Finite Group Databasebeta

← #13250499

Subgroups — #13250499

Order 1792.

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

normal subgroups only

This group has 470 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#11740768C211yesmaximal, characteristic
2#11731597C211yesmaximal, characteristic
3C7×#1518C211yesmaximal, characteristic
4#11729594C211yesmaximal, characteristic
5#11728532C211yesmaximal, characteristic
6#11725543C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×#6450—21no—
10C2×#6432C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#382C2211yescharacteristic
13#89853C412yescomplemented
14#89851C412yescomplemented
15#89815—22no—
16#89792—22no—
17#89746—21no—
18#89704C2211yescharacteristic
19#89697C2211yescharacteristic
20#89696C2211yescharacteristic
21#89426—22no—
22#89426—22no—
23C7×#328—22no—
24#89327—21no—
25#89327—21no—
26#89281—22no—
27#9296—71nomaximal, Sylow
28C22×Dic14—21no—
29C7×C4⋊Q8C2×C411yescharacteristic
30C7×C4⋊Q8C2×C411yescharacteristic
31C7×#125—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C7×#117—21no—
37C2×C7×C4⋊C4—21no—
38C2×C42×C7C2311yescharacteristic
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7D412yes—
41C2×Dic7⋊C4—22no—
42C2×C4×Dic7—22no—
43C2×C4×Dic7D411yescharacteristic
44C2×C4×Dic7D411yescharacteristic
45C2×#1136—21no—
46#6450—42no—
47#6450—42no—
48#6432—21no—
49#6432—21no—
50#6432C2×C412yes—
51#6432D412yes—
52C7×C4⋊C8—21no—
53C7×C22⋊C8—21no—
54C4×C7×C8—21no—
55C7×#95—21no—
56C7×#95D412yes—
57#6409—24no—
58C7⋊#95—24no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C22⋊(C7⋊C8)—21no—
62#6382—24no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4×C7⋊C8C2×C412yes—
67#3229—71no—
68#2327—71no—
69#1995—71no—
70#1546—71no—
71#1518D711yescharacteristic
72#1286—71no—
73#1274—71no—
74C2×C7×Q8—21no—
75C2×C7×Q8—21no—
76C22×C4×C7—22no—
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×Dic7—21no—
81C22×Dic7—22no—
82C22×Dic7—21no—
83C2×Dic14—22no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C2×Dic14—41no—
88C2×Dic14—41no—
89C2×C7×C8—41no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C22⋊C4—41no—
99C42×C7—41no—
100C42×C7C22×C411yescharacteristic
101C42×C7C2×D411yescharacteristic
102C42×C7#4111yescharacteristic
103C42×C7#4111yescharacteristic
104C4⋊Dic7—22no—
105C4⋊Dic7—22no—
106C4⋊Dic7—22no—
107C4⋊Dic7C22⋊C414yes—
108Dic7⋊C4—44no—
109C4×Dic7—21no—
110C4×Dic7—21no—
111C4×Dic7—21no—
112C4×Dic7—44no—
113C4×Dic7SD1612yes—
114#1136—41no—
115#1136—41no—
116C2×C7⋊C8—22no—
117C2×C7⋊C8—41no—
118#533C2×D711yescharacteristic
119C2×C4⋊Q8—71no—
120C2×C4×Q8—141no—
121#498—72no—
122#487—72no—
123#483—142no—
124#475—142no—
125#431—71no—
126#431C2×D711yescharacteristic
127#423—71no—
128#391—141no—
129#387—142no—
130#387—142no—
131#382C2×D711yescharacteristic
132#376—71no—
133#339—141no—
134#339—141no—
135#332—142no—
136#328—22no—
137C23×C7#12011yescharacteristic
138C7×Q8—42no—
139C7×Q8—81no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—42no—
147C2×C4×C7C2×C22⋊C411yescharacteristic
148C2×C4×C7C4×D411yescharacteristic
149C2×C4×C7C4×D411yescharacteristic
150C2×C4×C7#12111yescharacteristic
151C2×C4×C7#12111yescharacteristic
152C2×C4×C7#12311yescharacteristic
153C2×C4×C7C2×SD1611yescharacteristic
154C2×C4×C7#13511yescharacteristic
155C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C7#13711yescharacteristic
157C2×Dic7—24no—
158C2×Dic7—22no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—42no—
162C2×Dic7—42no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166Dic14—42no—
167Dic14—42no—
168Dic14—42no—
169Dic14—42no—
