Finite Group Databasebeta

← #28338486

Subgroups — #28338486

Order 13696.

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

normal subgroups only

This group has 236 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#20293343C211yesmaximal, characteristic, complemented
2#20292800C212yesmaximal, complemented
3#20292426C211yesmaximal, characteristic, complemented
4#20292423C212yesmaximal, complemented
5#456×C107C211yesmaximal, characteristic, complemented
6#16812134—22no—
7#16812098—21no—
8#16812098C2212yes—
9#16812084—22no—
10C4×D428—21no—
11#16812067—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×D428—21no—
20C2×D428—21no—
21C2×D428D412yes—
22C2×C4×D107—22no—
23#13133805—42no—
24#13133803—42no—
25#13133802—22no—
26C4⋊Dic107—21no—
27(C2×D107)⋊C4—22no—
28(C2×D107)⋊C4—22no—
29(C2×D107)⋊C4—22no—
30C4×Dic107—21no—
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
43C22×D107—22no—
44Q8×C107—21no—
45Q8×C107—22no—
46Q8×C107—41no—
47Q8×C107—41no—
48D4×C107—21no—
49D4×C107—22no—
50D4×C107—41no—
51C2×C4×C107—21no—
52C2×C4×C107—21no—
53C2×C4×C107—42no—
54C2×C4×C107—41no—
55C2×C4×C107C2×D412yes—
56C2×C4×C107C2×D411yescharacteristic
57C2×Dic107—22no—
58D428—42no—
59D428—42no—
60D428—81no—
61C4×D107—44no—
62C8×C107—41no—
63C107⋊C8—42no—
64C22×C107—41no—
65C22×C107#12011yescharacteristic
66C2×D107—42no—
67C2×D107—42no—
68C2×D107—42no—
69C4×C107—22no—
70C4×C107—22no—
71C4×C107—21no—
72C4×C107—41no—
73C4×C107—41no—
74Dic107—42no—
75C2×C107—21no—
76C2×C107—81no—
77C2×C107B(4,𝔽2)11yescharacteristic
78D107—82no—
79#2204—1071nomaximal, Sylow
80C107#220411yescharacteristic, complemented, Sylow
81#555—1071no—
82#456D10711yescharacteristic, complemented
83#454—1072no—
84#356—1071no—
85#355—1072no—
862−1+4C2×D10711yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—2142no—
89C4⋊Q8C2×D10711yescharacteristic
90#124—1072no—
91#124—2141no—
92#121—2142no—
93C4×D4—2141no—
94#117—2142no—
95#104—21no—
96#104—2142no—
97#101—1072no—
98#101C2×D10711yescharacteristic
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8C22⋊D10711yescharacteristic
104C2×Q8C22×D10711yescharacteristic
105C2×D4—1072no—
106C2×D4—2141no—
107C2×D4—2141no—
108C22×C4—2142no—
109Q16—41no—
110SD16—41no—
111SD16—4282no—
112D8—4282no—
113M16—21no—
114M16—2142no—
115C4⋊C4—22no—
116C4⋊C4—2141no—
117C22⋊C4—2142no—
118C22⋊C4—2142no—
119C22⋊C4—2142no—
120C42—1072no—
121C42—2141no—
122C42C22⋊D10711yescharacteristic
123C23—2142no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—41no—
131D4—8561no—
132D4—4282no—
133D4—4282no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—4284no—
139C2×C4—2142no—
140C2×C4D4×D10712yes—
141C2×C4C2×C22⋊D10711yescharacteristic
142C8—41no—
143C8—4282no—
144C22—41no—
145C22—4282no—
146C22—4282no—
147C22—4282no—
148C22#1681207711yescharacteristic
149C4—22no—
150C4—21no—
151C4—22no—
152C4—41no—
153C4—41no—
154C4—4282no—
155C2—21no—
156C2—81no—
157C2—8562no—
158C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×Q8×C107C23
FrattiniC2×Q8C22×D107
Fitting#456×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#456D107
hypocenterC107#2204
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. #20293343C2semidirect
2. #20292800C2semidirect
3. #20292426C2semidirect
4. #20292423C2semidirect
5. #456×C107C2semidirect
80. C107#2204semidirect
82. #456D107semidirect

Maximal subgroups

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

subgroupindex
#202933432
#202928002
#202924262
#202924232
#456×C1072
#2204107