Finite Group Databasebeta

← #28378448

Subgroups — #28378448

Order 13952.

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

normal subgroups only

This group has 286 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#20295998C211yesmaximal, characteristic, complemented
2#20295968C211yesmaximal, characteristic, complemented
3#20295879C212yesmaximal, complemented
4#20295877C212yesmaximal, complemented
5#20295536C211yesmaximal, characteristic
6C2×#13182038—21no—
7C2×C4⋊#89137—21no—
8C2×C22⋊#89137—21no—
9#16812756—21no—
10#16812756C2212yes—
11(C2×#89137)⋊C4C2212yes—
12C2×#13182028C411yescharacteristic, complemented
13C22×C4×D109C2211yescharacteristic
14#16812681—21no—
15#16812679C2211yescharacteristic
16#16812678C411yescharacteristic, complemented
17#16812670C2211yescharacteristic
18#16812667—21no—
19D4×Dic109—21no—
20#16812659—21no—
21#16812657C411yescharacteristic, complemented
22#121×C109C411yescharacteristic, complemented
23C22×#89137—22no—
24#13182038—42no—
25C2×#11724786—22no—
26C4⋊#89137—42no—
27C22⋊#89137—42no—
28C23×D109D411yescharacteristic
29#13182028—42no—
30C2×Dic218C2×C411yescharacteristic
31C2×C22⋊D109—21no—
32C2×C22⋊D109—21no—
33C22×Dic109—21no—
34C22×Dic109D411yescharacteristic
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—
43C22⋊Dic109—21no—
44C22⋊Dic109—21no—
45#13182006—21no—
46#13182006—21no—
47C4×Dic109—21no—
48C2×D4×C109—21no—
49C2×D4×C109C2×C411yescharacteristic
50C22×C4×C109C2×C411yescharacteristic
51C4⋊C4×C109C2×C411yescharacteristic
52C22⋊C4×C109—21no—
53C23×C109—21no—
54C23×C109C22⋊C411yescharacteristic
55C22×D109—21no—
56C22×D109—21no—
57C22×D109—41no—
58C22×D109—41no—
59C22×D109C22⋊C411yescharacteristic
60C22×D109C2×D411yescharacteristic
61C2×#89137—42no—
62C2×#89137—42no—
63C2×#89137—42no—
64D4×C109—41no—
65D4×C109—42no—
66C2×C4×C109—21no—
67C2×C4×C109—41no—
68C2×C4×C109C22⋊C411yescharacteristic
69C2×C4×C109C22×C411yescharacteristic
70C22⋊D109—42no—
71C22⋊D109—41no—
72C22⋊D109—41no—
73C2×Dic109—22no—
74C2×Dic109—21no—
75C2×Dic109—21no—
76C2×Dic109—42no—
77C2×Dic109C22⋊C411yescharacteristic
78C2×Dic109C2×D411yescharacteristic
79C4×D109—22no—
80C4×D109—22no—
81C4×D109—41no—
82C4×D109—41no—
83Dic218—41no—
84#11724786—42no—
85C22×C109—21no—
86C22×C109—21no—
87C22×C109—21no—
88C22×C109—41no—
89C22×C109—41no—
90C22×C109—41no—
91C22×C109C2×C22⋊C411yescharacteristic
92C2×D109—21no—
93C2×D109—21no—
94C2×D109—21no—
95C2×D109—21no—
96C2×D109—21no—
97C2×D109—42no—
98C2×D109—42no—
99C2×D109AGL(1,ℤ/8)11yescharacteristic
100C2×D109#13711yescharacteristic
101#89137—82no—
102C4×C109—21no—
103C4×C109—21no—
104C4×C109—41no—
105Dic109—21no—
106Dic109—41no—
107Dic109—41no—
108Dic109#9912yes—
109C2×C109—41no—
110C2×C109—81no—
111C2×C109C2×#9911yescharacteristic
112C2×C109#41711yescharacteristic
113C2×C109#42111yescharacteristic
