Finite Group Databasebeta

← #11740963

Subgroups — #11740963

Order 896.

376 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×#547C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic, complemented
3#89863C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89814C211yesmaximal, characteristic, complemented
6#89813C211yesmaximal, characteristic, complemented
7#89807C211yesmaximal, characteristic, complemented
8#89805C211yesmaximal, characteristic, complemented
9#89790C211yesmaximal, characteristic, complemented
10#89783C211yesmaximal, characteristic, complemented
11#89781C211yesmaximal, characteristic, complemented
12#89743C211yesmaximal, characteristic, complemented
13#89742C211yesmaximal, characteristic, complemented
14#89736C211yesmaximal, characteristic, complemented
15C2×#6381C211yesmaximal, characteristic, complemented
16C2×C7×#41—21no—
17C7×C4⋊Q8C2211yescharacteristic
18C7×#123—21no—
19C7×#123—21no—
20C7×C22⋊Q8—21no—
21C7×#121C2211yescharacteristic
22C7×#121C2211yescharacteristic
23C4×C7×D4C2211yescharacteristic
24C4×C7×D4C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27#6498—21no—
28C2×#1142C2211yescharacteristic
29#6489C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31#6487C2211yescharacteristic
32#6487C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6452—21no—
36#6442—21no—
37#6440—21no—
38#6433C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40#6406C2211yescharacteristic
41#6406C2211yescharacteristic
42#6406C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
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#3343—71nomaximal, Sylow
60C7×#41—44no—
61C2×C7×Q8—21no—
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4D411yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C22×Dic7—21no—
72C2×Dic14C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4D411yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4D411yescharacteristic
84C7×C22⋊C4D411yescharacteristic
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89#1142—22no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101#1136—22no—
102C2×C7⋊C8—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#552—71no—
108#547D711yescharacteristic
109#501—71no—
110#484—71no—
111#483—71no—
112#482—71no—
113#480—71no—
114#478—71no—
115#477—71no—
116#468—71no—
117#465—71no—
118#440—71no—
119#435—71no—
120#428—71no—
121C2×#107—71no—
122C23×C7—21no—
123C23×C7C2×D411yescharacteristic
124C23×C7C2×D411yescharacteristic
125C7×Q8—42no—
126C7×D4—22no—
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—42no—
130C7×D4—42no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—42no—
151Dic14—21no—
152Dic14—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C7⋊C8#4112yescomplemented
157C2×#41—21no—
158C2×SD16—71no—
159C2×M16—71no—
160C22×C8—71no—
161C4⋊Q8C2×D711yescharacteristic
162#126—141no—
163#125—71no—
164#125—141no—
165#123—21no—
166#123—21no—
167#123—141no—
168C22⋊Q8—21no—
169C22⋊Q8—71no—
170C22⋊Q8—71no—
171#121C2×D711yescharacteristic
172#121C2×D711yescharacteristic
173C4×Q8—71no—
174C4×D4C2×D711yescharacteristic
175C4×D4C2×D711yescharacteristic
176#117—71no—
177#117—141no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4C2×D711yescharacteristic
180C2×C4⋊C4C2×D711yescharacteristic
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197C22⋊C8—71no—
198C22⋊C8—71no—
199#97—71no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7AGL(1,ℤ/8)12yes—
209C22×C7C22×D411yescharacteristic
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7C2×#4111yescharacteristic
218C4×C72−1+411yescharacteristic
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223#41—44no—
224C2×Q8—21no—
225C2×Q8—71no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4C22⋊D711yescharacteristic
229C2×D4C22×D711yescharacteristic
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—141no—
234C22×C4C22×D711yescharacteristic
235C22×C4C22×D711yescharacteristic
236SD16—142no—
237M16—142no—
238C2×C8—71no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8—71no—
242C2×C8—142no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4C22⋊D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C4⋊C4C22×D711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4C22⋊D711yescharacteristic
266C22⋊C4C22⋊D711yescharacteristic
267C42—141no—
268C42—141no—
269C42C22×D711yescharacteristic
270C2×C7—22no—
271C2×C7—41no—
272C2×C7—81no—
273C2×C7#54711yescharacteristic
274C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
275C2×C7#57811yescharacteristic
276C23—21no—
277C23C2×C22⋊D711yescharacteristic
278C23C2×C22⋊D711yescharacteristic
279Q8—42no—
280Q8—141no—
281Q8—141no—
282D4—22no—
283D4—21no—
284D4—21no—
285D4—42no—
286D4—42no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—282no—
302C2×C4C2×C22⋊D711yescharacteristic
303C2×C4C2×C22⋊D711yescharacteristic
304C2×C4C2×C22⋊D711yescharacteristic
305C2×C4C2×C22⋊D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—72no—
308C8—141no—
309C8—141no—
310C8—141no—
311C7#334311yescharacteristic, complemented, Sylow
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#649512yes—
321C22C22×C22⋊D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—281no—
330C4—281no—
331C4—281no—
332C4—281no—
333C4D7×#4111yescharacteristic
334C4#655411yescharacteristic
335C2—22no—
336C2—41no—
337C2—81no—
338C2C2×#649511yescharacteristic
339C2#9051511yescharacteristic
340C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#547C2
socleC22×C7C22×D4
hypercenter#547D7
hypocenterC7#3343
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
2. #90232C2semidirect
3. #89863C2semidirect
4. #89852C2semidirect
5. #89814C2semidirect
6. #89813C2semidirect
7. #89807C2semidirect
8. #89805C2semidirect
9. #89790C2semidirect
10. #89783C2semidirect
11. #89781C2semidirect
12. #89743C2semidirect
13. #89742C2semidirect
14. #89736C2semidirect
15. C2×#6381C2semidirect
156. C7⋊C8#41semidirect
311. C7#3343semidirect

Maximal subgroups

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

subgroupindex
C7×#5472
#902322
#898632
#898522
#898142
#898132
#898072
#898052
#897902
#897832
#897812
#897432
#897422
#897362
C2×#63812
#33437