Finite Group Databasebeta

← C109×#12308 #42130302

Subgroups — C109×#12308 #42130302

Order 27904.

274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C109×#3255C211yesmaximal, characteristic
2C109×#2239C212yesmaximal
3C109×#2102C211yesmaximal, characteristic
4C109×#1340C212yesmaximal
5C109×#1327C211yesmaximal, characteristic
6#533×C109—21no—
7#475×C109—22no—
8#470×C109—21no—
9#470×C109C2212yes—
10#462×C109—22no—
11#430×C109C2211yescharacteristic
12#423×C109C2211yescharacteristic
13#365×C109—22no—
14#362×C109—22no—
15#349×C109C2212yes—
16#344×C109—21no—
17#339×C109—21no—
18H(3,ℤ/4)×C109C2211yescharacteristic
19C2×Q16×C109—21no—
20C2×Q16×C109D412yes—
21C2×M16×C109—21no—
22C2×M16×C109D411yescharacteristic
23C2×M16×C109C2311yescharacteristic
24C4⋊Q8×C109—21no—
25C4⋊Q8×C109—21no—
26#125×C109—41no—
27C22⋊Q8×C109—22no—
28C22⋊Q8×C109—41no—
29C4×Q8×C109—22no—
30C4×Q8×C109—41no—
31#117×C109—21no—
32#117×C109—41no—
33C2×C42×C109D411yescharacteristic
34M32×C109—22no—
35#108×C109—22no—
36#106×C109—21no—
37#106×C109D412yes—
38C4⋊C8×C109—22no—
39#103×C109—22no—
40#103×C109—22no—
41C2×Q8×C109—22no—
42C2×Q8×C109—41no—
43C22×C4×C109—21no—
44C22×C4×C109C2×D411yescharacteristic
45Q16×C109—42no—
46Q16×C109—81no—
47M16×C109—41no—
48M16×C109—41no—
49M16×C109—41no—
50M16×C109—41no—
51C2×C8×C109—21no—
52C2×C8×C109—41no—
53C2×C8×C109C2×D412yes—
54C4⋊C4×C109—22no—
55C4⋊C4×C109—41no—
56C4⋊C4×C109—41no—
57C4⋊C4×C109—42no—
58C4⋊C4×C109—41no—
59C4⋊C4×C109—41no—
60C4⋊C4×C109—42no—
61C22⋊C4×C109—42no—
62C22⋊C4×C109—41no—
63C42×C109—21no—
64C42×C109—42no—
65C42×C109—41no—
66C42×C109SD1611yescharacteristic
67C42×C109SD1611yescharacteristic
68C16×C109—42no—
69C23×C109AGL(1,ℤ/8)11yescharacteristic
70Q8×C109—42no—
71Q8×C109—42no—
72Q8×C109—82no—
73C2×C4×C109—21no—
74C2×C4×C109—21no—
75C2×C4×C109—21no—
76C2×C4×C109—21no—
77C2×C4×C109—42no—
78C2×C4×C109—42no—
79C2×C4×C109—42no—
80C2×C4×C109—81no—
81C2×C4×C109#12011yescharacteristic
82C2×C4×C109C2×SD1611yescharacteristic
83C8×C109—22no—
84C8×C109—41no—
85C8×C109—81no—
86C22×C109—21no—
87C22×C109—41no—
88C22×C109#45011yescharacteristic
89C4×C109—22no—
90C4×C109—21no—
91C4×C109—41no—
92C4×C109—82no—
93C4×C109—82no—
94C4×C109—161no—
95C4×C109#45411yescharacteristic
96C4×C109#45811yescharacteristic
97#12308C10911yesmaximal, characteristic, direct product factor, Sylow
98C2×C109—21no—
99C2×C109—41no—
100C2×C109#161611yescharacteristic
101#3255C2×C10911yescharacteristic
102#2239C2×C10912yes—
103#2102C2×C10911yescharacteristic
104#1340C2×C10912yes—
105#1327C2×C10911yescharacteristic
106C109#1230811yescharacteristic, direct product factor, Sylow
107#533—21no—
108#475—22no—
109#470—21no—
110#470C22×C10912yes—
111#462—22no—
112#430C22×C10911yescharacteristic
113#423C22×C10911yescharacteristic
114#365—22no—
115#362—22no—
116#349C22×C10912yes—
117#344—21no—
118#339—21no—
119H(3,ℤ/4)C22×C10911yescharacteristic
120C2×Q16—21no—
121C2×Q16D4×C10912yes—
122C2×M16—21no—
123C2×M16D4×C10911yescharacteristic
124C2×M16C23×C10911yescharacteristic
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—41no—
132#117—21no—
133#117—41no—
134C2×C42D4×C10911yescharacteristic
135M32—22no—
136#108—22no—
137#106—21no—
138#106D4×C10912yes—
139C4⋊C8—22no—
140#103—22no—
141#103—22no—
142C2×Q8—22no—
143C2×Q8—41no—
144C22×C4—21no—
145C22×C4C2×D4×C10911yescharacteristic
146Q16—42no—
147Q16—81no—
148M16—41no—
149M16—41no—
150M16—41no—
151M16—41no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8C2×D4×C10912yes—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—41no—
164C42—21no—
165C42—42no—
166C42—41no—
167C42SD16×C10911yescharacteristic
168C42SD16×C10911yescharacteristic
169C16—42no—
170C23AGL(1,ℤ/8)×C10911yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—81no—
182C2×C4#120×C10911yescharacteristic
183C2×C4C2×SD16×C10911yescharacteristic
184C8—22no—
185C8—41no—
186C8—81no—
187C22—21no—
188C22—41no—
189C22#450×C10911yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—161no—
196C4#454×C10911yescharacteristic
197C4#458×C10911yescharacteristic
198C2—21no—
199C2—41no—
200C2C109×#161611yescharacteristic
subgroupG/H
center, socleC2×C109#1616
commutator, Frattini, cosocleC2×M16C23×C109
prime radicalC23×C109AGL(1,ℤ/8)
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×M16×C109 ◁ G

Complements

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

NH ≅ G/Ntype
97. #12308C109direct (N × H)
106. C109#12308direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32552
C109×#22392
C109×#21022
C109×#13402
C109×#13272
#12308109