Finite Group Databasebeta

← #22589124

Subgroups — #22589124

Order 9088.

196 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#14556063C211yesmaximal, characteristic
2#17986041C211yesmaximal, characteristic
3#17986039C211yesmaximal, characteristic
4C2×#14556061C211yesmaximal, characteristic
5#17985957C211yesmaximal, characteristic
6#17985767C211yesmaximal, characteristic
7#343×C71C211yesmaximal, characteristic
8C2×#11765849C2211yescharacteristic
9C22×C71⋊C8C2211yescharacteristic
10#14556087C2211yescharacteristic
11C2×C4⋊Dic71C2211yescharacteristic
12#14556063C414yescomplemented
13#14556062—21no—
14#14556061—21no—
15#14556061—21no—
16#14556061—21no—
17#14556060—21no—
18#14556059—21no—
19#14556038—21no—
20C8×Dic71—21no—
21C2×M16×C71C2211yescharacteristic
22C2×M16×C71C2211yescharacteristic
23C2×C4⋊C4×C71C2211yescharacteristic
24C22×Dic71—21no—
25#11765849—21no—
26#11765849C2×C412yes—
27C2×C71⋊C8—21no—
28C2×C71⋊C8C2×C412yes—
29C2×C71⋊C8D411yescharacteristic
30C2×C71⋊C8D411yescharacteristic
31C2×C71⋊C8Q812yes—
32C4⋊Dic71—21no—
33C4⋊Dic71—21no—
34C4⋊Dic71D411yescharacteristic
35C4⋊Dic71D411yescharacteristic
36C22⋊Dic71—21no—
37C4×Dic71—21no—
38C22×C4×C71—21no—
39C22×C4×C71C2311yescharacteristic
40M16×C71—21no—
41M16×C71—21no—
42M16×C71C2×C412yes—
43M16×C71Q812yes—
44C2×C8×C71—21no—
45C2×C8×C71—21no—
46C4⋊C4×C71—21no—
47C4⋊C4×C71—21no—
48C23×C71#4111yescharacteristic
49C2×C4×C71—41no—
50C2×C4×C71—81no—
51C2×C4×C71C22×C411yescharacteristic
52C2×C4×C71C2×D411yescharacteristic
53C2×C4×C71C2×D411yescharacteristic
54C2×C4×C71C2×Q811yescharacteristic
55C2×C4×C71C2×Q811yescharacteristic
56C2×C4×C71#4111yescharacteristic
57C2×Dic71—21no—
58C2×Dic71—21no—
59C2×Dic71—41no—
60C2×Dic71—81no—
61C8×C71—22no—
62C8×C71—22no—
63C71⋊C8—22no—
64C71⋊C8C4⋊C412yes—
65C71⋊C8C4⋊C412yes—
66C22×C71—21no—
67C22×C71—21no—
68C22×C71C4×Q811yescharacteristic
69C22×C71#12111yescharacteristic
70C22×C71#12511yescharacteristic
71C4×C71—81no—
72C4×C71C2×C4⋊C411yescharacteristic
73C4×C71C4×D411yescharacteristic
74C4×C71C22⋊Q811yescharacteristic
75C4×C71C4⋊Q811yescharacteristic
76Dic71—41no—
77Dic71—41no—
78Dic71—81no—
79C2×C71—21no—
80C2×C71—21no—
81C2×C71#38911yescharacteristic
82C2×C71#47211yescharacteristic
83C2×C71#47311yescharacteristic
84#1952—711nomaximal, Sylow
85C71#195211yescharacteristic, complemented, Sylow
86C2×#108—711no—
87#428—711no—
88#428—711no—
89C2×#107—711no—
90#405—711no—
91#343—711no—
92#343D7111yescharacteristic
93C2×M16—711no—
94C2×M16C2×D7111yescharacteristic
95C2×M16C2×D7111yescharacteristic
96C22×C8—711no—
97#117—711no—
98C2×C4⋊C4—711no—
99C2×C4⋊C4C2×D7111yescharacteristic
100#108—714no—
101#107—1421no—
102#107—1421no—
103#107—1421no—
104#107—1421no—
105#106—1421no—
106#106—1421no—
107#97—1421no—
108C4×C8—1421no—
109C22×C4—21no—
110C22×C4—1421no—
111C22×C4C22×D7111yescharacteristic
112M16—21no—
113M16—21no—
114M16—712no—
115M16—1421no—
116M16Dic14212yes—
117M16C4×D7112yes—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—712no—
121C2×C8—711no—
122C2×C8—711no—
123C2×C8—712no—
124C2×C8—1421no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—711no—
128C4⋊C4—711no—
129C4⋊C4—1421no—
130C4⋊C4—1421no—
131C22⋊C4—1421no—
132C42—1421no—
133C23#1176586511yescharacteristic
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—2841no—
137C2×C4—5681no—
138C2×C4—1421no—
139C2×C4—1421no—
140C2×C4C2×C4×D7111yescharacteristic
141C2×C4D4×D7111yescharacteristic
142C2×C4D4×D7111yescharacteristic
143C2×C4Q8×D7111yescharacteristic
144C2×C4C2×Dic14211yescharacteristic
145C2×C4#1176586411yescharacteristic
146C8—22no—
147C8—22no—
148C8—712no—
149C8—712no—
150C8—1422no—
151C22—21no—
152C22—21no—
153C22C4×Dic14211yescharacteristic
154C22#1455610111yescharacteristic
155C22#1455611811yescharacteristic
156C4—81no—
157C4—2841no—
158C4—2841no—
159C4—5681no—
160C4C4⋊C4×D7111yescharacteristic
161C4(C22⋊D71)⋊C411yescharacteristic
162C4#1455610511yescharacteristic
163C4#1455612711yescharacteristic
164C2—21no—
165C2—21no—
166C2#1798589611yescharacteristic
167C2#1798626811yescharacteristic
168C2#1798627311yescharacteristic
subgroupG/H
centerC22#14556101
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#343×C71C2
socleC22×C71#121
hypercenter#343D71
hypocenterC71#1952
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
12. #14556063C4semidirect
85. C71#1952semidirect