Finite Group Databasebeta

← #13242039

Subgroups — #13242039

Order 1792.

730 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 105 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#11733923C211yesmaximal, characteristic
2#11726414C212yesmaximal, complemented
3C7×#1272C211yesmaximal, characteristic, complemented
4#11725448C211yesmaximal, characteristic, complemented
5#11725285C212yesmaximal, complemented
6C22×#1161C411yescharacteristic
7C22×#1131—22no—
8C2×#6455C411yescharacteristic
9C2×C42×D7C2211yescharacteristic
10C7×#391C2211yescharacteristic
11#90002C411yescharacteristic
12#89760C411yescharacteristic
13#89711C2211yescharacteristic
14#89438—22no—
15#89437—22no—
16C7×#336C2212yes—
17#89345—22no—
18#89319C2212yes—
19#7631—71nomaximal, Sylow
20C22×C7×Q8C2×C411yescharacteristic
21C2×#1161—22no—
22C2×#1161—22no—
23C2×#1161—42no—
24C2×#1161—42no—
25C22×D28—21no—
26C22×D28C2×C411yescharacteristic
27C22×C4×D7—21no—
28C22×C4×D7C2×C411yescharacteristic
29C22×C4×D7D412yes—
30C22×C7×C8—22no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C42×C7C2311yescharacteristic
33C2×(C2×D7)⋊C4—21no—
34C2×(C2×D7)⋊C4—22no—
35C2×(C2×D7)⋊C4—22no—
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×C4×Dic7—21no—
38C2×C4×Dic7C2×C411yescharacteristic
39C2×C4×Dic7D412yes—
40C22×C7⋊C8—22no—
41C2×#1131—28no—
42C2×#1131—44no—
43C2×#1131—44no—
44#6455—42no—
45C42×D7—42no—
46C7×#95—22no—
47C7⋊#95—22no—
48#2297—71no—
49#1728—72no—
50#1272—71no—
51#1272D711yescharacteristic, complemented
52#1270—72no—
53C23×D7—21no—
54C23×D7C22⋊C411yescharacteristic
55C2×C7×Q8—21no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C2×C7×Q8M1612yescomplemented
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D412yes—
63C22×Dic7—21no—
64C22×Dic7C22⋊C411yescharacteristic
65C22×Dic7C22⋊C412yes—
66#1161—48no—
67#1161—48no—
68C2×D28—24no—
69C2×D28—21no—
70C2×D28—41no—
71C2×D28—41no—
72C2×D28—42no—
73C2×D28—42no—
74C2×D28M1612yescomplemented
75C2×C4×D7—24no—
76C2×C4×D7—22no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7—22no—
82C2×C4×D7—24no—
83C2×C4×D7—24no—
84C2×C4×D7—42no—
85C2×C4×D7—42no—
86C2×C7×C8—24no—
87C2×C7×C8—42no—
88C2×C7×C8—42no—
89C7×C4⋊C4—41no—
90C42×C7—41no—
91(C2×D7)⋊C4—42no—
92(C2×D7)⋊C4—42no—
93(C2×D7)⋊C4—42no—
94(C2×D7)⋊C4—44no—
95C4⋊Dic7—41no—
96C4×Dic7—41no—
97C4×Dic7—42no—
98C4×Dic7—42no—
99C2×C7⋊C8—24no—
100C2×C7⋊C8—42no—
101C2×C7⋊C8—42no—
102#1131—416no—
103C22×#41—71no—
104C22×M16—142no—
105C2×#117—71no—
106C22×C42—71no—
107#391—71no—
108#391C2×D711yescharacteristic
109#390—71no—
110#390—71no—
111#386—142no—
112C4×C22⋊C4—142no—
113#336—72no—
114#336—142no—
115#336C2×D712yes—
116C23×C7C2×C22⋊C411yescharacteristic
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—22no—
120C22×D7—22no—
121C22×D7—22no—
122C22×D7—41no—
123C22×D7—41no—
124C22×D7—42no—
125C22×D7—42no—
126C22×D7#9912yes—
127C7×Q8—42no—
128C7×Q8—42no—
129C2×C4×C7—22no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7C2×M1611yescharacteristic
139C2×C4×C7C2×M1611yescharacteristic
140C2×C4×C7AGL(1,ℤ/8)12yes—
141C2×C4×C7#13712yes—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—22no—
145C2×Dic7—22no—
146C2×Dic7—41no—
147C2×Dic7—42no—
148C2×Dic7#10012yes—
149C2×Dic7#10414yes—
150D28—42no—
151D28—42no—
152D28—48no—
153C4×D7—28no—
154C4×D7—28no—
155C4×D7—24no—
156C4×D7—24no—
157C4×D7—48no—
158C7×C8—44no—
159C7⋊C8—44no—
160C2×#41—142no—
161C2×#41—142no—
162C2×#41—282no—
163C2×#41—282no—
164C22×Q8C4×D711yescharacteristic
165C22×D4—71no—
166C22×D4—141no—
167C23×C4—71no—
168C23×C4—72no—
169C23×C4—141no—
170C2×M16—148no—
171C2×M16—284no—
