Finite Group Databasebeta

← #13251275

Subgroups — #13251275

Order 1792.

380 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 85 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#11740793C211yesmaximal, characteristic
2#11736762C211yesmaximal, characteristic
3C7×#1515C211yesmaximal, characteristic
4#11729577C211yesmaximal, characteristic
5#11727320C211yesmaximal, characteristic
6#11725560C211yesmaximal, characteristic
7#11725439C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9#90161C2211yescharacteristic
10C7×#432C2211yescharacteristic
11C7×#383C2211yescharacteristic
12#89865C411yescharacteristic, complemented
13#89864C411yescharacteristic, complemented
14#89853C411yescharacteristic, complemented
15#89851C411yescharacteristic, complemented
16#89824—22no—
17#89802—22no—
18#89698C2211yescharacteristic
19#89698C2211yescharacteristic
20C4×#1136C2211yescharacteristic
21#89600—22no—
22#89492—22no—
23#89473—22no—
24C7×#328—22no—
25#89338—22no—
26#89247—22no—
27#9416—71nomaximal, Sylow
28C7×C4⋊Q8C2×C411yescharacteristic
29C7×C4⋊Q8C2×C411yescharacteristic
30C7×#125—21no—
31C7×C22⋊Q8—21no—
32C7×C22⋊Q8—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8—21no—
35C7×#117—21no—
36C2×C42×C7C2311yescharacteristic
37C2×#1144—22no—
38#6489—21no—
39#6487—21no—
40#6487D412yes—
41C2×#1136—21no—
42#6433—21no—
43#6432C2×C411yescharacteristic
44#6432C2×C411yescharacteristic
45C4×Dic14—21no—
46C4×Dic14D412yes—
47C7×C4⋊C8—21no—
48C7×C22⋊C8D412yes—
49C7×#97—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53#6409—22no—
54C22⋊(C7⋊C8)—21no—
55C22⋊(C7⋊C8)—21no—
56#6389—22no—
57Dic7⋊C8—24no—
58C8×Dic7—22no—
59#6384—22no—
60#6382—22no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63#6377—21no—
64#6377—21no—
65#6377D412yes—
66C4×C7⋊C8C2×C411yescharacteristic
67C4×C7⋊C8C2×C411yescharacteristic
68#3185—71no—
69#2987—71no—
70#1537—71no—
71#1515—71no—
72#1515D711yescharacteristic
73#1291—71no—
74#1288—71no—
75C2×C7×Q8—21no—
76C2×C7×Q8—21no—
77C22×C4×C7—41no—
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7#4111yescharacteristic
81C2×Dic14—21no—
82C2×Dic14C22⋊C412yes—
83C2×C7×C8—22no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4—21no—
87C7×C4⋊C4—21no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C22⋊C4—41no—
91C42×C7—41no—
92C42×C7C22×C411yescharacteristic
93C42×C7C2×D411yescharacteristic
94C42×C7C2×D411yescharacteristic
95C42×C7#4111yescharacteristic
96C22⋊Dic7—22no—
97#1144—44no—
98C4⋊Dic7—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C22⋊C412yes—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104C4×Dic7—22no—
105#1136—42no—
106C2×C7⋊C8—21no—
107C2×C7⋊C8—21no—
108C2×C7⋊C8—22no—
109C2×C7⋊C8—41no—
110#533—71no—
111#533C2×D711yescharacteristic
112#498—71no—
113#494—71no—
114#487—71no—
115#487—71no—
116#470—142no—
117#446—142no—
118#446—142no—
119#441—142no—
120#433—142no—
121#432—71no—
122#432—71no—
123#432C2×D711yescharacteristic
124C4×M16—71no—
125#383C2×D711yescharacteristic
126#330—142no—
127#328—22no—
128#328—142no—
129C23×C7#12111yescharacteristic
130C7×Q8—42no—
131C7×Q8—81no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—81no—
138C2×C4×C7—81no—
139C2×C4×C7—81no—
140C2×C4×C7C2×C22⋊C411yescharacteristic
