Finite Group Databasebeta

← #11741441

Subgroups — #11741441

Order 896.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#137C211yesmaximal, characteristic, complemented
2#90515C211yesmaximal, characteristic
3Q8×C22⋊D7C211yesmaximal, characteristic, complemented
4C2×#6478C211yesmaximal, characteristic, complemented
5#90011C211yesmaximal, characteristic, complemented
6#90000C211yesmaximal, characteristic, complemented
7#89961C211yesmaximal, characteristic, complemented
8#89958C211yesmaximal, characteristic, complemented
9#89954C211yesmaximal, characteristic, complemented
10#89953C211yesmaximal, characteristic, complemented
11#89945C211yesmaximal, characteristic, complemented
12#89934C211yesmaximal, characteristic, complemented
13SD16×Dic7C211yesmaximal, characteristic, complemented
14#89909C211yesmaximal, characteristic, complemented
15#89900C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C2×Q8×D7C2211yescharacteristic
19C2×#1159C2211yescharacteristic
20C7×#137—24no—
21C2×C7×Q16C2211yescharacteristic
22C2×C7×Q16C2211yescharacteristic
23C2×C7×SD16C2211yescharacteristic
24C2×C7×SD16C2211yescharacteristic
25C2×C7×M16C2211yescharacteristic
26#6509—21no—
27#6509C2211yescharacteristic
28#6509C2211yescharacteristic
29Q8×Dic7C2211yescharacteristic
30#6507—21no—
31#6507C2211yescharacteristic
32#6507C2211yescharacteristic
33C2×#1144C2211yescharacteristic
34C2×#1144C2211yescharacteristic
35C2×#1144C2211yescharacteristic
36#6502—21no—
37#6498—21no—
38D4×Dic7C2211yescharacteristic
39C2×#1142C2211yescharacteristic
40#6492—21no—
41C4×C22⋊D7—21no—
42C4×C22⋊D7C2211yescharacteristic
43#6487—21no—
44#6487C2211yescharacteristic
45#6487C2211yescharacteristic
46C2×Dic7⋊C4—21no—
47#6478—24no—
48C2×Dic28C2211yescharacteristic
49C2×#1131C2211yescharacteristic
50#6409C2211yescharacteristic
51#6409C2211yescharacteristic
52#6409C2211yescharacteristic
53#6406C2211yescharacteristic
54C22⋊(C7⋊C8)C2211yescharacteristic
55(C2×D7)⋊C8C2211yescharacteristic
56#6391C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59Dic7⋊C8C2211yescharacteristic
60C8×Dic7C2211yescharacteristic
61#3276—71nomaximal, Sylow
62C7×#41—22no—
63C7×#41—41no—
64C2×C7×Q8—21no—
65C2×C7×Q8—22no—
66C2×C7×Q8—41no—
67C2×C7×Q8C2311yescharacteristic
68C2×C7×Q8C2311yescharacteristic
69C2×C7×Q8C2311yescharacteristic
70C2×C7×D4—21no—
71C2×C7×D4C2311yescharacteristic
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C2311yescharacteristic
75C2×C22⋊D7—21no—
76C2×C22⋊D7D411yescharacteristic
77C22×Dic7—21no—
78C22×Dic7—21no—
79Q8×D7—22no—
80Q8×D7—41no—
81#1159—22no—
82#1159—41no—
83C2×C4×D7—21no—
84C2×C4×D7C2311yescharacteristic
85C2×Dic14—21no—
86C2×Dic14C2311yescharacteristic
87C2×Dic14C2311yescharacteristic
88C7×Q16—22no—
89C7×Q16—22no—
90C7×SD16—22no—
91C7×SD16D414yescomplemented
92C7×M16—22no—
93C2×C7×C8C2311yescharacteristic
94C2×C7×C8C2311yescharacteristic
95C22⋊Dic7—21no—
96C22⋊Dic7—21no—
97C22⋊Dic7—21no—
