Finite Group Databasebeta

← C109×#12312 #42130306

Subgroups — C109×#12312 #42130306

Order 27904.

372 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3292C211yesmaximal, characteristic
2C109×#2214C212yesmaximal, complemented
3C109×#1709C211yesmaximal, characteristic, complemented
4C109×#1357C211yesmaximal, characteristic, complemented
5C109×#1335C212yesmaximal, complemented
6#576×C109—22no—
7#510×C109—22no—
8#494×C109C2211yescharacteristic, complemented
9#493×C109C2211yescharacteristic
10#486×C109—22no—
11#472×C109—22no—
12#453×C109—22no—
13#443×C109C2212yes—
14#364×C109C2212yescomplemented
15#362×C109—22no—
16C82×C109C2211yescharacteristic, complemented
17H(5,𝔽2)×C109—22no—
18AGL(1,ℤ/8)×C109—24no—
19AGL(1,ℤ/8)×C109—44no—
20#135×C109—42no—
21#135×C109D412yes—
22C2×Q16×C109—22no—
23C2×D8×C109—22no—
24C2×D8×C109—42no—
25#131×C109—22no—
26C4⋊Q8×C109—21no—
27#127×C109D412yes—
28Q32×C109—42no—
29SD32×C109—42no—
30M32×C109—22no—
31#108×C109D412yes—
32#104×C109—22no—
33#102×C109—44no—
34#100×C109—22no—
35C4×C8×C109—21no—
36C4×C8×C109C2311yescharacteristic
37#41×C109—22no—
38#41×C109—44no—
39C2×Q8×C109—41no—
40C2×D4×C109—22no—
41C2×D4×C109—42no—
42C2×D4×C109—44no—
43C2×D4×C109—82no—
44Q16×C109—42no—
45Q16×C109—42no—
46Q16×C109D812yes—
47SD16×C109—22no—
48SD16×C109—44no—
49D8×C109—44no—
50D8×C109—42no—
51D8×C109—42no—
52D8×C109D812yes—
53M16×C109—22no—
54M16×C109—42no—
55C2×C8×C109—22no—
56C2×C8×C109—42no—
57C2×C8×C109C2×D412yes—
58C4⋊C4×C109—42no—
59C42×C109C2×D411yescharacteristic
60C16×C109—42no—
61C23×C109—42no—
62C23×C109—82no—
63Q8×C109—22no—
64Q8×C109—82no—
65D4×C109—22no—
66D4×C109—42no—
67D4×C109—44no—
68D4×C109—84no—
69D4×C109—84no—
70C2×C4×C109—21no—
71C2×C4×C109—42no—
72C2×C4×C109—82no—
73C2×C4×C109—81no—
74C2×C4×C109#12011yescharacteristic
75C8×C109—42no—
76C8×C109—42no—
77C8×C109C2×D812yes—
78C8×C109AGL(1,ℤ/8)12yes—
79C22×C109—42no—
80C22×C109—82no—
81C22×C109—162no—
82C22×C109—322no—
83C22×C109#45311yescharacteristic
84C4×C109—22no—
85C4×C109—42no—
86C4×C109—161no—
87C4×C109#44712yes—
88#12312C10911yesmaximal, characteristic, direct product factor, Sylow
89C2×C109—21no—
90C2×C109—82no—
91C2×C109—162no—
92C2×C109#161911yescharacteristic
93#3292C2×C10911yescharacteristic
94#2214C2×C10912yescomplemented
95#1709C2×C10911yescharacteristic, complemented
96#1357C2×C10911yescharacteristic, complemented
97#1335C2×C10912yescomplemented
98C109#1231211yescharacteristic, direct product factor, Sylow
99#576—22no—
100#510—22no—
101#494C22×C10911yescharacteristic, complemented
102#493C22×C10911yescharacteristic
103#486—22no—
104#472—22no—
105#453—22no—
106#443C22×C10912yes—
107#364C22×C10912yescomplemented
108#362—22no—
109C82C22×C10911yescharacteristic, complemented
110H(5,𝔽2)—22no—
111AGL(1,ℤ/8)—24no—
112AGL(1,ℤ/8)—44no—
113#135—42no—
114#135D4×C10912yes—
115C2×Q16—22no—
116C2×D8—22no—
117C2×D8—42no—
118#131—22no—
119C4⋊Q8—21no—
120#127D4×C10912yes—
121Q32—42no—
122SD32—42no—
123M32—22no—
124#108D4×C10912yes—
125#104—22no—
126#102—44no—
127#100—22no—
128C4×C8—21no—
129C4×C8C23×C10911yescharacteristic
130#41—22no—
131#41—44no—
132C2×Q8—41no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—82no—
137Q16—42no—
138Q16—42no—
139Q16D8×C10912yes—
140SD16—22no—
141SD16—44no—
142D8—44no—
143D8—42no—
144D8—42no—
145D8D8×C10912yes—
146M16—22no—
147M16—42no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8C2×D4×C10912yes—
151C4⋊C4—42no—
152C42C2×D4×C10911yescharacteristic
153C16—42no—
154C23—42no—
155C23—82no—
156Q8—22no—
157Q8—82no—
158D4—22no—
159D4—42no—
160D4—44no—
161D4—84no—
162D4—84no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4#120×C10911yescharacteristic
168C8—42no—
169C8—42no—
170C8C2×D8×C10912yes—
171C8AGL(1,ℤ/8)×C10912yes—
172C22—42no—
173C22—82no—
174C22—162no—
175C22—322no—
176C22#453×C10911yescharacteristic
177C4—22no—
178C4—42no—
179C4—161no—
180C4#447×C10912yes—
181C2—21no—
182C2—82no—
183C2—162no—
184C2C109×#161911yescharacteristic
subgroupG/H
center, socleC2×C109#1619
commutator, Frattini, cosocleC4×C8C23×C109
prime radicalC109×#3292C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C4×C8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2214C2semidirect
3. C109×#1709C2semidirect
4. C109×#1357C2semidirect
5. C109×#1335C2semidirect
8. #494×C109C22semidirect
14. #364×C109C22semidirect
16. C82×C109C22semidirect
88. #12312C109direct (N × H)
94. #2214C2×C109semidirect
95. #1709C2×C109semidirect
96. #1357C2×C109semidirect
97. #1335C2×C109semidirect
98. C109#12312direct (N × H)
101. #494C22×C109semidirect
107. #364C22×C109semidirect
109. C82C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32922
C109×#22142
C109×#17092
C109×#13572
C109×#13352
#12312109