Finite Group Databasebeta

← #13252124

Subgroups — #13252124

Order 1792.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11728211C211yesmaximal, characteristic
2#11728207C211yesmaximal, characteristic, complemented
3C2×#89280C211yesmaximal, characteristic
4#11727526C211yesmaximal, characteristic
5C8×C4⋊Dic7C211yesmaximal, characteristic, complemented
6C7×#1277C211yesmaximal, characteristic, complemented
7#11725508C211yesmaximal, characteristic, complemented
8C2×#6437C2211yescharacteristic
9C2×C7×C4⋊C8C2211yescharacteristic
10C2×C7×C4⋊C8C2211yescharacteristic
11C2×C4×C7×C8C2211yescharacteristic
12C2×#6395—21no—
13C2×#6395—21no—
14C2×#6395—21no—
15C2×#6395—21no—
16C2×#6392—21no—
17C2×C8×Dic7—21no—
18#89707C2211yescharacteristic
19C4×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21#89346—21no—
22#89345—21no—
23#89280C414yescomplemented
24#9563—71nomaximal, Sylow
25C22×D28—21no—
26C22×D28—21no—
27C22×D28—21no—
28C22×C7×C8—21no—
29C22×C7×C8—21no—
30C22×C7×C8D411yescharacteristic
31C22×C7×C8D411yescharacteristic
32C2×C42×C7C2311yescharacteristic
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7D411yescharacteristic
37C2×C4⋊Dic7D411yescharacteristic
38C2×C4×Dic7—21no—
39C22×C7⋊C8—21no—
40#6437—21no—
41#6437—21no—
42#6437—21no—
43#6437C2×C412yes—
44C7×C4⋊C8—21no—
45C7×C4⋊C8—21no—
46C7×C4⋊C8—21no—
47C7×C4⋊C8C2×C412yes—
48C4×C7×C8—21no—
49C4×C7×C8—21no—
50C7⋊#95—21no—
51#6395—42no—
52#6395—42no—
53#6395—42no—
54#6395—42no—
55#6392—42no—
56C8×Dic7—42no—
57C4⋊(C7⋊C8)—21no—
58C4⋊(C7⋊C8)C2×C412yes—
59#1982—71no—
60#1957—71no—
61#1947—71no—
62C8×C4⋊C4—71no—
63C2×#331—71no—
64#1296—71no—
65#1277D711yescharacteristic, complemented
66C23×D7—41no—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7#4111yescharacteristic
70C22×Dic7—21no—
71C22×Dic7—21no—
72C22×Dic7—41no—
73C2×D28—22no—
74C2×D28—22no—
75C2×D28—22no—
76C2×D28—41no—
77C2×D28—41no—
78C2×D28—41no—
79C2×D28—41no—
80C2×D28—41no—
81C2×D28—41no—
82C2×D28—41no—
83C2×D28—41no—
84C2×D28—41no—
85C2×D28—41no—
86C2×D28—41no—
87C2×D28—81no—
88C2×D28—81no—
89C2×C7×C8—21no—
90C2×C7×C8—21no—
91C2×C7×C8—22no—
92C2×C7×C8—21no—
93C2×C7×C8—21no—
94C2×C7×C8—21no—
95C2×C7×C8—41no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8D812yes—
101C42×C7C22×C411yescharacteristic
102C42×C7#4111yescharacteristic
103C42×C7#4111yescharacteristic
104C42×C7#4111yescharacteristic
105C4⋊Dic7—21no—
106C4⋊Dic7—21no—
107C4⋊Dic7—41no—
108C4⋊Dic7—41no—
109C4⋊Dic7—41no—
110C4⋊Dic7—41no—
111C4⋊Dic7—81no—
112C4⋊Dic7D812yes—
113C4⋊Dic7SD1612yes—
114C4×Dic7—41no—
115C4×Dic7—41no—
116C2×C7⋊C8—22no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119C2×#127—71no—
120C2×#107—141no—
121C2×C4⋊C8—71no—
122C2×C4⋊C8C2×D711yescharacteristic
123C2×C4⋊C8C2×D711yescharacteristic
124C2×#102—141no—
125C2×#102—141no—
126C2×#102—141no—
127C2×#102—141no—
128C2×C4×C8—141no—
129C2×C4×C8C2×D711yescharacteristic
130#384—71no—
131C4×C4⋊C4—71no—
132#340—141no—
133#336—141no—
134#331—74no—
135C23×C7#12111yescharacteristic
