Finite Group Databasebeta

← #11732907

Subgroups — #11732907

Order 896.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6448C211yesmaximal, characteristic
2C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
3C2×C4×Dic14C211yesmaximal, characteristic, complemented
4C7×#388C211yesmaximal, characteristic, complemented
5#89761C211yesmaximal, characteristic, complemented
6#89748C211yesmaximal, characteristic, complemented
7#89729C211yesmaximal, characteristic
8#89722C211yesmaximal, characteristic, complemented
9#89718C211yesmaximal, characteristic, complemented
10#89716C211yesmaximal, characteristic, complemented
11#89712C211yesmaximal, characteristic, complemented
12#89442C211yesmaximal, characteristic, complemented
13#89440C211yesmaximal, characteristic, complemented
14#89420C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43#6448C418yescomplemented
44(C22⋊D7)⋊C4—24no—
45C4×Dic14—24no—
46C7×#95C2211yescharacteristic
47C7×#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54#2326—71nomaximal, Sylow
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—22no—
64C2×C22⋊D7—22no—
65C2×C22⋊D7C2×C414yes—
66C22×Dic7—22no—
67C22×Dic7—41no—
68C22×Dic7—41no—
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C2×C4×D7—22no—
77C2×C4×D7—41no—
78C2×C4×D7—41no—
79C2×Dic14—22no—
80C2×Dic14—22no—
81C2×Dic14C2×C414yescomplemented
82C7×C4⋊C4—41no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4C2×C414yescomplemented
86C42×C7—22no—
87C22⋊Dic7—41no—
88C22⋊Dic7C2×C414yescomplemented
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—41no—
92(C2×D7)⋊C4C2×C414yescomplemented
93(C2×D7)⋊C4C2×C414yescomplemented
94C4⋊Dic7—22no—
95C4⋊Dic7—41no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—41no—
100Dic7⋊C4—41no—
101C4×Dic7—22no—
102C4×Dic7—22no—
103C4×Dic7C2×C414yescomplemented
104C4×Dic7D414yes—
105C2×#124—71no—
106C2×C4×Q8—71no—
107C2×C4×D4—71no—
108#391—71no—
109#390—71no—
110#388—71no—
111#388—71no—
112#388—71no—
113#388D711yescharacteristic, complemented
114#385—71no—
115#385—71no—
116#385—71no—
117#385—71no—
118#384—71no—
119#376—71no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7C22×C412yes—
127C23×C7C2411yescharacteristic
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7C22×C412yes—
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—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7C22×C412yes—
151C2×C4×C7C22×C412yes—
152C2×C4×C7C2×D412yes—
153C22⋊D7—28no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
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—21no—
169C2×Dic7—44no—
170C2×Dic7C22×C412yes—
171C2×Dic7C22×C412yes—
172C2×Dic7C22×C412yes—
173C2×Dic7C2×D412yes—
174C2×Dic7C2×D412yes—
175C2×Dic7#4112yes—
176C4×D7—44no—
177Dic14—28no—
178C22×Q8—71no—
179C22×D4—71no—
180C23×C4—71no—
181C23×C4—71no—
182#124—78no—
183C4×Q8—144no—
184C4×D4—144no—
185C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
193C2×C22⋊C4—71no—
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4C2×D711yescharacteristic
201C2×C22⋊C4C2×D711yescharacteristic
202C2×C22⋊C4C2×D711yescharacteristic
203C2×C42—71no—
204C2×C42—71no—
205C2×C42—71no—
206C2×C42—71no—
207C2×C42C2×D711yescharacteristic
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95C2×D711yescharacteristic
215#95C2×D711yescharacteristic
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C23×C411yescharacteristic
224C22×C7C22×D411yescharacteristic
225C22×C7C2×#4111yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C22×C7H(5,𝔽2)11yescharacteristic
228C22×C72−1+411yescharacteristic
229C22×C72−1+411yescharacteristic
