Finite Group Databasebeta

← #28338481

Subgroups — #28338481

Order 13696.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#20293377C211yesmaximal, characteristic, complemented
2#20292810C212yesmaximal, complemented
3#20292425C212yesmaximal, complemented
4#20292424C211yesmaximal, characteristic, complemented
5#454×C107C211yesmaximal, characteristic
6#16812139—22no—
7#16812107C2212yes—
8#16812102—22no—
9#16812101—21no—
10#16812090—21no—
11#16812086—22no—
12#16812036—22no—
13#16812028C2212yes—
142−1+4×C107C2211yescharacteristic
15AGL(1,ℤ/8)×C107—21no—
16#124×C107C2211yescharacteristic
17#104×C107—21no—
18#101×C107C2211yescharacteristic
19C2×Dic214D412yes—
20C22×Dic107—21no—
21#13133808—42no—
22#13133806—42no—
23#13133802—22no—
24C4⋊Dic107—21no—
25C4⋊Dic107—21no—
26Dic107⋊C4—22no—
27Dic107⋊C4—22no—
28Dic107⋊C4—22no—
29C22⋊Dic107—22no—
30C22⋊Dic107—21no—
31C4×Dic107D412yes—
32#41×C107—22no—
33#41×C107—21no—
34#41×C107—41no—
35C2×Q8×C107—41no—
36C2×Q8×C107C2311yescharacteristic
37C2×D4×C107D411yescharacteristic
38SD16×C107—41no—
39D8×C107—41no—
40M16×C107—21no—
41C22⋊C4×C107—22no—
42C42×C107D411yescharacteristic
43C23×C107—21no—
44Q8×C107—21no—
45Q8×C107—22no—
46Q8×C107—41no—
47D4×C107—21no—
48D4×C107—22no—
49D4×C107—41no—
50D4×C107—41no—
51C2×C4×C107—21no—
52C2×C4×C107—42no—
53C2×C4×C107—41no—
54C2×C4×C107C2×D412yes—
55C2×C4×C107C2×D411yescharacteristic
56C2×Dic107—22no—
57C2×Dic107—22no—
58C2×Dic107—81no—
59Dic214—42no—
60C8×C107—41no—
61C107⋊C8—42no—
62C22×C107—42no—
63C22×C107—41no—
64C22×C107—41no—
65C22×C107#12011yescharacteristic
66C4×C107—22no—
67C4×C107—21no—
68C4×C107—41no—
69C4×C107—41no—
70Dic107—42no—
71Dic107—42no—
72C2×C107—21no—
73C2×C107—81no—
74C2×C107—81no—
75C2×C107B(4,𝔽2)11yescharacteristic
76#2203—1071nomaximal, Sylow
77C107#220311yescharacteristic, complemented, Sylow
78#563—1071no—
79#456—1072no—
80#454D10711yescharacteristic
81#356—1072no—
82#355—1071no—
832−1+4C2×D10711yescharacteristic
84#137—2142no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1072no—
87#126—2142no—
88#125—2141no—
89#124C2×D10711yescharacteristic
90#123—2141no—
91C22⋊Q8—2142no—
92#104—21no—
93#104—2142no—
94#101—1072no—
95#101C2×D10711yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1072no—
101C2×Q8C22×D10711yescharacteristic
102C2×D4C22⋊D10711yescharacteristic
103C22×C4—2141no—
104Q16—4282no—
105SD16—41no—
106SD16—4282no—
107D8—41no—
108M16—21no—
109M16—2142no—
110C4⋊C4—2141no—
111C4⋊C4—2142no—
112C4⋊C4—2141no—
113C4⋊C4—2142no—
114C4⋊C4—2142no—
115C22⋊C4—22no—
116C22⋊C4—2142no—
117C22⋊C4—2141no—
118C42—1072no—
119C42C22⋊D10711yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—4282no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—8561no—
133C2×C4—2142no—
134C2×C4—2142no—
135C2×C4D4×D10712yes—
136C2×C4C2×C22⋊D10711yescharacteristic
137C8—41no—
138C8—4282no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1681207711yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—4282no—
148C4—4282no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×Q8×C107C23
FrattiniC2×Q8C22×D107
Fitting, prime radical#454×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#454D107
hypocenterC107#2203
LCSG ▷ C2×Q8×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20293377C2semidirect
2. #20292810C2semidirect
3. #20292425C2semidirect
4. #20292424C2semidirect
77. C107#2203semidirect

Maximal subgroups

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

subgroupindex
#202933772
#202928102
#202924252
#202924242
#454×C1072
#2203107