Finite Group Databasebeta

← #11740997

Subgroups — #11740997

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×Q8C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic
3#90000C211yesmaximal, characteristic, complemented
4#90000C211yesmaximal, characteristic, complemented
5#89997C211yesmaximal, characteristic, complemented
6C2×#6409C211yesmaximal, characteristic, complemented
7#89862C211yesmaximal, characteristic, complemented
8#89854C211yesmaximal, characteristic, complemented
9#89824C211yesmaximal, characteristic, complemented
10#89823C211yesmaximal, characteristic, complemented
11#89817C211yesmaximal, characteristic, complemented
12#89815C211yesmaximal, characteristic, complemented
13#89802C211yesmaximal, characteristic, complemented
14#89794C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic
16C22×C7×Q8C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C7×C4⋊Q8—21no—
19C7×C4⋊Q8C2211yescharacteristic
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4—21no—
26C4×C7×D4C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30#6498—21no—
31#6489C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33#6487C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6452—21no—
38#6442—21no—
39#6440—21no—
40#6433C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6409C2211yescharacteristic
50#6409C2211yescharacteristic
51C22⋊(C7⋊C8)C2211yescharacteristic
52C22⋊(C7⋊C8)C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3312—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—21no—
62C2×C7×Q8—22no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C2×C7×Q8C2311yescharacteristic
67C2×C7×Q8C2311yescharacteristic
68C2×C7×Q8C2311yescharacteristic
69C2×C7×D4D411yescharacteristic
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C22×Dic7—21no—
75C2×Dic14C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4D411yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C7×C22⋊C4D411yescharacteristic
86C7×C22⋊C4D411yescharacteristic
87C42×C7—21no—
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92#1144—22no—
93#1144—41no—
94#1144—41no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#552—71no—
113D4×Q8D711yescharacteristic
114#491—71no—
115#485—71no—
116#483—71no—
117#479—71no—
118#470—71no—
119#467—71no—
120#462—71no—
121#451—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#419—71no—
126C2×#103—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—42no—
135C7×Q8—42no—
136C7×Q8#4112yes—
137C7×D4—22no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2411yescharacteristic
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—42no—
158Dic14—21no—
159Dic14—21no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C22×Q8C2×D711yescharacteristic
165C22×Q8C2×D711yescharacteristic
166C2×Q16—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×M16—71no—
170C22×C8—71no—
171C4⋊Q8—21no—
172C4⋊Q8C2×D711yescharacteristic
173#126—141no—
174#125—71no—
175#125—141no—
176#123—141no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8C2×D711yescharacteristic
182C22⋊Q8C2×D711yescharacteristic
183C4×Q8—71no—
184C4×Q8C2×D711yescharacteristic
185C4×D4—21no—
186C4×D4C2×D711yescharacteristic
187#117—71no—
188#117—141no—
189C2×C4⋊C4—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204C22⋊C8—71no—
205C22⋊C8—71no—
206#97—71no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7#13712yes—
213C22×C7C22×D411yescharacteristic
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—22no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C7H(5,𝔽2)11yescharacteristic
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—71no—
235C2×Q8C22×D711yescharacteristic
236C2×Q8C22×D711yescharacteristic
237C2×Q8C22×D711yescharacteristic
238C2×D4C22⋊D711yescharacteristic
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—141no—
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244Q16—142no—
245Q16—281no—
246Q16—281no—
247M16—281no—
248C2×C8—71no—
249C2×C8—71no—
250C2×C8—71no—
251C2×C8—71no—
252C2×C8—281no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
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⋊C4C22⋊D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4C22⋊D711yescharacteristic
275C22⋊C4C22⋊D711yescharacteristic
276C42—21no—
277C42—141no—
278C42—141no—
279C42C22×D711yescharacteristic
280C2×C7—22no—
281C2×C7—41no—
282C2×C7#54611yescharacteristic
283C2×C7C2×#13711yescharacteristic
284C2×C7#57811yescharacteristic
285C23C2×C22⋊D711yescharacteristic
286C23C2×C22⋊D711yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—42no—
293Q8—42no—
294Q8—141no—
295Q8—141no—
296Q8#115912yes—
297D4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—282no—
313C2×C4C2×C22⋊D711yescharacteristic
314C2×C4C2×C22⋊D711yescharacteristic
315C2×C4C2×C22⋊D711yescharacteristic
316C2×C4C2×C22⋊D711yescharacteristic
317C2×C4C23×D711yescharacteristic
318C8—141no—
319C8—141no—
320C8—141no—
321C8—141no—
322C7#331211yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#650512yes—
329C22C22×C22⋊D711yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4C2×#115911yescharacteristic
343C4#654811yescharacteristic
344C2—22no—
345C2—41no—
346C2#9049711yescharacteristic
347C2C2×#650511yescharacteristic
348C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×D4×Q8C2
socleC22×C7C22×D4
prime radicalC2×C7×D4D4
hypercenterD4×Q8D7
hypocenterC7#3312
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
3. #90000C2semidirect
4. #90000C2semidirect
5. #89997C2semidirect
6. C2×#6409C2semidirect
7. #89862C2semidirect
8. #89854C2semidirect
9. #89824C2semidirect
10. #89823C2semidirect
11. #89817C2semidirect
12. #89815C2semidirect
13. #89802C2semidirect
14. #89794C2semidirect
322. C7#3312semidirect

Maximal subgroups

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

subgroupindex
C7×D4×Q82
#902322
#900002
#899972
C2×#64092
#898622
#898542
#898242
#898232
#898172
#898152
#898022
#897942
#897832
#33127