Finite Group Databasebeta

← #13264406

Subgroups — #13264406

Order 1792.

590 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 87 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
2C7×#1466C211yesmaximal, characteristic
3C4×#6408C211yesmaximal, characteristic
4#11725255C214yesmaximal, complemented
5C23×Dic14C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487—21no—
8C7×#432—21no—
9#89992—21no—
10C2×#6408—21no—
11#89984C411yescharacteristic, complemented
12C4×C22⋊Dic7—21no—
13#89751C412yescomplemented
14#89719C2212yes—
15#89718—21no—
16#89701—21no—
17C7×#323—21no—
18C7×#323C2212yes—
19#89321C2212yes—
20#11785—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×Dic7—21no—
25C22×Dic14—21no—
26C22×Dic14—44no—
27C22×Dic14—41no—
28C22×Dic14—41no—
29C22×Dic14C2×C412yes—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C22⋊C4D411yescharacteristic
34C2×C7×C22⋊C4C2311yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C22⋊Dic7—22no—
37C2×C22⋊Dic7—21no—
38C2×C22⋊Dic7—22no—
39C2×C4⋊Dic7—21no—
40#6487—44no—
41C2×Dic7⋊C4—22no—
42C2×C4×Dic7—41no—
43C7×C4⋊C8—41no—
44C7×C22⋊C8—22no—
45C7×#97—41no—
46#6408—24no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—41no—
50C7⋊#95C2×C412yes—
51#2409—71no—
52C4×#99—71no—
53#1466D711yescharacteristic
54#1351—74no—
55C24×C7C22⋊C411yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7—44no—
74C22×Dic7—42no—
75C22×Dic7—42no—
76C2×Dic14—24no—
77C2×Dic14—42no—
78C2×Dic14—48no—
79C2×Dic14—44no—
80C2×Dic14—41no—
81C2×Dic14—41no—
82C2×Dic14—42no—
83C2×Dic14—44no—
84C2×Dic14—44no—
85C2×C7×C8—42no—
86C7×C4⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C42×C7—21no—
90C42×C7—21no—
91C22⋊Dic7—24no—
92C22⋊Dic7—44no—
93C22⋊Dic7—42no—
94C22⋊Dic7—42no—
95C4⋊Dic7—42no—
96Dic7⋊C4—44no—
97C4×Dic7—82no—
98C23×Q8—71no—
99C2×C22⋊Q8—141no—
100C2×C4×D4C2×D711yescharacteristic
101#432—21no—
102C2×#99—141no—
103#395—72no—
104#393—72no—
105#391—141no—
106#386—141no—
107#385—141no—
108#380—71no—
109C4×C22⋊C4—141no—
110#342—72no—
111#323—21no—
112#323C2×D712yes—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7#10412yes—
122C23×C7C2×C22⋊C411yescharacteristic
123C23×C7#13712yes—
124C7×D4—44no—
125C2×C4×C7—22no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—81no—
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7#9911yescharacteristic
141C2×Dic7—24no—
142C2×Dic7—42no—
143C2×Dic7—48no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—44no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—82no—
150C2×Dic7—82no—
151Dic14—44no—
152Dic14—44no—
153Dic14—44no—
154Dic14—44no—
155C7×C8—82no—
156C22×Q8—72no—
157C22×Q8—141no—
158C22×Q8—281no—
159C22×Q8—281no—
160C22×Q8—284no—
161C22×D4D2811yescharacteristic
162C23×C4—141no—
163C23×C4C4×D711yescharacteristic
164C23×C4D2811yescharacteristic
165C22⋊Q8—284no—
166C4×D4—41no—
167C4×D4—41no—
168C2×C4⋊C4—142no—
169C2×C4⋊C4—141no—
170C2×C4⋊C4C4×D711yescharacteristic
171C2×C22⋊C4—142no—
172C2×C22⋊C4—141no—
173C2×C22⋊C4—142no—
174C2×C22⋊C4C22⋊D711yescharacteristic
175C2×C22⋊C4C22×D711yescharacteristic
176C2×C42—281no—
177C2×C42C22⋊D711yescharacteristic
178C4⋊C8—41no—
179#99—144no—
180C22⋊C8—22no—
181#97—41no—
182#95—72no—
183#95—142no—
184#95—142no—
185#95—281no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—22no—
189C22×C7—22no—
190C22×C7—42no—
191C22×C7—42no—
192C22×C7—42no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—81no—
198C22×C7C2×#9911yescharacteristic
199C22×C7#41911yescharacteristic
200C22×C7C2×#10411yescharacteristic
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7AΣL(1,𝔽16)12yes—
207Dic7—44no—
208Dic7—82no—
209Dic7—84no—
210C24(C2×D7)⋊C411yescharacteristic
211C24C2×D2811yescharacteristic
212C2×Q8—144no—
213C2×Q8—282no—
214C2×Q8—282no—
215C2×Q8—284no—
216C2×Q8—284no—
217C2×Q8—288no—
218C2×Q8—284no—
219C2×Q8—281no—
220C2×Q8—281no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—142no—
231C22×C4—142no—
232C22×C4—284no—
233C22×C4—282no—
234C22×C4—282no—
235C22×C4(C2×D7)⋊C411yescharacteristic
236C22×C4(C2×D7)⋊C411yescharacteristic
237C22×C4(C2×D7)⋊C411yescharacteristic
238C22×C4C2×C4×D711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—282no—
243C4⋊C4—284no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—144no—
247C22⋊C4—284no—
248C22⋊C4—282no—
249C22⋊C4—282no—
250C42—21no—
251C42—21no—
252C42—562no—
253C2×C7—21no—
254C2×C7—21no—
255C2×C7—42no—
256C2×C7—42no—
257C2×C7#150811yescharacteristic
258C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
259C2×C7#212911yescharacteristic
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#639912yes—
269C23#647212yes—
270C23C2×(C2×D7)⋊C411yescharacteristic
271Q8—284no—
272Q8—284no—
273Q8—284no—
274Q8—284no—
275D4—44no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—144no—
291C2×C4—282no—
292C2×C4—288no—
293C2×C4—282no—
294C2×C4—284no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—562no—
299C2×C4—562no—
300C2×C4#638011yescharacteristic
301C2×C4#638011yescharacteristic
302C8—82no—
303C7#1178511yescharacteristic, complemented, Sylow
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#638011yescharacteristic
317C22#8989211yescharacteristic
318C22C2×#639911yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—284no—
325C4—562no—
326C4—564no—
327C4#8927012yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#1172773411yescharacteristic
333C2#1172781311yescharacteristic
334C2C2×#8927011yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#1466C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#1466D7
hypocenterC7#11785
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
4. #11725255C2semidirect
5. C23×Dic14C4semidirect
11. #89984C4semidirect
13. #89751C4semidirect
303. C7#11785semidirect

Maximal subgroups

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

subgroupindex
#117397942
C7×#14662
C4×#64082
#117252552
#117857