Finite Group Databasebeta

← #329×D71 #22586919

Subgroups — #329×D71 #22586919

Order 9088.

284 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 67 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#124×D71C211yesmaximal, characteristic
2#97×D71C212yesmaximal, complemented
3#17985739C211yesmaximal, characteristic, complemented
4#17985738C212yesmaximal, complemented
5#329×C71C211yesmaximal, characteristic, complemented
6C2×Q8×D71C411yescharacteristic
7C2×D4×D71C411yescharacteristic
8C2×C8×D71—22no—
9#14556117C411yescharacteristic
10#14556116C2211yescharacteristic
11#14556115C411yescharacteristic
12#14556114—22no—
13C22⋊C4×D71—22no—
14C42×D71C2211yescharacteristic
15#14556039C2212yescomplemented
16#14556038—22no—
17#124×C71C2211yescharacteristic
18#97×C71C2212yescomplemented
19C23×D71—21no—
20C2×Dic142—21no—
21C2×Dic142C2×C411yescharacteristic
22Q8×D71—42no—
23C2×D284C2×C411yescharacteristic
24C2×C22⋊D71—21no—
25D4×D71—42no—
26C2×C4×D71—21no—
27C2×C4×D71C2×C411yescharacteristic
28C2×C4×D71D412yes—
29C2×C71⋊C8—22no—
30C8×D71—28no—
31C22⋊Dic71—22no—
32(C2×D71)⋊C4—22no—
33(C2×D71)⋊C4—22no—
34C4×Dic71C2×C411yescharacteristic
35C4×Dic71D412yes—
36C2×Q8×C71C2×C411yescharacteristic
37C2×D4×C71C2×C411yescharacteristic
38C2×C8×C71—22no—
39C22⋊C4×C71—22no—
40C42×C71C2311yescharacteristic
41C23×C71—21no—
42C22×D71—21no—
43C22×D71—42no—
44C22×D71—44no—
45C22×D71C22⋊C411yescharacteristic
46Q8×C71—41no—
47D4×C71—41no—
48C2×C4×C71—21no—
49C2×C4×C71C22×C411yescharacteristic
50C2×C4×C71C2×D412yes—
51C22⋊D71—42no—
52C2×Dic71—21no—
53C2×Dic71C22⋊C412yes—
54C2×Dic71C22⋊C411yescharacteristic
55D284—41no—
56C4×D71—22no—
57C4×D71—24no—
58C4×D71—42no—
59Dic142—42no—
60Dic142—41no—
61C8×C71—24no—
62C71⋊C8—24no—
63C22×C71—41no—
64C22×C71—42no—
65C22×C71C2×C22⋊C411yescharacteristic
66C2×D71—41no—
67C2×D71—42no—
68C2×D71—84no—
69C2×D71#10012yes—
70C2×D71#10414yes—
71C4×C71—21no—
72C4×C71—22no—
73C4×C71—41no—
74Dic71—21no—
75Dic71—22no—
76Dic71—41no—
77C2×C71—81no—
78C2×C71C2×#10011yescharacteristic
79C2×C71C2×#10412yes—
80D71—81no—
81D71#32914yesdirect product factor
82C2×#329—711nomaximal, Sylow
83C71C2×#32911yescharacteristic, complemented, Sylow
84C2×#124—711no—
85C2×#97—712no—
86#329—711no—
87#329—712no—
88#329D7111yescharacteristic, direct product factor
89C22×Q8—711no—
90C22×D4—711no—
91C22×C8—1422no—
92#124—711no—
93#124—711no—
94#124—711no—
95#124—1422no—
96#124C2×D7111yescharacteristic
97C2×C22⋊C4—1422no—
98C2×C42—711no—
99#97—712no—
100#97—1422no—
101#97C2×D7112yescomplemented
102C24—1421no—
103C2×Q8—2842no—
104C2×Q8—711no—
105C2×Q8—1421no—
106C2×Q8C4×D7111yescharacteristic
107C2×D4—2842no—
108C2×D4—711no—
109C2×D4—1421no—
110C2×D4C4×D7111yescharacteristic
111C22×C4—711no—
112C22×C4—712no—
113C22×C4—1421no—
114C2×C8—22no—
115C2×C8—1428no—
116C2×C8—1422no—
117C22⋊C4—22no—
118C22⋊C4—1422no—
119C22⋊C4—1422no—
120C22⋊C4—1422no—
121C42—711no—
122C42—712no—
123C42C22×D7111yescharacteristic
124C23—21no—
125C23—2842no—
126C23—2844no—
127C23—711no—
128C23—1421no—
129Q8—41no—
130Q8—2841no—
131Q8—2842no—
132D4—41no—
133D4—2841no—
134D4—2842no—
135C2×C4—21no—
136C2×C4—2842no—
137C2×C4—712no—
138C2×C4—711no—
139C2×C4—1424no—
140C2×C4—1422no—
141C2×C4—1421no—
142C2×C4C2×C4×D7111yescharacteristic
143C2×C4D4×D7112yes—
144C8—24no—
145C8—1424no—
146C22—41no—
147C22—42no—
148C22—2841no—
149C22—2842no—
150C22—5684no—
151C22—712no—
152C22—714no—
153C22C22⋊C4×D7111yescharacteristic
154C4—21no—
155C4—22no—
156C4—41no—
157C4—2841no—
158C4—1421no—
159C4—1422no—
160C2—81no—
161C2—5681no—
162C2—714no—
163C2#100×D7111yescharacteristic
164C2#104×D7112yes—
subgroupG/H
centerC22C22⋊C4×D71
commutatorC2×C4×C71C22×C4
FrattiniC42C22×D71
Fitting#329×C71C2
socleC22×C71C2×C22⋊C4
hypercenter#329D71
hypocenterC71C2×#329
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #97×D71C2semidirect
3. #17985739C2semidirect
4. #17985738C2semidirect
5. #329×C71C2semidirect
15. #14556039C22semidirect
18. #97×C71C22semidirect
81. D71#329direct (N × H)
83. C71C2×#329semidirect
88. #329D71direct (N × H)
101. #97C2×D71semidirect