Finite Group Databasebeta

← #11738512

Subgroups — #11738512

Order 896.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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
1D7×C4⋊Q8C211yesmaximal, characteristic, complemented
2#90248C211yesmaximal, characteristic
3C7×#476C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89780C211yesmaximal, characteristic, complemented
6#89634C212yesmaximal, complemented
7#89628C212yesmaximal, complemented
8#89622C211yesmaximal, characteristic, complemented
9#89607C211yesmaximal, characteristic, complemented
10#89546C212yesmaximal
11#89537C212yesmaximal, complemented
12C2×Q8×D7—21no—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×D4C2211yescharacteristic
15#6507—21no—
16#6501C2211yescharacteristic
17#6499C2211yescharacteristic
18C4×C22⋊D7—21no—
19#6489—21no—
20#6460C2211yescharacteristic
21D7×C4⋊C4—21no—
22D7×C4⋊C4C2212yes—
23#6453C2211yescharacteristic
24#6451—21no—
25#6451C2212yes—
26#6446—21no—
27#6442—21no—
28C42×D7C2211yescharacteristic
29#6432C2211yescharacteristic
30C4×Dic14C2211yescharacteristic
31C7×#106C2212yes—
32C7×C4⋊C8C2211yescharacteristic
33C7×#102C2212yes—
34#6406C2212yes—
35(C2×D7)⋊C8C2212yes—
36#6392C2212yes—
37#6390C2212yes—
38#6389C2212yes—
39#6384C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3238—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48Q8×D7—44no—
49C2×C4×D7—22no—
50C2×C4×D7D411yescharacteristic
51C2×C4×D7D411yescharacteristic
52C2×C4×D7C2311yescharacteristic
53C2×Dic14—21no—
54C2×Dic14—22no—
55C2×Dic14C2311yescharacteristic
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4C2312yes—
59C7×C4⋊C4C2311yescharacteristic
60C7×C22⋊C4—21no—
61C42×C7C2311yescharacteristic
62C22⋊Dic7—21no—
63C22⋊Dic7—21no—
64C22⋊Dic7—21no—
65(C2×D7)⋊C4—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80C2×C4⋊Q8—71no—
81#529—71no—
82#501—72no—
83#489—72no—
84#483—72no—
85#480—72no—
86#477—71no—
87#477—71no—
88#476—71no—
89#476D711yescharacteristic, complemented
90#432—71no—
91C23×C7—21no—
92C22×D7C2×D411yescharacteristic
93C7×Q8—42no—
94C7×D4—21no—
95C7×D4—21no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—42no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2×D411yescharacteristic
101C2×C4×C7C2411yescharacteristic
102C22⋊D7—42no—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C4×D7—22no—
110C4×D7—22no—
111C4×D7—21no—
112C4×D7—21no—
113C4×D7—44no—
114Dic14—21no—
115Dic14—21no—
116Dic14—44no—
117Dic14—42no—
118C7×C8—22no—
119C7⋊C8—22no—
120C22×Q8—141no—
121C4⋊Q8—72no—
122C4⋊Q8—71no—
123C4⋊Q8—141no—
124C4⋊Q8—141no—
125C4⋊Q8C2×D711yescharacteristic
126#126—141no—
127#125—71no—
128#124—71no—
129#123—141no—
130C22⋊Q8—71no—
131#121—71no—
132C4×Q8—71no—
133C4×D4—141no—
134C4×D4C2×D711yescharacteristic
135#117—141no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—141no—
138C2×C42—71no—
139#107—72no—
140#107—72no—
141#106—72no—
142#106C2×D712yes—
143C4⋊C8—71no—
144C4⋊C8C2×D711yescharacteristic
145#103—72no—
146#103—72no—
147#102—72no—
148#102C2×D712yes—
149C22⋊C8—72no—
150#97—72no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7C22×D411yescharacteristic
155C2×D7—21no—
156C2×D7—21no—
157C2×D7AGL(1,ℤ/8)12yes—
158C4×C7—21no—
159C4×C7—42no—
160C4×C7—41no—
161C4×C7#13712yes—
162C4×C72−1+411yescharacteristic
163C4×C72−1+411yescharacteristic
164Dic7—21no—
165Dic7—21no—
166Dic7—21no—
167Dic7—41no—
168Dic7—42no—
169Dic7—41no—
170C2×Q8—21no—
171C2×Q8—71no—
172C2×Q8—142no—
173C2×Q8—141no—
174C2×Q8—284no—
175C2×D4—141no—
176C2×D4C22×D711yescharacteristic
177C22×C4—21no—
178C22×C4—71no—
179C22×C4—71no—
180C22×C4—71no—
181C22×C4—142no—
182C2×C8—72no—
183C2×C8C22×D712yes—
184C4⋊C4—21no—
185C4⋊C4—72no—
186C4⋊C4—71no—
187C4⋊C4—142no—
188C4⋊C4—142no—
189C4⋊C4—141no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4—141no—
193C4⋊C4—141no—
194C4⋊C4C22×D712yes—
195C4⋊C4C22×D711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—141no—
198C22⋊C4—141no—
199C22⋊C4—141no—
200C22⋊C4—141no—
201C42—71no—
202C42—71no—
203C42—71no—
204C42—141no—
205C42C22×D711yescharacteristic
206C2×C7—81no—
207C2×C7#53611yescharacteristic
208C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C7C2×#13711yescharacteristic
210D7—22no—
211C23—21no—
212C23—71no—
213Q8—42no—
214Q8—141no—
215Q8—141no—
216Q8—282no—
217Q8—284no—
218D4—21no—
219D4—21no—
220D4—282no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—71no—
225C2×C4—71no—
226C2×C4—71no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—141no—
230C2×C4—141no—
231C2×C4—141no—
232C2×C4—141no—
233C2×C4—142no—
234C2×C4—284no—
235C2×C4D4×D711yescharacteristic
236C2×C4D4×D711yescharacteristic
237C2×C4C23×D711yescharacteristic
238C8—22no—
239C8—142no—
240C7#323811yescharacteristic, complemented, Sylow
241C22—41no—
242C22—41no—
243C22—41no—
244C22—72no—
245C22—141no—
246C22—141no—
247C22C2×D4×D711yescharacteristic
248C4—21no—
249C4—41no—
250C4—42no—
251C4—141no—
252C4—141no—
253C4—141no—
254C4—282no—
255C4—281no—
256C4—281no—
257C4#647812yes—
258C4#655111yescharacteristic
259C4#655411yescharacteristic
260C2—81no—
261C2—142no—
262C2#9031911yescharacteristic
263C2C2×#647811yescharacteristic
264C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#476C2
socleC22×C7C22×D4
prime radical#6501C22
hypercenter#476D7
hypocenterC7#3238
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. D7×C4⋊Q8C2semidirect
3. C7×#476C2semidirect
4. #89863C2semidirect
5. #89780C2semidirect
6. #89634C2semidirect
7. #89628C2semidirect
8. #89622C2semidirect
9. #89607C2semidirect
11. #89537C2semidirect
89. #476D7semidirect
240. C7#3238semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#902482
C7×#4762
#898632
#897802
#896342
#896282
#896222
#896072
#895462
#895372
#32387