98C22⋊Dic7D411yescharacteristic
99#1144—22no—
100#1144—41no—
101#1144—41no—
102#1142—22no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4D411yescharacteristic
106C4⋊Dic7—21no—
107C4⋊Dic7C2311yescharacteristic
108C4⋊Dic7C2311yescharacteristic
109Dic7⋊C4—21no—
110Dic7⋊C4—21no—
111Dic7⋊C4—21no—
112Dic7⋊C4—21no—
113Dic7⋊C4—21no—
114Dic7⋊C4—21no—
115Dic7⋊C4—21no—
116Dic7⋊C4—21no—
117Dic7⋊C4D411yescharacteristic
118C4×Dic7—21no—
119C4×Dic7C2311yescharacteristic
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122Dic28—22no—
123#1131—22no—
124C2×#137—71no—
125C2×#137D711yescharacteristic, complemented
126D4×Q8—71no—
127#547—71no—
128#494—71no—
129#470—71no—
130#470—71no—
131#462—71no—
132#461—71no—
133#452—71no—
134#452—71no—
135#451—71no—
136#451—71no—
137C4×SD16—71no—
138#436—71no—
139C23×C7—21no—
140C23×C7C2×D411yescharacteristic
141C22×D7C2×D411yescharacteristic
142C7×Q8—21no—
143C7×Q8—21no—
144C7×Q8—21no—
145C7×Q8—21no—
146C7×Q8—21no—
147C7×Q8—42no—
148C7×Q8C2×D412yes—
149C7×D4—21no—
150C7×D4—42no—
151C7×D4C2×D412yes—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7C2×D411yescharacteristic
160C2×C4×C7C2411yescharacteristic
161C22⋊D7—22no—
162C22⋊D7—42no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—42no—
168C2×Dic7—42no—
169C2×Dic7C2×D411yescharacteristic
170C2×Dic7C2×D411yescharacteristic
171C2×Dic7C2×D411yescharacteristic
172C4×D7—21no—
173C4×D7—21no—
174C4×D7—42no—
175Dic14—21no—
176Dic14—21no—
177Dic14—21no—
178Dic14—21no—
179Dic14—42no—
180C7×C8—21no—
181C7×C8C2×D412yes—
182C7⋊C8—21no—
183C7⋊C8—21no—
184C2×#41—71no—
185C2×#41C2×D711yescharacteristic
186C22×Q8—71no—
187C22×Q8C2×D711yescharacteristic
188#137—24no—
189#137—144no—
190C2×Q16—71no—
191C2×Q16—71no—
192C2×Q16—71no—
193C2×Q16—71no—
194C2×Q16C2×D711yescharacteristic
195C2×Q16C2×D711yescharacteristic
196C2×SD16—71no—
197C2×SD16C2×D711yescharacteristic
198C2×SD16C2×D711yescharacteristic
199C2×M16—71no—
200C2×M16C2×D711yescharacteristic
201C4⋊Q8—71no—
202C4⋊Q8—71no—
203C4⋊Q8—141no—
204#123—141no—
205#123—141no—
206C22⋊Q8—71no—
207C22⋊Q8—71no—
208C22⋊Q8—71no—
209C22⋊Q8—71no—
210C22⋊Q8—141no—
211C22⋊Q8—141no—
212#121—141no—
213C4×Q8—71no—
214C4×D4—71no—
215C4×D4—71no—
216C4×D4—141no—
217C2×C4⋊C4—141no—
218#106—71no—
219C4⋊C8—71no—
220#103—71no—
221#103—71no—
222#103—71no—
223#103—71no—
224#103—71no—
225#102—71no—
226C22⋊C8—71no—
227C22⋊C8—71no—
228C4×C8—71no—
229C22×C7—22no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7C22×D411yescharacteristic
236C2×D7—21no—
237C2×D7—21no—
238C2×D7—21no—
239C4×C7—22no—
240C4×C7—21no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7C22×D411yescharacteristic
244C4×C7H(5,𝔽2)11yescharacteristic
245Dic7—21no—
246Dic7—21no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7#13712yescomplemented
252#41—22no—
253#41—41no—
254#41—142no—
255#41—281no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—71no—
260C2×Q8—71no—
261C2×Q8—141no—
262C2×Q8—142no—
