Finite Group Databasebeta

← #28338490

Subgroups — #28338490

Order 13696.

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

normal subgroups only

This group has 224 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#20293357C211yesmaximal, characteristic, complemented
2#20292810C212yesmaximal
3#20292426C211yesmaximal, characteristic, complemented
4#20292425C212yesmaximal, complemented
5#456×C107C211yesmaximal, characteristic
6#16812139—22no—
7#16812108—22no—
8#16812107—21no—
9#16812107C2212yes—
10C4×Dic214—21no—
11#16812074—22no—
12#16812036—22no—
13#16812028C2212yes—
142−1+4×C107C2211yescharacteristic
15#137×C107—21no—
16C4⋊Q8×C107C2211yescharacteristic
17#104×C107—21no—
18#101×C107C2211yescharacteristic
19C2×Dic214—21no—
20C2×Dic214—21no—
21C2×Dic214D412yes—
22#13133808—42no—
23#13133806—42no—
24#13133802—22no—
25C4⋊Dic107—21no—
26Dic107⋊C4—22no—
27Dic107⋊C4—22no—
28Dic107⋊C4—22no—
29C4×Dic107—21no—
30C4×Dic107—22no—
31C4×Dic107D412yes—
32#41×C107—22no—
33#41×C107—21no—
34#41×C107—41no—
35C2×Q8×C107—41no—
36C2×Q8×C107D411yescharacteristic
37C2×Q8×C107C2311yescharacteristic
38Q16×C107—41no—
39SD16×C107—41no—
40M16×C107—21no—
41C4⋊C4×C107—22no—
42C42×C107D411yescharacteristic
43Q8×C107—21no—
44Q8×C107—22no—
45Q8×C107—41no—
46Q8×C107—41no—
47D4×C107—21no—
48D4×C107—22no—
49D4×C107—41no—
50C2×C4×C107—21no—
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—
58Dic214—24no—
59Dic214—42no—
60Dic214—81no—
61C8×C107—41no—
62C107⋊C8—42no—
63C22×C107—41no—
64C22×C107#12011yescharacteristic
65C4×C107—22no—
66C4×C107—22no—
67C4×C107—21no—
68C4×C107—41no—
69C4×C107—41no—
70Dic107—24no—
71Dic107—42no—
72C2×C107—21no—
73C2×C107—81no—
74C2×C107B(4,𝔽2)11yescharacteristic
75#2205—1071nomaximal, Sylow
76C107#220511yescharacteristic, complemented, Sylow
77Q82—1071no—
78#456—1072no—
79#456D10711yescharacteristic
80#356—1071no—
81#356—1072no—
822−1+4C2×D10711yescharacteristic
83#137—21no—
84#137—2142no—
85C4⋊Q8—1072no—
86C4⋊Q8—2142no—
87C4⋊Q8—2141no—
88C4⋊Q8C2×D10711yescharacteristic
89C4×Q8—2142no—
90C4×Q8—2141no—
91#104—21no—
92#104—2142no—
93#101—1072no—
94#101C2×D10711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—1072no—
100C2×Q8—2141no—
101C2×Q8—2141no—
102C2×Q8C22⋊D10711yescharacteristic
103C2×Q8C22×D10711yescharacteristic
104Q16—41no—
105Q16—4282no—
106SD16—41no—
107SD16—4282no—
108M16—21no—
109M16—2142no—
110C4⋊C4—22no—
111C4⋊C4—2142no—
112C4⋊C4—2141no—
113C4⋊C4—2142no—
114C4⋊C4—2142no—
115C42—1072no—
116C42—2142no—
117C42—2141no—
118C42C22⋊D10711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—8561no—
124Q8—4282no—
125Q8—2144no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—2142no—
134C2×C4—2142no—
135C2×C4D4×D10712yes—
136C2×C4C2×C22⋊D10711yescharacteristic
137C8—41no—
138C8—4282no—
139C22—41no—
140C22#1681207711yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—4282no—
147C4—2144no—
148C2—21no—
149C2—81no—
150C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×Q8×C107C23
FrattiniC2×Q8C22×D107
Fitting#456×C107C2
socleC2×C107B(4,𝔽2)
prime radical2−1+4×C107C22
hypercenter#456D107
hypocenterC107#2205
LCSG ▷ C2×Q8×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #20293357C2semidirect
3. #20292426C2semidirect
4. #20292425C2semidirect
76. C107#2205semidirect

Maximal subgroups

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

subgroupindex
#202933572
#202928102
#202924262
#202924252
#456×C1072
#2205107