Finite Group Databasebeta

← #13273399

Subgroups — #13273399

Order 1792.

446 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6374C211yesmaximal, characteristic
2C2×Dic7⋊C16C211yesmaximal, characteristic, complemented
3#11728382C211yesmaximal, characteristic
4#11728381C211yesmaximal, characteristic, complemented
5C2×#89287C211yesmaximal, characteristic, complemented
6C2×#89286C211yesmaximal, characteristic
7C7×#1380C211yesmaximal, characteristic, complemented
8C22×Dic28C2211yescharacteristic
9C22×D56C2211yescharacteristic
10C22×C7×C16C2211yescharacteristic
11C2×C7×#107C2211yescharacteristic
12C2×C7×#107C2211yescharacteristic
13C2×C8×Dic7C2211yescharacteristic
14C22×C7⋊C16C2211yescharacteristic
15#89760—21no—
16#89746—21no—
17C2×#6384—21no—
18C2×#6383—21no—
19C2×#6374—22no—
20C2×#6374—22no—
21C2×#6374—22no—
22Dic7⋊C16—22no—
23#89287—22no—
24#89286—22no—
25#13449—71nomaximal, Sylow
26C22×D28—21no—
27C22×Dic14—21no—
28C22×C7×C8C2311yescharacteristic
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×(C2×D7)⋊C4—41no—
32C2×(C2×D7)⋊C4—41no—
33C2×C4×Dic7D411yescharacteristic
34C22×C7⋊C8D411yescharacteristic
35C2×Dic28—21no—
36C2×Dic28—21no—
37C2×Dic28—21no—
38C2×Dic28—21no—
39C2×Dic28—21no—
40C2×Dic28D412yes—
41C2×D56—21no—
42C2×D56—21no—
43C2×D56—21no—
44C2×D56—21no—
45C2×D56—21no—
46C2×D56D412yes—
47C2×C7×C16—21no—
48C2×C7×C16—21no—
49C2×C7×C16D412yes—
50C7×#107—21no—
51C7×#107—21no—
52C7×#107—21no—
53C7×#107—21no—
54C7⋊#95—41no—
55C7⋊#95—41no—
56C8×Dic7—21no—
57C8×Dic7—21no—
58C2×C7⋊C16—21no—
59C2×C7⋊C16—21no—
60C2×C7⋊C16—21no—
61#6384—42no—
62#6383—42no—
63#6374—28no—
64C22×SD32—71no—
65C2×C4⋊C16—71no—
66C2×#358—71no—
67C2×#357—71no—
68#1970—71no—
69#1969—71no—
70#1380D711yescharacteristic, complemented
71C23×D7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—41no—
76C22×Dic7D811yescharacteristic
77C22×Dic7D811yescharacteristic
78C2×D28—22no—
79C2×D28—41no—
80C2×D28—41no—
81C2×D28—41no—
82C2×D28—41no—
83C2×D28—41no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C2×Dic14—41no—
88C2×Dic14—41no—
89C2×Dic14—41no—
90C2×C7×C8C2×D411yescharacteristic
91C2×C7×C8C2×D411yescharacteristic
92C2×C7×C8C2×D411yescharacteristic
93C2×C7×C8#4111yescharacteristic
94C2×C7×C8#4111yescharacteristic
95C2×C7×C8#4111yescharacteristic
96C7×C4⋊C4—41no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100(C2×D7)⋊C4—82no—
101(C2×D7)⋊C4—82no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C2×C7⋊C8—21no—
105C2×C7⋊C8—21no—
106C2×C7⋊C8—21no—
107Dic28—22no—
108Dic28—22no—
109Dic28—41no—
110Dic28—41no—
111D56—22no—
112D56—22no—
113D56—41no—
114D56—41no—
115C7×C16—22no—
116C7⋊C16—41no—
117C22×Q16—71no—
118C22×D8—71no—
119C2×SD32—142no—
120C2×SD32—142no—
121C2×SD32—142no—
122C22×C16—71no—
123C22×C16C2×D711yescharacteristic
124C2×#107C2×D711yescharacteristic
125C2×#107C2×D711yescharacteristic
126C2×#103—141no—
127C2×#102—141no—
128C2×C4×C8—71no—
129#391—141no—
130#390—141no—
131C4⋊C16—142no—
132#358—142no—
133#357—142no—
134C23×C7C2×D811yescharacteristic
135C22×D7—42no—
136C22×D7—81no—
137C22×D7—81no—
138C22×D7—81no—
139C22×D7—81no—
140C22×D7—81no—
141C22×D7—81no—
142C2×C4×C7—81no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7—81no—
147C2×C4×C7—81no—
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12111yescharacteristic
150C2×C4×C7#12111yescharacteristic
151C2×C4×C7#13511yescharacteristic
152C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—42no—
160C2×Dic7—81no—
161C2×Dic7—81no—
162C2×Dic7SD3212yescomplemented
163D28—42no—
164D28—42no—
165D28—81no—
166D28—81no—
167Dic14—42no—
168Dic14—42no—
169Dic14—81no—
170Dic14—81no—
171C7×C8#12011yescharacteristic
172C7×C8#12311yescharacteristic
173C7×C8#12311yescharacteristic
174C7×C8#12411yescharacteristic
175C7⋊C8—41no—
176C22×Q8—141no—
177C22×D4—141no—
178C2×Q16—72no—
179C2×Q16—141no—
180C2×Q16—141no—
181C2×Q16—141no—
182C2×Q16—141no—
183C2×Q16—141no—
184C2×D8—72no—
185C2×D8—141no—
186C2×D8—141no—
187C2×D8—141no—
188C2×D8—141no—
189C2×D8—141no—
190C22×C8—71no—
191C22×C8C22×D711yescharacteristic