114D109—21no—
115D109—21no—
116D109—41no—
117#1513—1091nomaximal, Sylow
118C109#151311yescharacteristic, complemented, Sylow
119#525—1091no—
120#407—1091no—
121#380—1091no—
122#328—1092no—
123#327—1092no—
124C2×#41—1091no—
125C23×C4—1091no—
126C2×M16—2181no—
127#123—2181no—
128C22⋊Q8—1091no—
129C22⋊Q8—1091no—
130#121—1091no—
131#121#8913711yescharacteristic, complemented
132#120—2181no—
133C4×D4—2181no—
134C4×D4—2181no—
135#117—1091no—
136C2×C4⋊C4—2181no—
137C2×C22⋊C4—2181no—
138C22⋊C8—1092no—
139C22⋊C8—2181no—
140#95—1092no—
141C24—1091no—
142#41—4362no—
143C2×Q8—1091no—
144C2×D4—21no—
145C2×D4—2181no—
146C2×D4—2181no—
147C2×D4C2×#8913711yescharacteristic
148C22×C4—1091no—
149C22×C4—1091no—
150C22×C4—1091no—
151C22×C4—1091no—
152C22×C4—1091no—
153C22×C4—4362no—
154C22×C4—2181no—
155C22×C4—2182no—
156C22×C4C2×#8913711yescharacteristic
157M16—4362no—
158C2×C8—2182no—
159C4⋊C4—1091no—
160C4⋊C4—4362no—
161C4⋊C4—2181no—
162C4⋊C4—2181no—
163C4⋊C4C2×#8913711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—4362no—
166C22⋊C4—2181no—
167C22⋊C4—2181no—
168C22⋊C4—2181no—
169C22⋊C4—2181no—
170C42—2181no—
171C23—21no—
172C23—1091no—
173C23—1091no—
174C23—4361no—
175C23—4361no—
176C23—2181no—
177C23—2181no—
178C23C22⋊#8913711yescharacteristic
179Q8—4361no—
180D4—41no—
181D4—42no—
182D4—4361no—
183D4—4361no—
184D4—4362no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—1091no—
188C2×C4—1091no—
189C2×C4—4361no—
190C2×C4—4361no—
191C2×C4—4362no—
192C2×C4—4362no—
193C2×C4—4362no—
194C2×C4—4362no—
195C2×C4—2182no—
196C2×C4—2182no—
197C2×C4—2182no—
198C2×C4—2181no—
199C2×C4—2181no—
200C2×C4C22⋊#8913711yescharacteristic
201C2×C4C22×#8913711yescharacteristic
202C8—4362no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—1091no—
210C22—1091no—
211C22—4362no—
212C22—4362no—
213C22—2181no—
214C22—2181no—
215C22—2181no—
216C22—2181no—
217C22—2181no—
218C22C2×C22⋊#8913711yescharacteristic
219C4—21no—
220C4—21no—
221C4—41no—
222C4—8722no—
223C4—1092no—
224C4—4361no—
225C4—4361no—
226C4—2181no—
227C2—41no—
228C2—81no—
229C2—4361no—
230C2—2181no—
231C2—2181no—
232C2D109⋊#9911yescharacteristic
233C2#2029601711yescharacteristic
234C2#2029602411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#121×C109C4
socleC22×C109C2×C22⋊C4
prime radical#20295536C2
cosocleC2×C4×D109C23
hypercenter#121#89137
hypocenterC109#1513
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20295998C2semidirect
2. #20295968C2semidirect
3. #20295879C2semidirect
4. #20295877C2semidirect
12. C2×#13182028C4semidirect
16. #16812678C4semidirect
21. #16812657C4semidirect
22. #121×C109C4semidirect
118. C109#1513semidirect
131. #121#89137semidirect

Maximal subgroups

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

subgroupindex
#202959982
#202959682
#202958792
#202958772
#202955362
#1513109