170C7×C8—81no—
171C7⋊C8—42no—
172C7⋊C8—81no—
173C22×Q8—141no—
174C2×M16—141no—
175C4⋊Q8—72no—
176C4⋊Q8—72no—
177C4⋊Q8—141no—
178C4⋊Q8—141no—
179C4⋊Q8C4×D711yescharacteristic
180C4⋊Q8C4×D711yescharacteristic
181#125—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—282no—
187C4×Q8—282no—
188#117—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—72no—
191C2×C4⋊C4—141no—
192C2×C4⋊C4—142no—
193C2×C42—71no—
194C2×C42—71no—
195C2×C42—142no—
196C2×C42C22×D711yescharacteristic
197#106—144no—
198C4⋊C8—21no—
199C4⋊C8—141no—
200C4⋊C8—141no—
201C4⋊C8—141no—
202#103—144no—
203C22⋊C8—21no—
204C22⋊C8—141no—
205C4×C8—21no—
206C4×C8—72no—
207#95—21no—
208#95—144no—
209#95—141no—
210#95—141no—
211#95C22⋊D712yes—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7#38611yescharacteristic
215C22×C7#45011yescharacteristic
216C22×C7#45211yescharacteristic
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—82no—
222C4×C7—81no—
223C4×C7—81no—
224C4×C7C4×SD1611yescharacteristic
225C4×C7#44011yescharacteristic
226C4×C7#46111yescharacteristic
227C4×C7#46311yescharacteristic
228Dic7—42no—
229Dic7—44no—
230Dic7—81no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—142no—
234C2×Q8—142no—
235C2×Q8—281no—
236C2×Q8—281no—
237C2×Q8—281no—
238C2×Q8—281no—
239C22×C4—22no—
240C22×C4—141no—
241C22×C4—142no—
242C22×C4—141no—
243C22×C4D4×D711yescharacteristic
244C22×C4D4×D711yescharacteristic
245C22×C4C2×C22⋊D711yescharacteristic
246M16—281no—
247M16—281no—
248C2×C8—41no—
249C2×C8—142no—
250C2×C8—281no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—74no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—284no—
264C22⋊C4—41no—
265C42—41no—
266C42—72no—
267C42—141no—
268C42—141no—
269C42—141no—
270C42—284no—
271C42C2×C4×D711yescharacteristic
272C42#115711yescharacteristic
273C42D4×D711yescharacteristic
274C42#115911yescharacteristic
275C2×C7—21no—
276C2×C7—21no—
277C2×C7#187711yescharacteristic
278C2×C7#190511yescharacteristic
279C2×C7#191211yescharacteristic
280C23#650011yescharacteristic
281Q8—42no—
282Q8—81no—
283Q8—282no—
284Q8—282no—
285Q8—282no—
286Q8—282no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—142no—
295C2×C4—144no—
296C2×C4—281no—
297C2×C4—281no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4D7×C22⋊C411yescharacteristic
304C2×C4(C22⋊D7)⋊C411yescharacteristic
305C2×C4#644611yescharacteristic
306C2×C4#644711yescharacteristic
307C2×C4D7×SD1611yescharacteristic
308C2×C4#647711yescharacteristic
309C2×C4#648111yescharacteristic
310C2×C4#648211yescharacteristic
311C2×C4C4×C22⋊D711yescharacteristic
312C2×C4#650111yescharacteristic
313C8—81no—
314C8—282no—
315C8—561no—
316C7#929611yescharacteristic, complemented, Sylow
317C22—21no—
318C22—21no—
319C22#8972811yescharacteristic
320C22#8994111yescharacteristic
321C22#8995911yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—282no—
330C4—284no—
331C4—561no—
332C4#8956011yescharacteristic
333C4#8958911yescharacteristic
334C4#8959011yescharacteristic
335C4#8959111yescharacteristic
336C2—21no—
337C2—21no—
338C2#1172780611yescharacteristic
339C2#1172803011yescharacteristic
340C2#1172816511yescharacteristic
subgroupG/H
centerC22#89728
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1518C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#1518D7
hypocenterC7#9296
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
13. #89853C4semidirect
14. #89851C4semidirect
316. C7#9296semidirect

Maximal subgroups

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

subgroupindex
#117407682
#117315972
C7×#15182
#117295942
#117285322
#117255432
#117254372
#92967