Finite Group Databasebeta

← #11733741

Subgroups — #11733741

Order 896.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic
2C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
3C2×#6440C211yesmaximal, characteristic, complemented
4C7×#390C211yesmaximal, characteristic, complemented
5#89992C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7#89744C211yesmaximal, characteristic, complemented
8#89729C211yesmaximal, characteristic, complemented
9#89729C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89718C211yesmaximal, characteristic, complemented
12#89717C211yesmaximal, characteristic, complemented
13#89714C211yesmaximal, characteristic, complemented
14#89712C211yesmaximal, characteristic, complemented
15C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C42×C7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32#6502C418yescomplemented
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊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
45C2×C4×Dic7C2211yescharacteristic
46(C22⋊D7)⋊C4—24no—
47#6440—24no—
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54#2324—71nomaximal, Sylow
55C24×C7C2311yescharacteristic
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C2×C7×D4C2×C414yescomplemented
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—22no—
67C2×C22⋊D7—22no—
68C2×C22⋊D7—41no—
69C2×C22⋊D7—41no—
70C2×C22⋊D7C2×C414yes—
71C2×C22⋊D7C2×C414yescomplemented
72C22×Dic7—22no—
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—41no—
76C22×Dic7—41no—
77C22×Dic7C2×C414yescomplemented
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C2×C4×D7—22no—
86C2×C4×D7—41no—
87C2×C4×D7—41no—
88C7×C4⋊C4—41no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C42×C7—41no—
94C22⋊Dic7—22no—
95C22⋊Dic7—41no—
96C22⋊Dic7—41no—
97C22⋊Dic7C2×C414yescomplemented
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—41no—
101(C2×D7)⋊C4C2×C414yescomplemented
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106Dic7⋊C4—41no—
107Dic7⋊C4C2×C414yescomplemented
108C4×Dic7—22no—
109C4×Dic7—41no—
110C4×Dic7D414yescomplemented
111C2×#121—71no—
112C2×C4×D4—71no—
113C2×#117—71no—
114#390—71no—
115#390—71no—
116#390D711yescharacteristic, complemented
117#389—71no—
118#388—71no—
119#388—71no—
120#386—71no—
121#386—71no—
122#385—71no—
123#385—71no—
124#380—71no—
125C4×C4⋊C4—71no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7C22×C412yes—
139C23×C7C22×C412yes—
140C23×C7C2411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7C22×C412yes—
148C7×D4—44no—
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×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7C22×C412yes—
163C2×C4×C7C2×D412yes—
164C22⋊D7—28no—
165C22⋊D7—44no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—28no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—44no—
183C2×Dic7C22×C412yes—
184C2×Dic7C22×C412yes—
185C2×Dic7C22×C412yes—
186C2×Dic7C2×D412yes—
187C2×Dic7C2×D412yes—
188C2×Dic7#4112yes—
189C4×D7—44no—
190C22×D4—71no—
191C22×D4—71no—
192C22×D4C2×D711yescharacteristic
193C23×C4—71no—
194C23×C4—71no—
195C23×C4—71no—
196#121—78no—
197C4×D4—144no—
198#117—144no—
199C2×C4⋊C4—71no—
200C2×C4⋊C4—71no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—71no—
205C2×C4⋊C4C2×D711yescharacteristic
206C2×C22⋊C4—71no—
207C2×C22⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4C2×D711yescharacteristic
215C2×C22⋊C4C2×D711yescharacteristic
216C2×C22⋊C4C2×D711yescharacteristic
217C2×C22⋊C4C2×D711yescharacteristic
218C2×C42—71no—
219C2×C42—71no—
220C2×C42—71no—
221C2×C42C2×D711yescharacteristic
222#95—71no—
223#95—71no—
224#95—71no—
225#95—71no—
226#95—71no—
227#95—71no—
228C22×C7—22no—
229C22×C7—22no—
230C22×C7—22no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—22no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7C23×C411yescharacteristic
243C22×C7C22×D411yescharacteristic
244C22×C7C2×#4111yescharacteristic
245C22×C7H(5,𝔽2)11yescharacteristic
246C22×C7H(5,𝔽2)11yescharacteristic
