Finite Group Databasebeta

← #28338456

Subgroups — #28338456

Order 13696.

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

normal subgroups only

This group has 246 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#20293363C211yesmaximal, characteristic, complemented
2#20292817C212yesmaximal, complemented
3#20292420C211yesmaximal, characteristic, complemented
4#20292418C212yesmaximal, complemented
5#453×C107C211yesmaximal, characteristic
6#16812096—22no—
7#16812090—22no—
8#16812090—21no—
9#16812090C2212yes—
10C4×Dic214—21no—
11D4×Dic107—22no—
12#16812024—22no—
13#16812024C2212yes—
14H(5,𝔽2)×C107C2211yescharacteristic
15AGL(1,ℤ/8)×C107—21no—
16#127×C107C2211yescharacteristic
17#104×C107—21no—
18#100×C107C2211yescharacteristic
19C2×Dic214—21no—
20C22×Dic107—21no—
21C22×Dic107D412yes—
22C4⋊Dic107—21no—
23Dic107⋊C4—22no—
24Dic107⋊C4—42no—
25C22⋊Dic107—22no—
26C22⋊Dic107—21no—
27C22⋊Dic107—22no—
28C22⋊Dic107—22no—
29C22⋊Dic107—42no—
30C22⋊Dic107D412yes—
31C4×Dic107—22no—
32#41×C107—21no—
33#41×C107—41no—
34C2×D4×C107—22no—
35C2×D4×C107—22no—
36C2×D4×C107—41no—
37C2×D4×C107D411yescharacteristic
38C2×D4×C107C2311yescharacteristic
39SD16×C107—41no—
40D8×C107—41no—
41M16×C107—21no—
42C42×C107D411yescharacteristic
43C23×C107—22no—
44C23×C107—21no—
45C23×C107C2×D412yes—
46Q8×C107—21no—
47D4×C107—21no—
48D4×C107—41no—
49D4×C107—41no—
50D4×C107—44no—
51D4×C107—42no—
52D4×C107—41no—
53C2×C4×C107—21no—
54C2×C4×C107—22no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×Dic107—22no—
58C2×Dic107—22no—
59C2×Dic107—42no—
60C2×Dic107—42no—
61C2×Dic107—81no—
62Dic214—42no—
63C8×C107—41no—
64C22×C107—22no—
65C22×C107—42no—
66C22×C107—42no—
67C22×C107—41no—
68C22×C107—41no—
69C22×C107—82no—
70C22×C107#12011yescharacteristic
71C4×C107—22no—
72C4×C107—21no—
73C4×C107—41no—
74Dic107—42no—
75Dic107—42no—
76Dic107—82no—
77C2×C107—21no—
78C2×C107—42no—
79C2×C107—81no—
80C2×C107—81no—
81C2×C107B(4,𝔽2)11yescharacteristic
82#2198—1071nomaximal, Sylow
83C107#219811yescharacteristic, complemented, Sylow
84#559—1071no—
85#458—1072no—
86#453D10711yescharacteristic
87#353—1071no—
88#352—1072no—
89H(5,𝔽2)C2×D10711yescharacteristic
90AGL(1,ℤ/8)—21no—
91#127C2×D10711yescharacteristic
92#124—2142no—
93#123—1072no—
94#123—2142no—
95#123—2141no—
96C4×Q8—2141no—
97C4×D4—2142no—
98#104—21no—
99#100C2×D10711yescharacteristic
100#99—1072no—
101#99—2142no—
102#41—21no—
103#41—41no—
104C2×Q8—2141no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—41no—
108C2×D4C22⋊D10711yescharacteristic
109C2×D4C22×D10711yescharacteristic
110C22×C4—1072no—
111C22×C4—2141no—
112SD16—41no—
113D8—41no—
114M16—21no—
115C4⋊C4—4282no—
116C4⋊C4—2141no—
117C4⋊C4—2142no—
118C22⋊C4—1072no—
119C22⋊C4—4282no—
120C22⋊C4—2142no—
121C22⋊C4—2142no—
122C22⋊C4—2141no—
123C22⋊C4—2142no—
124C42—2142no—
125C42C22⋊D10711yescharacteristic
126C23—22no—
127C23—21no—
128C23D4×D10712yes—
129Q8—21no—
130Q8—4282no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—44no—
135D4—42no—
136D4—41no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—8561no—
141C2×C4—4282no—
142C2×C4—4282no—
143C2×C4—2142no—
144C2×C4—2142no—
145C2×C4C2×C22⋊D10711yescharacteristic
146C8—41no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—82no—
153C22#1681207711yescharacteristic
154C4—22no—
155C4—21no—
156C4—41no—
157C4—8562no—
158C4—4282no—
159C4—4282no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—81no—
164C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×D4×C107C23
FrattiniC2×D4C22×D107
Fitting, prime radical#453×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#453D107
hypocenterC107#2198
LCSG ▷ C2×D4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20293363C2semidirect
2. #20292817C2semidirect
3. #20292420C2semidirect
4. #20292418C2semidirect
83. C107#2198semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202928172
#202924202
#202924182
#453×C1072
#2198107