Finite Group Databasebeta

← #11737925

Subgroups — #11737925

Order 896.

384 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 123 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
2#90269C211yesmaximal, characteristic
3C7×#462C211yesmaximal, characteristic, complemented
4#89958C212yesmaximal
5#89953C212yesmaximal, complemented
6#89862C211yesmaximal, characteristic, complemented
7#89803C211yesmaximal, characteristic, complemented
8#89621C211yesmaximal, characteristic, complemented
9#89605C211yesmaximal, characteristic
10#89580C212yesmaximal
11#89564C212yesmaximal, complemented
12C2×Q8×D7C2212yes—
13C2×C7×Q16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16#6509C2211yescharacteristic
17#6507C2212yes—
18#6507C2211yescharacteristic
19C2×#1144C2212yes—
20C2×#1144C2212yes—
21C2×Dic28C2212yes—
22#6460C2211yescharacteristic
23#6459—21no—
24D7×C4⋊C4—21no—
25D7×C4⋊C4—21no—
26#6452—21no—
27#6451—21no—
28#6451—21no—
29#6451C2211yescharacteristic
30#6435—21no—
31C42×D7C2211yescharacteristic
32#6432C2211yescharacteristic
33C4×Dic14—21no—
34C4×Dic14C2211yescharacteristic
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6409C2212yes—
38(C2×D7)⋊C8C2212yes—
39#6390C2212yes—
40C8×Dic7C2212yes—
41#6384C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3209—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45C2×C7×Q8C2311yescharacteristic
46Q8×D7—42no—
47Q8×D7—42no—
48C2×C4×D7—22no—
49C2×C4×D7D411yescharacteristic
50C2×C4×D7D411yescharacteristic
51C2×C4×D7C2311yescharacteristic
52C2×Dic14—21no—
53C2×Dic14—22no—
54C2×Dic14C2311yescharacteristic
55C2×Dic14C2312yes—
56C7×Q16—42no—
57C2×C7×C8C2312yes—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2311yescharacteristic
62C42×C7—21no—
63C42×C7C2311yescharacteristic
64#1144—42no—
65#1144—42no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7C2311yescharacteristic
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—22no—
78Dic7⋊C4—22no—
79C4×Dic7—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84Dic28—42no—
85#533—71no—
86C2×C4⋊Q8—71no—
87#494—72no—
88#487—72no—
89#462—71no—
90#462—71no—
91#462—71no—
92#462D711yescharacteristic, complemented
93#451—72no—
94#451—72no—
95#431—71no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—21no—
98C7×Q8—22no—
99C7×Q8—21no—
100C7×Q8—22no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2411yescharacteristic
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113C4×D7—21no—
114C4×D7—21no—
115C4×D7—22no—
116C4×D7—44no—
117C4×D7Q1614yescomplemented
118Dic14—21no—
119Dic14—22no—
120Dic14—21no—
121Dic14—22no—
122Dic14—44no—
123Dic14—42no—
124C7×C8—22no—
125C7⋊C8—22no—
126C22×Q8—72no—
127C2×Q16—72no—
128C2×Q16—72no—
129C2×Q16—72no—
130C2×Q16C2×D712yes—
131C4⋊Q8—71no—
132C4⋊Q8—72no—
133C4⋊Q8—71no—
134C4⋊Q8—71no—
135C4⋊Q8—141no—
136C4⋊Q8—141no—
137C4⋊Q8C2×D711yescharacteristic
138#125—141no—
139C22⋊Q8—71no—
140C22⋊Q8—71no—
141C22⋊Q8—141no—
142C4×Q8—71no—
143C4×Q8—141no—
144C4×Q8C2×D711yescharacteristic
145#117—141no—
146C2×C4⋊C4—141no—
147C2×C4⋊C4—141no—
148C2×C42—71no—
149C4⋊C8—71no—
150C4⋊C8C2×D711yescharacteristic
151#103—72no—
152#103—72no—
153#103—72no—
154#103C2×D712yes—
155C22⋊C8—72no—
156C4×C8—72no—
157C22×C7C22×D411yescharacteristic
158C2×D7—21no—
159C2×D7—21no—
160C2×D7C2×Q1612yes—
161C4×C7—21no—
162C4×C7—42no—
163C4×C7—41no—
164C4×C7—41no—
165C4×C7C2×Q1612yes—
166C4×C7H(5,𝔽2)11yescharacteristic
167C4×C7H(5,𝔽2)11yescharacteristic
168Dic7—21no—
169Dic7—42no—
170Dic7—41no—
171Dic7—41no—
172Dic7C2×Q1612yes—
173Dic7#13712yes—
174C2×Q8—72no—
175C2×Q8—71no—
176C2×Q8—141no—
177C2×Q8—142no—
178C2×Q8—282no—
179C2×Q8—282no—
180C2×Q8C22×D712yes—
181C2×Q8C22×D711yescharacteristic
182C22×C4—71no—
183C22×C4—71no—
184C22×C4—71no—
185C22×C4—142no—
186Q16—42no—
187Q16—282no—
188Q16—282no—
189Q16—282no—
190C2×C8—72no—
191C2×C8C22×D712yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—71no—
196C4⋊C4—142no—
197C4⋊C4—142no—
198C4⋊C4—141no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4C22×D711yescharacteristic
207C22⋊C4—141no—
208C22⋊C4—141no—
209C42—21no—
210C42—71no—
211C42—71no—
212C42—71no—
213C42—141no—
214C42C22×D711yescharacteristic
215C2×C7#53511yescharacteristic
216C2×C7C22×Q1611yescharacteristic
217C2×C7C2×#13711yescharacteristic
218D7—22no—
219C23—71no—
220Q8—21no—
221Q8—22no—
222Q8—21no—
223Q8—22no—
224Q8—141no—
225Q8—142no—
226Q8—141no—
227Q8—142no—
228Q8—282no—
229Q8—284no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—71no—
234C2×C4—71no—
235C2×C4—74no—
236C2×C4—71no—
237C2×C4—142no—
238C2×C4—141no—
239C2×C4—141no—
240C2×C4—141no—
241C2×C4—141no—
242C2×C4—142no—
243C2×C4—284no—
244C2×C4D4×D711yescharacteristic
245C2×C4D4×D711yescharacteristic
246C2×C4C23×D711yescharacteristic
247C8—22no—
248C8—142no—
249C7#320911yescharacteristic, complemented, Sylow
250C22—72no—
251C22—141no—
252C22—141no—
253C22C2×D4×D711yescharacteristic
254C4—21no—
255C4—41no—
256C4—42no—
257C4—41no—
258C4—72no—
259C4—72no—
260C4—141no—
261C4—281no—
262C4—282no—
263C4—281no—
264C4D7×Q1612yes—
265C4#654811yescharacteristic
266C4#655311yescharacteristic
267C2—142no—
268C2#9029811yescharacteristic
269C2C2×D7×Q1611yescharacteristic
270C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#462D7
hypocenterC7#3209
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
5. #89953C2semidirect
6. #89862C2semidirect
7. #89803C2semidirect
8. #89621C2semidirect
11. #89564C2semidirect
92. #462D7semidirect
117. C4×D7Q16semidirect
249. C7#3209semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#902692
C7×#4622
#899582
#899532
#898622
#898032
#896212
#896052
#895802
#895642
#32097