Finite Group Databasebeta

← #28338482

Subgroups — #28338482

Order 13696.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#20292809C212yesmaximal, complemented
2#20292799C211yesmaximal, characteristic
3#20292426C212yesmaximal, complemented
4#20292424C211yesmaximal, characteristic, complemented
5#563×C107C211yesmaximal, characteristic, complemented
6#16812157C2211yescharacteristic
7#16812140—22no—
8#16812135—21no—
9#16812037—21no—
10#16812037—22no—
11#16812028C2212yes—
12#16812028C2211yescharacteristic
13C4⋊Q8×C107C2212yes—
14#126×C107—22no—
15#125×C107—21no—
16#124×C107C2211yescharacteristic
17#123×C107—21no—
18C22⋊Q8×C107—22no—
19#13133819—41no—
20#13133818—21no—
21#13133818—22no—
22Q8×D107—41no—
23#13133808—42no—
24#13133806—41no—
25#13133805—42no—
26#13133803—41no—
27#13133802—21no—
28#13133802—22no—
29C2×Q8×C107D412yes—
30C2×Q8×C107C2311yescharacteristic
31C2×D4×C107D411yescharacteristic
32C22×C4×C107—21no—
33C4⋊C4×C107—21no—
34C4⋊C4×C107—22no—
35C4⋊C4×C107—21no—
36C4⋊C4×C107—22no—
37C4⋊C4×C107—22no—
38C22⋊C4×C107—22no—
39C22⋊C4×C107—21no—
40C22⋊C4×C107—22no—
41C42×C107D411yescharacteristic
42C42×C107D412yes—
43C23×C107—21no—
44Q8×C107—41no—
45Q8×C107—42no—
46D4×C107—41no—
47C2×C4×C107—21no—
48C2×C4×C107—22no—
49C2×C4×C107—22no—
50C2×C4×C107—81no—
51C2×C4×C107C2×D412yes—
52C2×C4×C107C2×D411yescharacteristic
53C22⋊D107—41no—
54D428—21no—
55D428—22no—
56C4×D107—41no—
57C4×D107—42no—
58Dic214—21no—
59Dic214—22no—
60C107⋊C8—41no—
61C107⋊C8—42no—
62C22×C107—42no—
63C22×C107—41no—
64C22×C107#12011yescharacteristic
65C2×D107—41no—
66C4×C107—21no—
67C4×C107—22no—
68C4×C107—41no—
69C4×C107—42no—
70C4×C107—42no—
71Dic107—41no—
72C2×C107—21no—
73C2×C107—81no—
74C2×C107B(4,𝔽2)11yescharacteristic
75D107—81no—
76#2203—1071nomaximal, Sylow
77C107#220311yescharacteristic, complemented, Sylow
78#563D10711yescharacteristic, complemented
79#456—1072no—
80#454—1071no—
81#356—1072no—
82#355—1071no—
832−1+4—1071no—
84#137—2142no—
85AGL(1,ℤ/8)—2141no—
86C4⋊Q8C2×D10712yes—
87#126—22no—
88#125—21no—
89#124C2×D10711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—2141no—
93#104—2142no—
94#101—1072no—
95#101—1071no—
96#41—4281no—
97#41—2141no—
98#41—2142no—
99C2×Q8—4281no—
100C2×Q8C22⋊D10712yes—
101C2×Q8C22×D10711yescharacteristic
102C2×D4C22⋊D10711yescharacteristic
103C22×C4—21no—
104Q16—4282no—
105SD16—4281no—
106SD16—4282no—
107D8—4281no—
108M16—2141no—
109M16—2142no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D10711yescharacteristic
119C42C22⋊D10712yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—2141no—
124Q8—2142no—
125D4—41no—
126D4—4281no—
127D4—2141no—
128D4—2142no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—4281no—
134C2×C4—4282no—
135C2×C4C2×C22⋊D10712yes—
136C2×C4C2×C22⋊D10711yescharacteristic
137C8—4281no—
138C8—4282no—
139C22—42no—
140C22—41no—
141C22—4281no—
142C22#1681207811yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—4281no—
149C2—21no—
150C2—81no—
151C2—8561no—
152C2#2029281411yescharacteristic
subgroupG/H
centerC2#20292814
commutator, cosocleC2×Q8×C107C23
FrattiniC2×Q8C22×D107
Fitting#563×C107C2
socleC2×C107B(4,𝔽2)
prime radical#20292799C2
hypercenter#563D107
hypocenterC107#2203
LCSG ▷ C2×Q8×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #20292809C2semidirect
3. #20292426C2semidirect
4. #20292424C2semidirect
5. #563×C107C2semidirect
77. C107#2203semidirect
78. #563D107semidirect

Maximal subgroups

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

subgroupindex
#202928092
#202927992
#202924262
#202924242
#563×C1072
#2203107