Finite Group Databasebeta

← #28378441

Subgroups — #28378441

Order 13952.

274 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 58 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#20295997C211yesmaximal, characteristic
2#20295969C211yesmaximal, characteristic
3#20295880C212yesmaximal, complemented
4#20295879C212yesmaximal, complemented
5C22⋊Q8×D109C211yesmaximal, characteristic
6C2×#13182039—21no—
7C2×C22⋊#89137—21no—
8#16812756C2212yes—
9C22⋊#11724786—21no—
10(C2×#89137)⋊C4—21no—
11(C2×#89137)⋊C4C2212yes—
12C2×Q8×D109C411yescharacteristic, complemented
13C22×C4×D109C2211yescharacteristic
14#16812679C2211yescharacteristic
15#16812678C411yescharacteristic, complemented
16#16812677—21no—
17#16812676—21no—
18#16812675C2211yescharacteristic
19C4⋊C4×D109—21no—
20C4⋊C4×D109C411yescharacteristic, complemented
21C22⋊C4×D109—21no—
22C22⋊Q8×C109C411yescharacteristic, complemented
23C22×#89137—22no—
24#13182039—42no—
25C2×#11724786—22no—
26C22⋊#89137—42no—
27C23×D109D411yescharacteristic
28C2×Dic218—21no—
29C2×Dic218C2×C411yescharacteristic
30Q8×D109—42no—
31C22×Dic109D411yescharacteristic
32C2×C4×D109—21no—
33C2×C4×D109—21no—
34C2×C4×D109—42no—
35C2×C4×D109C2×C411yescharacteristic
36C2×C4×D109D411yescharacteristic
37C2×C4×D109D411yescharacteristic
38C2×C4×D109C2311yescharacteristic
39C4⋊Dic109—21no—
40C4⋊Dic109C2×C411yescharacteristic
41#13182008—21no—
42#13182008—21no—
43#13182008—21no—
44C22⋊Dic109—21no—
45#13182006—21no—
46#13182006—21no—
47C2×Q8×C109C2×C411yescharacteristic
48C22×C4×C109C2×C411yescharacteristic
49C4⋊C4×C109—21no—
50C4⋊C4×C109C2×C411yescharacteristic
51C22⋊C4×C109—21no—
52C23×C109C22⋊C411yescharacteristic
53C22×D109—22no—
54C22×D109—22no—
55C22×D109—41no—
56C22×D109C22⋊C411yescharacteristic
57C22×D109C2×D411yescharacteristic
58C2×#89137—42no—
59C2×#89137—42no—
60C2×#89137—42no—
61Q8×C109—41no—
62C2×C4×C109—21no—
63C2×C4×C109—21no—
64C2×C4×C109—41no—
65C2×C4×C109C22⋊C411yescharacteristic
66C2×C4×C109C22×C411yescharacteristic
67C2×Dic109—21no—
68C2×Dic109—21no—
69C2×Dic109—41no—
70C2×Dic109C22⋊C411yescharacteristic
71C2×Dic109C2×D411yescharacteristic
72C4×D109—22no—
73C4×D109—22no—
74C4×D109—41no—
75C4×D109—41no—
76C4×D109—42no—
77C4×D109—42no—
78Dic218—42no—
79Dic218—41no—
80#11724786—42no—
81C22×C109—21no—
82C22×C109—21no—
83C22×C109—21no—
84C22×C109C2×C22⋊C411yescharacteristic
85C2×D109—21no—
86C2×D109—21no—
87C2×D109—21no—
88C2×D109—21no—
89C2×D109—44no—
90C2×D109#9912yes—
91C2×D109#13712yes—
92#89137—82no—
93C4×C109—21no—
94C4×C109—21no—
95C4×C109—41no—
96C4×C109—41no—
97Dic109—21no—
98Dic109—21no—
99Dic109—41no—
100Dic109—41no—
101C2×C109—41no—
102C2×C109C2×#9911yescharacteristic
103C2×C109#41911yescharacteristic
104C2×C109#42111yescharacteristic
105D109—22no—
106D109—22no—
107#1511—1091nomaximal, Sylow
108C109#151111yescharacteristic, complemented, Sylow
109C2×C22⋊Q8—1091no—
110#407—1091no—
111#381—1091no—
112#328—1092no—
113#328—1092no—
114C22×Q8—1091no—
115C23×C4—1091no—
116C2×M16—2181no—
117C22⋊Q8—1091no—
118C22⋊Q8—1091no—
119C22⋊Q8—1091no—
120C22⋊Q8—2181no—
121C22⋊Q8—2181no—
122C22⋊Q8#8913711yescharacteristic, complemented
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—
130#95—2181no—
131C24—1091no—
132C2×Q8—1091no—
133C2×Q8—4362no—
134C2×Q8—2181no—
135C2×Q8C2×#8913711yescharacteristic
136C22×C4—1091no—
137C22×C4—1091no—
138C22×C4—1091no—
139C22×C4—1091no—
140C22×C4—1091no—
141C22×C4—4362no—
142C22×C4—2181no—
143C22×C4—2181no—
144C22×C4—2182no—
145C22×C4C2×#8913711yescharacteristic
146M16—4362no—
147C2×C8—2182no—
148C4⋊C4—21no—
149C4⋊C4—1091no—
150C4⋊C4—2181no—
151C4⋊C4—2181no—
152C4⋊C4—2181no—
153C4⋊C4—2181no—
154C4⋊C4C2×#8913711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—4362no—
157C22⋊C4—2181no—
158C22⋊C4—2181no—
159C22⋊C4—2181no—
160C23—1091no—
161C23—1091no—
162C23—4361no—
163C23—2182no—
164C23—2182no—
165C23C22⋊#8913711yescharacteristic
166Q8—41no—
167Q8—4361no—
168Q8—4362no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—1091no—
173C2×C4—1091no—
174C2×C4—4361no—
175C2×C4—4361no—
176C2×C4—4362no—
177C2×C4—4362no—
178C2×C4—4362no—
179C2×C4—4362no—
180C2×C4—4362no—
181C2×C4—4361no—
182C2×C4—2182no—
183C2×C4—2182no—
184C2×C4—2181no—
185C2×C4—2181no—
186C2×C4C22⋊#8913711yescharacteristic
187C2×C4C22×#8913711yescharacteristic
188C8—4362no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—1092no—
193C22—1092no—
194C22—4364no—
195C22—2181no—
196C22—2181no—
197C22—2181no—
198C22—2181no—
199C22C2×C22⋊#8913711yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—8722no—
205C4—4361no—
206C4—4361no—
207C4—2181no—
208C4—2181no—
209C2—41no—
210C2—2182no—
211C2—2182no—
212C2D109⋊#9911yescharacteristic
213C2#2029601911yescharacteristic
214C2#2029602311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22⋊Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC23×D109D4
cosocleC2×C4×D109C23
hypercenterC22⋊Q8#89137
hypocenterC109#1511
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20295880C2semidirect
4. #20295879C2semidirect
12. C2×Q8×D109C4semidirect
15. #16812678C4semidirect
20. C4⋊C4×D109C4semidirect
22. C22⋊Q8×C109C4semidirect
108. C109#1511semidirect
122. C22⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202959972
#202959692
#202958802
#202958792
C22⋊Q8×D1092
#1511109