Finite Group Databasebeta

← #11737906

Subgroups — #11737906

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90422C211yesmaximal, characteristic
2#90263C211yesmaximal, characteristic
3C7×#462C211yesmaximal, characteristic, complemented
4#89958C211yesmaximal, characteristic
5#89958C211yesmaximal, characteristic
6#89956C211yesmaximal, characteristic, complemented
7Q16×Dic7C211yesmaximal, characteristic, complemented
8#89862C211yesmaximal, characteristic, complemented
9#89794C211yesmaximal, characteristic, complemented
10#89612C211yesmaximal, characteristic
11#89601C211yesmaximal, characteristic, complemented
12#89587C211yesmaximal, characteristic
13#89586C211yesmaximal, characteristic
14#89562C211yesmaximal, characteristic, complemented
15#89561C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C7×C4⋊Q8C2211yescharacteristic
20C4×C7×Q8C2211yescharacteristic
21#6509C2211yescharacteristic
22#6509C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25Q8×Dic7C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28#6461—21no—
29#6461—21no—
30#6455—21no—
31#6455C2211yescharacteristic
32#6452—21no—
33#6452—21no—
34#6450—21no—
35#6450C2211yescharacteristic
36#6435—21no—
37#6435C2211yescharacteristic
38#6433C2211yescharacteristic
39C4×Dic14—21no—
40C4×Dic14C2211yescharacteristic
41C7×C4⋊C8C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54#6389C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3127—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63Q8×D7—22no—
64Q8×D7—22no—
65C2×C4×D7—21no—
66C2×C4×D7C2311yescharacteristic
67C2×Dic14—21no—
68C2×Dic14C2311yescharacteristic
69C7×Q16—22no—
70C7×Q16—22no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79#1144—22no—
80#1144—22no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85(C2×D7)⋊C4D411yescharacteristic
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#541—71no—
110#519—71no—
111#485—71no—
112#485—71no—
113#479—71no—
114#479—71no—
115#462—71no—
116#462D711yescharacteristic, complemented
117#451—71no—
118#451—71no—
119#441—71no—
120#441—71no—
121C4×Q16—71no—
122C4×Q16—71no—
123#433—71no—
124C22×D7C2×D411yescharacteristic
125C7×Q8—21no—
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8#4112yes—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C4×D7—24no—
146C4×D7—21no—
147C4×D7—21no—
148Dic14—24no—
149Dic14—21no—
150Dic14#4112yes—
151C7×C8—21no—
152C7×C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C22×Q8—71no—
156C2×Q16—71no—
157C2×Q16—71no—
158C2×Q16C2×D711yescharacteristic
159C2×Q16C2×D711yescharacteristic
160C4⋊Q8C2×D711yescharacteristic
161#126—141no—
162#126—141no—
163#125—71no—
164#125—141no—
165#125—141no—
166C22⋊Q8—71no—
167C22⋊Q8—71no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8—141no—
175C4×Q8C2×D711yescharacteristic
176#117—71no—
177#117—71no—
178#117—141no—
179#117—141no—
180#107—71no—
181#107—71no—
182#106—71no—
183#106—71no—
184C4⋊C8—71no—
185C4⋊C8C2×D711yescharacteristic
186#103—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#103C2×D711yescharacteristic
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197C4×C8—71no—
198C22×C7C22×D411yescharacteristic
199C2×D7—21no—
200C2×D7—21no—
201C2×D7—21no—
202C4×C7—22no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7C2×#4111yescharacteristic
209C4×C7C2×#4111yescharacteristic
210Dic7—21no—
211Dic7—22no—
212Dic7—21no—
213Dic7—21no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217C2×Q8—71no—
218C2×Q8—141no—
219C2×Q8—142no—
220C2×Q8—142no—
221C2×Q8C22×D711yescharacteristic
222C2×Q8C22×D711yescharacteristic
223C2×Q8C22×D711yescharacteristic
224C22×C4—71no—
225C22×C4—141no—
226Q16—22no—
227Q16—22no—
228Q16—142no—
229Q16—142no—
230C2×C8—71no—
231C2×C8—71no—
232C2×C8C22×D711yescharacteristic
233C2×C8C22×D711yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4C22×D711yescharacteristic
252C22⋊C4—71no—
253C22⋊C4—71no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C42—21no—
258C42—71no—
259C42—141no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7#52511yescharacteristic
267C2×C7#57811yescharacteristic
268C2×C7#57811yescharacteristic
269D7—41no—
270C23—71no—
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—72no—
277Q8—144no—
278Q8—141no—
279Q8#115912yes—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—71no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—144no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4D4×D711yescharacteristic
295C2×C4D4×D711yescharacteristic
296C2×C4C23×D711yescharacteristic
297C8—21no—
298C8—21no—
299C8—141no—
300C8—141no—
301C7#312711yescharacteristic, complemented, Sylow
302C22—141no—
303C22—141no—
304C22—141no—
305C22C2×D4×D711yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—141no—
313C4—141no—
314C4—141no—
315C4—142no—
316C4—281no—
317C4—281no—
318C4—281no—
319C4C2×#115911yescharacteristic
320C4D7×#4111yescharacteristic
321C2—281no—
322C2#9029711yescharacteristic
323C2#9045711yescharacteristic
324C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#462D7
hypocenterC7#3127
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. C7×#462C2semidirect
6. #89956C2semidirect
7. Q16×Dic7C2semidirect
8. #89862C2semidirect
9. #89794C2semidirect
11. #89601C2semidirect
14. #89562C2semidirect
15. #89561C2semidirect
116. #462D7semidirect
301. C7#3127semidirect

Maximal subgroups

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

subgroupindex
#904222
#902632
C7×#4622
#899582
#899562
Q16×Dic72
#898622
#897942
#896122
#896012
#895872
#895862
#895622
#895612
#31277