Finite Group Databasebeta

← #11736350

Subgroups — #11736350

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90522C211yesmaximal, characteristic
2#90204C211yesmaximal, characteristic, complemented
3C7×#417C211yesmaximal, characteristic, complemented
4#90011C211yesmaximal, characteristic, complemented
5#89909C211yesmaximal, characteristic, complemented
6#89900C211yesmaximal, characteristic, complemented
7#89742C211yesmaximal, characteristic, complemented
8#89580C211yesmaximal, characteristic, complemented
9#89574C211yesmaximal, characteristic, complemented
10#89571C211yesmaximal, characteristic, complemented
11#89562C211yesmaximal, characteristic, complemented
12#89552C211yesmaximal, characteristic, complemented
13#89551C211yesmaximal, characteristic, complemented
14#89540C211yesmaximal, characteristic, complemented
15#89539C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22#6501—21no—
23#6501C2211yescharacteristic
24#6499—21no—
25#6499C2211yescharacteristic
26C2×#1142C2211yescharacteristic
27#6492—21no—
28C4×C22⋊D7—21no—
29C4×C22⋊D7C2211yescharacteristic
30#6489—21no—
31#6487—21no—
32#6487C2211yescharacteristic
33#6487C2211yescharacteristic
34C2×Dic28C2211yescharacteristic
35C2×#1131C2211yescharacteristic
36#6461—21no—
37#6460C2211yescharacteristic
38#6455C2211yescharacteristic
39#6453C2211yescharacteristic
40#6452—21no—
41#6450C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6391C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3327—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7D411yescharacteristic
70Q8×D7—22no—
71Q8×D7—41no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C7×M16—41no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7D411yescharacteristic
89#1144—22no—
90#1142—41no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
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#552—71no—
113#548—71no—
114#488—71no—
115#484—71no—
116#482—71no—
117#479—71no—
118#470—71no—
119#468—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123C4×Q16—71no—
124#436—71no—
125#419—71no—
126#417D711yescharacteristic, complemented
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8#4112yes—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—42no—
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×C7—42no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C22⋊D7—41no—
145C22⋊D7—42no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—42no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41C2×D711yescharacteristic
166C22×Q8—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×SD16—71no—
170C2×M16—71no—
171C2×M16C2×D711yescharacteristic
172#126—141no—
173#125—71no—
174#125—141no—
175#124—71no—
176#124—141no—
177#123—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—71no—
183#121—141no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×D4—71no—
187C4×D4—141no—
188#117—71no—
189#117—141no—
190C2×C4⋊C4C2×D711yescharacteristic
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic
201#103C2×D711yescharacteristic
202#102—71no—
203#102C2×D711yescharacteristic
204#102C2×D711yescharacteristic
205C22⋊C8—71no—
206C22⋊C8—71no—
207C4×C8—71no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7C22×D411yescharacteristic
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C4×C7—22no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C7H(5,𝔽2)11yescharacteristic
224Dic7—21no—
225Dic7—21no—
226Dic7—21no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—22no—
232#41—41no—
233C2×Q8—71no—
234C2×Q8—71no—
235C2×Q8—141no—
236C2×Q8—142no—
237C2×Q8—281no—
238C2×Q8C22×D711yescharacteristic
239C2×D4—21no—
240C2×D4—71no—
241C2×D4—141no—
242C2×D4C22×D711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—71no—
246C22×C4—141no—
247C22×C4C22×D711yescharacteristic
248Q16—142no—
249Q16—281no—
250SD16—281no—
251M16—41no—
252M16—281no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4C22×D711yescharacteristic
269C4⋊C4C22×D711yescharacteristic
270C22⋊C4—71no—
271C22⋊C4—71no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C42—71no—
280C42—141no—
281C42—141no—
282C42—141no—
283C2×C7—41no—
284C2×C7—81no—
285C2×C7#54611yescharacteristic
286C2×C7#57511yescharacteristic
287C2×C7#57811yescharacteristic
288D7—41no—
289C23—21no—
290C23—71no—
291C23D4×D711yescharacteristic
292Q8—21no—
293Q8—141no—
294Q8—141no—
295Q8—141no—
296Q8—141no—
297Q8—282no—
298Q8#115712yes—
299D4—21no—
300D4—21no—
301D4—42no—
302D4—281no—
303D4—282no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—71no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4D4×D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—141no—
326C8—141no—
327C7#332711yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—141no—
335C22—141no—
336C22—141no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—141no—
343C4—141no—
344C4—141no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4C2×#115711yescharacteristic
350C4#654811yescharacteristic
351C2—41no—
352C2—81no—
353C2—281no—
354C2#9018111yescharacteristic
355C2#9046411yescharacteristic
356C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
prime radical#90522C2
hypercenter#417D7
hypocenterC7#3327
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #90204C2semidirect
3. C7×#417C2semidirect
4. #90011C2semidirect
5. #89909C2semidirect
6. #89900C2semidirect
7. #89742C2semidirect
8. #89580C2semidirect
9. #89574C2semidirect
10. #89571C2semidirect
11. #89562C2semidirect
12. #89552C2semidirect
13. #89551C2semidirect
14. #89540C2semidirect
15. #89539C2semidirect
126. #417D7semidirect
327. C7#3327semidirect

Maximal subgroups

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

subgroupindex
#905222
#902042
C7×#4172
#900112
#899092
#899002
#897422
#895802
#895742
#895712
#895622
#895522
#895512
#895402
#895392
#33277