Finite Group Databasebeta

← #28379285

Subgroups — #28379285

Order 13952.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#20296012C211yesmaximal, characteristic, complemented
2#20296005C211yesmaximal, characteristic, complemented
3#20295929C212yesmaximal, complemented
4#20295927C212yesmaximal, complemented
5#20295538C211yesmaximal, characteristic
6D109⋊M16—21no—
7#16812778—21no—
8#16812760—21no—
9#16812759C2212yes—
10#16812758—21no—
11#16812757C2212yes—
12C2×#13182029C2211yescharacteristic
13C22×C4×D109C411yescharacteristic, complemented
14#16812683C411yescharacteristic, complemented
15#16812683C2211yescharacteristic
16#16812679—21no—
17#16812674—21no—
18#16812669C411yescharacteristic, complemented
19#16812669C2211yescharacteristic
20C4×D436—21no—
21C4×D436—21no—
22#117×C109C411yescharacteristic, complemented
23#13182039—41no—
24#13182038—22no—
25D109⋊C8—41no—
26C4⋊#89137—41no—
27C22⋊#89137—22no—
28C4×#89137—41no—
29C23×D109C2×C411yescharacteristic
30#13182029—41no—
31#13182029—41no—
32C2×Dic218D411yescharacteristic
33C2×D436—21no—
34C2×D436—21no—
35C2×D436D411yescharacteristic
36C2×C22⋊D109C2×C411yescharacteristic
37C2×C22⋊D109C2311yescharacteristic
38C22×Dic109C2×C411yescharacteristic
39C2×C4×D109—21no—
40C2×C4×D109—21no—
41C2×C4×D109—41no—
42C2×C4×D109—41no—
43C2×C4×D109D411yescharacteristic
44C2×C4×D109D411yescharacteristic
45C4⋊Dic109—21no—
46C4⋊Dic109—21no—
47#13182006—21no—
48#13182006—21no—
49#13182006—21no—
50#13182006—21no—
51C22×C4×C109C2×C411yescharacteristic
52C4⋊C4×C109—21no—
53C22⋊C4×C109C2×C411yescharacteristic
54C22⋊C4×C109C2×C411yescharacteristic
55C42×C109—21no—
56C23×C109C22×C411yescharacteristic
57C22×D109—21no—
58C22×D109—42no—
59C22×D109—41no—
60C22×D109C22⋊C411yescharacteristic
61C22×D109C2×D411yescharacteristic
62C2×#89137—42no—
63C2×C4×C109—21no—
64C2×C4×C109—21no—
65C2×C4×C109—21no—
66C2×C4×C109—21no—
67C2×C4×C109C22⋊C411yescharacteristic
68C2×C4×C109C22⋊C411yescharacteristic
69C22⋊D109—41no—
70C22⋊D109—41no—
71C2×Dic109—21no—
72C2×Dic109—41no—
73C2×Dic109C22⋊C411yescharacteristic
74C2×Dic109C2×D411yescharacteristic
75D436—41no—
76D436—42no—
77D436—42no—
78C4×D109—21no—
79C4×D109—21no—
80C4×D109—21no—
81C4×D109—21no—
82C4×D109—42no—
83C4×D109—41no—
84C4×D109—41no—
85Dic218—41no—
86#11724786—42no—
87C22×C109—21no—
88C22×C109—41no—
89C22×C109C2×C22⋊C411yescharacteristic
90C2×D109—22no—
91C2×D109—21no—
92C2×D109—21no—
93C2×D109—41no—
94C2×D109—42no—
95C2×D109—41no—
96C2×D109—41no—
97C2×D109—41no—
98C2×D109—42no—
99#89137—82no—
100C4×C109—21no—
101C4×C109—41no—
102C4×C109—41no—
103C4×C109#9911yescharacteristic
104C4×C109#9911yescharacteristic
105Dic109—21no—
106Dic109—21no—
107Dic109—41no—
108C2×C109—21no—
109C2×C109—41no—
110C2×C109C2×#9911yescharacteristic
111D109—41no—
112D109—41no—
