4⋊C4C2311yescharacteristic
31C2×C42×C7C2×C411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C42×C7D411yescharacteristic
34#6507—22no—
35#6507—22no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—21no—
38C2×Dic7⋊C4—21no—
39C2×Dic7⋊C4—21no—
40C2×C4×Dic7—21no—
41C2×C4×Dic7C2×C411yescharacteristic
42C2×C4×Dic7C2×C411yescharacteristic
43C4×Dic14—41no—
44C4×Dic14—41no—
45C7×C4⋊C8—41no—
46C7×C22⋊C8—22no—
47C7×#97—41no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2327—71no—
60#1467—71no—
61#1467D711yescharacteristic
62#1324—72no—
63#1319—72no—
64C2×C7×Q8—21no—
65C2×C7×Q8—21no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79C2×Dic14—21no—
80C2×Dic14—41no—
81C2×Dic14—41no—
82C2×Dic14M1612yes—
83C2×C7×C8—42no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93C4⋊Dic7—41no—
94Dic7⋊C4—24no—
95Dic7⋊C4—24no—
96Dic7⋊C4—41no—
97Dic7⋊C4—41no—
98C4×Dic7—21no—
99C4×Dic7—42no—
100C4×Dic7—41no—
101C4×Dic7M1612yes—
102C2×C7⋊C8—42no—
103C2×C4⋊Q8—71no—
104C2×C4×Q8—71no—
105C2×C4×Q8C2×D711yescharacteristic
106#432—21no—
107#432—141no—
108#391—71no—
109#391—71no—
110#387—141no—
111#387—141no—
112#387—141no—
113#387—141no—
114#382—71no—
115#376—71no—
116#324—21no—
117#324—72no—
118#324—141no—
119#324C2×D712yes—
120C23×C7C2×C22⋊C411yescharacteristic
121C7×Q8—42no—
122C7×Q8—42no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7#9911yescharacteristic
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7C2×M1611yescharacteristic
141C2×C4×C7C2×M1611yescharacteristic
142C2×Dic7—22no—
143C2×Dic7—22no—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—42no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153Dic14—42no—
154Dic14—42no—
155C7×C8—82no—
156C7⋊C8—82no—
157C22×Q8—71no—
158C22×Q8C4×D711yescharacteristic
159C4⋊Q8—142no—
160C4⋊Q8—142no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—281no—
164C4×Q8—281no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—141no—
167C2×C4⋊C4—141no—
168C2×C4⋊C4—141no—
169C2×C4⋊C4D2811yescharacteristic
170C2×C4⋊C4C22⋊D711yescharacteristic
171C2×C4⋊C4C22×D711yescharacteristic
172C2×C42—71no—
173C2×C42—71no—
174C2×C42—141no—
175C2×C42C4×D711yescharacteristic
176C2×C42D2811yescharacteristic
177C2×C42C22⋊D711yescharacteristic
178C4⋊C8—41no—
179C4⋊C8—281no—
180C22⋊C8—22no—
181C22⋊C8—142no—
182#97—41no—
183#97—281no—
184#95—141no—
185#95—141no—
186#95—141no—
187#95—141no—
188#95—141no—
189#95—141no—
190#95—141no—
191#95—141no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7#40711yescharacteristic
195C22×C7C2×#9911yescharacteristic
196C22×C7#41311yescharacteristic
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206Dic7—42no—
207Dic7—42no—
208Dic7—81no—
209Dic7—81no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—72no—
215C2×Q8—141no—
216C2×Q8—281no—
217C2×Q8—281no—
218C22×C4—141no—
219C22×C4—141no—
220C22×C4—141no—
221C22×C4—141no—
222C22×C4(C2×D7)⋊C411yescharacteristic
223C22×C4(C2×D7)⋊C411yescharacteristic
224C22×C4(C2×D7)⋊C411yescharacteristic
225C22×C4(C2×D7)⋊C411yescharacteristic
226C22×C4C2×C4×D711yescharacteristic
227C22×C4C2×D2811yescharacteristic
228C22×C4C2×C22⋊D711yescharacteristic
229C2×C8—42no—
230C2×C8—282no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—144no—
235C4⋊C4—144no—
236C4⋊C4—281no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—72no—
246C42—141no—
247C42—281no—
248C42—282no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7#146511yescharacteristic
252C2×C7#212911yescharacteristic
253C2×C7#213311yescharacteristic
254C23C2×(C2×D7)⋊C411yescharacteristic
255Q8—42no—
256Q8—42no—
257Q8—282no—
258Q8—282no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—142no—
275C2×C4—142no—
276C2×C4—281no—
277C2×C4—281no—
278C2×C4—281no—
279C2×C4—281no—
280C2×C4—281no—
281C2×C4—281no—
282C2×C4—282no—
283C2×C4—281no—
284C2×C4—281no—
285C2×C4#638011yescharacteristic
286C2×C4#638011yescharacteristic
287C2×C4D7×M1611yescharacteristic
288C2×C4D7×M1611yescharacteristic
289C8—82no—
290C8—562no—
291C7#1153611yescharacteristic, complemented, Sylow
292C22—21no—
293C22—21no—
294C22C2×#638011yescharacteristic
295C22#8989811yescharacteristic
296C22#8990411yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—282no—
307C4—282no—
308C4—561no—
309C4—561no—
310C2—21no—
311C2—21no—
312C2#1172772211yescharacteristic
313C2#1172781311yescharacteristic
314C2#1172788211yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1467C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1467D7
hypocenterC7#11536
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
291. C7#11536semidirect

Maximal subgroups

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

subgroupindex
#117399012
C7×#14672
#117293792
#117252612
#117252582
#115367