247C22×C7H(5,𝔽2)11yescharacteristic
248C22×C72−1+411yescharacteristic
249C2×D7—22no—
250C2×D7—22no—
251C2×D7—22no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C4×C7—22no—
257C4×C7—22no—
258C4×C7—41no—
259C4×C7—41no—
260C4×C7—41no—
261Dic7—22no—
262Dic7—22no—
263Dic7—22no—
264Dic7—41no—
265Dic7—41no—
266Dic7—41no—
267Dic7C4×D414yes—
268C24—71no—
269C24C22×D711yescharacteristic
270C24C22×D711yescharacteristic
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—74no—
274C2×D4—74no—
275C2×D4—142no—
276C2×D4—142no—
277C2×D4—281no—
278C2×D4—281no—
279C2×D4C4×D714yescomplemented
280C22×C4—71no—
281C22×C4—74no—
282C22×C4—71no—
283C22×C4—71no—
284C22×C4—71no—
285C22×C4—71no—
286C22×C4—71no—
287C22×C4—71no—
288C22×C4—142no—
289C22×C4—142no—
290C22×C4—142no—
291C22×C4—142no—
292C22×C4—281no—
293C22×C4—281no—
294C22×C4—281no—
295C22×C4—281no—
296C22×C4C22×D711yescharacteristic
297C22×C4C22×D711yescharacteristic
298C22×C4C22×D711yescharacteristic
299C22×C4C22×D711yescharacteristic
300C22×C4C22×D711yescharacteristic
301C4⋊C4—41no—
302C4⋊C4—74no—
303C4⋊C4—142no—
304C4⋊C4—142no—
305C4⋊C4—142no—
306C4⋊C4—142no—
307C4⋊C4—281no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—74no—
313C22⋊C4—74no—
314C22⋊C4—142no—
315C22⋊C4—142no—
316C22⋊C4—281no—
317C22⋊C4—281no—
318C22⋊C4—281no—
319C22⋊C4—281no—
320C42—41no—
321C42—74no—
322C42—142no—
323C42—281no—
324C2×C7—42no—
325C2×C7—42no—
326C2×C7C2×C4×D411yescharacteristic
327C2×C7#51811yescharacteristic
328C2×C7#52011yescharacteristic
329C2×C7#53511yescharacteristic
330C2×C7#53711yescharacteristic
331C2×C7#54011yescharacteristic
332C2×C7#54211yescharacteristic
333D7—42no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—72no—
347C23—141no—
348C23—141no—
349C23—141no—
350C23—141no—
351C23—141no—
352C23—141no—
353C23C2×C4×D712yes—
354C23C2×C4×D712yes—
355C23C23×D711yescharacteristic
356D4—44no—
357D4—148no—
358D4—284no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—72no—
373C2×C4—72no—
374C2×C4—72no—
375C2×C4—72no—
376C2×C4—72no—
377C2×C4—72no—
378C2×C4—148no—
379C2×C4—141no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—141no—
393C2×C4—141no—
394C2×C4—284no—
395C2×C4—284no—
396C2×C4C2×C4×D712yes—
397C2×C4D4×D712yes—
398C7#232411yescharacteristic, complemented, Sylow
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—142no—
414C22—142no—
415C22—142no—
416C22—281no—
417C22—281no—
418C22—281no—
419C22—281no—
420C22C22×C4×D711yescharacteristic
421C22C2×D4×D711yescharacteristic
422C22#654811yescharacteristic
423C22#654811yescharacteristic
424C22#654811yescharacteristic
425C22D7×#4111yescharacteristic
426C22#655411yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—74no—
433C4—142no—
434C4—142no—
435C4—142no—
436C4—281no—
437C4—281no—
438C4—281no—
439C2—42no—
440C2—42no—
441C2—282no—
442C2#9017711yescharacteristic
443C2C4×D4×D711yescharacteristic
444C2#9023711yescharacteristic
445C2#9028811yescharacteristic
446C2#9030611yescharacteristic
447C2#9036011yescharacteristic
448C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#390C2
socle, cosocleC23×C7C24
prime radicalC2×#6502C2
hypercenter#390D7
hypocenterC7#2324
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D7)⋊C4C2semidirect
3. C2×#6440C2semidirect
4. C7×#390C2semidirect
5. #89992C2semidirect
6. #89761C2semidirect
7. #89744C2semidirect
8. #89729C2semidirect
9. #89729C2semidirect
10. #89728C2semidirect
11. #89718C2semidirect
12. #89717C2semidirect
13. #89714C2semidirect
14. #89712C2semidirect
15. C4×Dic7⋊C4C2semidirect
32. #6502C4semidirect
60. C2×C7×D4C2×C4semidirect
71. C2×C22⋊D7C2×C4semidirect
77. C22×Dic7C2×C4semidirect
97. C22⋊Dic7C2×C4semidirect
101. (C2×D7)⋊C4C2×C4semidirect
107. Dic7⋊C4C2×C4semidirect
110. C4×Dic7D4semidirect
116. #390D7semidirect
279. C2×D4C4×D7semidirect
398. C7#2324semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×(C22⋊D7)⋊C42
C2×#64402
C7×#3902
#899922
#897612
#897442
#897292
#897282
#897182
#897172
#897142
#897122
C4×Dic7⋊C42
#23247