Finite Group Databasebeta

← #13241893

Subgroups — #13241893

Order 1792.

654 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 117 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#11726416C212yesmaximal, complemented
3C7×#1272C211yesmaximal, characteristic, complemented
4#11725448C211yesmaximal, characteristic, complemented
5#11725285C212yesmaximal, complemented
6C22×#1161C411yescharacteristic
7C2×#6455C411yescharacteristic
8C2×C42×D7C2211yescharacteristic
9C7×#391C2211yescharacteristic
10#90002C411yescharacteristic
11C2×(C2×D7)⋊C8—22no—
12C2×Dic7⋊C8—22no—
13#89760C411yescharacteristic
14#89711C2211yescharacteristic
15#89438—22no—
16#89437—22no—
17C7×#336C2212yes—
18#89319C2212yes—
19#7597—71nomaximal, Sylow
20C22×C7×Q8C2×C411yescharacteristic
21C2×#1161—22no—
22C2×#1161—42no—
23C2×#1161—42no—
24C2×#1161C814yescomplemented
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—
41#6455—42no—
42C42×D7—42no—
43C7×#95—22no—
44C7⋊#95—22no—
45(C2×D7)⋊C8—44no—
46Dic7⋊C8—44no—
47#2297—71no—
48#1797—72no—
49#1272—71no—
50#1272D711yescharacteristic, complemented
51#1270—72no—
52C23×D7—21no—
53C23×D7C22⋊C411yescharacteristic
54C2×C7×Q8—21no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8C2×C812yes—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7C22×C411yescharacteristic
61C22×C4×C7C2×D412yes—
62C22×Dic7—21no—
63C22×Dic7C22⋊C412yes—
64C22×Dic7C22⋊C411yescharacteristic
65#1161—48no—
66#1161—48no—
67C2×D28—24no—
68C2×D28—21no—
69C2×D28—41no—
70C2×D28—41no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28C2×C812yes—
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—24no—
79C2×C4×D7—42no—
80C2×C4×D7—42no—
81C2×C4×D7—42no—
82C2×C4×D7—42no—
83C2×C4×D7C2×C812yes—
84C2×C4×D7M1612yes—
85C2×C7×C8—42no—
86C2×C7×C8—42no—
87C2×C7×C8—42no—
88C7×C4⋊C4—41no—
89C42×C7—41no—
90(C2×D7)⋊C4—44no—
91(C2×D7)⋊C4—42no—
92(C2×D7)⋊C4—42no—
93(C2×D7)⋊C4—42no—
94C4⋊Dic7—41no—
95C4×Dic7—24no—
96C4×Dic7—41no—
97C4×Dic7—42no—
98C2×C7⋊C8—42no—
99C2×C7⋊C8—42no—
100C2×C7⋊C8—42no—
101C22×#41—71no—
102C2×#117—71no—
103C22×C42—71no—
104C2×C4⋊C8—142no—
105C2×C22⋊C8—142no—
106#391—71no—
107#391C2×D711yescharacteristic
108#390—71no—
109#390—71no—
110#386—142no—
111C4×C22⋊C4—142no—
112#336—72no—
113#336C2×D712yes—
114C23×C7C2×C22⋊C411yescharacteristic
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—22no—
118C22×D7—22no—
119C22×D7—22no—
120C22×D7—41no—
121C22×D7—41no—
122C22×D7—42no—
123C22×D7—42no—
124C22×D7C22⋊C812yes—
125C7×Q8—42no—
126C7×Q8—42no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7C22×C811yescharacteristic
137C2×C4×C7C2×M1611yescharacteristic
138C2×C4×C7AGL(1,ℤ/8)12yes—
139C2×C4×C7#13712yes—
140C2×Dic7—22no—
141C2×Dic7—22no—
142C2×Dic7—22no—
143C2×Dic7—41no—
144C2×Dic7—42no—
145C2×Dic7C22⋊C812yes—
146C2×Dic7#9912yes—
147C2×Dic7#10012yes—
148D28—42no—
149D28—42no—
150D28—48no—
151C4×D7—24no—
152C4×D7—24no—
153C4×D7—44no—
154C4×D7—44no—
155C4×D7—48no—
156C7×C8—82no—
157C7⋊C8—82no—
158C2×#41—74no—
159C2×#41—142no—
160C2×#41—282no—
161C2×#41—282no—
162C22×Q8C4×D711yescharacteristic
163C22×D4—71no—
164C22×D4—141no—
165C23×C4—71no—
166C23×C4—72no—
167C23×C4—141no—
168C22×C8—22no—
169C22×C8—142no—
170#117—282no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4C4×D711yescharacteristic
173C2×C22⋊C4—142no—
174C2×C22⋊C4—141no—
175C2×C22⋊C4—142no—
176C2×C42—72no—
177C2×C42—71no—
178C2×C42—141no—
