Finite Group Databasebeta

← #11739049

Subgroups — #11739049

Order 896.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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
2D7×C4⋊Q8C211yesmaximal, characteristic
3C7×#487C211yesmaximal, characteristic, complemented
4#89865C211yesmaximal, characteristic, complemented
5#89864C211yesmaximal, characteristic, complemented
6#89580C214yesmaximal
7#89565C214yesmaximal, complemented
8#89471C211yesmaximal, characteristic, complemented
9#89457C211yesmaximal, characteristic
10C2×Q8×D7C2212yes—
11C7×C4⋊Q8C2211yescharacteristic
12C7×C4⋊Q8C2211yescharacteristic
13#6509—21no—
14Q8×Dic7—21no—
15#6507C2212yes—
16C2×#1144C2214yes—
17C2×Dic28C2214yes—
18#6460C2212yes—
19#6455—21no—
20D7×C4⋊C4—21no—
21D7×C4⋊C4—21no—
22#6453—21no—
23#6451—21no—
24#6451—21no—
25#6450C2212yes—
26C42×D7C2211yescharacteristic
27#6432C2211yescharacteristic
28#6432C2211yescharacteristic
29C7×#103C2214yes—
30C4×C7×C8C2211yescharacteristic
31(C2×D7)⋊C8C2212yes—
32Dic7⋊C8C2212yes—
33#6384C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3209—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8C2312yes—
38Q8×D7—44no—
39C2×C4×D7—22no—
40C2×C4×D7D411yescharacteristic
41C2×C4×D7D411yescharacteristic
42C2×C4×D7C2311yescharacteristic
43C2×Dic14—22no—
44C2×Dic14C2312yes—
45C2×Dic14C2312yes—
46C2×C7×C8C2312yes—
47C7×C4⋊C4—21no—
48C7×C4⋊C4—21no—
49C7×C4⋊C4—21no—
50C7×C4⋊C4C2312yes—
51C42×C7C2311yescharacteristic
52#1144—44no—
53(C2×D7)⋊C4—22no—
54C4⋊Dic7—21no—
55C4⋊Dic7—21no—
56C4⋊Dic7—21no—
57C4⋊Dic7—21no—
58C4⋊Dic7—22no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61Dic7⋊C4—22no—
62C4×Dic7—22no—
63C4×Dic7D411yescharacteristic
64C4×Dic7D411yescharacteristic
65C4×Dic7C2311yescharacteristic
66C2×C7⋊C8C2312yes—
67Dic28—44no—
68#533—71no—
69C2×C4⋊Q8—71no—
70#494—71no—
71#494—71no—
72#487—71no—
73#487D711yescharacteristic, complemented
74#462—74no—
75#451—74no—
76#431—71no—
77C22×D7C2×D411yescharacteristic
78C7×Q8—24no—
79C7×Q8—42no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2411yescharacteristic
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7C2×D411yescharacteristic
88C2×Dic7C2×D411yescharacteristic
89C2×Dic7C2×D411yescharacteristic
90C4×D7—22no—
91C4×D7—22no—
92C4×D7—44no—
93C4×D7Q1614yescomplemented
94Dic14—24no—
95Dic14—24no—
96Dic14—44no—
97C7×C8—22no—
98C7⋊C8—22no—
99C22×Q8—72no—
100C2×Q16—74no—
101C2×Q16—74no—
102C4⋊Q8—72no—
103C4⋊Q8—71no—
104C4⋊Q8—71no—
105C4⋊Q8—141no—
106C4⋊Q8—141no—
107C4⋊Q8C2×D711yescharacteristic
108C4⋊Q8C2×D711yescharacteristic
109#125—141no—
110C22⋊Q8—72no—
111C22⋊Q8—141no—
112C4×Q8—72no—
113C4×Q8—141no—
114#117—141no—
115C2×C4⋊C4—141no—
116C2×C4⋊C4—141no—
117C2×C42—71no—
118C4⋊C8—72no—
119#103—74no—
120#103C2×D714yes—
121C22⋊C8—72no—
122C4×C8—71no—
123C4×C8C2×D711yescharacteristic
124C22×C7C22×D411yescharacteristic
125C2×D7—21no—
126C2×D7—21no—
127C2×D7C2×Q1612yes—
128C4×C7—42no—
129C4×C7—42no—
130C4×C7C2×Q1612yes—
131C4×C7#13712yes—
132C4×C7H(5,𝔽2)12yes—
133Dic7—21no—
134Dic7—21no—
135Dic7—42no—
136Dic7—42no—
137Dic7C2×Q1612yes—
138C2×Q8—21no—
139C2×Q8—72no—
140C2×Q8—72no—
141C2×Q8—142no—
142C2×Q8—284no—
143C2×Q8C22×D712yes—
144C22×C4—71no—
145C22×C4—71no—
146C22×C4—71no—
147C22×C4—142no—
148Q16—284no—
149Q16—284no—
150C2×C8—72no—
151C2×C8C22×D712yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—142no—
156C4⋊C4—142no—
157C4⋊C4—142no—
158C4⋊C4—142no—
159C4⋊C4—141no—
160C4⋊C4—141no—
161C4⋊C4—141no—
162C4⋊C4—141no—
163C4⋊C4C22×D712yes—
164C22⋊C4—142no—
165C42—71no—
166C42—71no—
167C42—71no—
168C42—142no—
169C42C22×D711yescharacteristic
170C2×C7#53511yescharacteristic
171C2×C7C22×Q1611yescharacteristic
172C2×C7C2×#13711yescharacteristic
173D7—22no—
174C23—71no—
175Q8—24no—
176Q8—42no—
177Q8—144no—
178Q8—144no—
179Q8—284no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—71no—
183C2×C4—74no—
184C2×C4—71no—
185C2×C4—71no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—284no—
191C2×C4D4×D711yescharacteristic
192C2×C4D4×D711yescharacteristic
193C2×C4C23×D711yescharacteristic
194C8—22no—
195C8—142no—
196C7#320911yescharacteristic, complemented, Sylow
197C22—72no—
198C22—141no—
199C22—141no—
200C22C2×D4×D711yescharacteristic
201C4—42no—
202C4—42no—
203C4—72no—
204C4—141no—
205C4—141no—
206C4—282no—
207C4—282no—
208C4#647812yes—
209C4D7×Q1612yes—
210C4#655312yes—
211C2—142no—
212C2#9036511yescharacteristic
213C2C2×#647811yescharacteristic
214C2C2×D7×Q1611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#487D7
hypocenterC7#3209
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. C7×#487C2semidirect
4. #89865C2semidirect
5. #89864C2semidirect
7. #89565C2semidirect
8. #89471C2semidirect
73. #487D7semidirect
93. C4×D7Q16semidirect
196. C7#3209semidirect

Maximal subgroups

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

subgroupindex
#904192
D7×C4⋊Q82
C7×#4872
#898652
#898642
#895802
#895652
#894712
#894572
#32097