Finite Group Databasebeta

← #13272967

Subgroups — #13272967

Order 1792.

446 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89659C211yesmaximal, characteristic
2C2×C7×#367C211yesmaximal, characteristic, complemented
3C2×#89298C211yesmaximal, characteristic, complemented
4C2×(C2×D7)⋊C16C211yesmaximal, characteristic
5#11728384C211yesmaximal, characteristic
6C2×#89287C211yesmaximal, characteristic, complemented
7#11725528C211yesmaximal, characteristic, complemented
8C22×Dic28C2211yescharacteristic
9C22×C8×D7C2211yescharacteristic
10C2×#6460—21no—
11C22×C7×C16C2211yescharacteristic
12C2×C7×#107C2211yescharacteristic
13C2×C7×#107C2211yescharacteristic
14C2×#6392C2211yescharacteristic
15C22×C7⋊C16C2211yescharacteristic
16#89759—21no—
17C2×#6384—21no—
18#89659—22no—
19#89659—22no—
20C7×#367—22no—
21#89324—21no—
22(C2×D7)⋊C16—22no—
23#89298—22no—
24#89287—22no—
25#13382—71nomaximal, Sylow
26C22×C4×D7D411yescharacteristic
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⋊Dic7—21no—
34C2×C4⋊Dic7—41no—
35C22×C7⋊C8D411yescharacteristic
36C2×Dic28—21no—
37C2×Dic28—21no—
38C2×Dic28—21no—
39C2×Dic28—21no—
40C2×Dic28—21no—
41C2×Dic28D412yes—
42C2×C8×D7—21no—
43C2×C8×D7—21no—
44C2×C8×D7—21no—
45C2×C8×D7—21no—
46C2×C8×D7—21no—
47C2×C8×D7—21no—
48#6460—24no—
49#6460—42no—
50C2×C7×C16—21no—
51C2×C7×C16—21no—
52C2×C7×C16D412yes—
53C7×#107—21no—
54C7×#107—21no—
55C7×#107—21no—
56C7×#107D412yes—
57C7⋊#95—41no—
58#6392—21no—
59#6392—21no—
60C2×C7⋊C16—21no—
61C2×C7⋊C16—21no—
62C2×C7⋊C16—21no—
63#6384—24no—
64C2×#467—71no—
65C2×#367D711yescharacteristic, complemented
66C2×#358—71no—
67C2×#358—71no—
68C2×C22⋊C16—71no—
69#1808—71no—
70#1380—71no—
71C23×D7D811yescharacteristic
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—41no—
76C22×Dic7—41no—
77C22×Dic7D811yescharacteristic
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×C4×D7—21no—
83C2×C4×D7—21no—
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—22no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100(C2×D7)⋊C4—44no—
101(C2×D7)⋊C4—82no—
102C4⋊Dic7—44no—
103C4⋊Dic7—41no—
104C4⋊Dic7—41no—
105C2×C7⋊C8—21no—
106C2×C7⋊C8—21no—
107C2×C7⋊C8—21no—
108Dic28—22no—
109Dic28—22no—
110Dic28—41no—
111Dic28—41no—
112C8×D7—41no—
113C8×D7—41no—
114C8×D7—41no—
115C8×D7—41no—
116C7×C16—22no—
117C7⋊C16—41no—
118C22×Q16—71no—
119C23×C8—71no—
120C2×C22⋊Q8—141no—
121C22×C16—71no—
122C22×C16C2×D711yescharacteristic
123#467—142no—
124#467—142no—
125C2×#107—71no—
126C2×#107C2×D711yescharacteristic
127C2×#107C2×D711yescharacteristic
128C2×#103—141no—
129#380—141no—
130#367—22no—
131#358—142no—
132#358—142no—
133C22⋊C16—142no—
134#340—141no—
135C23×C7C2×D811yescharacteristic
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7SD3212yes—
143C2×C4×C7—42no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7—81no—
147C2×C4×C7—81no—
148C2×C4×C7—81no—
149C2×C4×C7#12011yescharacteristic
150C2×C4×C7#12111yescharacteristic
151C2×C4×C7#12311yescharacteristic
152C2×C4×C7#13511yescharacteristic
153C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C7#13711yescharacteristic
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—42no—
159C2×Dic7—81no—
160C2×Dic7—81no—
161C2×Dic7—81no—
162C2×Dic7—81no—
163C2×Dic7—81no—
164C4×D7—41no—
165C4×D7—41no—
166C4×D7—41no—
167C4×D7—41no—
168Dic14—42no—
169Dic14—42no—
170Dic14—81no—
171Dic14—81no—
172C7×C8#12111yescharacteristic
173C7×C8#12111yescharacteristic
174C7×C8#12311yescharacteristic
175C7×C8#12411yescharacteristic
176C7⋊C8—41no—
177C22×Q8—141no—
178C23×C4—71no—
179C2×Q16—72no—
180C2×Q16—141no—
181C2×Q16—141no—
182C2×Q16—141no—
183C2×Q16—141no—
184C2×Q16—141no—
185C22×C8—71no—
186C22×C8—141no—
187C22×C8—141no—
188C22×C8—141no—
189C22×C8—141no—
190C22×C8—141no—
191C22×C8—141no—
192C22×C8C22×D711yescharacteristic
193C22⋊Q8—144no—
194C22⋊Q8—282no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—281no—
199C2×C22⋊C4—281no—