113D109—42no—
114#2120—1091nomaximal, Sylow
115C109#212011yescharacteristic, complemented, Sylow
116#525—1091no—
117#413—1091no—
118#410—1091no—
119#352—1092no—
120AΣL(1,𝔽16)—1092no—
121C2×#41—1091no—
122C23×C4—1091no—
123C2×M16—2181no—
124#123—1091no—
125#123—1091no—
126C22⋊Q8—2181no—
127#121—2181no—
128#120—1091no—
129#120—1091no—
130C4×D4—2181no—
131C4×D4—2181no—
132#117—2181no—
133#117#8913711yescharacteristic, complemented
134#101—2181no—
135#100—1092no—
136#99—1092no—
137#99—2181no—
138C24—1091no—
139#41—4361no—
140#41—4361no—
141C2×Q8—1091no—
142C2×D4—1091no—
143C2×D4—1091no—
144C2×D4—1091no—
145C2×D4—2181no—
146C2×D4—2181no—
147C22×C4—1091no—
148C22×C4—1091no—
149C22×C4—1091no—
150C22×C4—4361no—
151C22×C4—4361no—
152C22×C4—2181no—
153C22×C4—2181no—
154C22×C4C2×#8913711yescharacteristic
155M16—4361no—
156M16—2182no—
157C2×C8—4361no—
158C4⋊C4—21no—
159C4⋊C4—4361no—
160C4⋊C4—2181no—
161C4⋊C4—2181no—
162C22⋊C4—2181no—
163C22⋊C4—2181no—
164C22⋊C4—2181no—
165C22⋊C4—2181no—
166C22⋊C4—2182no—
167C22⋊C4C2×#8913711yescharacteristic
168C22⋊C4C2×#8913711yescharacteristic
169C42—21no—
170C42—4361no—
171C23—1091no—
172C23—1091no—
173C23—4361no—
174C23—4362no—
175C23—2181no—
176C23C22×#8913711yescharacteristic
177Q8—4361no—
178D4—4361no—
179D4—4362no—
180D4—4362no—
181D4—4361no—
182D4—4361no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—1091no—
188C2×C4—1091no—
189C2×C4—4362no—
190C2×C4—4362no—
191C2×C4—4361no—
192C2×C4—4361no—
193C2×C4—4361no—
194C2×C4—2181no—
195C2×C4—2181no—
196C2×C4—2181no—
197C2×C4—2181no—
198C2×C4—2181no—
199C2×C4C22⋊#8913711yescharacteristic
200C2×C4C22⋊#8913711yescharacteristic
201C8—4362no—
202C22—21no—
203C22—41no—
204C22—4361no—
205C22—4361no—
206C22—4361no—
207C22—4362no—
208C22—4362no—
209C22—4361no—
210C22—2182no—
211C22—2181no—
212C22—2181no—
213C22C2×C22⋊#8913711yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—8722no—
218C4—4361no—
219C4—2181no—
220C4—2181no—
221C4#1681275811yescharacteristic
222C4#1681275811yescharacteristic
223C2—21no—
224C2—41no—
225C2—4361no—
226C2—4362no—
227C2—4361no—
228C2C2×#1681275811yescharacteristic
subgroupG/H
centerC4#16812758
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
Fitting#117×C109C4
socleC2×C109C2×#99
prime radical#20295538C2
cosocleC2×C22⋊D109C23
hypercenter#117#89137
hypocenterC109#2120
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20296012C2semidirect
2. #20296005C2semidirect
3. #20295929C2semidirect
4. #20295927C2semidirect
13. C22×C4×D109C4semidirect
14. #16812683C4semidirect
18. #16812669C4semidirect
22. #117×C109C4semidirect
115. C109#2120semidirect
133. #117#89137semidirect

Maximal subgroups

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

subgroupindex
#202960122
#202960052
#202959292
#202959272
#202955382
#2120109