Finite Group Databasebeta

← #13242221

Subgroups — #13242221

Order 1792.

550 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 111 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#11739525C211yesmaximal, characteristic
2C7×#1749C211yesmaximal, characteristic, complemented
3C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4#11725285C212yesmaximal, complemented
5#11725283C212yesmaximal, complemented
6C22×C42×C7C2211yescharacteristic
7C2×#6493C411yescharacteristic
8C2×#6492—21no—
9C2×C7×C22⋊C8—21no—
10C2×C7×#97—21no—
11#89985—21no—
12#89984C411yescharacteristic
13C2×C22⋊(C7⋊C8)—21no—
14#89712—21no—
15#89712—21no—
16#89711C411yescharacteristic
17#89711C2211yescharacteristic
18#89700C411yescharacteristic
19#89700C2211yescharacteristic
20C2×C4×C7⋊C8—21no—
21C7×#336C2212yes—
22#89319C2212yes—
23#7674—71nomaximal, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×C22⋊D7—21no—
28C22×D28C2×C411yescharacteristic
29C22×C7×C8—22no—
30C2×C42×C7—42no—
31C2×C42×C7C2×C411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C42×C7D411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×C22⋊Dic7—21no—
36#6493—42no—
37#6492—44no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×(C2×D7)⋊C4—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×C4⋊Dic7C2×C411yescharacteristic
44C2×C4⋊Dic7C2×C411yescharacteristic
45C2×C4⋊Dic7C2×C411yescharacteristic
46C2×Dic7⋊C4—21no—
47C2×Dic7⋊C4—21no—
48C22×C7⋊C8—22no—
49C7×C22⋊C8—24no—
50C7×#97—42no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C22⋊(C7⋊C8)—24no—
55C4×C7⋊C8—42no—
56#2854—71no—
57#1749D711yescharacteristic, complemented
58C4×C22⋊C8—71no—
59#1271—72no—
60#1270—72no—
61C24×C7C22⋊C411yescharacteristic
62C23×D7—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—41no—
70C22×C4×C7—41no—
71C22×C4×C7—41no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C2×C22⋊D7—42no—
79C2×C22⋊D7—42no—
80C2×C22⋊D7—42no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C22×Dic7—21no—
84C2×D28—22no—
85C2×D28—41no—
86C2×D28—41no—
87C2×C7×C8—24no—
88C2×C7×C8—42no—
89C2×C7×C8—42no—
90C42×C7—22no—
91C42×C7—22no—
92C42×C7—41no—
93C42×C7—41no—
94C22⋊Dic7—42no—
95(C2×D7)⋊C4—42no—
96(C2×D7)⋊C4—42no—
97(C2×D7)⋊C4—42no—
98(C2×D7)⋊C4—42no—
99(C2×D7)⋊C4—42no—
100C4⋊Dic7—22no—
101C4⋊Dic7—41no—
102C4⋊Dic7—41no—
103Dic7⋊C4—42no—
104Dic7⋊C4—42no—
105C2×C7⋊C8—24no—
106C2×C7⋊C8—42no—
107C2×C7⋊C8—42no—
108C2×#123—141no—
109C2×#121—71no—
110C22×C42C2×D711yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—141no—
113C2×#97—21no—
114C2×C4×C8—141no—
115#390—71no—
116#390—71no—
117#389—71no—
118#389—71no—
119#388—141no—
120#388—141no—
121#386—141no—
122#380—71no—
123#336—72no—
124#336C2×D712yes—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7#10412yes—
132C23×C7C2×C22⋊C411yescharacteristic
133C22×D7—41no—
134C22×D7—41no—
135C22×D7—41no—
136C22×D7—41no—
137C22×D7—41no—
138C22×D7—41no—
139C22×D7—41no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—44no—
154C2×C4×C7—44no—
155C2×C4×C7—44no—
156C2×C4×C7#9912yes—
157C2×C4×C7#10412yes—
158C2×C4×C7#13112yes—
159C2×C4×C7#13512yes—
160C2×C4×C7AGL(1,ℤ/8)12yes—
161C22⋊D7—84no—
162C2×Dic7—41no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—41no—
171D28—82no—
172C7×C8—44no—
173C7⋊C8—44no—
174C22×D4—71no—
175C22×D4—141no—
176C23×C4C4×D711yescharacteristic
177C23×C4D2811yescharacteristic
178C23×C4C22⋊D711yescharacteristic
179C22×C8—22no—
180C22×C8—142no—
181#123—284no—
182#121—282no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—141no—
187C2×C4⋊C4—141no—
188C2×C22⋊C4—141no—
189C2×C22⋊C4—141no—
190C2×C22⋊C4—141no—
191C2×C22⋊C4—141no—
192C2×C22⋊C4—141no—
193C2×C22⋊C4—141no—
