Finite Group Databasebeta

← #11731861

Subgroups — #11731861

Order 896.

608 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6487C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic
3C2×#6451C211yesmaximal, characteristic, complemented
4C2×#6441C211yesmaximal, characteristic
5C7×#385C211yesmaximal, characteristic
6#89756C211yesmaximal, characteristic, complemented
7#89751C211yesmaximal, characteristic, complemented
8#89748C211yesmaximal, characteristic, complemented
9#89722C211yesmaximal, characteristic
10#89714C211yesmaximal, characteristic
11C22⋊C4×Dic7C211yesmaximal, characteristic
12#89429C211yesmaximal, characteristic, complemented
13#89428C211yesmaximal, characteristic, complemented
14#89419C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31#6487—24no—
32C2×Dic7⋊C4—24no—
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45#6451—24no—
46#6441—24no—
47C7×#95C2211yescharacteristic
48C7×#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2440—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—41no—
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×Dic7—22no—
67C22×Dic7—22no—
68C22×Dic7—22no—
69C22×Dic7—22no—
70C22×Dic7—41no—
71C22×Dic7Q814yescomplemented
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C2×Dic14—22no—
81C2×Dic14—22no—
82C2×Dic14—22no—
83C2×Dic14—22no—
84C2×Dic14—41no—
85C2×Dic14—41no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—41no—
88C7×C22⋊C4—22no—
89C7×C22⋊C4Q814yescomplemented
90C22⋊Dic7—22no—
91C22⋊Dic7D414yes—
92C4⋊Dic7—22no—
93C4⋊Dic7—22no—
94C4⋊Dic7—22no—
95C4⋊Dic7—41no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4D414yes—
106C4×Dic7—22no—
107C4×Dic7—22no—
108C2×C4⋊Q8—71no—
109C2×C22⋊Q8—71no—
110C2×C22⋊Q8—71no—
111C22×C4⋊C4—71no—
112#398—71no—
113#398—71no—
114#395—71no—
115#395—71no—
116#389—71no—
117#389—71no—
118#385—71no—
119#385D711yescharacteristic
120#384—71no—
121#380—71no—
122C4×C22⋊C4—71no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7C2×D412yes—
128C23×C7C2×Q812yes—
129C23×C7C2×Q812yes—
130C23×C7C2411yescharacteristic
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
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—44no—
147C2×C4×C7C2×D412yes—
148C2×C4×C7C2×Q812yes—
149C2×C4×C7C2×Q812yes—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—28no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—44no—
169C2×Dic7C2×D412yes—
170C2×Dic7C2×D412yes—
171C2×Dic7C2×D412yes—
172C2×Dic7C2×D412yes—
173C2×Dic7C2×Q812yes—
174C2×Dic7C2×Q812yes—
175C2×Dic7#4112yes—
176Dic14—44no—
177Dic14—44no—
178C22×Q8—71no—
179C22×Q8—71no—
180C23×C4—71no—
181C23×C4—71no—
182C23×C4C2×D711yescharacteristic
183C4⋊Q8—144no—
184C22⋊Q8—144no—
185C22⋊Q8—144no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C4⋊C4—71no—
200C2×C4⋊C4—144no—
201C2×C4⋊C4C2×D711yescharacteristic
202C2×C4⋊C4C2×D711yescharacteristic
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4C2×D711yescharacteristic
206C2×C22⋊C4C2×D711yescharacteristic
207C2×C42—71no—
208C2×C42—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95C2×D711yescharacteristic
216#95C2×D711yescharacteristic
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7C4⋊Q814yescomplemented
224C22×C7C22×D411yescharacteristic
225C22×C7C22×D411yescharacteristic
226C22×C7C22×Q811yescharacteristic
227C22×C7C22×Q811yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C7H(5,𝔽2)11yescharacteristic
230C22×C72−1+411yescharacteristic
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—22no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237Dic7—22no—
238Dic7—22no—
239Dic7—22no—
240Dic7—22no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7C22⋊Q814yescomplemented
245C24C22×D711yescharacteristic
246C2×Q8—142no—
247C2×Q8—142no—
248C2×Q8—142no—
249C2×Q8—142no—
250C2×Q8—281no—
251C2×Q8—281no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—71no—
256C22×C4—74no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—281no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—74no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—142no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C4⋊C4—142no—
289C4⋊C4—142no—
290C4⋊C4—281no—
291C22⋊C4—22no—
292C22⋊C4—74no—
293C22⋊C4—142no—
294C22⋊C4Dic1414yescomplemented
295C42—142no—
296C42—142no—
297C2×C7—24no—
298C2×C7C2×C22⋊Q811yescharacteristic
299C2×C7C2×C4⋊Q811yescharacteristic
300C2×C7#54411yescharacteristic
301C2×C7D4211yescharacteristic
302C2×C7#54711yescharacteristic
303C2×C7D4×Q811yescharacteristic
304C2×C7#55411yescharacteristic
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×Dic1412yes—
310C23D4×D712yes—
311C23Q8×D712yes—
312C23C23×D711yescharacteristic
313Q8—284no—
314Q8—284no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—72no—
332C2×C4—72no—
333C2×C4—72no—
334C2×C4—72no—
335C2×C4—72no—
336C2×C4—72no—
337C2×C4—72no—
338C2×C4—148no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—284no—
357C2×C4C2×Dic1412yes—
358C2×C4C2×Dic1412yes—
359C2×C4D4×D712yes—
360C7#244011yescharacteristic, complemented, Sylow
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22#645114yescomplemented
368C22C22×Dic1411yescharacteristic
369C22C2×D4×D711yescharacteristic
370C22C2×D4×D711yescharacteristic
371C22#654811yescharacteristic
372C22C2×Q8×D711yescharacteristic
373C22D7×#4111yescharacteristic
374C22#655411yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—74no—
382C4—142no—
383C4—142no—
384C4—142no—
385C4—142no—
386C4—281no—
387C4—281no—
388C4—281no—
389C2—24no—
390C2C2×#645111yescharacteristic
391C2D4×Dic1411yescharacteristic
392C2#9023011yescharacteristic
393C2#9028111yescharacteristic
394C2#9031411yescharacteristic
395C2D7×C22⋊Q811yescharacteristic
396C2#9034611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×C7C23
hypercenter#385D7
hypocenterC7#2440
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C2×#6451C2semidirect
6. #89756C2semidirect
7. #89751C2semidirect
8. #89748C2semidirect
12. #89429C2semidirect
13. #89428C2semidirect
14. #89419C2semidirect
15. #89419C2semidirect
71. C22×Dic7Q8semidirect
89. C7×C22⋊C4Q8semidirect
223. C22×C7C4⋊Q8semidirect
244. Dic7C22⋊Q8semidirect
294. C22⋊C4Dic14semidirect
360. C7#2440semidirect
367. C22#6451semidirect

Maximal subgroups

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

subgroupindex
C2×#64872
C22×Dic7⋊C42
C2×#64512
C2×#64412
C7×#3852
#897562
#897512
#897482
#897222
#897142
C22⋊C4×Dic72
#894292
#894282
#894192
#24407