Finite Group Databasebeta

← #11739185

Subgroups — #11739185

Order 896.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#90416C211yesmaximal, characteristic
2D7×#125C211yesmaximal, characteristic, complemented
3C7×#491C211yesmaximal, characteristic, complemented
4#89854C211yesmaximal, characteristic, complemented
5#89839C211yesmaximal, characteristic, complemented
6#89575C214yesmaximal, complemented
7#89572C214yesmaximal, complemented
8#89485C211yesmaximal, characteristic, complemented
9#89479C211yesmaximal, characteristic, complemented
10C2×#1161C2212yes—
11C7×C4⋊Q8C2211yescharacteristic
12C7×#125C2211yescharacteristic
13#6510C2212yes—
14C2×#1131C2212yes—
15#6458—22no—
16#6456—22no—
17D7×C4⋊C4—22no—
18D7×C4⋊C4C2212yes—
19#6453C2212yes—
20#6452—22no—
21#6437C2211yescharacteristic
22C42×D7C2211yescharacteristic
23#6433C2211yescharacteristic
24C7×#103C2214yes—
25C7×#97C2211yescharacteristic
26#6409C2214yes—
27#6395C2214yes—
28C8×Dic7C2212yes—
29#6383C2214yes—
30#6377C2211yescharacteristic
31#3142—71nomaximal, Sylow
32C2×C7×Q8C2312yes—
33#1161—28no—
34C2×D28—22no—
35C2×D28—22no—
36C2×D28C2312yes—
37C2×C4×D7—22no—
38C2×C4×D7—22no—
39C2×C4×D7D412yes—
40C2×C4×D7C2311yescharacteristic
41C2×C7×C8C2312yes—
42C7×C4⋊C4—22no—
43C7×C4⋊C4—22no—
44C7×C4⋊C4C2312yes—
45C42×C7C2311yescharacteristic
46(C2×D7)⋊C4—24no—
47C4⋊Dic7—22no—
48C4⋊Dic7C2312yes—
49Dic7⋊C4—22no—
50Dic7⋊C4—22no—
51Dic7⋊C4—22no—
52C4×Dic7D412yes—
53C4×Dic7C2311yescharacteristic
54C2×C7⋊C8C2312yes—
55#1131—24no—
56#532—71no—
57C2×#125—71no—
58#491—71no—
59#491D711yescharacteristic, complemented
60#490—71no—
61#490—71no—
62#488—74no—
63#417—74no—
64C4×M16—71no—
65C22×D7—22no—
66C22×D7C2×D411yescharacteristic
67C7×Q8—24no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7C2×D412yes—
71C2×C4×C7C2411yescharacteristic
72C2×Dic7—22no—
73C2×Dic7C2×D412yes—
74C2×Dic7C2×D411yescharacteristic
75D28—24no—
76D28—44no—
77D28—44no—
78C4×D7—42no—
79C4×D7—44no—
80C4×D7—44no—
81C4×D7#4114yes—
82C7×C8—22no—
83C7⋊C8—22no—
84C2×#41—72no—
85C2×M16—72no—
86C4⋊Q8C2×D711yescharacteristic
87#127—71no—
88#125—72no—
89#125—71no—
90#125—142no—
91#125C2×D711yescharacteristic
92#124—72no—
93#121—142no—
94C4×D4—142no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—142no—
97C2×C42—71no—
98#103—74no—
99#103C2×D714yes—
100#102—74no—
101#102—74no—
102#97—71no—
103#97C2×D711yescharacteristic
104C4×C8—72no—
105C22×C7C22×D411yescharacteristic
106C2×D7—22no—
107C2×D7—42no—
108C2×D7—44no—
109C2×D7#12412yes—
110C4×C7—22no—
111C4×C7—42no—
112C4×C7—42no—
113C4×C7C2×#4112yes—
114Dic7—22no—
115Dic7—42no—
116Dic7#12412yes—
117#41—148no—
118C2×Q8C22×D712yes—
119C2×D4—72no—
120C2×D4—142no—
121C2×D4—142no—
122C22×C4—71no—
123C22×C4—72no—
124C22×C4—142no—
125C22×C4—142no—
126M16—144no—
127C2×C8—72no—
128C2×C8C22×D712yes—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—72no—
132C4⋊C4—142no—
133C4⋊C4—142no—
134C4⋊C4—142no—
135C4⋊C4—142no—
136C4⋊C4C22×D712yes—
137C22⋊C4—144no—
138C42—72no—
139C42—71no—
140C42C22×D711yescharacteristic
141C2×C7C2×#12411yescharacteristic
142C2×C7#57512yes—
143D7—22no—
144D7—82no—
145C23—71no—
146C23—142no—
147Q8—24no—
148D4—144no—
149D4—284no—
150D4—284no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—71no—
154C2×C4—74no—
155C2×C4—72no—
156C2×C4—142no—
157C2×C4—282no—
158C2×C4—284no—
159C2×C4—284no—
160C2×C4D4×D712yes—
161C2×C4C23×D711yescharacteristic
162C8—22no—
163C8—142no—
164C7#314211yescharacteristic, complemented, Sylow
165C22—72no—
166C22—142no—
167C22—284no—
168C22—282no—
169C22C2×D4×D711yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—72no—
174C4—142no—
175C4—282no—
176C4D7×#4112yes—
177C2—142no—
178C2—562no—
179C2D7×#12411yescharacteristic
180C2#9046812yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
prime radical#90416C2
hypercenter#491D7
hypocenterC7#3142
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. D7×#125C2semidirect
3. C7×#491C2semidirect
4. #89854C2semidirect
5. #89839C2semidirect
6. #89575C2semidirect
7. #89572C2semidirect
8. #89485C2semidirect
9. #89479C2semidirect
59. #491D7semidirect
164. C7#3142semidirect

Maximal subgroups

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

subgroupindex
#904162
D7×#1252
C7×#4912
#898542
#898392
#895752
#895722
#894852
#894792
#31427