Finite Group Databasebeta

← C109×#2204 #28359458

Subgroups — C109×#2204 #28359458

Order 13952.

236 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 56 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#555×C109C211yesmaximal, characteristic, complemented
2#456×C109C211yesmaximal, characteristic, complemented
3#454×C109C212yesmaximal, complemented
4#356×C109C211yesmaximal, characteristic, complemented
5#355×C109C212yesmaximal, complemented
62−1+4×C109C2211yescharacteristic
7#137×C109—21no—
8AGL(1,ℤ/8)×C109—22no—
9C4⋊Q8×C109C2211yescharacteristic
10#124×C109—21no—
11#124×C109C2212yes—
12#121×C109—22no—
13C4×D4×C109—21no—
14#117×C109—22no—
15#104×C109—22no—
16#104×C109—21no—
17#101×C109C2211yescharacteristic
18#101×C109C2212yes—
19#41×C109—22no—
20#41×C109—21no—
21#41×C109—41no—
22C2×Q8×C109—41no—
23C2×Q8×C109D411yescharacteristic
24C2×Q8×C109C2311yescharacteristic
25C2×D4×C109—21no—
26C2×D4×C109—21no—
27C2×D4×C109D412yes—
28C22×C4×C109—22no—
29Q16×C109—41no—
30SD16×C109—41no—
31SD16×C109—42no—
32D8×C109—42no—
33M16×C109—22no—
34M16×C109—21no—
35C4⋊C4×C109—21no—
36C4⋊C4×C109—22no—
37C22⋊C4×C109—22no—
38C22⋊C4×C109—22no—
39C22⋊C4×C109—22no—
40C42×C109—21no—
41C42×C109D412yes—
42C42×C109D411yescharacteristic
43C23×C109—22no—
44Q8×C109—22no—
45Q8×C109—21no—
46Q8×C109—41no—
47Q8×C109—41no—
48D4×C109—22no—
49D4×C109—21no—
50D4×C109—42no—
51D4×C109—42no—
52D4×C109—41no—
53D4×C109—81no—
54C2×C4×C109—21no—
55C2×C4×C109—22no—
56C2×C4×C109—21no—
57C2×C4×C109—44no—
58C2×C4×C109—42no—
59C2×C4×C109—41no—
60C2×C4×C109C2×D411yescharacteristic
61C2×C4×C109C2×D412yes—
62C8×C109—42no—
63C8×C109—41no—
64C22×C109—42no—
65C22×C109—42no—
66C22×C109—41no—
67C22×C109—42no—
68C22×C109#12011yescharacteristic
69C4×C109—22no—
70C4×C109—22no—
71C4×C109—21no—
72C4×C109—42no—
73C4×C109—41no—
74C4×C109—41no—
75C2×C109—21no—
76C2×C109—81no—
77C2×C109—82no—
78C2×C109B(4,𝔽2)11yescharacteristic
79#2204C10911yesmaximal, characteristic, direct product factor, Sylow
80C109#220411yescharacteristic, direct product factor, Sylow
81#555C2×C10911yescharacteristic, complemented
82#456C2×C10911yescharacteristic, complemented
83#454C2×C10912yescomplemented
84#356C2×C10911yescharacteristic, complemented
85#355C2×C10912yescomplemented
862−1+4C22×C10911yescharacteristic
87#137—21no—
88AGL(1,ℤ/8)—22no—
89C4⋊Q8C22×C10911yescharacteristic
90#124—21no—
91#124C22×C10912yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—22no—
96#104—21no—
97#101C22×C10911yescharacteristic
98#101C22×C10912yes—
99#41—22no—
100#41—21no—
101#41—41no—
102C2×Q8—41no—
103C2×Q8D4×C10911yescharacteristic
104C2×Q8C23×C10911yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4D4×C10912yes—
108C22×C4—22no—
109Q16—41no—
110SD16—41no—
111SD16—42no—
112D8—42no—
113M16—22no—
114M16—21no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42D4×C10912yes—
122C42D4×C10911yescharacteristic
123C23—22no—
124Q8—22no—
125Q8—21no—
126Q8—41no—
127Q8—41no—
128D4—22no—
129D4—21no—
130D4—42no—
131D4—42no—
132D4—41no—
133D4—81no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4C2×D4×C10911yescharacteristic
141C2×C4C2×D4×C10912yes—
142C8—42no—
143C8—41no—
144C22—42no—
145C22—42no—
146C22—41no—
147C22—42no—
148C22#120×C10911yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—41no—
155C2—21no—
156C2—82no—
157C2—81no—
158C2B(4,𝔽2)×C10911yescharacteristic
subgroupG/H
center, socleC2×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C23×C109
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C22×C109 ◁ C2×Q8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. #555×C109C2semidirect
2. #456×C109C2semidirect
3. #454×C109C2semidirect
4. #356×C109C2semidirect
5. #355×C109C2semidirect
79. #2204C109direct (N × H)
80. C109#2204direct (N × H)
81. #555C2×C109semidirect
82. #456C2×C109semidirect
83. #454C2×C109semidirect
84. #356C2×C109semidirect
85. #355C2×C109semidirect

Maximal subgroups

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

subgroupindex
#555×C1092
#456×C1092
#454×C1092
#356×C1092
#355×C1092
#2204109