Finite Group Databasebeta

← #11734228

Subgroups — #11734228

Order 896.

620 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6509C211yesmaximal, characteristic, complemented
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
4C2×#6441C211yesmaximal, characteristic, complemented
5C7×#393C211yesmaximal, characteristic, complemented
6#89729C211yesmaximal, characteristic
7#89729C211yesmaximal, characteristic
8#89720C211yesmaximal, characteristic, complemented
9#89718C211yesmaximal, characteristic, complemented
10#89448C211yesmaximal, characteristic, complemented
11#89445C211yesmaximal, characteristic, complemented
12#89440C211yesmaximal, characteristic, complemented
13#89421C211yesmaximal, characteristic, complemented
14#89419C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26#6509—24no—
27#6507—24no—
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
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×Dic7⋊C4C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46C2×C4×Dic7C2211yescharacteristic
47(C22⋊D7)⋊C4—24no—
48#6441—24no—
49C7×#95C2211yescharacteristic
50C7×#95C2211yescharacteristic
51C7×#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2452—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—41no—
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7—22no—
68C2×C22⋊D7—22no—
69C2×C22⋊D7Q814yescomplemented
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—41no—
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C2×C4×D7—22no—
81C2×C4×D7—22no—
82C2×C4×D7—41no—
83C2×Dic14—22no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4—22no—
88C7×C22⋊C4—41no—
89C22⋊Dic7—22no—
90C22⋊Dic7—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—41no—
96C4⋊Dic7—22no—
97C4⋊Dic7—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4D414yes—
106Dic7⋊C4D414yescomplemented
107C4×Dic7—22no—
108C4×Dic7—22no—
109C4×Dic7—22no—
110C2×C4⋊Q8—71no—
111C2×C22⋊Q8—71no—
112C2×C22⋊Q8—71no—
113C2×C4×D4—71no—
114#396—71no—
115#396—71no—
116#393—71no—
117#393D711yescharacteristic, complemented
118#390—71no—
119#390—71no—
120#389—71no—
121#389—71no—
122#385—71no—
123#385—71no—
124C4×C4⋊C4—71no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7C2×Q812yes—
132C23×C7C2411yescharacteristic
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7C2×Q812yes—
140C7×Q8—44no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7C2×D412yes—
158C2×C4×C7C2×D412yes—
159C22⋊D7—28no—
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—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—44no—
175C2×Dic7C2×D412yes—
176C2×Dic7C2×D412yes—
177C2×Dic7C2×D412yes—
178C2×Dic7C2×D412yes—
179C2×Dic7C2×Q812yes—
180C2×Dic7#4112yes—
181C2×Dic7#4112yes—
182C4×D7—44no—
183Dic14—44no—
184C22×Q8—71no—
185C22×Q8C2×D711yescharacteristic
186C22×D4—71no—
187C23×C4—71no—
188C23×C4—71no—
189C4⋊Q8—144no—
190C22⋊Q8—144no—
191C22⋊Q8—144no—
192C4×D4—144no—
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—71no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4—71no—
207C2×C22⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
211C2×C22⋊C4C2×D711yescharacteristic
212C2×C22⋊C4C2×D711yescharacteristic
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C42—71no—
215C2×C42—71no—
216C2×C42—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95C2×D711yescharacteristic
221#95C2×D711yescharacteristic
222#95C2×D711yescharacteristic
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—22no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7C22×D411yescharacteristic
231C22×C7C22×D411yescharacteristic
232C22×C7C22×Q811yescharacteristic
233C22×C7C2×#4111yescharacteristic
