Finite Group Databasebeta

← #28378468

Subgroups — #28378468

Order 13952.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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
1D109⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#20295968C211yesmaximal, characteristic, complemented
3#20295877C214yesmaximal, complemented
4C2×#16812670C211yesmaximal, characteristic
5C2×D109⋊C8—21no—
6C2×C4⋊#89137—21no—
7C2×C22⋊#89137—21no—
8#16812756C2212yescomplemented
9C22⋊#11724786—21no—
10(C2×#89137)⋊C4C2212yescomplemented
11C22×C22⋊D109—21no—
12C22×C4×D109C2211yescharacteristic
13#16812670—21no—
14#16812670—21no—
15#16812670C412yescomplemented
16#16812670C2212yes—
17C2×C4⋊Dic109C411yescharacteristic, complemented
18C2×C22⋊Dic109—21no—
19C22×D4×C109C411yescharacteristic, complemented
20C22×#89137—22no—
21D109⋊C8—42no—
22C2×#11724786—22no—
23C4⋊#89137—42no—
24C22⋊#89137—42no—
25C23×D109D411yescharacteristic
26C2×C22⋊D109—22no—
27C2×C22⋊D109—22no—
28C2×C22⋊D109—42no—
29C2×C22⋊D109—42no—
30C22×Dic109—21no—
31C22×Dic109D411yescharacteristic
32C2×C4×D109—42no—
33C2×C4×D109C2×C411yescharacteristic
34C2×C4×D109D411yescharacteristic
35C2×C4×D109D411yescharacteristic
36C2×C4×D109C2311yescharacteristic
37C4⋊Dic109—21no—
38C4⋊Dic109C2×C412yes—
39C22⋊Dic109—22no—
40C22⋊Dic109—22no—
41C24×C109—21no—
42C2×D4×C109—21no—
43C2×D4×C109—42no—
44C2×D4×C109C2×C412yes—
45C22×C4×C109C2×C411yescharacteristic
46C23×C109—22no—
47C23×C109—42no—
48C23×C109—42no—
49C23×C109—42no—
50C23×C109C22⋊C411yescharacteristic
51C22×D109—41no—
52C22×D109—41no—
53C22×D109C22⋊C411yescharacteristic
54C22×D109D812yes—
55C22×D109SD1612yes—
56C22×D109C2×D411yescharacteristic
57C2×#89137—42no—
58C2×#89137—42no—
59C2×#89137—42no—
60D4×C109—42no—
61D4×C109—42no—
62C2×C4×C109—22no—
63C2×C4×C109C22⋊C411yescharacteristic
64C2×C4×C109C22×C411yescharacteristic
65C22⋊D109—44no—
66C22⋊D109—42no—
67C22⋊D109—42no—
68C2×Dic109—22no—
69C2×Dic109—41no—
70C2×Dic109—42no—
71C2×Dic109C22⋊C411yescharacteristic
72C2×Dic109C2×D411yescharacteristic
73C4×D109—22no—
74C4×D109—22no—
75C4×D109—41no—
76C4×D109—41no—
77#11724786—42no—
78C22×C109—21no—
79C22×C109—21no—
80C22×C109—42no—
81C22×C109—42no—
82C22×C109—42no—
83C22×C109—84no—
84C22×C109#10212yes—
85C22×C109C2×C22⋊C411yescharacteristic
86C2×D109—21no—
87C2×D109—21no—
88C2×D109—21no—
89C2×D109—21no—
90C2×D109—42no—
91C2×D109—42no—
92C2×D109#10212yes—
93C2×D109C2×D811yescharacteristic
94C2×D109C2×SD1611yescharacteristic
95#89137—82no—
96C4×C109—21no—
97C4×C109#9912yes—
98Dic109—21no—
99Dic109—21no—
100Dic109—42no—
101C2×C109—22no—
102C2×C109—82no—
103C2×C109C2×#9911yescharacteristic
104C2×C109C2×#10211yescharacteristic
105C2×C109#42111yescharacteristic
106D109—21no—
107D109—22no—
108D109—21no—
109#1519—1091nomaximal, Sylow
110C109#151911yescharacteristic, complemented, Sylow
111C2×#121—1091no—
112C2×C22⋊C8—1091no—
113#380—1091no—
114#327—1094no—
115C22×D4—2181no—
116C22×D4#8913711yescharacteristic, complemented
117C23×C4—1091no—
118C22×C8—2181no—
119#121—1092no—
120#121—1092no—
121#121—2181no—
122#121—2181no—
123C2×C4⋊C4—1091no—
124C2×C4⋊C4—2181no—
125C2×C22⋊C4—2181no—
126C2×C22⋊C4—2181no—
127C22⋊C8—1092no—
128C22⋊C8—2181no—
129#95—1092no—
130C24—21no—
131C24—1091no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—4362no—
135C2×D4—4362no—
136C2×D4—2182no—
137C2×D4—2182no—
138C2×D4C2×#8913712yes—
139C22×C4—1091no—
140C22×C4—1091no—
141C22×C4—1091no—
142C22×C4—1091no—
143C22×C4—1091no—
144C22×C4—4362no—
145C22×C4—2182no—
146C22×C4—2181no—
147C22×C4C2×#8913711yescharacteristic
148C2×C8—4362no—
149C2×C8—2182no—
150C4⋊C4—1092no—
151C4⋊C4—4362no—
152C4⋊C4—2181no—
153C22⋊C4—4362no—
154C22⋊C4—2182no—
155C22⋊C4—2182no—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—1092no—
161C23—1092no—
162C23—1091no—
163C23—1091no—
164C23—4361no—
165C23—4361no—
166C23C22⋊#8913711yescharacteristic
167D4—42no—
168D4—42no—
169D4—4362no—
170D4—4364no—
171D4—4362no—
172C2×C4—22no—
173C2×C4—1091no—
174C2×C4—1091no—
175C2×C4—4361no—
176C2×C4—4361no—
177C2×C4—4362no—
178C2×C4—4362no—
179C2×C4—4362no—
180C2×C4—4362no—
181C2×C4—4361no—
182C2×C4—2182no—
183C2×C4—2182no—
184C2×C4—2182no—
185C2×C4C22⋊#8913711yescharacteristic
186C2×C4C22×#8913711yescharacteristic
187C8—4362no—
188C22—21no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—84no—
194C22—1091no—
195C22—1092no—
196C22—1091no—
197C22—4362no—
198C22—4362no—
199C22—2181no—
200C22—2181no—
201C22—2181no—
202C22—2181no—
203C22#1681276112yes—
204C22C2×C22⋊#8913711yescharacteristic
205C4—21no—
206C4—8722no—
207C4—4362no—
208C4—2181no—
209C4—2181no—
210C4#1681275712yes—
211C2—22no—
212C2—82no—
213C2—2181no—
214C2—2182no—
215C2—2181no—
216C2C2×#1681275711yescharacteristic
217C2C2×#1681276111yescharacteristic
218C2#2029602411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22×D4×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC2×#16812670C2
cosocleC2×C4×D109C23
hypercenterC22×D4#89137
hypocenterC109#1519
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. D109⋊(C22⋊C8)C2semidirect
2. #20295968C2semidirect
3. #20295877C2semidirect
8. #16812756C22semidirect
10. (C2×#89137)⋊C4C22semidirect
15. #16812670C4semidirect
17. C2×C4⋊Dic109C4semidirect
19. C22×D4×C109C4semidirect
110. C109#1519semidirect
116. C22×D4#89137semidirect

Maximal subgroups

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

subgroupindex
D109⋊(C22⋊C8)2
#202959682
#202958772
C2×#168126702
#1519109