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C22⋊C4—281no—
195C2×C22⋊C4—281no—
196C2×C42—71no—
197SD32—148no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—141no—
201C2×C16—141no—
202C2×C16—141no—
203C2×C16D2812yes—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#103—282no—
209#102—282no—
210C4×C8—141no—
211C4×C8—141no—
212#95—281no—
213#95—281no—
214C22×C7#45911yescharacteristic
215C22×C7#45911yescharacteristic
216C22×C7#46611yescharacteristic
217C22×C7C2×SD3211yescharacteristic
218C22×C7#50811yescharacteristic
219C22×C7#50911yescharacteristic
220C22×C7#51011yescharacteristic
221C2×D7—82no—
222C2×D7—82no—
223C2×D7—82no—
224C2×D7—161no—
225C2×D7—161no—
226C2×D7—161no—
227C2×D7—161no—
228C4×C7—161no—
229C4×C7—161no—
230C4×C7#39411yescharacteristic
231C4×C7#46311yescharacteristic
232C4×C7#46311yescharacteristic
233C4×C7#46911yescharacteristic
234Dic7—22no—
235Dic7—41no—
236Dic7—82no—
237C24—281no—
238C2×Q8—142no—
239C2×Q8—281no—
240C2×Q8—281no—
241C2×Q8—281no—
242C2×Q8—281no—
243C2×Q8—281no—
244C2×D4—142no—
245C2×D4—281no—
246C2×D4—281no—
247C2×D4—281no—
248C2×D4—281no—
249C2×D4—281no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—281no—
255C22×C4D4×D711yescharacteristic
256Q16—142no—
257Q16—142no—
258Q16—281no—
259Q16—281no—
260D8—142no—
261D8—142no—
262D8—281no—
263D8—281no—
264C2×C8—141no—
265C2×C8—141no—
266C2×C8—141no—
267C2×C8C2×D2811yescharacteristic
268C2×C8#115711yescharacteristic
269C2×C8#115711yescharacteristic
270C2×C8D4×D711yescharacteristic
271C2×C8D4×D711yescharacteristic
272C2×C8#116111yescharacteristic
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C22⋊C4—562no—
278C22⋊C4—562no—
279C42—141no—
280C42—141no—
281C16—22no—
282C16—281no—
283C2×C7#205411yescharacteristic
284C2×C7#220711yescharacteristic
285C2×C7#220811yescharacteristic
286C2×C7#221011yescharacteristic
287C2×C7#221111yescharacteristic
288C2×C7#221711yescharacteristic
289C2×C7#222011yescharacteristic
290D7—162no—
291C23—282no—
292C23—561no—
293C23—561no—
294C23—561no—
295C23—561no—
296C23—561no—
297C23—561no—
298C23D7×D811yescharacteristic
299Q8—282no—
300Q8—282no—
301Q8—561no—
302Q8—561no—
303D4—282no—
304D4—282no—
305D4—561no—
306D4—561no—
307C2×C4—72no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—282no—
320C2×C4—561no—
321C2×C4—561no—
322C2×C4#644711yescharacteristic
323C2×C4#644711yescharacteristic
324C2×C4#645811yescharacteristic
325C2×C4#647711yescharacteristic
326C2×C4#647711yescharacteristic
327C2×C4#648211yescharacteristic
328C8—281no—
329C8#643811yescharacteristic
330C8#644511yescharacteristic
331C8#645711yescharacteristic
332C8#645711yescharacteristic
333C7#1344911yescharacteristic, complemented, Sylow
334C22—562no—
335C22—562no—
336C22—562no—
337C22—1121no—
338C22—1121no—
339C22—1121no—
340C22—1121no—
341C22#8955811yescharacteristic
342C22#8955811yescharacteristic
343C22#8965511yescharacteristic
344C22#8968311yescharacteristic
345C22D7×SD3211yescharacteristic
346C22#8968811yescharacteristic
347C22#8969011yescharacteristic
348C4—142no—
349C4—161no—
350C4—161no—
351C4—281no—
352C4—562no—
353C4#8944411yescharacteristic
354C4#8959111yescharacteristic
355C4#8959111yescharacteristic
356C4#8963711yescharacteristic
357C2—1122no—
358C2#1172661611yescharacteristic
359C2#1172695711yescharacteristic
360C2#1172695811yescharacteristic
361C2#1172698811yescharacteristic
362C2#1172699111yescharacteristic
363C2#1172708311yescharacteristic
364C2#1172710111yescharacteristic
subgroupG/H
centerC23D7×D8
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#1380C2
socleC23×C7C2×D8
prime radicalC22×D56C22
hypercenter#1380D7
hypocenterC7#13449
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×Dic7⋊C16C2semidirect
4. #11728381C2semidirect
5. C2×#89287C2semidirect
7. C7×#1380C2semidirect
70. #1380D7semidirect
162. C2×Dic7SD32semidirect
333. C7#13449semidirect

Maximal subgroups

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

subgroupindex
C22×#63742
C2×Dic7⋊C162
#117283822
#117283812
C2×#892872
C2×#892862
C7×#13802
#134497