Finite Group Databasebeta

← C109×#12301 #42130295

Subgroups — C109×#12301 #42130295

Order 27904.

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

normal subgroups only

This group has 342 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×#3199C211yesmaximal, characteristic
2C109×#2238C212yesmaximal, complemented
3C109×#2098C211yesmaximal, characteristic, complemented
4C109×#1339C212yesmaximal, complemented
5C109×#1321C211yesmaximal, characteristic, complemented
6#535×C109—21no—
7#482×C109—22no—
8#469×C109C2212yes—
9#466×C109—21no—
10#447×C109—22no—
11#430×C109C2211yescharacteristic, complemented
12#424×C109—21no—
13#421×C109—21no—
14#408×C109C2211yescharacteristic
15#365×C109—22no—
16#361×C109—22no—
17#349×C109C2212yescomplemented
18H(3,ℤ/4)×C109C2211yescharacteristic, complemented
19C2×#41×C109D411yescharacteristic
20C22×D4×C109—21no—
21C2×D8×C109—21no—
22C2×D8×C109D412yes—
23C2×M16×C109—21no—
24C2×M16×C109C2311yescharacteristic
25C22×C8×C109D411yescharacteristic
26#127×C109—41no—
27#124×C109—41no—
28#121×C109—22no—
29#121×C109—22no—
30#120×C109—41no—
31#120×C109—41no—
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#102×C109—22no—
41#102×C109—22no—
42C22⋊C8×C109—22no—
43C24×C109—41no—
44#41×C109—41no—
45#41×C109—41no—
46C2×Q8×C109SD1611yescharacteristic
47C2×D4×C109—21no—
48C2×D4×C109—22no—
49C2×D4×C109—41no—
50C2×D4×C109—42no—
51C2×D4×C109—42no—
52C2×D4×C109—81no—
53C2×D4×C109—81no—
54C2×D4×C109SD1611yescharacteristic
55C22×C4×C109—21no—
56C22×C4×C109C2×D411yescharacteristic
57D8×C109—42no—
58D8×C109—81no—
59M16×C109—41no—
60M16×C109—41no—
61M16×C109—41no—
62C2×C8×C109—21no—
63C2×C8×C109—41no—
64C2×C8×C109—41no—
65C2×C8×C109C2×D412yes—
66C4⋊C4×C109—22no—
67C4⋊C4×C109—41no—
68C22⋊C4×C109—42no—
69C22⋊C4×C109—41no—
70C22⋊C4×C109—41no—
71C22⋊C4×C109—42no—
72C42×C109—22no—
73C42×C109—41no—
74C16×C109—42no—
75C23×C109—21no—
76C23×C109—42no—
77C23×C109—82no—
78C23×C109—162no—
79C23×C109AGL(1,ℤ/8)11yescharacteristic
80Q8×C109—41no—
81D4×C109—41no—
82D4×C109—42no—
83D4×C109—42no—
84D4×C109—81no—
85D4×C109—82no—
86D4×C109—162no—
87D4×C109—162no—
88C2×C4×C109—21no—
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—41no—
92C2×C4×C109—41no—
93C2×C4×C109—42no—
94C2×C4×C109—42no—
95C2×C4×C109#12011yescharacteristic
96C2×C4×C109C2×SD1611yescharacteristic
97C8×C109—22no—
98C8×C109—41no—
99C8×C109—81no—
100C22×C109—21no—
101C22×C109—41no—
102C22×C109—42no—
103C22×C109—41no—
104C22×C109—82no—
105C22×C109—82no—
106C22×C109—82no—
107C22×C109—164no—
108C22×C109#45011yescharacteristic
109C4×C109—21no—
110C4×C109—41no—
111C4×C109—82no—
112C4×C109—82no—
113C4×C109#45311yescharacteristic
114C4×C109#45811yescharacteristic
115#12301C10911yesmaximal, characteristic, direct product factor, Sylow
116C2×C109—21no—
117C2×C109—41no—
118C2×C109—81no—
119C2×C109—162no—
120C2×C109#161311yescharacteristic
121#3199C2×C10911yescharacteristic
122#2238C2×C10912yescomplemented
123#2098C2×C10911yescharacteristic, complemented
124#1339C2×C10912yescomplemented
125#1321C2×C10911yescharacteristic, complemented
126C109#1230111yescharacteristic, direct product factor, Sylow
127#535—21no—
128#482—22no—
129#469C22×C10912yes—
130#466—21no—
131#447—22no—
132#430C22×C10911yescharacteristic, complemented
133#424—21no—
134#421—21no—
135#408C22×C10911yescharacteristic
136#365—22no—
137#361—22no—
138#349C22×C10912yescomplemented
139H(3,ℤ/4)C22×C10911yescharacteristic, complemented
140C2×#41D4×C10911yescharacteristic
141C22×D4—21no—
142C2×D8—21no—
143C2×D8D4×C10912yes—
144C2×M16—21no—
145C2×M16C23×C10911yescharacteristic
146C22×C8D4×C10911yescharacteristic
147#127—41no—
148#124—41no—
149#121—22no—
150#121—22no—
151#120—41no—
152#120—41no—
153#117—21no—
154#117—21no—
155M32—22no—
156#108—22no—
157#107—21no—
158#106D4×C10912yes—
159C4⋊C8—42no—
160#104—42no—
161#102—22no—
162#102—22no—
163C22⋊C8—22no—
164C24—41no—
165#41—41no—
166#41—41no—
167C2×Q8SD16×C10911yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—81no—
174C2×D4—81no—
175C2×D4SD16×C10911yescharacteristic
176C22×C4—21no—
177C22×C4C2×D4×C10911yescharacteristic
178D8—42no—
179D8—81no—
180M16—41no—
181M16—41no—
182M16—41no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×D4×C10912yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C22⋊C4—42no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C42—22no—
194C42—41no—
195C16—42no—
196C23—21no—
197C23—42no—
198C23—82no—
199C23—162no—
200C23AGL(1,ℤ/8)×C10911yescharacteristic
201Q8—41no—
202D4—41no—
203D4—42no—
204D4—42no—
205D4—81no—
206D4—82no—
207D4—162no—
208D4—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4#120×C10911yescharacteristic
217C2×C4C2×SD16×C10911yescharacteristic
218C8—22no—
219C8—41no—
220C8—81no—
221C22—21no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—164no—
229C22#450×C10911yescharacteristic
230C4—21no—
231C4—41no—
232C4—82no—
233C4—82no—
234C4#453×C10911yescharacteristic
235C4#458×C10911yescharacteristic
236C2—21no—
237C2—41no—
238C2—81no—
239C2—162no—
240C2C109×#161311yescharacteristic
subgroupG/H
center, socleC2×C109#1613
commutator, Frattini, cosocleC2×M16C23×C109
prime radicalC109×#3199C2
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×#2238C2semidirect
3. C109×#2098C2semidirect
4. C109×#1339C2semidirect
5. C109×#1321C2semidirect
11. #430×C109C22semidirect
17. #349×C109C22semidirect
18. H(3,ℤ/4)×C109C22semidirect
115. #12301C109direct (N × H)
122. #2238C2×C109semidirect
123. #2098C2×C109semidirect
124. #1339C2×C109semidirect
125. #1321C2×C109semidirect
126. C109#12301direct (N × H)
132. #430C22×C109semidirect
138. #349C22×C109semidirect
139. H(3,ℤ/4)C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#31992
C109×#22382
C109×#20982
C109×#13392
C109×#13212
#12301109