136C22×D7—42no—
137C22×D7—81no—
138C22×D7—81no—
139C22×D7—81no—
140C22×D7—81no—
141C22×D7—81no—
142C22×D7—81no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7#11711yescharacteristic
146C2×C4×C7C4×D411yescharacteristic
147C2×C4×C7C4×D411yescharacteristic
148C2×C4×C7#12311yescharacteristic
149C2×C4×C7#12411yescharacteristic
150C2×C4×C7#12611yescharacteristic
151C2×C4×C7C2×D811yescharacteristic
152C2×C4×C7C2×D811yescharacteristic
153C2×C4×C7C2×SD1611yescharacteristic
154C2×C4×C7#13511yescharacteristic
155C2×C4×C7#13511yescharacteristic
156C2×C4×C7#13511yescharacteristic
157C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C7#13711yescharacteristic
159C2×Dic7—22no—
160C2×Dic7—22no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—81no—
166C2×Dic7—81no—
167C2×Dic7—81no—
168D28—42no—
169D28—42no—
170D28—42no—
171D28—42no—
172D28—82no—
173D28—82no—
174D28—81no—
175D28—81no—
176D28—81no—
177D28—81no—
178C7×C8—22no—
179C7×C8—42no—
180C7×C8—41no—
181C7×C8—81no—
182C7⋊C8—42no—
183C22×D4—141no—
184C22×D4—141no—
185C22×D4—141no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—141no—
189C22×C8D2811yescharacteristic
190C22×C8D2811yescharacteristic
191#127—72no—
192#127—141no—
193#127—141no—
194#127—141no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—141no—
199C2×C4⋊C4—141no—
200C2×C42—141no—
201C2×C42C22×D711yescharacteristic
202#107—282no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—72no—
207C4⋊C8—141no—
208C4⋊C8C4×D712yes—
209#102—282no—
210#102—282no—
211#102—282no—
212#102—282no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—282no—
216#95—141no—
217C22×C7#38811yescharacteristic
218C22×C7#45911yescharacteristic
219C22×C7#46411yescharacteristic
220C22×C7#46511yescharacteristic
221C22×C7#46611yescharacteristic
222C22×C7#47111yescharacteristic
223C22×C7#47211yescharacteristic
224C2×D7—82no—
225C2×D7—82no—
226C2×D7—82no—
227C2×D7—161no—
228C2×D7—161no—
229C2×D7—161no—
230C2×D7—161no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7C4×D811yescharacteristic
234C4×C7C4×D811yescharacteristic
235C4×C7C4×SD1611yescharacteristic
236C4×C7#44011yescharacteristic
237C4×C7#48011yescharacteristic
238C4×C7#48211yescharacteristic
239C4×C7#48611yescharacteristic
240C4×C7#48811yescharacteristic
241Dic7—42no—
242Dic7—42no—
243Dic7—161no—
244C24—281no—
245C2×D4—142no—
246C2×D4—142no—
247C2×D4—142no—
248C2×D4—281no—
249C2×D4—281no—
250C2×D4—281no—
251C2×D4—281no—
252C2×D4—281no—
253C2×D4—281no—
254C2×D4—281no—
255C2×D4—281no—
256C2×D4—281no—
257C2×D4—281no—
258C2×D4—281no—
259C2×D4—561no—
260C2×D4—561no—
261C22×C4—141no—
262C22×C4—141no—
263C22×C4—281no—
264C22×C4C2×D2811yescharacteristic
265C22×C4D4×D711yescharacteristic
266C22×C4#116111yescharacteristic
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—22no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—142no—
279C2×C8—281no—
280C2×C8—281no—
281C2×C8D5612yes—
282C4⋊C4—72no—
283C4⋊C4—72no—
284C4⋊C4—141no—
285C4⋊C4—141no—
286C4⋊C4—281no—
