Finite Group Databasebeta

← #11739038

Subgroups — #11739038

Order 896.

368 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 129 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
2#90419C211yesmaximal, characteristic
3C7×#487C211yesmaximal, characteristic, complemented
4#89864C211yesmaximal, characteristic, complemented
5#89850C211yesmaximal, characteristic, complemented
6#89585C214yesmaximal
7#89562C214yesmaximal, complemented
8C4×C8×D7C211yesmaximal, characteristic
9#89453C211yesmaximal, characteristic, complemented
10C7×C4⋊Q8C2211yescharacteristic
11C7×C4⋊Q8C2211yescharacteristic
12#6509—21no—
13#6509C2212yes—
14Q8×Dic7—21no—
15Q8×Dic7C2212yes—
16C2×#1144C2214yes—
17C2×C8×D7C2212yes—
18#6460—21no—
19#6460C2212yes—
20#6455—21no—
21#6455—21no—
22#6453—21no—
23#6453—21no—
24#6450—21no—
25#6450C2212yes—
26C42×D7C2211yescharacteristic
27#6432C2211yescharacteristic
28#6432C2211yescharacteristic
29C7×#103C2214yes—
30C4×C7×C8C2211yescharacteristic
31#6390C2214yes—
32C8×Dic7C2212yes—
33#6381C2214yes—
34C4×C7⋊C8C2211yescharacteristic
35#3170—71nomaximal, Sylow
36C2×C7×Q8—21no—
37C2×C7×Q8C2312yes—
38C2×C4×D7D411yescharacteristic
39C2×C4×D7D411yescharacteristic
40C2×C4×D7C2311yescharacteristic
41C2×Dic14—21no—
42C2×Dic14C2312yes—
43C2×C7×C8C2312yes—
44C7×C4⋊C4—21no—
45C7×C4⋊C4—21no—
46C7×C4⋊C4—21no—
47C7×C4⋊C4C2312yes—
48C42×C7C2311yescharacteristic
49#1144—28no—
50(C2×D7)⋊C4—22no—
51(C2×D7)⋊C4—22no—
52C4⋊Dic7—21no—
53C4⋊Dic7—21no—
54C4⋊Dic7—21no—
55C4⋊Dic7—22no—
56C4⋊Dic7—22no—
57C4⋊Dic7C2312yes—
58Dic7⋊C4—22no—
59Dic7⋊C4—22no—
60C4×Dic7—22no—
61C4×Dic7—22no—
62C4×Dic7D411yescharacteristic
63C4×Dic7D411yescharacteristic
64C4×Dic7C2311yescharacteristic
65C2×C7⋊C8C2312yes—
66C8×D7—24no—
67#533—71no—
68#533—71no—
69#500—71no—
70#494—71no—
71#487—71no—
72#487D711yescharacteristic, complemented
73#467—74no—
74C4×Q16—74no—
75C2×C4×C8—71no—
76C22×D7C2×D411yescharacteristic
77C7×Q8—24no—
78C7×Q8—42no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7C2×D411yescharacteristic
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2411yescharacteristic
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7C2×D411yescharacteristic
87C2×Dic7C2×D411yescharacteristic
88C2×Dic7C2×D411yescharacteristic
89C4×D7—22no—
90C4×D7—22no—
91C4×D7Q1614yescomplemented
92Dic14—24no—
93Dic14—42no—
94C7×C8—22no—
95C7⋊C8#4114yes—
96C2×Q16—74no—
97C22×C8—72no—
98C4⋊Q8—71no—
99C4⋊Q8—71no—
100C4⋊Q8C2×D711yescharacteristic
101C4⋊Q8C2×D711yescharacteristic
102#125—141no—
103#125—141no—
104C22⋊Q8—72no—
105C22⋊Q8—72no—
106C22⋊Q8—141no—
107C22⋊Q8—141no—
108C4×Q8—72no—
109C4×Q8—72no—
110C4×Q8—141no—
111C4×Q8—141no—
112#117—141no—
113#117—141no—
114C2×C42—71no—
115#107—74no—
116#103—74no—
117#103C2×D714yes—
118C4×C8—72no—
119C4×C8—71no—
120C4×C8C2×D711yescharacteristic
121C22×C7C22×D411yescharacteristic
122C2×D7—21no—
123C2×D7—21no—
124C2×D7C2×Q1612yes—
125C4×C7—42no—
126C4×C7—42no—
127C4×C7C2×Q1612yes—
128C4×C7#13512yes—
129C4×C7C2×#4112yes—
130Dic7—21no—
131Dic7—21no—
132Dic7—42no—
133Dic7—42no—
134Dic7C2×Q1612yes—
135C2×Q8—21no—
136C2×Q8—72no—
137C2×Q8—141no—
138C2×Q8C22×D712yes—
139C22×C4—71no—
140C22×C4—71no—
141C22×C4—71no—
142Q16—148no—
143C2×C8—72no—
144C2×C8—144no—
145C2×C8C22×D712yes—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—72no—
150C4⋊C4—142no—
151C4⋊C4—142no—
152C4⋊C4—142no—
153C4⋊C4—142no—
154C4⋊C4—141no—
155C4⋊C4—141no—
156C4⋊C4—141no—
157C4⋊C4C22×D712yes—
158C22⋊C4—142no—
159C22⋊C4—142no—
160C42—71no—
161C42—71no—
162C42—71no—
163C42—142no—
164C42—142no—
165C42C22×D711yescharacteristic
166C2×C7#53211yescharacteristic
167C2×C7C22×Q1611yescharacteristic
168C2×C7C2×#13511yescharacteristic
169D7—22no—
170C23—71no—
171Q8—24no—
172Q8—42no—
173Q8—144no—
174Q8—282no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—71no—
178C2×C4—74no—
179C2×C4—71no—
180C2×C4—71no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4D4×D711yescharacteristic
186C2×C4D4×D711yescharacteristic
187C2×C4C23×D711yescharacteristic
188C8—22no—
189C8—74no—
190C7#317011yescharacteristic, complemented, Sylow
191C22—72no—
192C22—141no—
193C22—141no—
194C22C2×D4×D711yescharacteristic
195C4—42no—
196C4—42no—
197C4—72no—
198C4—141no—
199C4—141no—
200C4—282no—
201C4—282no—
202C4#647912yes—
203C4D7×Q1612yes—
204C4D7×#4112yes—
205C2—142no—
206C2#9035911yescharacteristic
207C2C2×#647911yescharacteristic
208C2C2×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#3170
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. #89864C2semidirect
5. #89850C2semidirect
7. #89562C2semidirect
9. #89453C2semidirect
72. #487D7semidirect
91. C4×D7Q16semidirect
190. C7#3170semidirect

Maximal subgroups

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

subgroupindex
#904192
C7×#4872
#898642
#898502
#895852
#895622
C4×C8×D72
#894532
#31707