Finite Group Databasebeta

← C109×#12302 #42130296

Subgroups — C109×#12302 #42130296

Order 27904.

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

normal subgroups only

This group has 310 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×#3202C211yesmaximal, characteristic
2C109×#2239C212yesmaximal, complemented
3C109×#2098C211yesmaximal, characteristic
4C109×#1340C212yesmaximal, complemented
5C109×#1321C211yesmaximal, characteristic
6#536×C109—21no—
7#485×C109—22no—
8#470×C109C2212yes—
9#467×C109—21no—
10#451×C109—22no—
11#430×C109C2211yescharacteristic
12#424×C109—21no—
13#421×C109—21no—
14#408×C109C2211yescharacteristic
15#365×C109—22no—
16#362×C109—22no—
17#349×C109C2212yes—
18H(3,ℤ/4)×C109C2211yescharacteristic
19C2×#41×C109D411yescharacteristic
20C22×Q8×C109—21no—
21C2×Q16×C109—21no—
22C2×Q16×C109D412yes—
23C2×M16×C109—21no—
24C2×M16×C109C2311yescharacteristic
25C22×C8×C109D411yescharacteristic
26C4⋊Q8×C109—41no—
27#124×C109—41no—
28#123×C109—41no—
29#123×C109—41no—
30C22⋊Q8×C109—22no—
31C22⋊Q8×C109—22no—
32#117×C109—21no—
33#117×C109—21no—
34M32×C109—22no—
35#108×C109—22no—
36#107×C109—21no—
37#106×C109D412yes—
38C4⋊C8×C109—42no—
39#104×C109—42no—
40#103×C109—22no—
41#103×C109—22no—
42C22⋊C8×C109—22no—
43#41×C109—41no—
44#41×C109—41no—
45C2×Q8×C109—22no—
46C2×Q8×C109—41no—
47C2×Q8×C109—81no—
48C2×Q8×C109—81no—
49C2×Q8×C109SD1611yescharacteristic
50C2×D4×C109—21no—
51C2×D4×C109SD1611yescharacteristic
52C22×C4×C109—21no—
53C22×C4×C109—41no—
54C22×C4×C109C2×D411yescharacteristic
55Q16×C109—42no—
56Q16×C109—81no—
57M16×C109—41no—
58M16×C109—41no—
59M16×C109—41no—
60C2×C8×C109—21no—
61C2×C8×C109—41no—
62C2×C8×C109—41no—
63C2×C8×C109C2×D412yes—
64C4⋊C4×C109—22no—
65C4⋊C4×C109—41no—
66C4⋊C4×C109—42no—
67C4⋊C4×C109—42no—
68C22⋊C4×C109—41no—
69C22⋊C4×C109—42no—
70C22⋊C4×C109—41no—
71C22⋊C4×C109—42no—
72C42×C109—22no—
73C42×C109—41no—
74C16×C109—42no—
75C23×C109—21no—
76C23×C109AGL(1,ℤ/8)11yescharacteristic
77Q8×C109—41no—
78Q8×C109—42no—
79Q8×C109—42no—
80Q8×C109—82no—
81D4×C109—41no—
82D4×C109—81no—
83C2×C4×C109—21no—
84C2×C4×C109—21no—
85C2×C4×C109—21no—
86C2×C4×C109—41no—
87C2×C4×C109—41no—
88C2×C4×C109—42no—
89C2×C4×C109—42no—
90C2×C4×C109—42no—
91C2×C4×C109—82no—
92C2×C4×C109#12011yescharacteristic
93C2×C4×C109C2×SD1611yescharacteristic
94C8×C109—22no—
95C8×C109—41no—
96C8×C109—81no—
97C22×C109—21no—
98C22×C109—41no—
99C22×C109—42no—
100C22×C109—41no—
101C22×C109#45011yescharacteristic
102C4×C109—21no—
103C4×C109—41no—
104C4×C109—82no—
105C4×C109—82no—
106C4×C109—82no—
107C4×C109#45311yescharacteristic
108C4×C109#45811yescharacteristic
109#12302C10911yesmaximal, characteristic, direct product factor, Sylow
110C2×C109—21no—
111C2×C109—41no—
112C2×C109—81no—
113C2×C109#161311yescharacteristic
114#3202C2×C10911yescharacteristic
115#2239C2×C10912yescomplemented
116#2098C2×C10911yescharacteristic
117#1340C2×C10912yescomplemented
118#1321C2×C10911yescharacteristic
119C109#1230211yescharacteristic, direct product factor, Sylow
120#536—21no—
121#485—22no—
122#470C22×C10912yes—
123#467—21no—
124#451—22no—
125#430C22×C10911yescharacteristic
126#424—21no—
127#421—21no—
128#408C22×C10911yescharacteristic
129#365—22no—
130#362—22no—
131#349C22×C10912yes—
132H(3,ℤ/4)C22×C10911yescharacteristic
133C2×#41D4×C10911yescharacteristic
134C22×Q8—21no—
135C2×Q16—21no—
136C2×Q16D4×C10912yes—
137C2×M16—21no—
138C2×M16C23×C10911yescharacteristic
139C22×C8D4×C10911yescharacteristic
140C4⋊Q8—41no—
141#124—41no—
142#123—41no—
143#123—41no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146#117—21no—
147#117—21no—
148M32—22no—
149#108—22no—
150#107—21no—
151#106D4×C10912yes—
152C4⋊C8—42no—
153#104—42no—
154#103—22no—
155#103—22no—
156C22⋊C8—22no—
157#41—41no—
158#41—41no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8SD16×C10911yescharacteristic
164C2×D4—21no—
165C2×D4SD16×C10911yescharacteristic
166C22×C4—21no—
167C22×C4—41no—
168C22×C4C2×D4×C10911yescharacteristic
169Q16—42no—
170Q16—81no—
171M16—41no—
172M16—41no—
173M16—41no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8C2×D4×C10912yes—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C42—22no—
187C42—41no—
188C16—42no—
189C23—21no—
190C23AGL(1,ℤ/8)×C10911yescharacteristic
191Q8—41no—
192Q8—42no—
193Q8—42no—
194Q8—82no—
195D4—41no—
196D4—81no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—82no—
206C2×C4#120×C10911yescharacteristic
207C2×C4C2×SD16×C10911yescharacteristic
208C8—22no—
209C8—41no—
210C8—81no—
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#450×C10911yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4#453×C10911yescharacteristic
222C4#458×C10911yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2C109×#161311yescharacteristic
subgroupG/H
center, socleC2×C109#1613
commutator, Frattini, cosocleC2×M16C23×C109
prime radicalC2×#41×C109D4
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
2. C109×#2239C2semidirect
4. C109×#1340C2semidirect
109. #12302C109direct (N × H)
115. #2239C2×C109semidirect
117. #1340C2×C109semidirect
119. C109#12302direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32022
C109×#22392
C109×#20982
C109×#13402
C109×#13212
#12302109