Finite Group Databasebeta

← C109×#12207 #42130201

Subgroups — C109×#12207 #42130201

Order 27904.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2817C211yesmaximal, characteristic
2C109×#2030C211yesmaximal, characteristic
3C109×#1608C212yesmaximal
4C109×#1344C212yesmaximal
5C109×#1317C211yesmaximal, characteristic
6#543×C109C2211yescharacteristic
7C2×C22⋊Q8×C109—21no—
8#483×C109—22no—
9#400×C109—22no—
10#397×C109—21no—
11#397×C109C2212yes—
12#395×C109—22no—
13#381×C109—21no—
14#381×C109C2211yescharacteristic
15#342×C109—21no—
16#342×C109C2211yescharacteristic
17#328×C109—22no—
18#323×C109C2212yes—
19C22×Q8×C109—21no—
20C23×C4×C109D411yescharacteristic
21C22⋊Q8×C109—22no—
22C22⋊Q8×C109—42no—
23C22⋊Q8×C109—42no—
24C22⋊Q8×C109—42no—
25C2×C4⋊C4×C109—22no—
26C2×C4⋊C4×C109—21no—
27C2×C4⋊C4×C109—21no—
28C2×C4⋊C4×C109D412yes—
29C2×C22⋊C4×C109—22no—
30C2×C22⋊C4×C109—21no—
31C2×C22⋊C4×C109—21no—
32C2×C22⋊C4×C109D411yescharacteristic
33C2×C22⋊C4×C109C2311yescharacteristic
34#106×C109—42no—
35#103×C109—42no—
36C22⋊C8×C109—22no—
37#95×C109—22no—
38#95×C109—21no—
39#95×C109—22no—
40#95×C109—22no—
41#95×C109—42no—
42#95×C109D412yes—
43C24×C109C2×D411yescharacteristic
44C2×Q8×C109—22no—
45C2×Q8×C109—42no—
46C2×Q8×C109—81no—
47C2×Q8×C109—81no—
48C22×C4×C109—21no—
49C22×C4×C109—21no—
50C22×C4×C109—21no—
51C22×C4×C109—22no—
52C22×C4×C109—22no—
53C22×C4×C109—22no—
54C22×C4×C109—41no—
55C22×C4×C109—41no—
56C22×C4×C109—41no—
57C22×C4×C109SD1612yes—
58C22×C4×C109C2×D412yes—
59C2×C8×C109—42no—
60C4⋊C4×C109—22no—
61C4⋊C4×C109—42no—
62C4⋊C4×C109—42no—
63C4⋊C4×C109—44no—
64C4⋊C4×C109—42no—
65C4⋊C4×C109—81no—
66C22⋊C4×C109—41no—
67C22⋊C4×C109—41no—
68C22⋊C4×C109—41no—
69C22⋊C4×C109—42no—
70C22⋊C4×C109—42no—
71C22⋊C4×C109—41no—
72C22⋊C4×C109—41no—
73C22⋊C4×C109—41no—
74C23×C109—21no—
75C23×C109—21no—
76C23×C109—42no—
77C23×C109#12011yescharacteristic
78C23×C109C2×SD1611yescharacteristic
79C23×C109AGL(1,ℤ/8)11yescharacteristic
80Q8×C109—82no—
81Q8×C109—84no—
82C2×C4×C109—22no—
83C2×C4×C109—22no—
84C2×C4×C109—41no—
85C2×C4×C109—41no—
86C2×C4×C109—42no—
87C2×C4×C109—42no—
88C2×C4×C109—42no—
89C2×C4×C109—44no—
90C2×C4×C109—42no—
91C2×C4×C109—42no—
92C2×C4×C109—41no—
93C2×C4×C109—42no—
94C2×C4×C109—42no—
95C2×C4×C109—42no—
96C2×C4×C109—81no—
97C2×C4×C109—82no—
98C2×C4×C109—81no—
99C2×C4×C109—82no—
100C8×C109—82no—
101C22×C109—22no—
102C22×C109—21no—
103C22×C109—21no—
104C22×C109—41no—
105C22×C109—41no—
106C22×C109—42no—
107C22×C109—81no—
108C22×C109#45011yescharacteristic
109C22×C109#45511yescharacteristic
110C22×C109B(4,𝔽2)11yescharacteristic
111C4×C109—42no—
112C4×C109—42no—
113C4×C109—82no—
114C4×C109—81no—
115C4×C109—82no—
116C4×C109—82no—
117C4×C109—161no—
118#12207C10911yesmaximal, characteristic, direct product factor, Sylow
119C2×C109—21no—
120C2×C109—21no—
121C2×C109—41no—
122C2×C109—41no—
123C2×C109#160111yescharacteristic
124C2×C109#220112yes—
125#2817C2×C10911yescharacteristic
126#2030C2×C10911yescharacteristic
127#1608C2×C10912yes—
128#1344C2×C10912yes—
129#1317C2×C10911yescharacteristic
130C109#1220711yescharacteristic, direct product factor, Sylow
131#543C22×C10911yescharacteristic
132C2×C22⋊Q8—21no—
133#483—22no—
134#400—22no—
135#397—21no—
136#397C22×C10912yes—
137#395—22no—
138#381—21no—
139#381C22×C10911yescharacteristic
140#342—21no—
141#342C22×C10911yescharacteristic
142#328—22no—
143#323C22×C10912yes—
144C22×Q8—21no—
145C23×C4D4×C10911yescharacteristic
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4D4×C10912yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4D4×C10911yescharacteristic
158C2×C22⋊C4C23×C10911yescharacteristic
159#106—42no—
160#103—42no—
161C22⋊C8—22no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—42no—
167#95D4×C10912yes—
168C24C2×D4×C10911yescharacteristic
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—81no—
172C2×Q8—81no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4SD16×C10912yes—
183C22×C4C2×D4×C10912yes—
184C2×C8—42no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#120×C10911yescharacteristic
203C23C2×SD16×C10911yescharacteristic
204C23AGL(1,ℤ/8)×C10911yescharacteristic
205Q8—82no—
206Q8—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—82no—
225C8—82no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#450×C10911yescharacteristic
234C22#455×C10911yescharacteristic
235C22B(4,𝔽2)×C10911yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—82no—
242C4—161no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2C109×#160111yescharacteristic
248C2C109×#220112yes—
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C109
prime radicalC24×C109C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C22⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
118. #12207C109direct (N × H)
130. C109#12207direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#28172
C109×#20302
C109×#16082
C109×#13442
C109×#13172
#12207109