Finite Group Databasebeta

← #11732138

Subgroups — #11732138

Order 896.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic7C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
3C2×#6441C211yesmaximal, characteristic, complemented
4C2×#6441C211yesmaximal, characteristic, complemented
5C7×#386C211yesmaximal, characteristic
6#89992C211yesmaximal, characteristic
7#89984C211yesmaximal, characteristic, complemented
8#89720C211yesmaximal, characteristic, complemented
9#89720C211yesmaximal, characteristic, complemented
10#89714C211yesmaximal, characteristic, complemented
11#89714C211yesmaximal, characteristic, complemented
12#89433C211yesmaximal, characteristic, complemented
13#89429C211yesmaximal, characteristic, complemented
14#89419C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
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
30D4×Dic7—24no—
31C2×C4⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4—24no—
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
45#6441—24no—
46#6441—24no—
47C7×#95C2211yescharacteristic
48C7×#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2417—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C2×C7×D4—22no—
59C2×C7×D4—22no—
60C2×C7×D4Q814yescomplemented
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×Dic7—22no—
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—22no—
76C22×Dic7—41no—
77C22×Dic7D414yes—
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C2×Dic14—22no—
87C2×Dic14—22no—
88C2×Dic14—22no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—41no—
92C22⋊Dic7—22no—
93C22⋊Dic7—22no—
94C22⋊Dic7—22no—
95C22⋊Dic7—22no—
96C22⋊Dic7—41no—
97C4⋊Dic7—22no—
98C4⋊Dic7—22no—
99C4⋊Dic7—22no—
100C4⋊Dic7—41no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106Dic7⋊C4—22no—
107Dic7⋊C4—41no—
108Dic7⋊C4D414yescomplemented
109C4×Dic7—22no—
110C2×C22⋊Q8—71no—
111C2×C22⋊Q8—71no—
112C2×C4×D4—71no—
113C22×C4⋊C4—71no—
114#398—71no—
115#398—71no—
116#396—71no—
117#396—71no—
118#391—71no—
119#389—71no—
120#386—71no—
121#386D711yescharacteristic
122#380—71no—
123#380—71no—
124#380—71no—
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×C7—21no—
135C23×C7C2×D412yes—
136C23×C7C2×Q812yes—
137C23×C7C2×Q812yes—
138C23×C7#4112yes—
139C23×C7C2411yescharacteristic
140C7×D4—28no—
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—44no—
155C2×C4×C7C2×D412yes—
156C2×C4×C7C2×Q812yes—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—28no—
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—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—44no—
178C2×Dic7—44no—
179C2×Dic7C2×D412yes—
180C2×Dic7C2×D412yes—
181C2×Dic7C2×D412yes—
182C2×Dic7C2×D412yes—
183C2×Dic7#4112yes—
184Dic14—44no—
185C22×Q8—71no—
186C22×D4C2×D711yescharacteristic
187C23×C4—71no—
188C23×C4—71no—
189C23×C4—71no—
190C23×C4C2×D711yescharacteristic
191C22⋊Q8—144no—
192C22⋊Q8—144no—
193C4×D4—144no—
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×C4⋊C4—71no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—144no—
207C2×C22⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4C2×D711yescharacteristic
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C22⋊C4C2×D711yescharacteristic
215C2×C42—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95C2×D711yescharacteristic
223#95C2×D711yescharacteristic
224C22×C7—22no—
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—22no—
229C22×C7—22no—
230C22×C7—22no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7C22⋊Q814yescomplemented
238C22×C7C22×D411yescharacteristic
239C22×C7C22×D411yescharacteristic
240C22×C7C22×Q811yescharacteristic
241C22×C7C2×#4111yescharacteristic
242C22×C7C2×#4111yescharacteristic
243C22×C7H(5,𝔽2)11yescharacteristic
244C22×C72−1+411yescharacteristic
245C4×C7—22no—
246C4×C7—22no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—41no—
250Dic7—22no—
251Dic7—22no—
252Dic7—41no—
253Dic7—41no—
254Dic7—41no—
255Dic7—41no—
256Dic7—41no—
257Dic7#12114yes—
258C24C22×D711yescharacteristic
259C24C22×D711yescharacteristic
260C2×Q8—142no—
261C2×Q8—142no—
262C2×Q8—142no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4Dic1414yescomplemented
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—71no—
270C22×C4—74no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—71no—
275C22×C4—71no—
276C22×C4—71no—
277C22×C4—71no—
278C22×C4—142no—
279C22×C4—142no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4—142no—
283C22×C4—142no—
284C22×C4—142no—
285C22×C4—281no—
286C22×C4C22×D711yescharacteristic
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C4⋊C4—74no—
292C4⋊C4—142no—
293C4⋊C4—142no—
294C4⋊C4—142no—
295C4⋊C4—142no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C4⋊C4—142no—
299C4⋊C4—142no—
300C4⋊C4—142no—
301C4⋊C4—281no—
302C4⋊C4—281no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—41no—
306C22⋊C4—142no—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C22⋊C4—142no—
310C22⋊C4—281no—
311C42—142no—
312C2×C7—24no—
313C2×C7—42no—
314C2×C7C2×#12111yescharacteristic
315C2×C7C2×C22⋊Q811yescharacteristic
316C2×C7#53711yescharacteristic
317C2×C7#54611yescharacteristic
318C2×C7D4×Q811yescharacteristic
319C2×C7#55211yescharacteristic
320C2×C7#55411yescharacteristic
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23C2×Dic1412yes—
332C23C2×Dic1412yes—
333C23D4×D712yes—
334C23#115912yes—
335C23C23×D711yescharacteristic
336Q8—284no—
337D4—28no—
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—44no—
352C2×C4—72no—
353C2×C4—72no—
354C2×C4—72no—
355C2×C4—72no—
356C2×C4—72no—
357C2×C4—148no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
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—141no—
376C2×C4—141no—
377C2×C4—284no—
378C2×C4—284no—
379C2×C4C2×Dic1412yes—
380C2×C4D4×D712yes—
381C7#241711yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22#644114yescomplemented
396C22C22×Dic1411yescharacteristic
397C22C2×D4×D711yescharacteristic
398C22C2×D4×D711yescharacteristic
399C22C2×#115911yescharacteristic
400C22#654811yescharacteristic
401C22D7×#4111yescharacteristic
402C22#655411yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—74no—
409C4—142no—
410C4—142no—
411C4—281no—
412C4—281no—
413C4—281no—
414C4—281no—
415C4—281no—
416C2—24no—
417C2—42no—
418C2C2×#644111yescharacteristic
419C2D4×Dic1411yescharacteristic
420C2#9023011yescharacteristic
421C2#9028211yescharacteristic
422C2#9028811yescharacteristic
423C2D7×#12111yescharacteristic
424C2#9034111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2417
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C4C2semidirect
3. C2×#6441C2semidirect
4. C2×#6441C2semidirect
7. #89984C2semidirect
8. #89720C2semidirect
9. #89720C2semidirect
10. #89714C2semidirect
11. #89714C2semidirect
12. #89433C2semidirect
13. #89429C2semidirect
14. #89419C2semidirect
15. #89418C2semidirect
60. C2×C7×D4Q8semidirect
108. Dic7⋊C4D4semidirect
237. C22×C7C22⋊Q8semidirect
265. C2×D4Dic14semidirect
381. C7#2417semidirect
395. C22#6441semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C22×Dic7⋊C42
C2×#64412
C7×#3862
#899922
#899842
#897202
#897142
#894332
#894292
#894192
#894182
#24177