263C2×Q8—281no—
264C2×Q8C22×D711yescharacteristic
265C2×Q8C22×D711yescharacteristic
266C2×Q8C22×D711yescharacteristic
267C2×D4—21no—
268C2×D4—71no—
269C2×D4—141no—
270C2×D4C22×D711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—71no—
274C22×C4—141no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4C22×D711yescharacteristic
278Q16—22no—
279Q16—22no—
280Q16—142no—
281Q16—142no—
282Q16—281no—
283Q16—281no—
284SD16—22no—
285SD16—142no—
286SD16C22⋊D714yescomplemented
287M16—22no—
288M16—142no—
289C2×C8—71no—
290C2×C8—71no—
291C2×C8C22×D711yescharacteristic
292C2×C8C22×D711yescharacteristic
293C4⋊C4—71no—
294C4⋊C4—71no—
295C4⋊C4—71no—
296C4⋊C4—141no—
297C4⋊C4—141no—
298C4⋊C4—141no—
299C4⋊C4—141no—
300C4⋊C4—141no—
301C4⋊C4—141no—
302C4⋊C4—141no—
303C4⋊C4—141no—
304C4⋊C4—141no—
305C22⋊C4—71no—
306C22⋊C4—71no—
307C22⋊C4—141no—
308C22⋊C4—141no—
309C22⋊C4—141no—
310C22⋊C4—141no—
311C22⋊C4—141no—
312C42—71no—
313C42—141no—
314C2×C7—41no—
315C2×C7—42no—
316C2×C7D4211yescharacteristic
317C2×C7C2×#13711yescharacteristic
318C2×C7#57811yescharacteristic
319D7—41no—
320C23—21no—
321C23—71no—
322C23D4×D711yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—141no—
330Q8—141no—
331Q8—141no—
332Q8—141no—
333Q8—282no—
334Q8C2×C22⋊D712yes—
335D4—21no—
336D4—42no—
337D4—142no—
338D4—282no—
339D4C2×C22⋊D712yes—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—71no—
348C2×C4—71no—
349C2×C4—71no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—282no—
357C2×C4—282no—
358C2×C4—282no—
359C2×C4D4×D711yescharacteristic
360C2×C4C23×D711yescharacteristic
361C8—21no—
362C8—141no—
363C8—141no—
364C8C2×C22⋊D712yes—
365C7#327611yescharacteristic, complemented, Sylow
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—141no—
373C22—141no—
374C22—141no—
375C22C2×D4×D711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—72no—
381C4—141no—
382C4—141no—
383C4—281no—
384C4—281no—
385C4—281no—
386C4—281no—
387C4#654811yescharacteristic
388C4C22×C22⋊D711yescharacteristic
389C2—41no—
390C2—42no—
391C2—281no—
392C2D7×#13711yescharacteristic
393C2#9047011yescharacteristic
394C2D4×C22⋊D711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#137C2
socleC22×C7C22×D4
prime radical#90515C2
hypercenterC2×#137D7
hypocenterC7#3276
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×C7×#137C2semidirect
3. Q8×C22⋊D7C2semidirect
4. C2×#6478C2semidirect
5. #90011C2semidirect
6. #90000C2semidirect
7. #89961C2semidirect
8. #89958C2semidirect
9. #89954C2semidirect
10. #89953C2semidirect
11. #89945C2semidirect
12. #89934C2semidirect
13. SD16×Dic7C2semidirect
14. #89909C2semidirect
15. #89900C2semidirect
91. C7×SD16D4semidirect
125. C2×#137D7semidirect
251. Dic7#137semidirect
286. SD16C22⋊D7semidirect
365. C7#3276semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1372
#905152
Q8×C22⋊D72
C2×#64782
#900112
#900002
#899612
#899582
#899542
#899532
#899452
#899342
SD16×Dic72
#899092
#899002
#32767