Finite Group Databasebeta

← #13258209

Subgroups — #13258209

Order 1792.

560 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 119 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#11730806C211yesmaximal, characteristic
2C7×#1442C211yesmaximal, characteristic
3#11729309C212yesmaximal
4#11725419C211yesmaximal, characteristic
5#11725418C212yesmaximal
6C22×C7×C4⋊C4C2211yescharacteristic
7C2×#6487—21no—
8C2×C7×#95—21no—
9C2×C7×#95—21no—
10C2×C7×#95C2211yescharacteristic
11C2×C7×#95C2211yescharacteristic
12#89984—21no—
13#89984C2212yes—
14C2×C22⋊(C7⋊C8)C2212yes—
15C2×#6381—22no—
16#89433—22no—
17#89428—22no—
18#89324—22no—
19#10569—71nomaximal, Sylow
20C23×C4×C7—21no—
21C23×C4×C7—21no—
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C22×Dic14—21no—
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—42no—
28C2×C22⋊Dic7—22no—
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7D412yes—
33C2×C4⋊Dic7D412yes—
34#6487—42no—
35#6487—42no—
36C2×Dic7⋊C4—22no—
37C22×C7⋊C8—22no—
38C7×#95—22no—
39C7×#95—21no—
40C7×#95—42no—
41C7×#95—42no—
42C7×#95Q812yescomplemented
43C7⋊#95—22no—
44C7⋊#95—22no—
45C22⋊(C7⋊C8)—22no—
46C22⋊(C7⋊C8)Q814yescomplemented
47#6381—44no—
48#2659—71no—
49#1852—72no—
50#1442D711yescharacteristic
51#1294—72no—
52#1280—71no—
53C24×C7C2×D411yescharacteristic
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7—21no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7—41no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7—41no—
69C22×C4×C7—41no—
70C22×C4×C7—41no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7—41no—
77C22×C4×C7—41no—
78C22×C4×C7D812yes—
79C22×C4×C7Q1612yes—
80C22×C4×C7C2×D412yes—
81C22×C4×C7C2×Q812yes—
82C22×C4×C7C2×Q811yescharacteristic
83C22×C4×C7#4111yescharacteristic
84C22×Dic7—22no—
85C22×Dic7—22no—
86C2×Dic14—42no—
87C2×Dic14—81no—
88C2×Dic14—81no—
89C7×C4⋊C4—24no—
90C7×C4⋊C4—42no—
91C22⋊Dic7—42no—
92C22⋊Dic7—42no—
93C4⋊Dic7—42no—
94C4⋊Dic7—42no—
95C4⋊Dic7—42no—
96C4⋊Dic7—81no—
97C4⋊Dic7—81no—
98Dic7⋊C4—44no—
99C2×C7⋊C8—24no—
100C2×C7⋊C8—42no—
101C2×C7⋊C8—42no—
102C2×C22⋊Q8—141no—
103C22×C4⋊C4C2×D711yescharacteristic
104C2×#107—142no—
105C2×C22⋊C8—72no—
106#398—142no—
107#395—142no—
108#380—72no—
109#380—141no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95C2×D711yescharacteristic
113C2×#95C2×D711yescharacteristic
114#340—142no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7#12011yescharacteristic
120C23×C7C22⋊Q811yescharacteristic
121C23×C7C4⋊Q811yescharacteristic
122C23×C7C2×D811yescharacteristic
123C23×C7C2×Q1611yescharacteristic
124C23×C7AGL(1,ℤ/8)11yescharacteristic
125C23×C7#13711yescharacteristic
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—41no—
131C2×C4×C7—42no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—81no—
146C2×C4×C7—81no—
147C2×C4×C7—81no—
148C2×C4×C7—81no—
149C2×C4×C7—81no—
150C2×C4×C7—81no—
151C2×C4×C7—81no—
152C2×C4×C7—81no—
153C2×C4×C7—81no—
154C2×C4×C7—81no—
155C2×C4×C7C22⋊Q812yes—
156C2×C4×C7C4⋊Q812yes—
157C2×Dic7—42no—
158C2×Dic7—42no—
159C2×Dic7—42no—
160C2×Dic7—42no—
161C2×Dic7—42no—
162C2×Dic7—42no—
163Dic14—84no—
164C7⋊C8—44no—
165C22×Q8—141no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22⋊D711yescharacteristic
169C23×C4C22⋊D711yescharacteristic
170C23×C4C22×D711yescharacteristic
171C22×C8—142no—
172C22⋊Q8—282no—
173C22⋊Q8—282no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—72no—
