Finite Group Databasebeta

← #28378470

Subgroups — #28378470

Order 13952.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#16812756C211yesmaximal, characteristic
2#20295968C211yesmaximal, characteristic
3#20295880C212yesmaximal, complemented
4#20295879C212yesmaximal, complemented
5C22⋊Q8×D109C211yesmaximal, characteristic
6C22×#11724786—21no—
7C2×C4⋊#89137—21no—
8C2×C22⋊#89137—21no—
9#16812756—21no—
10#16812756C2212yes—
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—
24C2×#11724786—22no—
25C2×#11724786—42no—
26C4⋊#89137—42no—
27C22⋊#89137—42no—
28C23×D109D411yescharacteristic
29C2×Dic218—21no—
30C2×Dic218C2×C411yescharacteristic
31Q8×D109—42no—
32C22×Dic109D411yescharacteristic
33C2×C4×D109—21no—
34C2×C4×D109—21no—
35C2×C4×D109—42no—
36C2×C4×D109C2×C411yescharacteristic
37C2×C4×D109D411yescharacteristic
38C2×C4×D109D411yescharacteristic
39C2×C4×D109C2311yescharacteristic
40C4⋊Dic109—21no—
41C4⋊Dic109C2×C411yescharacteristic
42#13182008—21no—
43#13182008—21no—
44#13182008—21no—
45C22⋊Dic109—21no—
46#13182006—21no—
47#13182006—21no—
48C2×Q8×C109C2×C411yescharacteristic
49C22×C4×C109C2×C411yescharacteristic
50C4⋊C4×C109—21no—
51C4⋊C4×C109C2×C411yescharacteristic
52C22⋊C4×C109—21no—
53C23×C109C22⋊C411yescharacteristic
54C22×D109—41no—
55C22×D109—41no—
56C22×D109C22⋊C411yescharacteristic
57C22×D109SD1612yes—
58C22×D109Q1612yes—
59C22×D109C2×D411yescharacteristic
60C2×#89137—42no—
61C2×#89137—42no—
62C2×#89137—42no—
63Q8×C109—41no—
64C2×C4×C109—21no—
65C2×C4×C109—21no—
66C2×C4×C109—41no—
67C2×C4×C109C22⋊C411yescharacteristic
68C2×C4×C109C22×C411yescharacteristic
69C2×Dic109—22no—
70C2×Dic109—21no—
71C2×Dic109—21no—
72C2×Dic109C22⋊C411yescharacteristic
73C2×Dic109C2×D411yescharacteristic
74C4×D109—22no—
75C4×D109—22no—
76C4×D109—41no—
77C4×D109—41no—
78C4×D109—42no—
79C4×D109—42no—
80Dic218—42no—
81Dic218—41no—
82#11724786—42no—
83C22×C109—21no—
84C22×C109—21no—
85C22×C109#10312yes—
86C22×C109C2×C22⋊C411yescharacteristic
87C2×D109—21no—
88C2×D109—21no—
89C2×D109—21no—
90C2×D109—21no—
91C2×D109—42no—
92C2×D109—42no—
93C2×D109#10312yes—
94C2×D109C2×SD1611yescharacteristic
95C2×D109C2×Q1611yescharacteristic
96#89137—82no—
97C4×C109—21no—
98C4×C109—21no—
99C4×C109—41no—
100C4×C109—41no—
101Dic109—21no—
102Dic109—41no—
103Dic109—41no—
104Dic109#9912yes—
105C2×C109—22no—
106C2×C109C2×#9911yescharacteristic
107C2×C109C2×#10311yescharacteristic
108C2×C109#42111yescharacteristic
109D109—21no—
110D109—22no—
111D109—21no—
112#1520—1091nomaximal, Sylow
113C109#152011yescharacteristic, complemented, Sylow
114C2×C22⋊Q8—1091no—
115C2×C22⋊C8—1091no—
116#380—1091no—
117#328—1092no—
118#328—1092no—
119C22×Q8—1091no—
120C23×C4—1091no—
121C22×C8—2181no—
122C22⋊Q8—1091no—
123C22⋊Q8—1091no—
124C22⋊Q8—1091no—
125C22⋊Q8—2181no—
126C22⋊Q8—2181no—
127C22⋊Q8#8913711yescharacteristic, complemented
128C2×C4⋊C4—1091no—
129C2×C4⋊C4—2181no—
130C2×C4⋊C4—2181no—
131C2×C22⋊C4—2181no—
132C2×C22⋊C4—2181no—
133C22⋊C8—1092no—
134C22⋊C8—2181no—
135#95—1092no—
136C24—1091no—
137C2×Q8—1091no—
138C2×Q8—4362no—
139C2×Q8—2181no—
140C2×Q8C2×#8913711yescharacteristic
141C22×C4—1091no—
142C22×C4—1091no—
143C22×C4—1091no—
144C22×C4—1091no—
145C22×C4—1091no—
146C22×C4—4362no—
147C22×C4—2181no—
148C22×C4—2181no—
149C22×C4—2182no—
150C22×C4C2×#8913711yescharacteristic
151C2×C8—4362no—
152C2×C8—2182no—
153C4⋊C4—21no—
154C4⋊C4—1091no—
155C4⋊C4—4362no—
156C4⋊C4—2181no—
157C4⋊C4—2181no—
158C4⋊C4—2181no—
159C4⋊C4—2181no—
160C4⋊C4C2×#8913711yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—4362no—
163C22⋊C4—2181no—
164C22⋊C4—2181no—
165C22⋊C4—2181no—
166C23—1092no—
167C23—1092no—
168C23—1091no—
169C23—1091no—
170C23—4361no—
171C23—4361no—
172C23C22⋊#8913711yescharacteristic
173Q8—41no—
174Q8—4361no—
175Q8—4362no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—1091no—
180C2×C4—1091no—
181C2×C4—4361no—
182C2×C4—4361no—
183C2×C4—4362no—
184C2×C4—4362no—
185C2×C4—4362no—
186C2×C4—4362no—
187C2×C4—4362no—
188C2×C4—2182no—
189C2×C4—2182no—
190C2×C4—2182no—
191C2×C4—2181no—
192C2×C4—2181no—
193C2×C4C22⋊#8913711yescharacteristic
194C2×C4C22×#8913711yescharacteristic
195C8—4362no—
196C22—21no—
197C22—21no—
198C22—1091no—
199C22—1092no—
200C22—1091no—
201C22—4362no—
202C22—4362no—
203C22—2181no—
204C22—2181no—
205C22—2181no—
206C22—2181no—
207C22#1681276212yes—
208C22C2×C22⋊#8913711yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—8722no—
214C4—1092no—
215C4—4361no—
216C4—4361no—
217C4—2181no—
218C2—22no—
219C2—2181no—
220C2—2182no—
221C2—2181no—
222C2D109⋊#9911yescharacteristic
223C2C2×#1681276211yescharacteristic
224C2#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#1520
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
113. C109#1520semidirect
127. C22⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
C2×#168127562
#202959682
#202958802
#202958792
C22⋊Q8×D1092
#1520109