Finite Group Databasebeta

← #11741084

Subgroups — #11741084

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#554C211yesmaximal, characteristic, complemented
2#90276C211yesmaximal, characteristic, complemented
3#89861C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic, complemented
5#89840C211yesmaximal, characteristic, complemented
6#89839C211yesmaximal, characteristic, complemented
7#89838C211yesmaximal, characteristic, complemented
8#89835C211yesmaximal, characteristic, complemented
9#89796C211yesmaximal, characteristic, complemented
10#89790C211yesmaximal, characteristic, complemented
11#89788C211yesmaximal, characteristic, complemented
12#89788C211yesmaximal, characteristic, complemented
13#89787C211yesmaximal, characteristic
14#89786C211yesmaximal, characteristic, complemented
15C4×#1141C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125C2211yescharacteristic
18C7×#125C2211yescharacteristic
19C7×C22⋊Q8—21no—
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8—21no—
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4C2211yescharacteristic
24C4×C7×D4C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C2×C7×C4⋊C4—21no—
27#6510—21no—
28C2×#1142C2211yescharacteristic
29C2×#1141C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6457—21no—
32#6457—21no—
33#6456—21no—
34#6438C2211yescharacteristic
35#6438C2211yescharacteristic
36#6437C2211yescharacteristic
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40#6406C2211yescharacteristic
41#6384C2211yescharacteristic
42#6384C2211yescharacteristic
43#6384C2211yescharacteristic
44#6383C2211yescharacteristic
45#6383C2211yescharacteristic
46#6383C2211yescharacteristic
47#6383C2211yescharacteristic
48#6383C2211yescharacteristic
49#6383C2211yescharacteristic
50#6382C2211yescharacteristic
51#6381C2211yescharacteristic
52#6381C2211yescharacteristic
53C4⋊(C7⋊C8)C2211yescharacteristic
54C4⋊(C7⋊C8)C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56#6377C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3370—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8D411yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28C2311yescharacteristic
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×Dic14C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4D411yescharacteristic
79C7×C4⋊C4D411yescharacteristic
80C7×C4⋊C4D411yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C42×C7C2311yescharacteristic
88C42×C7C2311yescharacteristic
89C42×C7C2311yescharacteristic
90#1142—22no—
91#1141—22no—
92#1141—22no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#559—71no—
107#554D711yescharacteristic, complemented
108#490—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#474—71no—
114#463—71no—
115#463—71no—
116#459—71no—
117#446—71no—
118#442—71no—
119#440—71no—
120C4×D8—71no—
121C23×C7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C7×Q8—41no—
125C7×Q8—81no—
126C7×D4—21no—
127C7×D4#4112yes—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—42no—
132C2×C4×C7—81no—
133C2×C4×C7—81no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C2×Dic7—21no—
142C2×Dic7—21no—
143D28—21no—
144D28—21no—
145D28—21no—
146D28—21no—
147D28—42no—
148D28—81no—
149C4×D7—81no—
150C4×D7—81no—
151Dic14—21no—
152Dic14—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×SD16—71no—
158C2×D8—71no—
159C2×D8—71no—
160C4⋊Q8C2×D711yescharacteristic
161#127—71no—
162#125C2×D711yescharacteristic
163#125C2×D711yescharacteristic
164#124—71no—
165#124—71no—
166#124—141no—
167#123—141no—
168#123—141no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C4×Q8—71no—
173C4×Q8C2×D711yescharacteristic
174C4×D4—71no—
175C4×D4—71no—
176C4×D4—141no—
177C4×D4C2×D711yescharacteristic
178C4×D4C2×D711yescharacteristic
179C4×D4C2×D711yescharacteristic
180C2×C4⋊C4—21no—
181#107—71no—
182#107—71no—
183#106—71no—
184C4⋊C8—71no—
185C4⋊C8—71no—
186C4⋊C8—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#102—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#97—71no—
198#97—71no—
199C4×C8—71no—
200C22×C7—22no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C2×D7—41no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C72−1+411yescharacteristic
221Dic7—41no—
222Dic7—41no—
223C2×Q8—21no—
224C2×Q8—71no—
225C2×Q8C22⋊D711yescharacteristic
226C2×D4—71no—
227C2×D4—71no—
228C2×D4—141no—
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—141no—
235C22×C4—141no—
236SD16—142no—
237D8—142no—
238D8—142no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8—71no—
242C2×C8—71no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—71no—
249C4⋊C4—141no—
250C4⋊C4C22⋊D711yescharacteristic
251C4⋊C4C22⋊D711yescharacteristic
252C4⋊C4C22⋊D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—141no—
266C42C22×D711yescharacteristic
267C42C22×D711yescharacteristic
268C42C22×D711yescharacteristic
269C2×C7—42no—
270C2×C7#54811yescharacteristic
271C2×C7#57511yescharacteristic
272C2×C7#57611yescharacteristic
273D7—81no—
274D7—81no—
275C23—21no—
276C23—141no—
277C23—141no—
278Q8—41no—
279Q8—81no—
280Q8—141no—
281Q8—141no—
282D4—21no—
283D4—141no—
284D4—141no—
285D4—141no—
286D4—141no—
287D4—282no—
288D4—561no—
289D4#116112yes—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—561no—
299C2×C4—561no—
300C2×C4C2×C22⋊D711yescharacteristic
301C2×C4C2×C22⋊D711yescharacteristic
302C2×C4C2×C22⋊D711yescharacteristic
303C2×C4C2×C22⋊D711yescharacteristic
304C2×C4C2×C22⋊D711yescharacteristic
305C2×C4C2×C22⋊D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—141no—
308C8—141no—
309C8—141no—
310C8—141no—
311C7#337011yescharacteristic, complemented, Sylow
312C22—22no—
313C22—41no—
314C22—41no—
315C22—281no—
316C22—281no—
317C22—281no—
318C22—281no—
319C22—281no—
320C22—281no—
321C22C22×C22⋊D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—281no—
332C4—281no—
333C4C2×#116111yescharacteristic
334C4#655111yescharacteristic
335C2—42no—
336C2—561no—
337C2—561no—
338C2#9050811yescharacteristic
339C2#9051311yescharacteristic
340C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#554C2
socleC22×C7C22×D4
prime radical#89787C2
hypercenter#554D7
hypocenterC7#3370
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C7×#554C2semidirect
2. #90276C2semidirect
3. #89861C2semidirect
4. #89859C2semidirect
5. #89840C2semidirect
6. #89839C2semidirect
7. #89838C2semidirect
8. #89835C2semidirect
9. #89796C2semidirect
10. #89790C2semidirect
11. #89788C2semidirect
12. #89788C2semidirect
14. #89786C2semidirect
15. C4×#1141C2semidirect
107. #554D7semidirect
311. C7#3370semidirect

Maximal subgroups

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

subgroupindex
C7×#5542
#902762
#898612
#898592
#898402
#898392
#898382
#898352
#897962
#897902
#897882
#897872
#897862
C4×#11412
#33707