Finite Group Databasebeta

← #11736417

Subgroups — #11736417

Order 896.

374 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D7C211yesmaximal, characteristic
2#90224C211yesmaximal, characteristic
3C7×#419C211yesmaximal, characteristic, complemented
4#90000C211yesmaximal, characteristic, complemented
5#89909C211yesmaximal, characteristic, complemented
6#89900C211yesmaximal, characteristic
7#89768C211yesmaximal, characteristic, complemented
8#89588C211yesmaximal, characteristic, complemented
9#89587C211yesmaximal, characteristic, complemented
10#89580C211yesmaximal, characteristic, complemented
11#89574C211yesmaximal, characteristic, complemented
12#89565C211yesmaximal, characteristic, complemented
13#89564C211yesmaximal, characteristic, complemented
14#89563C211yesmaximal, characteristic, complemented
15#89562C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C7×#117C2211yescharacteristic
20#6509—21no—
21#6509C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507—21no—
24#6507C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C4×C22⋊D7—21no—
28C4×C22⋊D7C2211yescharacteristic
29#6487—21no—
30#6487C2211yescharacteristic
31#6487C2211yescharacteristic
32C2×Dic28C2211yescharacteristic
33C2×#1131C2211yescharacteristic
34#6460C2211yescharacteristic
35#6455C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6448—21no—
39#6442—21no—
40#6439—21no—
41C4×Dic14—21no—
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6391C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3328—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7D411yescharacteristic
68Q8×D7—22no—
69Q8×D7—41no—
70C2×C4×D7—21no—
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C2×Dic14C2311yescharacteristic
76C7×M16—41no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C42×C7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7D411yescharacteristic
86#1144—22no—
87#1144—41no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4D411yescharacteristic
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110Dic28—41no—
111#1131—41no—
112#556—71no—
113D4×Q8—71no—
114#487—71no—
115#485—71no—
116#485—71no—
117#475—71no—
118#470—71no—
119#470—71no—
120#462—71no—
121#451—71no—
122#451—71no—
123C4×Q16—71no—
124#436—71no—
125#419—71no—
126#419D711yescharacteristic, complemented
127C23×C7C2×D411yescharacteristic
128C22×D7C2×D411yescharacteristic
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—42no—
133C7×Q8#4112yes—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C22⋊D7—22no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C4×D7—21no—
152C4×D7—21no—
153C4×D7—42no—
154Dic14—21no—
155Dic14—21no—
156Dic14—21no—
157Dic14—21no—
158Dic14—42no—
159Dic14—42no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C22×Q8—71no—
165C22×Q8C2×D711yescharacteristic
166C2×Q16—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×M16—71no—
170C2×M16C2×D711yescharacteristic
171C4⋊Q8—71no—
172C4⋊Q8—71no—
173C4⋊Q8—141no—
174#126—141no—
175#126—141no—
176#124—141no—
177C22⋊Q8—71no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C4×Q8—71no—
184C4×Q8—71no—
185C4×Q8—141no—
186C4×D4—71no—
187C4×D4—141no—
188#117—71no—
189#117C2×D711yescharacteristic
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic
201#103C2×D711yescharacteristic
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204C22⋊C8—71no—
205C22⋊C8—71no—
206C4×C8—71no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C4×C7—22no—
215C4×C7—21no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C7H(5,𝔽2)11yescharacteristic
221Dic7—21no—
222Dic7—21no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7#13712yes—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—71no—
232C2×Q8—71no—
233C2×Q8—141no—
234C2×Q8—141no—
235C2×Q8—142no—
236C2×Q8—281no—
237C2×Q8C22×D711yescharacteristic
238C2×Q8C22×D711yescharacteristic
239C2×D4—71no—
240C22×C4—21no—
241C22×C4—71no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244Q16—142no—
245Q16—281no—
246Q16—281no—
247M16—41no—
248M16—281no—
249C2×C8—71no—
250C2×C8—71no—
251C2×C8C22×D711yescharacteristic
252C2×C8C22×D711yescharacteristic
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4C22×D711yescharacteristic
266C4⋊C4C22×D711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—71no—
269C22⋊C4—71no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C42—21no—
276C42—71no—
277C42—141no—
278C42—141no—
279C42—141no—
280C2×C7—41no—
281C2×C7#54611yescharacteristic
282C2×C7C2×#13711yescharacteristic
283C2×C7#57811yescharacteristic
284D7—41no—
285C23—71no—
286C23D4×D711yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—42no—
291Q8—141no—
292Q8—141no—
293Q8—141no—
294Q8—141no—
295Q8—282no—
296Q8—282no—
297Q8#115712yes—
298D4—142no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—71no—
307C2×C4—71no—
308C2×C4—71no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—282no—
316C2×C4D4×D711yescharacteristic
317C2×C4C23×D711yescharacteristic
318C8—21no—
319C8—21no—
320C8—141no—
321C8—141no—
322C7#332811yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—141no—
327C22—141no—
328C22—141no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—72no—
336C4—141no—
337C4—141no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4C2×#115711yescharacteristic
343C4#654811yescharacteristic
344C2—41no—
345C2—281no—
346C2#9018111yescharacteristic
347C2D7×#13711yescharacteristic
348C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#419C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#419D7
hypocenterC7#3328
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. C7×#419C2semidirect
4. #90000C2semidirect
5. #89909C2semidirect
7. #89768C2semidirect
8. #89588C2semidirect
9. #89587C2semidirect
10. #89580C2semidirect
11. #89574C2semidirect
12. #89565C2semidirect
13. #89564C2semidirect
14. #89563C2semidirect
15. #89562C2semidirect
126. #419D7semidirect
322. C7#3328semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D72
#902242
C7×#4192
#900002
#899092
#899002
#897682
#895882
#895872
#895802
#895742
#895652
#895642
#895632
#895622
#33287