287C4⋊C4—281no—
288C4⋊C4—281no—
289C4⋊C4—281no—
290C4⋊C4—561no—
291C42—281no—
292C42—281no—
293C42C2×C4×D711yescharacteristic
294C42#115711yescharacteristic
295C42#115911yescharacteristic
296C42#115911yescharacteristic
297C2×C7#166511yescharacteristic
298C2×C7#166711yescharacteristic
299C2×C7#167811yescharacteristic
300C2×C7#168211yescharacteristic
301C2×C7#192511yescharacteristic
302C2×C7#193111yescharacteristic
303C2×C7#193411yescharacteristic
304D7—162no—
305C23—282no—
306C23—561no—
307C23—561no—
308C23—561no—
309C23—561no—
310C23—561no—
311C23—561no—
312C23#645811yescharacteristic
313D4—282no—
314D4—282no—
315D4—282no—
316D4—282no—
317D4—562no—
318D4—562no—
319D4—561no—
320D4—561no—
321D4—561no—
322D4—561no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—142no—
326C2×C4—142no—
327C2×C4—281no—
328C2×C4—281no—
329C2×C4—281no—
330C2×C4—281no—
331C2×C4—561no—
332C2×C4—561no—
333C2×C4—561no—
334C2×C4C4×D2811yescharacteristic
335C2×C4(C22⋊D7)⋊C411yescharacteristic
336C2×C4#644811yescharacteristic
337C2×C4#644911yescharacteristic
338C2×C4#645511yescharacteristic
339C2×C4#646111yescharacteristic
340C2×C4C2×D5611yescharacteristic
341C2×C4#646711yescharacteristic
342C2×C4#647111yescharacteristic
343C2×C4#647211yescharacteristic
344C2×C4D7×D811yescharacteristic
345C2×C4D7×SD1611yescharacteristic
346C2×C4#647911yescharacteristic
347C2×C4#648211yescharacteristic
348C8—22no—
349C8—41no—
350C8—42no—
351C8—81no—
352C8—282no—
353C7#956311yescharacteristic, complemented, Sylow
354C22—562no—
355C22—562no—
356C22—562no—
357C22—1121no—
358C22—1121no—
359C22—1121no—
360C22—1121no—
361C22#8944211yescharacteristic
362C22#8961511yescharacteristic
363C22#8961711yescharacteristic
364C22#8963611yescharacteristic
365C22#8965511yescharacteristic
366C22#8966911yescharacteristic
367C22#8967011yescharacteristic
368C4—21no—
369C4—21no—
370C4—282no—
371C4—282no—
372C4—1121no—
373C4C4×D5611yescharacteristic
374C4#8948211yescharacteristic
375C4#8950811yescharacteristic
376C4#8950911yescharacteristic
377C4#8952811yescharacteristic
378C4#8953211yescharacteristic
379C4#8956011yescharacteristic
380C4#8957211yescharacteristic
381C2—1122no—
382C2#1172629011yescharacteristic
383C2#1172630211yescharacteristic
384C2#1172633011yescharacteristic
385C2#1172634411yescharacteristic
386C2#1172637911yescharacteristic
387C2#1172660711yescharacteristic
388C2#1172668511yescharacteristic
subgroupG/H
centerC23#6458
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1277C2
socleC23×C7#121
prime radicalC2×#6437C22
cosocleC2×C42×C7C23
hypercenter#1277D7
hypocenterC7#9563
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #11728207C2semidirect
5. C8×C4⋊Dic7C2semidirect
6. C7×#1277C2semidirect
7. #11725508C2semidirect
23. #89280C4semidirect
65. #1277D7semidirect
353. C7#9563semidirect

Maximal subgroups

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

subgroupindex
#117282112
#117282072
C2×#892802
#117275262
C8×C4⋊Dic72
C7×#12772
#117255082
#95637