172C2×M16—284no—
173C22×C8—22no—
174C22×C8—142no—
175#117—282no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4C4×D711yescharacteristic
178C2×C22⋊C4—142no—
179C2×C22⋊C4—141no—
180C2×C22⋊C4—142no—
181C2×C42—71no—
182C2×C42—72no—
183C2×C42—141no—
184C2×C42—282no—
185C2×C42C22×D711yescharacteristic
186#95—22no—
187#95—142no—
188C22×C7#40711yescharacteristic
189C22×C7C2×#9911yescharacteristic
190C22×C7C2×#10011yescharacteristic
191C22×C7#41712yes—
192C22×C7C2×#10412yes—
193C2×D7—22no—
194C2×D7—24no—
195C2×D7—41no—
196C2×D7—42no—
197C2×D7—41no—
198C2×D7—42no—
199C2×D7—44no—
200C2×D7—81no—
201C2×D7—82no—
202C2×D7—81no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—42no—
208C4×C7—81no—
209Dic7—22no—
210Dic7—24no—
211Dic7—81no—
212C24—71no—
213C24—141no—
214#41—288no—
215#41—288no—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8#113112yescomplemented
220C2×D4—72no—
221C2×D4—144no—
222C2×D4—141no—
223C2×D4—281no—
224C2×D4—281no—
225C2×D4—282no—
226C2×D4—282no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—72no—
230C22×C4—71no—
231C22×C4—142no—
232C22×C4—144no—
233C22×C4—144no—
234C22×C4—142no—
235C22×C4—144no—
236C22×C4—141no—
237C22×C4—141no—
238C22×C4—142no—
239C22×C4—142no—
240C22×C4—141no—
241C22×C4—282no—
242C22×C4—282no—
243C22×C4C2×C4×D711yescharacteristic
244C22×C4D4×D712yes—
245M16—2816no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—144no—
250C2×C8—282no—
251C2×C8—282no—
252C4⋊C4—41no—
253C4⋊C4—281no—
254C22⋊C4—284no—
255C22⋊C4—282no—
256C22⋊C4—282no—
257C22⋊C4—282no—
258C42—41no—
259C42—282no—
260C42—281no—
261C42—282no—
262C2×C7#146211yescharacteristic
263C2×C7#148912yes—
264C2×C7#150412yes—
265C2×C7#152511yescharacteristic
266C2×C7#156011yescharacteristic
267D7—42no—
268D7—82no—
269C23—72no—
270C23—141no—
271C23—141no—
272C23—142no—
273C23—142no—
274C23—142no—
275C23—281no—
276C23—281no—
277C23—282no—
278C23—282no—
279C23D7×C22⋊C411yescharacteristic
280Q8—42no—
281Q8—42no—
282D4—282no—
283D4—282no—
284D4—288no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—72no—
295C2×C4—74no—
296C2×C4—148no—
297C2×C4—148no—
298C2×C4—144no—
299C2×C4—144no—
300C2×C4—141no—
301C2×C4—142no—
302C2×C4—141no—
303C2×C4—142no—
304C2×C4—288no—
305C2×C4—281no—
306C2×C4—282no—
307C2×C4C2×#113111yescharacteristic
308C2×C4D7×M1611yescharacteristic
309C2×C4#647712yes—
310C2×C4#648112yes—
311C8—44no—
312C8—284no—
313C7#763111yescharacteristic, complemented, Sylow
314C22—144no—
315C22—142no—
316C22—281no—
317C22—281no—
318C22—282no—
319C22—282no—
320C22—284no—
321C22—562no—
322C22—561no—
323C22—561no—
324C22#8949811yescharacteristic
325C22D7×#9911yescharacteristic
326C22D7×#10011yescharacteristic
327C22#8957512yes—
328C22D7×#10412yes—
329C4—22no—
330C4—22no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—81no—
335C4—144no—
336C4—142no—
337C4—561no—
338C2—282no—
339C2—562no—
340C2#1172587411yescharacteristic
341C2#1172599212yes—
342C2#1172608812yes—
343C2#1172609611yescharacteristic
344C2#1172624311yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1272C2
socleC23×C7C2×C22⋊C4
prime radicalC22×#1161C4
cosocleC2×C42×C7C23
hypercenter#1272D7
hypocenterC7#7631
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #11726414C2semidirect
3. C7×#1272C2semidirect
4. #11725448C2semidirect
5. #11725285C2semidirect
51. #1272D7semidirect
58. C2×C7×Q8M16semidirect
74. C2×D28M16semidirect
219. C2×Q8#1131semidirect
313. C7#7631semidirect

Maximal subgroups

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

subgroupindex
#117339232
#117264142
C7×#12722
#117254482
#117252852
#76317