Finite Group Databasebeta

← #11738511

Subgroups — #11738511

Order 896.

366 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90419C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic
3C7×#476C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89779C211yesmaximal, characteristic, complemented
6#89638C212yesmaximal, complemented
7#89628C212yesmaximal, complemented
8#89618C211yesmaximal, characteristic, complemented
9#89607C211yesmaximal, characteristic, complemented
10#89556C212yesmaximal, complemented
11#89536C212yesmaximal, complemented
12C2×#1157—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×D4C2211yescharacteristic
15#6509—21no—
16Q8×Dic7—21no—
17#6503C2211yescharacteristic
18#6501C2211yescharacteristic
19C4×C22⋊D7—21no—
20#6460—21no—
21#6458C2211yescharacteristic
22#6455C2212yes—
23#6453C2212yes—
24#6451C2211yescharacteristic
25#6450—21no—
26#6448—21no—
27#6447—21no—
28C4×D28C2211yescharacteristic
29C42×D7C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#106C2212yes—
32C7×C4⋊C8C2211yescharacteristic
33C7×#102C2212yes—
34#6406C2212yes—
35#6395C2212yes—
36(C2×D7)⋊C8C2212yes—
37#6392C2212yes—
38#6389C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3235—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48C2×C22⋊D7—21no—
49C2×C22⋊D7—21no—
50#1157—44no—
51C2×D28—21no—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7D411yescharacteristic
55C2×C4×D7D411yescharacteristic
56C2×C4×D7C2311yescharacteristic
57C2×Dic14—21no—
58C2×Dic14—21no—
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2312yes—
62C7×C4⋊C4C2311yescharacteristic
63C7×C22⋊C4—21no—
64C42×C7C2311yescharacteristic
65C22⋊Dic7—21no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—22no—
71C4⋊Dic7C2312yes—
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—21no—
74Dic7⋊C4—22no—
75C4×Dic7—22no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#533—71no—
81#532—71no—
82#501—72no—
83#490—72no—
84#482—72no—
85#482—72no—
86#476—71no—
87#476D711yescharacteristic, complemented
88#474—71no—
89#474—71no—
90#432—71no—
91C23×C7—21no—
92C22×D7—21no—
93C22×D7C2×D411yescharacteristic
94C7×Q8—42no—
95C7×D4—21no—
96C7×D4—21no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—42no—
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2411yescharacteristic
103C22⋊D7—42no—
104C22⋊D7—42no—
105C22⋊D7—42no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111D28—21no—
112D28—21no—
113D28—42no—
114C4×D7—22no—
115C4×D7—21no—
116C4×D7—21no—
117C4×D7—41no—
118C4×D7—42no—
119Dic14—42no—
120Dic14—42no—
121C7×C8—22no—
122C7⋊C8—22no—
123C2×#41—141no—
124C4⋊Q8—71no—
125C4⋊Q8—71no—
126C4⋊Q8C2×D711yescharacteristic
127#127—71no—
128#125—72no—
129#124—141no—
130C22⋊Q8—141no—
131C22⋊Q8—141no—
132#121—71no—
133#121—71no—
134#121—141no—
135C4×Q8—141no—
136C4×Q8—141no—
137C4×D4—71no—
138C4×D4—141no—
139C4×D4C2×D711yescharacteristic
140#117—72no—
141C2×C42—71no—
142#107—72no—
143#107—72no—
144#106—72no—
145#106C2×D712yes—
146C4⋊C8—71no—
147C4⋊C8C2×D711yescharacteristic
148#102—72no—
149#102—72no—
150#102—72no—
151#102C2×D712yes—
152C22⋊C8—72no—
153#97—72no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7C22×D411yescharacteristic
158C2×D7—21no—
159C2×D7—21no—
160C2×D7—21no—
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—41no—
164C4×C7—21no—
165C4×C7—41no—
166C4×C7—42no—
167C4×C7AGL(1,ℤ/8)12yes—
168C4×C72−1+411yescharacteristic
169C4×C72−1+411yescharacteristic
170Dic7—21no—
171Dic7—21no—
172Dic7—21no—
173Dic7—41no—
174Dic7—42no—
175#41—284no—
176C2×Q8—21no—
177C2×Q8—141no—
178C2×Q8—141no—
179C2×D4—71no—
180C2×D4—141no—
181C2×D4—141no—
182C2×D4—141no—
183C2×D4—141no—
184C2×D4C22×D711yescharacteristic
185C22×C4—21no—
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—141no—
190C2×C8—72no—
191C2×C8C22×D712yes—
192C4⋊C4—21no—
193C4⋊C4—72no—
194C4⋊C4—71no—
195C4⋊C4—141no—
196C4⋊C4—142no—
197C4⋊C4—142no—
198C4⋊C4—141no—
199C4⋊C4C22×D712yes—
200C4⋊C4C22×D711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—141no—
203C22⋊C4—142no—
204C22⋊C4—141no—
205C22⋊C4—141no—
206C42—71no—
207C42—71no—
208C42—71no—
209C42—142no—
210C42C22×D711yescharacteristic
211C2×C7—81no—
212C2×C7#53611yescharacteristic
213C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C7#57511yescharacteristic
215D7—41no—
216D7—81no—
217C23—21no—
218C23—71no—
219C23—141no—
220Q8—42no—
221Q8—282no—
222Q8—282no—
223D4—21no—
224D4—21no—
225D4—141no—
226D4—141no—
227D4—282no—
228D4—282no—
229D4—282no—
230D4—282no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—71no—
235C2×C4—71no—
236C2×C4—71no—
237C2×C4—142no—
238C2×C4—141no—
239C2×C4—141no—
240C2×C4—142no—
241C2×C4—141no—
242C2×C4—281no—
243C2×C4—282no—
244C2×C4D4×D711yescharacteristic
245C2×C4D4×D711yescharacteristic
246C2×C4C23×D711yescharacteristic
247C8—22no—
248C8—142no—
249C7#323511yescharacteristic, complemented, Sylow
250C22—41no—
251C22—41no—
252C22—41no—
253C22—141no—
254C22—141no—
255C22—141no—
256C22—281no—
257C22—281no—
258C22—281no—
259C22C2×D4×D711yescharacteristic
260C4—21no—
261C4—41no—
262C4—42no—
263C4—141no—
264C4—141no—
265C4—141no—
266C4—281no—
267C4—282no—
268C4#647712yes—
269C4#655111yescharacteristic
270C4#655411yescharacteristic
271C2—81no—
272C2—281no—
273C2—561no—
274C2#9031911yescharacteristic
275C2C2×#647711yescharacteristic
276C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#476C2
socleC22×C7C22×D4
prime radical#90239C2
hypercenter#476D7
hypocenterC7#3235
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#476C2semidirect
4. #89856C2semidirect
5. #89779C2semidirect
6. #89638C2semidirect
7. #89628C2semidirect
8. #89618C2semidirect
9. #89607C2semidirect
10. #89556C2semidirect
11. #89536C2semidirect
87. #476D7semidirect
249. C7#3235semidirect

Maximal subgroups

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

subgroupindex
#904192
#902392
C7×#4762
#898562
#897792
#896382
#896282
#896182
#896072
#895562
#895362
#32357