Finite Group Databasebeta

← #13273398

Subgroups — #13273398

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×C7×C4⋊C16C211yesmaximal, characteristic, complemented
3C2×#89298C211yesmaximal, characteristic, complemented
4C2×#89304C211yesmaximal, characteristic
5#11727530C211yesmaximal, characteristic
6#11727529C211yesmaximal, characteristic, complemented
7#11725612C211yesmaximal, characteristic, complemented
8C22×Dic28C2211yescharacteristic
9C22×D56C2211yescharacteristic
10C22×C7×C16C2211yescharacteristic
11C22×C7×C16C2211yescharacteristic
12C2×C4×C7×C8C2211yescharacteristic
13C2×#6395—21no—
14C2×#6392C2211yescharacteristic
15C2×#6392C2211yescharacteristic
16C2×#6390—21no—
17#89711—21no—
18#89701—21no—
19C2×#6374—22no—
20C2×#6374—22no—
21C2×#6374—22no—
22C7×C4⋊C16—22no—
23#89304—22no—
24#89298—22no—
25#13449—71nomaximal, Sylow
26C22×D28—21no—
27C22×Dic14—21no—
28C22×C7×C8D411yescharacteristic
29C22×C7×C8C2311yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×(C2×D7)⋊C4—41no—
32C2×(C2×D7)⋊C4—41no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
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×C16—21no—
50C2×C7×C16—21no—
51C2×C7×C16—21no—
52C2×C7×C16D412yes—
53C4×C7×C8—21no—
54C4×C7×C8—21no—
55C7⋊#95—41no—
56C7⋊#95—41no—
57#6395—42no—
58#6392—21no—
59#6392—21no—
60#6392—21no—
61#6392—21no—
62#6390—42no—
63#6374—28no—
64C22×SD32—71no—
65C2×C4⋊C16D711yescharacteristic, complemented
66C2×#358—71no—
67C2×#357—71no—
68#1970—71no—
69#1969—71no—
70#1380—71no—
71C23×D7—41no—
72C22×C4×C7D811yescharacteristic
73C22×C4×C7D811yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—41no—
76C22×Dic7—41no—
77C22×Dic7—41no—
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×C8—21no—
91C2×C7×C8—21no—
92C2×C7×C8—21no—
93C2×C7×C8C2×D411yescharacteristic
94C2×C7×C8C2×D411yescharacteristic
95C2×C7×C8C2×D411yescharacteristic
96C2×C7×C8#4111yescharacteristic
97C2×C7×C8#4111yescharacteristic
98C2×C7×C8#4111yescharacteristic
99C42×C7—21no—
100C42×C7—21no—
101(C2×D7)⋊C4—82no—
102(C2×D7)⋊C4—82no—
103C4⋊Dic7—41no—
104C4⋊Dic7—41no—
105C4⋊Dic7—41no—
106C4⋊Dic7—41no—
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×C16C2×D711yescharacteristic
123C22×C16C2×D711yescharacteristic
124C2×#107—71no—
125C2×#107—71no—
126C2×#103—141no—
127C2×#102—141no—
128C2×C4×C8C2×D711yescharacteristic
129#391—141no—
130#390—141no—
131C4⋊C16—22no—
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—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7SD3212yescomplemented
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—42no—
155C2×Dic7—81no—
156C2×Dic7—81no—
157C2×Dic7—81no—
158C2×Dic7—81no—
159C2×Dic7—81no—
160C2×Dic7—81no—
161C2×Dic7—81no—
162C2×Dic7—81no—
163D28—42no—
164D28—42no—
165D28—81no—
166D28—81no—
167Dic14—42no—
168Dic14—42no—
169Dic14—81no—
170Dic14—81no—
171C7×C8—41no—
172C7×C8#12011yescharacteristic
173C7×C8#12311yescharacteristic
174C7×C8#12311yescharacteristic
175C7×C8#12411yescharacteristic
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×C8D2811yescharacteristic
191C22×C8C22×D711yescharacteristic
192C2×C4⋊C4—141no—
193C2×C4⋊C4—141no—
194C2×C22⋊C4—281no—
195C2×C22⋊C4—281no—
196C2×C42D2811yescharacteristic
197SD32—148no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16C22⋊D712yes—
