Finite Group Databasebeta

← #13272457

Subgroups — #13272457

Order 1792.

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

normal subgroups only

This group has 616 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#11739794C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11725500C214yesmaximal
5C23×Dic14C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487—21no—
8C2×C7×#99C2212yes—
9C4×C7×C22⋊C4—21no—
10#89992C2212yes—
11C2×#6407C2212yes—
12#89984C411yescharacteristic, complemented
13#89751—21no—
14#89719—21no—
15#89718C412yescomplemented
16#89701—21no—
17#89696—21no—
18C7×#342—21no—
19#89325—21no—
20#13296—71nomaximal, Sylow
21C22×C7×D4C2311yescharacteristic
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×Dic7C2×C412yes—
25C22×Dic14—21no—
26C22×Dic14—21no—
27C22×Dic14—44no—
28C22×Dic14—41no—
29C22×Dic14—41no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4C2×C411yescharacteristic
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C42×C7—41no—
37C2×C42×C7D411yescharacteristic
38C2×C22⋊Dic7—22no—
39C2×C22⋊Dic7C2×C412yes—
40C2×C4⋊Dic7—21no—
41#6487—44no—
42C2×Dic7⋊C4—22no—
43C2×#1136—22no—
44C7×#99—22no—
45C7×#99Q814yescomplemented
46C7×#95—41no—
47#6407—24no—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C7⋊#95—22no—
51C4⋊(C7⋊C8)—41no—
52C4⋊(C7⋊C8)—41no—
53#2409—71no—
54#1800—71no—
55C4×#99D711yescharacteristic
56#1390—74no—
57C24×C7C22×C411yescharacteristic
58C24×C7C2×D411yescharacteristic
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4C2×Q812yes—
62C2×C7×D4#4112yes—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—22no—
76C22×Dic7—22no—
77C22×Dic7—44no—
78C22×Dic7—42no—
79C2×Dic14—24no—
80C2×Dic14—44no—
81C2×Dic14—42no—
82C2×Dic14—42no—
83C2×Dic14—48no—
84C2×Dic14—44no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C2×Dic14—44no—
88C7×C4⋊C4—41no—
89C7×C22⋊C4—24no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—42no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7—82no—
96C22⋊Dic7—44no—
97C22⋊Dic7—42no—
98C4⋊Dic7—42no—
99Dic7⋊C4—44no—
100#1136—42no—
101#1136—42no—
102C2×C7⋊C8—42no—
103C23×Q8—71no—
104C2×C22⋊Q8—141no—
105C2×C4×D4C2×D711yescharacteristic
106#423—141no—
107C2×#100—72no—
108C2×#99C2×D712yes—
109#395—141no—
110#393—141no—
111#391—141no—
112#386—72no—
113#385—72no—
114#380—71no—
115C4×C22⋊C4—21no—
116#343—141no—
117#342—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7C2×C22⋊C411yescharacteristic
127C23×C7C4×Q812yes—
128C23×C7C22⋊Q812yes—
129C7×D4—42no—
130C7×D4—42no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—44no—
144C2×C4×C7—81no—
145C2×C4×C7—82no—
146C2×C4×C7—82no—
147C2×C4×C7#9911yescharacteristic
148C2×C4×C7#9911yescharacteristic
149C2×C4×C7#12411yescharacteristic
150C2×C4×C7C4⋊Q811yescharacteristic
151C2×Dic7—24no—
152C2×Dic7—42no—
153C2×Dic7—44no—
154C2×Dic7—42no—
155C2×Dic7—44no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159Dic14—44no—
160Dic14—48no—
161Dic14—44no—
162C7⋊C8—82no—
163C22×Q8—141no—
164C22×Q8—141no—
165C22×Q8—281no—
166C22×Q8—281no—
167C22×Q8—284no—
168C22×D4C22×D711yescharacteristic
169C23×C4—72no—
170C23×C4C4×D711yescharacteristic
171C23×C4C22⋊D711yescharacteristic
172C2×M16—142no—
173C22⋊Q8—284no—
174C4×D4—41no—
175C4×D4—41no—
176C2×C4⋊C4—142no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4D2811yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—72no—
181C2×C22⋊C4—142no—
182C2×C22⋊C4C4×D711yescharacteristic
183C2×C22⋊C4C22⋊D711yescharacteristic
184C2×C42—41no—
185C2×C42D2811yescharacteristic
186C4⋊C8—281no—
187C4⋊C8—281no—
188#100—144no—
189#99—22no—
190#99Dic1414yescomplemented
191#95—41no—
192#95—141no—
193#95—142no—
194#95—142no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—22no—
198C22×C7—22no—
199C22×C7—42no—
200C22×C7—42no—
201C22×C7—42no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—81no—
207C22×C7#39111yescharacteristic
208C22×C7C2×#9911yescharacteristic
209C22×C7#41011yescharacteristic
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—84no—
217C4×C7AΣL(1,𝔽16)12yes—
218Dic7—44no—
219Dic7—82no—
220C24C2×C4×D711yescharacteristic
221C24C2×C22⋊D711yescharacteristic
222C2×Q8—144no—
223C2×Q8—282no—
224C2×Q8—282no—
225C2×Q8—284no—
226C2×Q8—284no—
227C2×Q8—288no—
228C2×Q8—284no—
229C2×Q8—281no—
230C2×Q8—281no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4C2×Dic1412yes—
234C2×D4#115712yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—142no—
243C22×C4—142no—
244C22×C4—284no—
245C22×C4—282no—
246C22×C4(C2×D7)⋊C411yescharacteristic
247C22×C4(C2×D7)⋊C411yescharacteristic
248C22×C4(C2×D7)⋊C411yescharacteristic
249C22×C4(C2×D7)⋊C411yescharacteristic
250C22×C4C2×D2811yescharacteristic
251M16—282no—
252M16—282no—
253C2×C8—282no—
254C4⋊C4—41no—
255C4⋊C4—282no—
256C4⋊C4—284no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—284no—
262C22⋊C4—282no—
263C42—21no—
264C42—21no—
265C42—82no—
266C2×C7—21no—
267C2×C7—21no—
268C2×C7—42no—
269C2×C7—42no—
270C2×C7#199611yescharacteristic
271C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
272C2×C7#212311yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C4×Dic1412yes—
282C23#648712yes—
283C23C2×(C2×D7)⋊C411yescharacteristic
284Q8—284no—
285Q8—288no—
286Q8—284no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—144no—
306C2×C4—282no—
307C2×C4—284no—
308C2×C4—282no—
309C2×C4—284no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4—282no—
313C2×C4#638011yescharacteristic
314C2×C4#638011yescharacteristic
315C2×C4#643211yescharacteristic
316C2×C4#643811yescharacteristic
317C8—562no—
318C7#1329611yescharacteristic, complemented, Sylow
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#8970111yescharacteristic
332C22C2×#638011yescharacteristic
333C22#8977811yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—84no—
341C4—284no—
342C4—562no—
343C4#8926612yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2#1172721111yescharacteristic
349C2C2×#8926611yescharacteristic
350C2#1172781211yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC24×C7C22×C4
FrattiniC22×D4C22×D7
FittingC4×C7×#99C2
socleC22×C7C2×#99
prime radical, cosocleC22×C7×D4C23
hypercenterC4×#99D7
hypocenterC7#13296
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C23×Dic14C4semidirect
12. #89984C4semidirect
15. #89718C4semidirect
45. C7×#99Q8semidirect
190. #99Dic14semidirect
318. C7#13296semidirect

Maximal subgroups

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

subgroupindex
#117397942
C4×C7×#992
#117293712
#117255002
#132967