234C22×C7C2×#4111yescharacteristic
235C22×C7H(5,𝔽2)11yescharacteristic
236C22×C72−1+411yescharacteristic
237C2×D7—22no—
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C4×C7—22no—
245C4×C7—22no—
246C4×C7—41no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—41no—
250Dic7—22no—
251Dic7—22no—
252Dic7—22no—
253Dic7—22no—
254Dic7—41no—
255Dic7—41no—
256Dic7C22⋊Q814yescomplemented
257C24—71no—
258C24C22×D711yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—142no—
263C2×Q8—142no—
264C2×Q8—281no—
265C2×D4—74no—
266C2×D4—142no—
267C2×D4—142no—
268C22×C4—71no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—71no—
275C22×C4—142no—
276C22×C4—142no—
277C22×C4—142no—
278C22×C4—142no—
279C22×C4—281no—
280C22×C4—281no—
281C22×C4C22×D711yescharacteristic
282C22×C4C22×D711yescharacteristic
283C22×C4C22×D711yescharacteristic
284C22×C4C22×D711yescharacteristic
285C22×C4C22×D711yescharacteristic
286C22×C4C22×D711yescharacteristic
287C4⋊C4—74no—
288C4⋊C4—74no—
289C4⋊C4—142no—
290C4⋊C4—142no—
291C4⋊C4—142no—
292C4⋊C4—142no—
293C4⋊C4—142no—
294C4⋊C4—142no—
295C4⋊C4—142no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—41no—
301C22⋊C4—142no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—142no—
307C22⋊C4—281no—
308C42—142no—
309C42—142no—
310C42—142no—
311C2×C7—42no—
312C2×C7C2×C22⋊Q811yescharacteristic
313C2×C7#53211yescharacteristic
314C2×C7#54211yescharacteristic
315C2×C7D4211yescharacteristic
316C2×C7#54811yescharacteristic
317C2×C7D4×Q811yescharacteristic
318C2×C7#55411yescharacteristic
319D7—42no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—72no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23—141no—
333C23Q8×D712yes—
334C23C23×D711yescharacteristic
335Q8—44no—
336Q8—284no—
337D4—148no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—72no—
355C2×C4—72no—
356C2×C4—72no—
357C2×C4—72no—
358C2×C4—72no—
359C2×C4—72no—
360C2×C4—72no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—141no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—284no—
376C2×C4—284no—
377C2×C4D4×D712yes—
378C2×C4D4×D712yes—
379C7#245211yescharacteristic, complemented, Sylow
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—142no—
388C22—142no—
389C22—142no—
390C22—281no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22C2×D4×D711yescharacteristic
395C22C2×D4×D711yescharacteristic
396C22#654811yescharacteristic
397C22C2×Q8×D711yescharacteristic
398C22D7×#4111yescharacteristic
399C22D7×#4111yescharacteristic
400C22#655411yescharacteristic
401C4—22no—
402C4—22no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—74no—
408C4—142no—
409C4—142no—
410C4—142no—
411C4—142no—
412C4—281no—
413C4—281no—
414C2—42no—
415C2—282no—
416C2#9028111yescharacteristic
417C2D7×C22⋊Q811yescharacteristic
418C2#9032311yescharacteristic
419C2#9032711yescharacteristic
420C2#9035911yescharacteristic
421C2#9036011yescharacteristic
422C2#9036811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC22×C22⋊D7C22
hypercenter#393D7
hypocenterC7#2452
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#6509C2semidirect
2. C2×#6507C2semidirect
4. C2×#6441C2semidirect
5. C7×#393C2semidirect
8. #89720C2semidirect
9. #89718C2semidirect
10. #89448C2semidirect
11. #89445C2semidirect
12. #89440C2semidirect
13. #89421C2semidirect
14. #89419C2semidirect
15. #89419C2semidirect
69. C2×C22⋊D7Q8semidirect
106. Dic7⋊C4D4semidirect
117. #393D7semidirect
256. Dic7C22⋊Q8semidirect
379. C7#2452semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×#65072
C2×(C22⋊D7)⋊C42
C2×#64412
C7×#3932
#897292
#897202
#897182
#894482
#894452
#894402
#894212
#894192
#24527