141C2×C4×C7C4×D411yescharacteristic
142C2×C4×C7C4×D411yescharacteristic
143C2×C4×C7#12011yescharacteristic
144C2×C4×C7#12111yescharacteristic
145C2×C4×C7#12311yescharacteristic
146C2×Dic7—22no—
147C2×Dic7—22no—
148Dic14—42no—
149Dic14—81no—
150C7×C8—42no—
151C7⋊C8—22no—
152C7⋊C8—42no—
153C7⋊C8—41no—
154C7⋊C8—81no—
155C2×Q16—142no—
156C2×M16—141no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8C4×D711yescharacteristic
160C4⋊Q8C4×D711yescharacteristic
161#125—21no—
162#125—141no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—72no—
166C22⋊Q8—141no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—72no—
170C4×Q8—141no—
171#117—21no—
172#117—141no—
173C2×C42C22×D711yescharacteristic
174#106—142no—
175C4⋊C8—21no—
176C4⋊C8—144no—
177C4⋊C8—141no—
178C4⋊C8—141no—
179#103—142no—
180#103—142no—
181C22⋊C8—141no—
182C22⋊C8—141no—
183C22⋊C8C22⋊D712yes—
184#97—21no—
185#97—72no—
186#97—142no—
187#97—141no—
188#97—141no—
189C4×C8—71no—
190C4×C8—71no—
191C4×C8—142no—
192#95—21no—
193#95—21no—
194#95—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7#38611yescharacteristic
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—81no—
206C4×C7—81no—
207C4×C7—161no—
208Dic7—42no—
209Dic7—42no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—72no—
213C2×Q8—141no—
214C22×C4—41no—
215C22×C4D4×D711yescharacteristic
216C22×C4#115911yescharacteristic
217C22×C4C2×C22⋊D711yescharacteristic
218Q16—284no—
219M16—282no—
220C2×C8—22no—
221C2×C8—141no—
222C2×C8—141no—
223C2×C8—142no—
224C2×C8—281no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—72no—
232C4⋊C4—141no—
233C4⋊C4—142no—
234C4⋊C4—142no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C22⋊C4—41no—
238C22⋊C4—142no—
239C42—41no—
240C42—142no—
241C42C2×C4×D711yescharacteristic
242C42#115711yescharacteristic
243C42D4×D711yescharacteristic
244C42D4×D711yescharacteristic
245C2×C7—41no—
246C2×C7#188411yescharacteristic
247C2×C7#190211yescharacteristic
248C2×C7#190511yescharacteristic
249C23#650111yescharacteristic
250Q8—42no—
251Q8—81no—
252Q8—282no—
253Q8—561no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4D7×C22⋊C411yescharacteristic
265C2×C4(C22⋊D7)⋊C411yescharacteristic
266C2×C4#644611yescharacteristic
267C2×C4#644711yescharacteristic
268C2×C4C4×C22⋊D711yescharacteristic
269C2×C4#650011yescharacteristic
270C8—42no—
271C8—142no—
272C8—282no—
273C8—281no—
274C8—561no—
275C7#941611yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—21no—
279C22#8972811yescharacteristic
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—81no—
287C4—81no—
288C4—161no—
289C4—282no—
290C4—282no—
291C2—41no—
292C2#1172780611yescharacteristic
293C2#1172810311yescharacteristic
294C2#1172817111yescharacteristic
subgroupG/H
centerC22#89728
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1515C2
socleC22×C7#386
prime radicalC23×C7#121
cosocleC2×C42×C7C23
hypercenter#1515D7
hypocenterC7#9416
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
12. #89865C4semidirect
13. #89864C4semidirect
14. #89853C4semidirect
15. #89851C4semidirect
275. C7#9416semidirect

Maximal subgroups

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

subgroupindex
#117407932
#117367622
C7×#15152
#117295772
#117273202
#117255602
#117254392
#94167