Finite Group Databasebeta

← C109×#12288 #42130282

Subgroups — C109×#12288 #42130282

Order 27904.

354 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3245C211yesmaximal, characteristic
2C109×#2232C212yesmaximal, complemented
3C109×#2086C211yesmaximal, characteristic, complemented
4C109×#1347C212yesmaximal, complemented
5C109×#1322C211yesmaximal, characteristic, complemented
6#532×C109—21no—
7#474×C109—22no—
8#466×C109—21no—
9#466×C109C2212yes—
10#459×C109—22no—
11C2×#107×C109C2211yescharacteristic
12C2×C4⋊C8×C109C2211yescharacteristic
13#389×C109—21no—
14#366×C109—22no—
15#357×C109—22no—
16C22⋊C16×C109C2212yes—
17#340×C109—21no—
18#340×C109C2211yescharacteristic
19C2×#41×C109—41no—
20C2×D8×C109—21no—
21C2×D8×C109D412yes—
22C22×C8×C109D411yescharacteristic
23C22×C8×C109C2311yescharacteristic
24C4⋊Q8×C109—21no—
25#127×C109—21no—
26#124×C109—41no—
27#121×C109—22no—
28#121×C109—41no—
29C4×D4×C109—22no—
30C4×D4×C109—41no—
31C2×C4⋊C4×C109—21no—
32C2×C4⋊C4×C109—21no—
33C2×C4⋊C4×C109—41no—
34C2×C42×C109D411yescharacteristic
35C2×C16×C109—22no—
36#107×C109—21no—
37#107×C109—22no—
38#107×C109D412yes—
39C4⋊C8×C109—22no—
40#102×C109—22no—
41#102×C109—22no—
42#95×C109—41no—
43#41×C109—82no—
44#41×C109—82no—
45C2×Q8×C109—41no—
46C2×D4×C109—22no—
47C2×D4×C109—41no—
48C2×D4×C109—41no—
49C2×D4×C109—42no—
50C22×C4×C109—42no—
51C22×C4×C109—41no—
52C22×C4×C109—41no—
53C22×C4×C109D811yescharacteristic
54C22×C4×C109D811yescharacteristic
55C22×C4×C109C2×D411yescharacteristic
56D8×C109—42no—
57D8×C109—81no—
58C2×C8×C109—21no—
59C2×C8×C109—41no—
60C2×C8×C109—41no—
61C2×C8×C109C2×D412yes—
62C4⋊C4×C109—22no—
63C4⋊C4×C109—22no—
64C4⋊C4×C109—41no—
65C4⋊C4×C109—41no—
66C4⋊C4×C109—41no—
67C4⋊C4×C109—41no—
68C4⋊C4×C109—82no—
69C22⋊C4×C109—42no—
70C22⋊C4×C109—42no—
71C42×C109—21no—
72C42×C109—21no—
73C16×C109—42no—
74C23×C109—42no—
75C23×C109C2×D811yescharacteristic
76Q8×C109—82no—
77D4×C109—42no—
78D4×C109—42no—
79D4×C109—82no—
80D4×C109—82no—
81D4×C109—82no—
82D4×C109—82no—
83C2×C4×C109—21no—
84C2×C4×C109—21no—
85C2×C4×C109—21no—
86C2×C4×C109—21no—
87C2×C4×C109—21no—
88C2×C4×C109—41no—
89C2×C4×C109—42no—
90C2×C4×C109—42no—
91C2×C4×C109—84no—
92C2×C4×C109—81no—
93C2×C4×C109—81no—
94C2×C4×C109—81no—
95C2×C4×C109—81no—
96C2×C4×C109—81no—
97C2×C4×C109D1612yes—
98C2×C4×C109#12011yescharacteristic
99C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
100C8×C109—22no—
101C8×C109—41no—
102C22×C109—21no—
103C22×C109—21no—
104C22×C109—82no—
105C22×C109—82no—
106C22×C109—82no—
107C22×C109#44711yescharacteristic
108C22×C109C2×D1611yescharacteristic
109C22×C109#50911yescharacteristic
110C4×C109—21no—
111C4×C109—22no—
112C4×C109—41no—
113C4×C109—82no—
114C4×C109—82no—
115C4×C109—161no—
116C4×C109#45611yescharacteristic
117C4×C109#45811yescharacteristic
118#12288C10911yesmaximal, characteristic, direct product factor, Sylow
119C2×C109—21no—
120C2×C109—21no—
121C2×C109—162no—
122C2×C109#161011yescharacteristic
123C2×C109#218411yescharacteristic
124C2×C109#219111yescharacteristic
125#3245C2×C10911yescharacteristic
126#2232C2×C10912yescomplemented
127#2086C2×C10911yescharacteristic, complemented
128#1347C2×C10912yescomplemented
129#1322C2×C10911yescharacteristic, complemented
130C109#1228811yescharacteristic, direct product factor, Sylow
131#532—21no—
132#474—22no—
133#466—21no—
134#466C22×C10912yes—
135#459—22no—
136C2×#107C22×C10911yescharacteristic
137C2×C4⋊C8C22×C10911yescharacteristic
138#389—21no—
139#366—22no—
140#357—22no—
141C22⋊C16C22×C10912yes—
142#340—21no—
143#340C22×C10911yescharacteristic
144C2×#41—41no—
145C2×D8—21no—
146C2×D8D4×C10912yes—
147C22×C8D4×C10911yescharacteristic
148C22×C8C23×C10911yescharacteristic
149C4⋊Q8—21no—
150#127—21no—
151#124—41no—
152#121—22no—
153#121—41no—
154C4×D4—22no—
155C4×D4—41no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42D4×C10911yescharacteristic
160C2×C16—22no—
161#107—21no—
162#107—22no—
163#107D4×C10912yes—
164C4⋊C8—22no—
165#102—22no—
166#102—22no—
167#95—41no—
168#41—82no—
169#41—82no—
170C2×Q8—41no—
171C2×D4—22no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4D8×C10911yescharacteristic
179C22×C4D8×C10911yescharacteristic
180C22×C4C2×D4×C10911yescharacteristic
181D8—42no—
182D8—81no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×D4×C10912yes—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—82no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C16—42no—
199C23—42no—
200C23C2×D8×C10911yescharacteristic
201Q8—82no—
202D4—42no—
203D4—42no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4D16×C10912yes—
223C2×C4#120×C10911yescharacteristic
224C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22#447×C10911yescharacteristic
233C22C2×D16×C10911yescharacteristic
234C22#509×C10911yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—82no—
240C4—161no—
241C4#456×C10911yescharacteristic
242C4#458×C10911yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2C109×#161011yescharacteristic
247C2C109×#218411yescharacteristic
248C2C109×#219111yescharacteristic
subgroupG/H
center, socleC22×C109#447
commutator, Frattini, cosocleC22×C8C23×C109
prime radicalC109×#3245C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×C4⋊C8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2232C2semidirect
3. C109×#2086C2semidirect
4. C109×#1347C2semidirect
5. C109×#1322C2semidirect
118. #12288C109direct (N × H)
126. #2232C2×C109semidirect
127. #2086C2×C109semidirect
128. #1347C2×C109semidirect
129. #1322C2×C109semidirect
130. C109#12288direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32452
C109×#22322
C109×#20862
C109×#13472
C109×#13222
#12288109