Finite Group Databasebeta

← #11734381

Subgroups — #11734381

Order 896.

568 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic
2C2×#6440C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89994C211yesmaximal, characteristic, complemented
5#89762C211yesmaximal, characteristic, complemented
6#89748C211yesmaximal, characteristic, complemented
7#89733C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89729C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89723C211yesmaximal, characteristic, complemented
12#89717C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89422C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30#6502—24no—
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46#6440—24no—
47C7×#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2500—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C2×C7×D4—22no—
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—22no—
66C2×C22⋊D7—22no—
67C2×C22⋊D7—22no—
68C2×C22⋊D7—22no—
69C2×C22⋊D7—41no—
70C2×C22⋊D7—41no—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7D414yes—
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C7×C4⋊C4—41no—
82C7×C22⋊C4—22no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—41no—
86C22⋊Dic7—22no—
87C22⋊Dic7—22no—
88C22⋊Dic7—22no—
89C22⋊Dic7—22no—
90C22⋊Dic7—41no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—41no—
95(C2×D7)⋊C4—41no—
96C4⋊Dic7—41no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—41no—
101Dic7⋊C4—41no—
102C4×Dic7—22no—
103C4×Dic7—22no—
104C4×Dic7—22no—
105C4×Dic7—22no—
106C2×#121—71no—
107C2×#117—71no—
108#394—71no—
109#394—71no—
110#394—71no—
111#394D711yescharacteristic, complemented
112#390—71no—
113#390—71no—
114#388—71no—
115#388—71no—
116#388—71no—
117#388—71no—
118#384—71no—
119#382—71no—
120C4×C22⋊C4—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
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×C7C2×D412yes—
135C23×C7C2411yescharacteristic
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C7×D4—44no—
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×C7—21no—
158C2×C4×C7#4112yes—
159C22⋊D7—44no—
160C22⋊D7—44no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—28no—
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—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7C2×D412yes—
179C2×Dic7C2×D412yes—
180C2×Dic7#4112yes—
181C2×Dic7#4112yes—
182C2×Dic7#4112yes—
183C22×D4—71no—
184C22×D4—71no—
185C22×D4C2×D711yescharacteristic
186C23×C4—71no—
187#121—144no—
188#117—144no—
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⋊C4C2×D711yescharacteristic
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4C2×D711yescharacteristic
207C2×C22⋊C4C2×D711yescharacteristic
208C2×C22⋊C4C2×D711yescharacteristic
209C2×C22⋊C4C2×D711yescharacteristic
210C2×C42—71no—
211C2×C42—71no—
212C2×C42—71no—
213C2×C42—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95C2×D711yescharacteristic
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7C22×D411yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7C2×#4111yescharacteristic
237C22×C7C2×#4111yescharacteristic
238C22×C7C2×#4111yescharacteristic
239C22×C7H(5,𝔽2)11yescharacteristic
240C22×C7H(5,𝔽2)11yescharacteristic
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C4×C7—22no—
249C4×C7—41no—
250C4×C7—41no—
251C4×C7—41no—
252C4×C7—41no—
253Dic7—22no—
254Dic7—22no—
255Dic7—41no—
256Dic7—41no—
257Dic7—41no—
258Dic7—41no—
259Dic7#12314yes—
260C24—71no—
261C24C22×D711yescharacteristic
262C24C22×D711yescharacteristic
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—142no—
267C2×D4—142no—
268C2×D4—142no—
269C2×D4—142no—
270C2×D4—281no—
271C2×D4—281no—
272C22×C4—71no—
273C22×C4—74no—
274C22×C4—71no—
275C22×C4—71no—
276C22×C4—71no—
277C22×C4—71no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4C22×D711yescharacteristic
283C22×C4C22×D711yescharacteristic
284C22×C4C22×D711yescharacteristic
285C22×C4C22×D711yescharacteristic
286C22×C4C22×D711yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—142no—
289C4⋊C4—142no—
290C4⋊C4—142no—
291C4⋊C4—281no—
292C4⋊C4—281no—
293C4⋊C4—281no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—142no—
299C22⋊C4—142no—
300C22⋊C4—142no—
301C22⋊C4—142no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—281no—
306C22⋊C4—281no—
307C22⋊C4—281no—
308C42—142no—
309C42—142no—
310C42—142no—
311C42—142no—
312C2×C7—42no—
313C2×C7—81no—
314C2×C7C2×#12311yescharacteristic
315C2×C7#53211yescharacteristic
316C2×C7#53511yescharacteristic
317C2×C7#55311yescharacteristic
318C2×C7#55311yescharacteristic
319C2×C7#55511yescharacteristic
320C2×C7#55511yescharacteristic
321D7—81no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23D4×D712yes—
343C23C23×D711yescharacteristic
344D4—44no—
345D4—284no—
346D4—284no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—72no—
362C2×C4—72no—
363C2×C4—72no—
364C2×C4—72no—
365C2×C4—72no—
366C2×C4—148no—
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—141no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—141no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4#115912yes—
384C7#250011yescharacteristic, complemented, Sylow
385C22—22no—
386C22—22no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—281no—
400C22—281no—
401C22—281no—
402C22—281no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22C2×D4×D711yescharacteristic
407C22C2×#115911yescharacteristic
408C22#654811yescharacteristic
409C22#654811yescharacteristic
410C22D7×#4111yescharacteristic
411C22D7×#4111yescharacteristic
412C22D7×#4111yescharacteristic
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—74no—
419C4—142no—
420C4—142no—
421C4—281no—
422C4—281no—
423C4—281no—
424C4—281no—
425C2—42no—
426C2—81no—
427C2—561no—
428C2#9028311yescharacteristic
429C2#9028711yescharacteristic
430C2#9029211yescharacteristic
431C2#9030711yescharacteristic
432C2D7×#12311yescharacteristic
433C2#9035211yescharacteristic
434C2#9036111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6502C2
hypercenter#394D7
hypocenterC7#2500
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6440C2semidirect
3. C7×#394C2semidirect
4. #89994C2semidirect
5. #89762C2semidirect
6. #89748C2semidirect
7. #89733C2semidirect
8. #89731C2semidirect
9. #89729C2semidirect
10. #89729C2semidirect
11. #89723C2semidirect
12. #89717C2semidirect
13. C22⋊C4×Dic7C2semidirect
14. #89441C2semidirect
15. #89422C2semidirect
111. #394D7semidirect
384. C7#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#64402
C7×#3942
#899942
#897622
#897482
#897332
#897312
#897292
#897232
#897172
C22⋊C4×Dic72
#894412
#894222
#25007