Finite Group Databasebeta

← C109×#10584 #42128578

Subgroups — C109×#10584 #42128578

Order 27904.

374 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2844C211yesmaximal, characteristic
2C109×#1847C211yesmaximal, characteristic
3C109×#1836C212yesmaximal
4C109×#1295C212yesmaximal
5C109×#1295C211yesmaximal, characteristic
6C2×#107×C109—21no—
7C2×#106×C109—21no—
8C2×#97×C109C2212yes—
9C2×#97×C109C2211yescharacteristic
10#400×C109—22no—
11#400×C109—21no—
12#395×C109—22no—
13#395×C109—21no—
14#384×C109C2211yescharacteristic
15#382×C109C2212yes—
16#382×C109C2211yescharacteristic
17#340×C109—22no—
18#340×C109—22no—
19C22×Q8×C109—21no—
20C22×C8×C109—21no—
21C22×C8×C109—22no—
22C2×C4⋊C4×C109—22no—
23C2×C4⋊C4×C109—21no—
24C2×C4⋊C4×C109—21no—
25C2×C4⋊C4×C109—22no—
26C2×C4⋊C4×C109D412yes—
27C2×C4⋊C4×C109D412yes—
28C2×C4⋊C4×C109D411yescharacteristic
29C2×C4⋊C4×C109D411yescharacteristic
30C2×C42×C109C2311yescharacteristic
31#107×C109—42no—
32#106×C109—42no—
33#97×C109—21no—
34#97×C109—22no—
35#97×C109Q812yescomplemented
36#97×C109Q814yescomplemented
37#95×C109—22no—
38#95×C109—21no—
39#95×C109—21no—
40#95×C109—22no—
41#95×C109—22no—
42#95×C109—22no—
43C2×Q8×C109—81no—
44C2×Q8×C109—82no—
45C22×C4×C109—21no—
46C22×C4×C109—22no—
47C22×C4×C109—21no—
48C22×C4×C109—22no—
49C22×C4×C109—41no—
50C22×C4×C109C2×D411yescharacteristic
51C22×C4×C109C2×D412yes—
52C2×C8×C109—22no—
53C2×C8×C109—22no—
54C2×C8×C109—22no—
55C2×C8×C109—24no—
56C2×C8×C109—42no—
57C2×C8×C109—42no—
58C4⋊C4×C109—41no—
59C4⋊C4×C109—41no—
60C4⋊C4×C109—42no—
61C4⋊C4×C109—42no—
62C4⋊C4×C109—41no—
63C4⋊C4×C109—41no—
64C4⋊C4×C109—44no—
65C4⋊C4×C109—81no—
66C4⋊C4×C109—82no—
67C42×C109C2×Q811yescharacteristic
68C42×C109C2×Q812yes—
69C42×C109#4111yescharacteristic
70C23×C109#12011yescharacteristic
71Q8×C109—84no—
72C2×C4×C109—21no—
73C2×C4×C109—22no—
74C2×C4×C109—22no—
75C2×C4×C109—21no—
76C2×C4×C109—41no—
77C2×C4×C109—41no—
78C2×C4×C109—42no—
79C2×C4×C109—42no—
80C2×C4×C109—42no—
81C2×C4×C109—42no—
82C2×C4×C109—42no—
83C2×C4×C109—42no—
84C2×C4×C109—42no—
85C2×C4×C109—42no—
86C2×C4×C109—81no—
87C2×C4×C109—82no—
88C2×C4×C109C22⋊Q811yescharacteristic
89C2×C4×C109C22⋊Q812yes—
90C2×C4×C109C4⋊Q811yescharacteristic
91C2×C4×C109C4⋊Q812yes—
92C8×C109—24no—
93C8×C109—44no—
94C22×C109#39511yescharacteristic
95C22×C109#45311yescharacteristic
96C22×C109#45412yes—
97C22×C109#45512yes—
98C22×C109#45611yescharacteristic
99C4×C109—21no—
100C4×C109—21no—
101C4×C109—22no—
102C4×C109—22no—
103C4×C109—81no—
104C4×C109—81no—
105C4×C109—82no—
106C4×C109—82no—
107C4×C109—161no—
108#10584C10911yesmaximal, characteristic, direct product factor, Sylow
109C2×C109#165711yescharacteristic
110C2×C109#165911yescharacteristic
111C2×C109#166412yes—
112C2×C109#206011yescharacteristic
113C2×C109#206112yes—
114#2844C2×C10911yescharacteristic
115#1847C2×C10911yescharacteristic
116#1836C2×C10912yes—
117#1295C2×C10912yes—
118#1295C2×C10911yescharacteristic
119C109#1058411yescharacteristic, direct product factor, Sylow
120C2×#107—21no—
121C2×#106—21no—
122C2×#97C22×C10912yes—
123C2×#97C22×C10911yescharacteristic
124#400—22no—
125#400—21no—
126#395—22no—
127#395—21no—
128#384C22×C10911yescharacteristic
129#382C22×C10912yes—
130#382C22×C10911yescharacteristic
131#340—22no—
132#340—22no—
133C22×Q8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4D4×C10912yes—
141C2×C4⋊C4D4×C10912yes—
142C2×C4⋊C4D4×C10911yescharacteristic
143C2×C4⋊C4D4×C10911yescharacteristic
144C2×C42C23×C10911yescharacteristic
145#107—42no—
146#106—42no—
147#97—21no—
148#97—22no—
149#97Q8×C10912yescomplemented
150#97Q8×C10914yescomplemented
151#95—22no—
152#95—21no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4C2×D4×C10911yescharacteristic
165C22×C4C2×D4×C10912yes—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C42C2×Q8×C10911yescharacteristic
182C42C2×Q8×C10912yes—
183C42#41×C10911yescharacteristic
184C23#120×C10911yescharacteristic
185Q8—84no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—82no—
202C2×C4C22⋊Q8×C10911yescharacteristic
203C2×C4C22⋊Q8×C10912yes—
204C2×C4C4⋊Q8×C10911yescharacteristic
205C2×C4C4⋊Q8×C10912yes—
206C8—24no—
207C8—44no—
208C22#395×C10911yescharacteristic
209C22#453×C10911yescharacteristic
210C22#454×C10912yes—
211C22#455×C10912yes—
212C22#456×C10911yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—82no—
219C4—82no—
220C4—81no—
221C4—161no—
222C2C109×#165711yescharacteristic
223C2C109×#165911yescharacteristic
224C2C109×#166412yes—
225C2C109×#206011yescharacteristic
226C2C109×#206112yes—
subgroupG/H
center, socle, prime radicalC23×C109#120
commutator, Frattini, cosocleC2×C42C23×C109
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
35. #97×C109Q8semidirect
36. #97×C109Q8semidirect
108. #10584C109direct (N × H)
119. C109#10584direct (N × H)
149. #97Q8×C109semidirect
150. #97Q8×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#28442
C109×#18472
C109×#18362
C109×#12952
#10584109