177C2×C4⋊C4—72no—
178C2×C4⋊C4—142no—
179C2×C4⋊C4—141no—
180C2×C4⋊C4—141no—
181C2×C4⋊C4—141no—
182C2×C22⋊C4—142no—
183#107—284no—
184C22⋊C8—74no—
185C22⋊C8—142no—
186#95—21no—
187#95—22no—
188#95—42no—
189#95—42no—
190#95—142no—
191#95—142no—
192#95Dic1412yescomplemented
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7#39511yescharacteristic
206C22×C7#44711yescharacteristic
207C22×C7#45111yescharacteristic
208C22×C7#45411yescharacteristic
209C22×C7#45511yescharacteristic
210C22×C7B(4,𝔽2)11yescharacteristic
211C22×C7#45811yescharacteristic
212C22×C7#47411yescharacteristic
213C22×C7#47711yescharacteristic
214C22×C7#50011yescharacteristic
215C22×C7#50111yescharacteristic
216C4×C7—22no—
217C4×C7—22no—
218C4×C7—42no—
219C4×C7—42no—
220C4×C7—82no—
221C4×C7—82no—
222Dic7—82no—
223Dic7—82no—
224C24C2×C22⋊D711yescharacteristic
225C2×Q8—282no—
226C2×Q8—561no—
227C2×Q8—561no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—142no—
253C22×C4—142no—
254C22×C4#114112yes—
255C22×C4#114412yes—
256C22×C4C2×Dic1411yescharacteristic
257C22×C4#115711yescharacteristic
258C22×C4D4×D712yes—
259C22×C4Q8×D712yes—
260C2×C8—144no—
261C2×C8—282no—
262C2×C8—282no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—282no—
266C4⋊C4—282no—
267C4⋊C4—284no—
268C4⋊C4—282no—
269C4⋊C4—561no—
270C4⋊C4—561no—
271C22⋊C4—282no—
272C22⋊C4—282no—
273C2×C7—21no—
274C2×C7—21no—
275C2×C7—21no—
276C2×C7—21no—
277C2×C7#159811yescharacteristic
278C2×C7#160211yescharacteristic
279C2×C7#160711yescharacteristic
280C2×C7#161711yescharacteristic
281C2×C7#205711yescharacteristic
282C2×C7#206111yescharacteristic
283C2×C7#206211yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23#648711yescharacteristic
289C23C2×#114111yescharacteristic
290C23#650011yescharacteristic
291C23C2×#114411yescharacteristic
292C23#650711yescharacteristic
293C23#651211yescharacteristic
294C23#651411yescharacteristic
295Q8—564no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4—282no—
331C2×C4#645112yes—
332C2×C4#645912yes—
333C8—284no—
334C7#1056911yescharacteristic, complemented, Sylow
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#8975111yescharacteristic
348C22#8977911yescharacteristic
349C22#8979311yescharacteristic
350C22#8980411yescharacteristic
351C22#8980811yescharacteristic
352C22#8982211yescharacteristic
353C22#8982511yescharacteristic
354C22#8983311yescharacteristic
355C22#8983411yescharacteristic
356C22#8983711yescharacteristic
357C22#8985011yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—562no—
365C4—562no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#1172729711yescharacteristic
371C2#1172734811yescharacteristic
372C2#1172735811yescharacteristic
373C2#1172745511yescharacteristic
374C2#1172745711yescharacteristic
375C2#1172746211yescharacteristic
376C2#1172747311yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#1442C2
socleC23×C7#120
prime radicalC24×C7C2×D4
hypercenter#1442D7
hypocenterC7#10569
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1442

Complements

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

NH ≅ G/Ntype
42. C7×#95Q8semidirect
46. C22⋊(C7⋊C8)Q8semidirect
192. #95Dic14semidirect
334. C7#10569semidirect

Maximal subgroups

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

subgroupindex
#117308062
C7×#14422
#117293092
#117254192
#117254182
#105697