179C2×C42—282no—
180C2×C42C22×D711yescharacteristic
181C4⋊C8—284no—
182C22⋊C8—284no—
183#95—22no—
184#95—142no—
185C22×C7C2×C22⋊C811yescharacteristic
186C22×C7C2×#9911yescharacteristic
187C22×C7C2×#10011yescharacteristic
188C22×C7#41712yes—
189C22×C7#42112yes—
190C2×D7—22no—
191C2×D7—24no—
192C2×D7—41no—
193C2×D7—41no—
194C2×D7—42no—
195C2×D7—42no—
196C2×D7—44no—
197C2×D7—81no—
198C2×D7—81no—
199C2×D7—82no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—22no—
203C4×C7—22no—
204C4×C7—42no—
205C4×C7—81no—
206Dic7—42no—
207Dic7—81no—
208Dic7#32314yes—
209C24—71no—
210C24—141no—
211#41—288no—
212#41—288no—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8C8×D712yes—
217C2×D4—72no—
218C2×D4—144no—
219C2×D4—141no—
220C2×D4—281no—
221C2×D4—281no—
222C2×D4—282no—
223C2×D4—282no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—71no—
227C22×C4—72no—
228C22×C4—72no—
229C22×C4—72no—
230C22×C4—142no—
231C22×C4—142no—
232C22×C4—144no—
233C22×C4—142no—
234C22×C4—142no—
235C22×C4—141no—
236C22×C4—282no—
237C22×C4—282no—
238C22×C4—282no—
239C22×C4—282no—
240C22×C4C2×C4×D711yescharacteristic
241C22×C4D4×D712yes—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—282no—
246C2×C8—282no—
247C2×C8—282no—
248C4⋊C4—41no—
249C4⋊C4—281no—
250C22⋊C4—284no—
251C22⋊C4—282no—
252C22⋊C4—282no—
253C22⋊C4—282no—
254C42—41no—
255C42—144no—
256C42—281no—
257C42—282no—
258C2×C7C2×#32311yescharacteristic
259C2×C7#147712yes—
260C2×C7#151312yes—
261C2×C7#153611yescharacteristic
262C2×C7#156011yescharacteristic
263D7—42no—
264D7—82no—
265C23—72no—
266C23—141no—
267C23—141no—
268C23—142no—
269C23—142no—
270C23—142no—
271C23—281no—
272C23—281no—
273C23—282no—
274C23—282no—
275C23D7×C22⋊C411yescharacteristic
276Q8—42no—
277Q8—42no—
278D4—282no—
279D4—282no—
280D4—288no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—72no—
291C2×C4—72no—
292C2×C4—72no—
293C2×C4—144no—
294C2×C4—144no—
295C2×C4—142no—
296C2×C4—142no—
297C2×C4—142no—
298C2×C4—284no—
299C2×C4—284no—
300C2×C4—288no—
301C2×C4—281no—
302C2×C4—282no—
303C2×C4C2×C8×D711yescharacteristic
304C2×C4D7×M1611yescharacteristic
305C2×C4#647712yes—
306C2×C4#648112yes—
307C8—82no—
308C8—562no—
309C7#759711yescharacteristic, complemented, Sylow
310C22—142no—
311C22—144no—
312C22—282no—
313C22—281no—
314C22—281no—
315C22—282no—
316C22—284no—
317C22—562no—
318C22—561no—
319C22—561no—
320C22D7×C22⋊C811yescharacteristic
321C22D7×#9911yescharacteristic
322C22D7×#10011yescharacteristic
323C22#8957512yes—
324C22#8959312yes—
325C4—22no—
326C4—22no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—74no—
331C4—81no—
332C4—282no—
333C4—561no—
334C2—282no—
335C2—562no—
336C2D7×#32311yescharacteristic
337C2#1172598412yes—
338C2#1172609112yes—
339C2#1172610311yescharacteristic
340C2#1172624311yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC2×C4×C7C22×C8
FrattiniC2×C42C22×D7
FittingC7×#1272C2
socleC23×C7C2×C22⋊C4
prime radicalC22×#1161C4
cosocleC2×C42×C7C23
hypercenter#1272D7
hypocenterC7#7597
LCSG ▷ C2×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. #11726416C2semidirect
3. C7×#1272C2semidirect
4. #11725448C2semidirect
5. #11725285C2semidirect
24. C2×#1161C8semidirect
50. #1272D7semidirect
309. C7#7597semidirect

Maximal subgroups

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

subgroupindex
#117339232
#117264162
C7×#12722
#117254482
#117252852
#75977