200C2×C22⋊C4—281no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—141no—
204C2×C16—141no—
205C2×C16—141no—
206C2×C16D2812yes—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—141no—
211#107—141no—
212#107C22⋊D712yes—
213#103—144no—
214#95—281no—
215C22×C7#44711yescharacteristic
216C22×C7#46611yescharacteristic
217C22×C7#48011yescharacteristic
218C22×C7C2×SD3211yescharacteristic
219C22×C7#50811yescharacteristic
220C22×C7#51011yescharacteristic
221C22×C7#51011yescharacteristic
222C2×D7—22no—
223C2×D7—22no—
224C2×D7—22no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C4×C7—82no—
230C4×C7—161no—
231C4×C7#39411yescharacteristic
232C4×C7#45211yescharacteristic
233C4×C7#46811yescharacteristic
234C4×C7#48211yescharacteristic
235Dic7—41no—
236Dic7—82no—
237Dic7—161no—
238C24—71no—
239C2×Q8—142no—
240C2×Q8—281no—
241C2×Q8—281no—
242C2×Q8—281no—
243C2×Q8—281no—
244C2×Q8—281no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—281no—
255C22×C4—281no—
256C22×C4D4×D711yescharacteristic
257Q16—142no—
258Q16—142no—
259Q16—281no—
260Q16—281no—
261C2×C8—141no—
262C2×C8—141no—
263C2×C8—141no—
264C2×C8—281no—
265C2×C8—281no—
266C2×C8—281no—
267C2×C8—281no—
268C2×C8C2×D2811yescharacteristic
269C2×C8#115711yescharacteristic
270C2×C8D4×D711yescharacteristic
271C2×C8#116111yescharacteristic
272C2×C8#116111yescharacteristic
273C2×C8C2×C22⋊D711yescharacteristic
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—284no—
279C4⋊C4—281no—
280C4⋊C4—281no—
281C22⋊C4—284no—
282C22⋊C4—562no—
283C16—22no—
284C16—281no—
285C2×C7#203311yescharacteristic
286C2×C7#218511yescharacteristic
287C2×C7#218911yescharacteristic
288C2×C7#221711yescharacteristic
289C2×C7#221911yescharacteristic
290C2×C7#223511yescharacteristic
291C2×C7#223711yescharacteristic
292D7—42no—
293C23—72no—
294C23—141no—
295C23—141no—
296C23—141no—
297C23—141no—
298C23—141no—
299C23—141no—
300C23D7×D811yescharacteristic
301Q8—282no—
302Q8—282no—
303Q8—561no—
304Q8—561no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—281no—
315C2×C4—281no—
316C2×C4—281no—
317C2×C4—281no—
318C2×C4—282no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4—561no—
322C2×C4—561no—
323C2×C4—561no—
324C2×C4#645711yescharacteristic
325C2×C4#645811yescharacteristic
326C2×C4#647411yescharacteristic
327C2×C4#648111yescharacteristic
328C2×C4#648211yescharacteristic
329C2×C4#650011yescharacteristic
330C8—281no—
331C8#645711yescharacteristic
332C8#645811yescharacteristic
333C8#649311yescharacteristic
334C8#651011yescharacteristic
335C7#1338211yescharacteristic, complemented, Sylow
336C22—142no—
337C22—142no—
338C22—142no—
339C22—281no—
340C22—281no—
341C22—281no—
342C22—281no—
343C22#8965411yescharacteristic
344C22#8965511yescharacteristic
345C22D7×SD3211yescharacteristic
346C22#8968711yescharacteristic
347C22#8968711yescharacteristic
348C22#8968811yescharacteristic
349C22#8992811yescharacteristic
350C4—82no—
351C4—161no—
352C4—281no—
353C4—562no—
354C4—1121no—
355C4#8965711yescharacteristic
356C4#8965811yescharacteristic
357C4#8976311yescharacteristic
358C4#8995911yescharacteristic
359C2—282no—
360C2#1172709611yescharacteristic
361C2#1172710111yescharacteristic
362C2#1172710311yescharacteristic
363C2#1172710611yescharacteristic
364C2#1172838811yescharacteristic
365C2#1172923711yescharacteristic
366C2#1172924011yescharacteristic
subgroupG/H
centerC23D7×D8
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC2×C7×#367C2
socleC23×C7C2×D8
prime radicalC23×D7D8
hypercenterC2×#367D7
hypocenterC7#13382
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#367

Complements

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

NH ≅ G/Ntype
2. C2×C7×#367C2semidirect
3. C2×#89298C2semidirect
6. C2×#89287C2semidirect
7. #11725528C2semidirect
65. C2×#367D7semidirect
335. C7#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#896592
C2×C7×#3672
C2×#892982
C2×(C2×D7)⋊C162
#117283842
C2×#892872
#117255282
#133827