204#107—141no—
205#107—141no—
206#107—141no—
207#107—141no—
208#103—282no—
209#102—282no—
210C4×C8—21no—
211C4×C8—21no—
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—22no—
229C4×C7—41no—
230C4×C7#39411yescharacteristic
231C4×C7#46311yescharacteristic
232C4×C7#46311yescharacteristic
233C4×C7#46911yescharacteristic
234Dic7—82no—
235Dic7—161no—
236Dic7—161no—
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—281no—
251C22×C4—281no—
252C22×C4—281no—
253C22×C4D5611yescharacteristic
254C22×C4D5611yescharacteristic
255C22×C4C2×D2811yescharacteristic
256Q16—142no—
257Q16—142no—
258Q16—281no—
259Q16—281no—
260D8—142no—
261D8—142no—
262D8—281no—
263D8—281no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8#115711yescharacteristic
268C2×C8D4×D711yescharacteristic
269C2×C8D4×D711yescharacteristic
270C2×C8#116111yescharacteristic
271C2×C8#116111yescharacteristic
272C2×C8C2×C22⋊D711yescharacteristic
273C4⋊C4—281no—
274C4⋊C4—281no—
275C4⋊C4—281no—
276C4⋊C4—281no—
277C22⋊C4—562no—
278C22⋊C4—562no—
279C42—21no—
280C42—21no—
281C16—22no—
282C16—41no—
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—
298C23C2×D5611yescharacteristic
299Q8—282no—
300Q8—282no—
301Q8—561no—
302Q8—561no—
303D4—282no—
304D4—282no—
305D4—561no—
306D4—561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—282no—
313C2×C4—561no—
314C2×C4—561no—
315C2×C4—561no—
316C2×C4—561no—
317C2×C4—561no—
318C2×C4—561no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4#637412yescomplemented
322C2×C4#645811yescharacteristic
323C2×C4#645811yescharacteristic
324C2×C4#646711yescharacteristic
325C2×C4#647111yescharacteristic
326C2×C4#647111yescharacteristic
327C2×C4#649311yescharacteristic
328C8—41no—
329C8#645711yescharacteristic
330C8#645711yescharacteristic
331C8#650011yescharacteristic
332C8#651011yescharacteristic
333C7#1344911yescharacteristic, complemented, Sylow
334C22—562no—
335C22—562no—
336C22—562no—
337C22—1121no—
338C22—1121no—
339C22—1121no—
340C22—1121no—
341C22#8961511yescharacteristic
342C22#8961511yescharacteristic
343C22C2×#637411yescharacteristic
344C22#8967611yescharacteristic
345C22#8968011yescharacteristic
346C22#8968111yescharacteristic
347C22#8988711yescharacteristic
348C4—22no—
349C4—41no—
350C4—562no—
351C4—1121no—
352C4—1121no—
353C4#8961611yescharacteristic
354C4#8961611yescharacteristic
355C4#8976311yescharacteristic
356C4#8990711yescharacteristic
357C2—1122no—
358C2#1172704111yescharacteristic
359C2#1172704211yescharacteristic
360C2#1172704611yescharacteristic
361C2#1172704811yescharacteristic
362C2#1172823511yescharacteristic
363C2#1172916811yescharacteristic
364C2#1172919811yescharacteristic
subgroupG/H
centerC23C2×D56
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC2×C7×C4⋊C16C2
socleC23×C7C2×D8
prime radicalC22×D56C22
hypercenterC2×C4⋊C16D7
hypocenterC7#13449
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×C7×C4⋊C16C2semidirect
3. C2×#89298C2semidirect
6. #11727529C2semidirect
7. #11725612C2semidirect
65. C2×C4⋊C16D7semidirect
147. C2×C4×C7SD32semidirect
321. C2×C4#6374semidirect
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×C7×C4⋊C162
C2×#892982
C2×#893042
#117275302
#117275292
#117256122
#134497