Finite Group Databasebeta

← #28336996

Subgroups — #28336996

Order 13696.

314 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 85 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#20293201C211yesmaximal, characteristic
2#20293105C211yesmaximal, characteristic, complemented
3#20292643C212yesmaximal
4#20292384C212yesmaximal, complemented
5C2×#101×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C2211yescharacteristic
7#16812107C412yescomplemented
8#16812098C412yescomplemented
9#16812092—22no—
10#16812085—22no—
11Q8×Dic107—24no—
12#16812068—22no—
13#16812068C2211yescharacteristic
14#16812068C2212yes—
15#16812035—24no—
16C22×Q8×C107C2211yescharacteristic
17C2×M16×C107C2212yes—
18#101×C107—22no—
19#13133818—22no—
20#13133818—22no—
21C2×Dic214C2×C411yescharacteristic
22C2×D428C2×C411yescharacteristic
23C2×C22⋊D107—21no—
24C2×C4×D107—21no—
25C4⋊Dic107—24no—
26C4⋊Dic107—22no—
27C4⋊Dic107—21no—
28Dic107⋊C4—24no—
29C22⋊Dic107—22no—
30C22⋊Dic107—21no—
31(C2×D107)⋊C4—24no—
32C4×Dic107—24no—
33C4×Dic107C2×C412yes—
34C4×Dic107D414yes—
35C2×Q8×C107—22no—
36C2×Q8×C107—22no—
37C2×Q8×C107C2×C412yes—
38C2×Q8×C107D412yes—
39C22×C4×C107D412yes—
40C22×C4×C107C2311yescharacteristic
41M16×C107—22no—
42M16×C107—22no—
43C2×C8×C107—22no—
44C23×C107C2×D411yescharacteristic
45C22×D107—21no—
46Q8×C107—24no—
47Q8×C107—24no—
48C2×C4×C107—22no—
49C2×C4×C107—22no—
50C2×C4×C107C22⋊C414yes—
51C2×C4×C107C22×C411yescharacteristic
52C2×C4×C107C2×D412yes—
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107#4112yes—
55C22⋊D107—41no—
56C22⋊D107—41no—
57C2×Dic107—22no—
58C2×Dic107—21no—
59C2×Dic107—24no—
60D428—22no—
61C4×D107—42no—
62Dic214—22no—
63C8×C107—42no—
64C22×C107—41no—
65C22×C107C2×C22⋊C411yescharacteristic
66C22×C107#12011yescharacteristic
67C22×C107#12311yescharacteristic
68C2×D107—42no—
69C2×D107—41no—
70C4×C107—24no—
71C4×C107C4×D412yes—
72C4×C107#12112yes—
73Dic107—44no—
74Dic107—42no—
75Dic107—41no—
76C2×C107—21no—
77C2×C107—21no—
78C2×C107—21no—
79C2×C107#38611yescharacteristic
80D107—81no—
81#1912—1071nomaximal, Sylow
82C107#191211yescharacteristic, complemented, Sylow
83#536—1071no—
84#519—1071no—
85#421—1072no—
86C2×#101D10711yescharacteristic, complemented
87H(3,ℤ/4)—1072no—
88C2×#41—1071no—
89C22×Q8C2×D10711yescharacteristic
90C2×M16C2×D10712yes—
91C4⋊Q8—1072no—
92#124—1072no—
93#123—2142no—
94C22⋊Q8—2142no—
95C4×Q8—2144no—
96#117—1072no—
97#117—1071no—
98#117—2142no—
99#104—2144no—
100#101—22no—
101#41—2142no—
102#41—2142no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—1071no—
106C2×Q8C4×D10712yes—
107C2×Q8C22⋊D10712yes—
108C2×D4—1071no—
109C2×D4—2141no—
110C22×C4—2141no—
111C22×C4D42812yes—
112C22×C4C22×D10711yescharacteristic
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C4⋊C4—2144no—
117C4⋊C4—2144no—
118C4⋊C4—2142no—
119C4⋊C4—2141no—
120C22⋊C4—2142no—
121C22⋊C4—2141no—
122C22⋊C4—2144no—
123C42—1072no—
124C42—1074no—
125C42—2144no—
126C23—2141no—
127C23C2×D42811yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—2142no—
131D4—4281no—
132D4—4281no—
133D4—2142no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—4282no—
137C2×C4—2142no—
138C2×C4—2141no—
139C2×C4—2144no—
140C2×C4(C2×D107)⋊C414yes—
141C2×C4C2×C4×D10711yescharacteristic
142C2×C4D4×D10712yes—
143C2×C4C2×C22⋊D10711yescharacteristic
144C2×C4#1313381712yes—
145C8—42no—
146C22—41no—
147C22—4282no—
148C22—4281no—
149C22C2×(C2×D107)⋊C411yescharacteristic
150C22#1681207911yescharacteristic
151C22#1681209311yescharacteristic
152C4—24no—
153C4—4284no—
154C4—4282no—
155C4—4281no—
156C4(C22⋊D107)⋊C412yes—
157C4#1681208112yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—8561no—
162C2#2029250411yescharacteristic
subgroupG/H
centerC2#20292504
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#101×C107C2
socleC2×C107#386
prime radicalC2×#13133818C22
cosocleC22×C4×C107C23
hypercenterC2×#101D107
hypocenterC107#1912
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #20293105C2semidirect
4. #20292384C2semidirect
5. C2×#101×C107C2semidirect
7. #16812107C4semidirect
8. #16812098C4semidirect
82. C107#1912semidirect
86. C2×#101D107semidirect

Maximal subgroups

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

subgroupindex
#202932012
#202931052
#202926432
#202923842
C2×#101×C1072
#1912107