230C2×D7—22no—
231C2×D7—22no—
232C2×D7—22no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C4×C7—22no—
238C4×C7—22no—
239C4×C7—22no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—41no—
243Dic7—22no—
244Dic7—22no—
245Dic7—22no—
246Dic7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7C4×D414yes—
250C24—71no—
251C24C22×D711yescharacteristic
252C2×Q8—74no—
253C2×Q8—142no—
254C2×Q8—142no—
255C2×D4—74no—
256C2×D4—142no—
257C2×D4—142no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—281no—
268C22×C4—281no—
269C22×C4—281no—
270C22×C4—281no—
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C22×C4C22×D711yescharacteristic
276C22×C4C22×D711yescharacteristic
277C4⋊C4—41no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—281no—
283C4⋊C4—281no—
284C4⋊C4—281no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—74no—
288C22⋊C4—74no—
289C22⋊C4—74no—
290C22⋊C4—142no—
291C22⋊C4—281no—
292C22⋊C4—281no—
293C22⋊C4—281no—
294C22⋊C4C4×D714yescomplemented
295C42—22no—
296C42—74no—
297C42—74no—
298C42—142no—
299C42—142no—
300C2×C7—42no—
301C2×C7C2×C4×D411yescharacteristic
302C2×C7#52011yescharacteristic
303C2×C7#52011yescharacteristic
304C2×C7#53511yescharacteristic
305C2×C7#53611yescharacteristic
306C2×C7#54211yescharacteristic
307C2×C7#54211yescharacteristic
308D7—42no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—72no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23C2×C4×D712yes—
323C23C23×D711yescharacteristic
324Q8—148no—
325D4—148no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—72no—
342C2×C4—72no—
343C2×C4—72no—
344C2×C4—72no—
345C2×C4—72no—
346C2×C4—72no—
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—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—284no—
363C2×C4—284no—
364C2×C4C2×C4×D712yes—
365C2×C4C2×C4×D712yes—
366C2×C4D4×D712yes—
367C7#232611yescharacteristic, complemented, Sylow
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—142no—
376C22—142no—
377C22—142no—
378C22—281no—
379C22—281no—
380C22—281no—
381C22—281no—
382C22C22×C4×D711yescharacteristic
383C22C2×D4×D711yescharacteristic
384C22#654811yescharacteristic
385C22D7×#4111yescharacteristic
386C22#655311yescharacteristic
387C22#655411yescharacteristic
388C22#655411yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—74no—
396C4—142no—
397C4—142no—
398C4—142no—
399C4—281no—
400C4—281no—
401C4—281no—
402C2—42no—
403C2—282no—
404C2#9021411yescharacteristic
405C2C4×D4×D711yescharacteristic
406C2#9023711yescharacteristic
407C2#9029811yescharacteristic
408C2#9033911yescharacteristic
409C2#9036011yescharacteristic
410C2#9040111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#388C2
socle, cosocleC23×C7C24
prime radicalC22×C22⋊D7C22
hypercenter#388D7
hypocenterC7#2326
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic14C2semidirect
4. C7×#388C2semidirect
5. #89761C2semidirect
6. #89748C2semidirect
8. #89722C2semidirect
9. #89718C2semidirect
10. #89716C2semidirect
11. #89712C2semidirect
12. #89442C2semidirect
13. #89440C2semidirect
14. #89420C2semidirect
15. #89418C2semidirect
43. #6448C4semidirect
81. C2×Dic14C2×C4semidirect
85. C7×C22⋊C4C2×C4semidirect
88. C22⋊Dic7C2×C4semidirect
92. (C2×D7)⋊C4C2×C4semidirect
93. (C2×D7)⋊C4C2×C4semidirect
103. C4×Dic7C2×C4semidirect
113. #388D7semidirect
294. C22⋊C4C4×D7semidirect
367. C7#2326semidirect

Maximal subgroups

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

subgroupindex
C2×#64482
C2×(C22⋊D7)⋊C42
C2×C4×Dic142
C7×#3882
#897612
#897482
#897292
#897222
#897182
#897162
#897122
#894422
#894402
#894202
#894182
#23267