Finite Group Databasebeta

← C109×#1500 #28358754

Subgroups — C109×#1500 #28358754

Order 13952.

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#529×C109C211yesmaximal, characteristic
2#431×C109C211yesmaximal, characteristic, complemented
3#376×C109C211yesmaximal, characteristic, complemented
4#328×C109C212yesmaximal, complemented
5#327×C109C212yesmaximal
6#126×C109—21no—
7#125×C109C411yescharacteristic, complemented
8#124×C109C411yescharacteristic, complemented
9#123×C109—21no—
10C22⋊Q8×C109C2211yescharacteristic
11#121×C109C2211yescharacteristic
12C4×Q8×C109C411yescharacteristic, complemented
13C4×D4×C109—21no—
14C4×D4×C109C411yescharacteristic, complemented
15#117×C109—21no—
16C2×C42×C109—21no—
17C2×C42×C109C2211yescharacteristic
18C4⋊C8×C109—21no—
19C22⋊C8×C109C2212yes—
20C4×C8×C109—21no—
21#95×C109—21no—
22#95×C109C2212yes—
23C2×Q8×C109C2×C411yescharacteristic
24C2×D4×C109—21no—
25C2×D4×C109C2×C411yescharacteristic
26C22×C4×C109—22no—
27C22×C4×C109—21no—
28C22×C4×C109D411yescharacteristic
29C22×C4×C109D411yescharacteristic
30C22×C4×C109C2311yescharacteristic
31C2×C8×C109—22no—
32C4⋊C4×C109—21no—
33C4⋊C4×C109—21no—
34C4⋊C4×C109—21no—
35C4⋊C4×C109—21no—
36C4⋊C4×C109C2×C411yescharacteristic
37C4⋊C4×C109C2×C411yescharacteristic
38C22⋊C4×C109—21no—
39C22⋊C4×C109—21no—
40C22⋊C4×C109—21no—
41C22⋊C4×C109—21no—
42C22⋊C4×C109—21no—
43C42×C109—24no—
44C42×C109—21no—
45C42×C109C2×C411yescharacteristic
46C42×C109C2×C411yescharacteristic
47C42×C109D411yescharacteristic
48C42×C109D411yescharacteristic
49C23×C109—21no—
50C23×C109C2×D411yescharacteristic
51Q8×C109—41no—
52D4×C109—41no—
53D4×C109—42no—
54C2×C4×C109—22no—
55C2×C4×C109—22no—
56C2×C4×C109—24no—
57C2×C4×C109—21no—
58C2×C4×C109—21no—
59C2×C4×C109—21no—
60C2×C4×C109—41no—
61C2×C4×C109—42no—
62C2×C4×C109—42no—
63C2×C4×C109—42no—
64C2×C4×C109C22⋊C411yescharacteristic
65C2×C4×C109C22⋊C411yescharacteristic
66C2×C4×C109C22⋊C411yescharacteristic
67C2×C4×C109C22⋊C411yescharacteristic
68C2×C4×C109C22×C411yescharacteristic
69C2×C4×C109C2×D411yescharacteristic
70C8×C109—42no—
71C22×C109—21no—
72C22×C109—21no—
73C22×C109—41no—
74C22×C109—41no—
75C22×C109—41no—
76C22×C109C2×C22⋊C411yescharacteristic
77C22×C109#13512yes—
78C4×C109—21no—
79C4×C109—21no—
80C4×C109—21no—
81C4×C109—21no—
82C4×C109—21no—
83C4×C109—44no—
84C4×C109—41no—
85C4×C109—41no—
86C4×C109—41no—
87C4×C109#10412yes—
88C2×C109—22no—
89C2×C109—81no—
90C2×C109#41011yescharacteristic
91C2×C109#41611yescharacteristic
92C2×C109C2×#10411yescharacteristic
93#1500C10911yesmaximal, characteristic, direct product factor, Sylow
94C109#150011yescharacteristic, direct product factor, Sylow
95#529C2×C10911yescharacteristic
96#431C2×C10911yescharacteristic, complemented
97#376C2×C10911yescharacteristic, complemented
98#328C2×C10912yescomplemented
99#327C2×C10912yes—
100#126—21no—
101#125C4×C10911yescharacteristic, complemented
102#124C4×C10911yescharacteristic, complemented
103#123—21no—
104C22⋊Q8C22×C10911yescharacteristic
105#121C22×C10911yescharacteristic
106C4×Q8C4×C10911yescharacteristic, complemented
107C4×D4—21no—
108C4×D4C4×C10911yescharacteristic, complemented
109#117—21no—
110C2×C42—21no—
111C2×C42C22×C10911yescharacteristic
112C4⋊C8—21no—
113C22⋊C8C22×C10912yes—
114C4×C8—21no—
115#95—21no—
116#95C22×C10912yes—
117C2×Q8C2×C4×C10911yescharacteristic
118C2×D4—21no—
119C2×D4C2×C4×C10911yescharacteristic
120C22×C4—22no—
121C22×C4—21no—
122C22×C4D4×C10911yescharacteristic
123C22×C4D4×C10911yescharacteristic
124C22×C4C23×C10911yescharacteristic
125C2×C8—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4C2×C4×C10911yescharacteristic
131C4⋊C4C2×C4×C10911yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42—24no—
138C42—21no—
139C42C2×C4×C10911yescharacteristic
140C42C2×C4×C10911yescharacteristic
141C42D4×C10911yescharacteristic
142C42D4×C10911yescharacteristic
143C23—21no—
144C23C2×D4×C10911yescharacteristic
145Q8—41no—
146D4—41no—
147D4—42no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4C22⋊C4×C10911yescharacteristic
159C2×C4C22⋊C4×C10911yescharacteristic
160C2×C4C22⋊C4×C10911yescharacteristic
161C2×C4C22⋊C4×C10911yescharacteristic
162C2×C4C22×C4×C10911yescharacteristic
163C2×C4C2×D4×C10911yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×C22⋊C4×C10911yescharacteristic
171C22#135×C10912yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—44no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4#104×C10912yes—
182C2—22no—
183C2—81no—
184C2#410×C10911yescharacteristic
185C2#416×C10911yescharacteristic
186C2C2×#104×C10911yescharacteristic
subgroupG/H
centerC2×C4×C109C22⋊C4
commutatorC2×C4C22×C4×C109
Frattini, cosocleC22×C4C23×C109
socleC22×C109C2×C22⋊C4
prime radical#121×C109C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. #431×C109C2semidirect
3. #376×C109C2semidirect
4. #328×C109C2semidirect
7. #125×C109C4semidirect
8. #124×C109C4semidirect
12. C4×Q8×C109C4semidirect
14. C4×D4×C109C4semidirect
93. #1500C109direct (N × H)
94. C109#1500direct (N × H)
96. #431C2×C109semidirect
97. #376C2×C109semidirect
98. #328C2×C109semidirect
101. #125C4×C109semidirect
102. #124C4×C109semidirect
106. C4×Q8C4×C109semidirect
108. C4×D4C4×C109semidirect

Maximal subgroups

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

subgroupindex
#529×C1092
#431×C1092
#376×C1092
#328×C1092
#327×C1092
#1500109