Finite Group Databasebeta

← #11740211

Subgroups — #11740211

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×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D7C211yesmaximal, characteristic
4C2×#6459C211yesmaximal, characteristic, complemented
5#90003C211yesmaximal, characteristic, complemented
6#89982C211yesmaximal, characteristic, complemented
7#89766C211yesmaximal, characteristic, complemented
8#89761C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
10#89744C211yesmaximal, characteristic, complemented
11#89744C211yesmaximal, characteristic, complemented
12#89729C211yesmaximal, characteristic
13#89729C211yesmaximal, characteristic
14#89720C211yesmaximal, characteristic, complemented
15#89719C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C7×C22⋊Q8—24no—
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30#6507—24no—
31C4×C22⋊D7—24no—
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×C4⋊Dic7C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46C2×C4×Dic7C2211yescharacteristic
47C2×C4×Dic7C2211yescharacteristic
48#6459—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×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×C4×C7C2311yescharacteristic
70C2×C22⋊D7—22no—
71C2×C22⋊D7—22no—
72C2×C22⋊D7Q814yescomplemented
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×C4⋊C4—22no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4D414yescomplemented
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C22⋊Dic7—22no—
92C22⋊Dic7—22no—
93C22⋊Dic7—41no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—22no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—41no—
99C4⋊Dic7—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106Dic7⋊C4D414yes—
107C4×Dic7—22no—
108C4×Dic7—22no—
109C4×Dic7—22no—
110C2×C4⋊Q8—71no—
111C2×C22⋊Q8—71no—
112C2×C22⋊Q8D711yescharacteristic, complemented
113C2×C4×D4—71no—
114#396—71no—
115#396—71no—
116#393—71no—
117#393—71no—
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—44no—
156C2×C4×C7C2×D412yes—
157C2×C4×C7C2×D412yes—
158C2×C4×C7C2×D412yes—
159C2×C4×C7C2×D412yes—
160C22⋊D7—28no—
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—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
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×C4C2×D711yescharacteristic
189C4⋊Q8—144no—
190C22⋊Q8—24no—
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⋊C4C2×D711yescharacteristic
202C2×C4⋊C4C2×D711yescharacteristic
203C2×C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
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#95—71no—
221#95—71no—
222#95—71no—
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—22no—
247C4×C7—22no—
248C4×C7—41no—
249C4×C7—41no—
250Dic7—22no—
251Dic7—22no—
252Dic7—41no—
253Dic7—41no—
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—22no—
269C22×C4—22no—
270C22×C4—41no—
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—281no—
281C22×C4C22×D711yescharacteristic
282C22×C4C22×D711yescharacteristic
283C22×C4C22×D711yescharacteristic
284C22×C4C22×D711yescharacteristic
285C22×C4C22×D711yescharacteristic
286C22×C4C22×D711yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—74no—
290C4⋊C4—142no—
291C4⋊C4—142no—
292C4⋊C4—142no—
293C4⋊C4—142no—
294C4⋊C4—142no—
295C4⋊C4—142no—
296C4⋊C4—142no—
297C4⋊C4C22⋊D714yescomplemented
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—142no—
301C22⋊C4—142no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—281no—
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—44no—
353C2×C4—72no—
354C2×C4—72no—
355C2×C4—72no—
356C2×C4—72no—
357C2×C4—72no—
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—284no—
375C2×C4D4×D712yes—
376C2×C4C2×C22⋊D712yes—
377C2×C4C2×C22⋊D712yes—
378C2×C4C2×C22⋊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
395C22#654811yescharacteristic
396C22C2×Q8×D711yescharacteristic
397C22#655111yescharacteristic
398C22D7×#4111yescharacteristic
399C22D7×#4111yescharacteristic
400C22C22×C22⋊D711yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—74no—
408C4—142no—
409C4—142no—
410C4—281no—
411C4—281no—
412C4—281no—
413C4—281no—
414C2—42no—
415C2—282no—
416C2D7×C22⋊Q811yescharacteristic
417C2#9032411yescharacteristic
418C2#9032711yescharacteristic
419C2#9033111yescharacteristic
420C2D4×C22⋊D711yescharacteristic
421C2Q8×C22⋊D711yescharacteristic
422C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC22×C22⋊D7C22
hypercenterC2×C22⋊Q8D7
hypocenterC7#2452
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6507C2semidirect
4. C2×#6459C2semidirect
5. #90003C2semidirect
6. #89982C2semidirect
7. #89766C2semidirect
8. #89761C2semidirect
9. C4⋊C4×Dic7C2semidirect
10. #89744C2semidirect
11. #89744C2semidirect
14. #89720C2semidirect
15. #89719C2semidirect
72. C2×C22⋊D7Q8semidirect
88. C7×C4⋊C4D4semidirect
112. C2×C22⋊Q8D7semidirect
256. Dic7C22⋊Q8semidirect
297. C4⋊C4C22⋊D7semidirect
379. C7#2452semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65072
C2×C4×C22⋊D72
C2×#64592
#900032
#899822
#897662
#897612
C4⋊C4×Dic72
#897442
#897292
#897202
#897192
#24527