194C2×C42—42no—
195C2×C42C4×D711yescharacteristic
196C2×C42D2811yescharacteristic
197C2×C42C22⋊D711yescharacteristic
198C2×C42C22×D711yescharacteristic
199C22⋊C8—24no—
200C22⋊C8—144no—
201#97—42no—
202C4×C8—282no—
203#95—141no—
204#95—141no—
205#95—141no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7#40811yescharacteristic
214C22×C7C2×#9911yescharacteristic
215C22×C7#41311yescharacteristic
216C22×C7#41611yescharacteristic
217C22×C7#41811yescharacteristic
218C22×C7C2×#10411yescharacteristic
219C22×C7C2×#10411yescharacteristic
220C2×D7—81no—
221C2×D7—81no—
222C2×D7—81no—
223C2×D7—81no—
224C2×D7—81no—
225C2×D7—81no—
226C2×D7—81no—
227C4×C7—22no—
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—22no—
231C4×C7—22no—
232C4×C7—41no—
233Dic7—81no—
234Dic7—81no—
235Dic7—81no—
236C24—141no—
237C24(C2×D7)⋊C411yescharacteristic
238C2×D4—142no—
239C2×D4—281no—
240C2×D4—281no—
241C2×D4—282no—
242C2×D4—282no—
243C2×D4—282no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4—141no—
256C22×C4(C2×D7)⋊C411yescharacteristic
257C22×C4(C2×D7)⋊C411yescharacteristic
258C22×C4(C2×D7)⋊C411yescharacteristic
259C22×C4C2×C4×D711yescharacteristic
260C22×C4C2×D2811yescharacteristic
261C22×C4C2×C22⋊D711yescharacteristic
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—144no—
266C2×C8—282no—
267C2×C8—282no—
268C4⋊C4—142no—
269C4⋊C4—281no—
270C4⋊C4—281no—
271C4⋊C4—282no—
272C4⋊C4—282no—
273C22⋊C4—282no—
274C22⋊C4—282no—
275C22⋊C4—282no—
276C22⋊C4—282no—
277C22⋊C4—282no—
278C22⋊C4—282no—
279C42—22no—
280C42—22no—
281C42—41no—
282C42—41no—
283C2×C7—42no—
284C2×C7#146811yescharacteristic
285C2×C7#148411yescharacteristic
286C2×C7#149411yescharacteristic
287C2×C7#150111yescharacteristic
288C2×C7#150711yescharacteristic
289C2×C7#152411yescharacteristic
290C2×C7#155311yescharacteristic
291D7—161no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—281no—
299C23—281no—
300C23—281no—
301C23—281no—
302C23—281no—
303C23—281no—
304C23—281no—
305C23#639912yes—
306C23C2×(C2×D7)⋊C411yescharacteristic
307D4—562no—
308D4—564no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—281no—
326C2×C4—281no—
327C2×C4—281no—
328C2×C4—281no—
329C2×C4—281no—
330C2×C4—281no—
331C2×C4—281no—
332C2×C4—281no—
333C2×C4—281no—
334C2×C4#637912yes—
335C2×C4#638012yes—
336C2×C4#647012yes—
337C2×C4#647112yes—
338C2×C4#651312yes—
339C8—44no—
340C8—284no—
341C7#767411yescharacteristic, complemented, Sylow
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—561no—
350C22—561no—
351C22—561no—
352C22—561no—
353C22—561no—
354C22—561no—
355C22—561no—
356C22C2×#637911yescharacteristic
357C22C2×#638011yescharacteristic
358C22#8977411yescharacteristic
359C22#8990111yescharacteristic
360C22#8990311yescharacteristic
361C22#8990411yescharacteristic
362C22C2×#639911yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—561no—
370C4—561no—
371C4—561no—
372C2—42no—
373C2—1121no—
374C2#1172719711yescharacteristic
375C2#1172755111yescharacteristic
376C2#1172760711yescharacteristic
377C2#1172772311yescharacteristic
378C2#1172773111yescharacteristic
379C2#1172855411yescharacteristic
380C2#1172857111yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1749C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C4
cosocleC2×C42×C7C23
hypercenter#1749D7
hypocenterC7#7674
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C7×#1749C2semidirect
3. C4×C22⋊(C7⋊C8)C2semidirect
4. #11725285C2semidirect
5. #11725283C2semidirect
57. #1749D7semidirect
341. C7#7674semidirect

Maximal subgroups

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

subgroupindex
#117395252
C7×#17492
C4×C22⋊(C7⋊C8)2
#117252852
#117252832
#76747