Finite Group Databasebeta

← #11741070

Subgroups — #11741070

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#554C211yesmaximal, characteristic, complemented
2#90256C211yesmaximal, characteristic
3#89859C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89821C211yesmaximal, characteristic, complemented
6#89820C211yesmaximal, characteristic, complemented
7#89817C211yesmaximal, characteristic, complemented
8#89816C211yesmaximal, characteristic, complemented
9#89798C211yesmaximal, characteristic, complemented
10#89794C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89740C211yesmaximal, characteristic, complemented
13#89739C211yesmaximal, characteristic, complemented
14#89736C211yesmaximal, characteristic, complemented
15C2×#6381C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#125—21no—
18C7×C22⋊Q8—21no—
19C7×C22⋊Q8—21no—
20C7×C22⋊Q8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4—21no—
24C4×C7×D4C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28#6502—21no—
29#6493C2211yescharacteristic
30#6493C2211yescharacteristic
31#6489C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6461—21no—
36#6446—21no—
37(C22⋊D7)⋊C4—21no—
38C4×D28C2211yescharacteristic
39#6433C2211yescharacteristic
40#6409C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6383C2211yescharacteristic
46#6383C2211yescharacteristic
47#6383C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6381C2211yescharacteristic
52#6381C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3346—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4D411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C7×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4D411yescharacteristic
85C7×C22⋊C4D411yescharacteristic
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89#1143—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7—21no—
100#1136—22no—
101C2×C7⋊C8—22no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#554D711yescharacteristic, complemented
107#553—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#474—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435—71no—
119#428—71no—
120C2×#107—71no—
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—81no—
127C7×D4—22no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—81no—
137C2×C4×C7—81no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C22⋊D7—42no—
144C22⋊D7—81no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—81no—
149D28—21no—
150D28—21no—
151C4×D7—81no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C7⋊C8#4112yes—
156C2×SD16—71no—
157C2×M16—71no—
158C22×C8—71no—
159C4⋊Q8C2×D711yescharacteristic
160#126—141no—
161#125—21no—
162#125—71no—
163#123—141no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8C2×D711yescharacteristic
167C22⋊Q8C2×D711yescharacteristic
168#121—71no—
169#121—71no—
170#121—141no—
171C4×Q8C2×D711yescharacteristic
172C4×D4—21no—
173C4×D4—71no—
174C4×D4—141no—
175C4×D4C2×D711yescharacteristic
176#117—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4C2×D711yescharacteristic
179C2×C4⋊C4C2×D711yescharacteristic
180#107—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#106—71no—
187#106—71no—
188#106—71no—
189C4⋊C8—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196C22⋊C8—71no—
197C22⋊C8—71no—
198#97—71no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7#13712yes—
205C22×C7C22×D411yescharacteristic
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7C2×#4111yescharacteristic
218C4×C72−1+411yescharacteristic
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—21no—
223C2×Q8C22×D711yescharacteristic
224C2×D4—71no—
225C2×D4—141no—
226C2×D4—141no—
227C2×D4C22⋊D711yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4C22×D711yescharacteristic
233C22×C4C22×D711yescharacteristic
234SD16—142no—
235M16—142no—
236C2×C8—71no—
237C2×C8—71no—
238C2×C8—71no—
239C2×C8—71no—
240C2×C8—142no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22⋊D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4C22⋊D711yescharacteristic
264C22⋊C4C22⋊D711yescharacteristic
265C42—21no—
266C42—141no—
267C42C22×D711yescharacteristic
268C2×C7—22no—
269C2×C7—41no—
270C2×C7#54711yescharacteristic
271C2×C7C2×#13711yescharacteristic
272C2×C7#57611yescharacteristic
273D7—81no—
274C23—141no—
275C23C2×C22⋊D711yescharacteristic
276C23C2×C22⋊D711yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—81no—
280D4—22no—
281D4—141no—
282D4—141no—
283D4—282no—
284D4—561no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—141no—
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×C4C23×D711yescharacteristic
305C8—72no—
306C8—141no—
307C8—141no—
308C8—141no—
309C7#334611yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—281no—
316C22—281no—
317C22—281no—
318C22#650512yes—
319C22C22×C22⋊D711yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—281no—
329C4—281no—
330C4—281no—
331C4D7×#4111yescharacteristic
332C4#655411yescharacteristic
333C2—22no—
334C2—41no—
335C2—561no—
336C2C2×#650511yescharacteristic
337C2#9051511yescharacteristic
338C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#554C2
socleC22×C7C22×D4
prime radical#90256C2
hypercenter#554D7
hypocenterC7#3346
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C7×#554C2semidirect
3. #89859C2semidirect
4. #89852C2semidirect
5. #89821C2semidirect
6. #89820C2semidirect
7. #89817C2semidirect
8. #89816C2semidirect
9. #89798C2semidirect
10. #89794C2semidirect
11. #89783C2semidirect
12. #89740C2semidirect
13. #89739C2semidirect
14. #89736C2semidirect
15. C2×#6381C2semidirect
106. #554D7semidirect
309. C7#3346semidirect

Maximal subgroups

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

subgroupindex
C7×#5542
#902562
#898592
#898522
#898212
#898202
#898172
#898162
#897982
#897942
#897832
#897402
#897392
#897362
C2×#63812
#33467