Finite Group Databasebeta

← #11740910

Subgroups — #11740910

Order 896.

358 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
1C7×#542C211yesmaximal, characteristic, complemented
2#90226C211yesmaximal, characteristic, complemented
3#89859C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89829C211yesmaximal, characteristic, complemented
6#89829C211yesmaximal, characteristic, complemented
7#89821C211yesmaximal, characteristic, complemented
8#89820C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic, complemented
10#89810C211yesmaximal, characteristic, complemented
11#89799C211yesmaximal, characteristic, complemented
12#89794C211yesmaximal, characteristic, complemented
13#89787C211yesmaximal, characteristic, complemented
14#89781C211yesmaximal, characteristic, complemented
15#89771C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#126—21no—
18C7×#124—21no—
19C7×#124C2211yescharacteristic
20C7×#123—21no—
21C7×C22⋊Q8—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C7×#117C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1141C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6493C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6457—21no—
36#6457—21no—
37#6447—21no—
38#6447—21no—
39#6437C2211yescharacteristic
40C4×D28C2211yescharacteristic
41C4×D28C2211yescharacteristic
42#6433C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6383C2211yescharacteristic
48#6383C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3257—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×D28—21no—
69C2×D28—21no—
70C2×D28C2311yescharacteristic
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4D411yescharacteristic
78C7×C4⋊C4D411yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C7×C22⋊C4D411yescharacteristic
87C7×C22⋊C4D411yescharacteristic
88C42×C7—21no—
89C42×C7C2311yescharacteristic
90C42×C7C2311yescharacteristic
91#1143—41no—
92#1143—41no—
93#1141—41no—
94#1141—41no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103#1136—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#542D711yescharacteristic, complemented
110#540—71no—
111#479—71no—
112#478—71no—
113#476—71no—
114#474—71no—
115#469—71no—
116#468—71no—
117#466—71no—
118#465—71no—
119#463—71no—
120#463—71no—
121#460—71no—
122#459—71no—
123#424—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C22×D7—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8—81no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—81no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—81no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—81no—
147C22⋊D7—81no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150D28—21no—
151D28—21no—
152D28—21no—
153D28—21no—
154D28—81no—
155D28—81no—
156C4×D7—81no—
157C4×D7—81no—
158C7⋊C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C2×SD16—71no—
163C2×SD16—71no—
164C2×D8—71no—
165C2×D8—71no—
166C2×M16—71no—
167C22×C8—71no—
168C4⋊Q8C2×D711yescharacteristic
169#127—71no—
170#126—21no—
171#125—71no—
172#124—21no—
173#124C2×D711yescharacteristic
174#123—21no—
175#123—141no—
176#123—141no—
177C22⋊Q8—21no—
178C22⋊Q8C2×D711yescharacteristic
179#121—71no—
180#121—71no—
181#121—141no—
182#121—141no—
183#121C2×D711yescharacteristic
184C4×Q8C2×D711yescharacteristic
185C4×D4—71no—
186C4×D4—71no—
187C4×D4C2×D711yescharacteristic
188#117C2×D711yescharacteristic
189#107—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194C4⋊C8—71no—
195C4⋊C8—71no—
196C4⋊C8—71no—
197#103—71no—
198#103—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C22×D411yescharacteristic
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7H(5,𝔽2)11yescharacteristic
227C4×C72−1+411yescharacteristic
228Dic7—41no—
229Dic7—41no—
230C2×Q8—21no—
231C2×Q8C22×D711yescharacteristic
232C2×D4—21no—
233C2×D4—71no—
234C2×D4—71no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4—141no—
239C2×D4C22×D711yescharacteristic
240C22×C4—21no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244SD16—281no—
245SD16—281no—
246D8—281no—
247D8—281no—
248M16—281no—
249C2×C8—71no—
250C2×C8—71no—
251C2×C8—71no—
252C2×C8—71no—
253C2×C8—281no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4C22⋊D711yescharacteristic
262C4⋊C4C22⋊D711yescharacteristic
263C4⋊C4C22×D711yescharacteristic
264C4⋊C4C22×D711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4C22⋊D711yescharacteristic
275C22⋊C4C22⋊D711yescharacteristic
276C42—21no—
277C42C22×D711yescharacteristic
278C42C22×D711yescharacteristic
279C2×C7—41no—
280C2×C7—81no—
281C2×C7#53711yescharacteristic
282C2×C7#57611yescharacteristic
283C2×C7#57711yescharacteristic
284D7—81no—
285D7—81no—
286C23—21no—
287C23—141no—
288C23—141no—
289C23C2×C22⋊D711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—81no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—141no—
297D4—141no—
298D4—141no—
299D4—141no—
300D4—561no—
301D4—561no—
302D4—561no—
303D4—561no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—561no—
313C2×C4—561no—
314C2×C4C2×C22⋊D711yescharacteristic
315C2×C4C2×C22⋊D711yescharacteristic
316C2×C4C2×C22⋊D711yescharacteristic
317C2×C4C2×C22⋊D711yescharacteristic
318C2×C4C2×C22⋊D711yescharacteristic
319C2×C4C23×D711yescharacteristic
320C8—141no—
321C8—141no—
322C8—141no—
323C8—141no—
324C7#325711yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—281no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C22×C22⋊D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—281no—
347C4—281no—
348C4#655311yescharacteristic
349C4#655411yescharacteristic
350C2—41no—
351C2—81no—
352C2—561no—
353C2—561no—
354C2#9052411yescharacteristic
355C2#9052611yescharacteristic
356C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#542C2
socleC22×C7C22×D4
hypercenter#542D7
hypocenterC7#3257
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. C7×#542C2semidirect
2. #90226C2semidirect
3. #89859C2semidirect
4. #89856C2semidirect
5. #89829C2semidirect
6. #89829C2semidirect
7. #89821C2semidirect
8. #89820C2semidirect
9. #89811C2semidirect
10. #89810C2semidirect
11. #89799C2semidirect
12. #89794C2semidirect
13. #89787C2semidirect
14. #89781C2semidirect
15. #89771C2semidirect
109. #542D7semidirect
324. C7#3257semidirect

Maximal subgroups

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

subgroupindex
C7×#5422
#902262
#898592
#898562
#898292
#898212
#898202
#898112
#898102
#897992
#897942
#897872
#897812
#897712
#32577