Finite Group Databasebeta

← #28365653

Subgroups — #28365653

Order 13952.

600 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 229 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×#16812657C211yesmaximal, characteristic
2C2×#16812656C211yesmaximal, characteristic
3C2×C4⋊C4×D109C211yesmaximal, characteristic
4#20294921C214yesmaximal, complemented
5#20294897C214yesmaximal
6#20294895C211yesmaximal, characteristic, complemented
7#20294892C211yesmaximal, characteristic, complemented
8#20294891C211yesmaximal, characteristic, complemented
9#384×C109C211yesmaximal, characteristic, complemented
10C22×C4×D109C2212yes—
11C22×C4×D109C2211yescharacteristic
12#16812657—24no—
13#16812656C418yescomplemented
14C2×C4⋊Dic109C2211yescharacteristic
15C2×C4⋊Dic109C2211yescharacteristic
16C2×#13182008C2212yes—
17C2×#13182008C2214yes—
18C2×#13182008C2212yes—
19C4⋊C4×D109—24no—
20C2×#13182006C2212yes—
21C2×#13182006C2212yes—
22C2×C4×Dic109C2212yes—
23C2×C4×Dic109C2211yescharacteristic
24#16812606C2214yes—
25#16812606C2214yes—
26C2×C4⋊C4×C109C2211yescharacteristic
27C2×C4⋊C4×C109C2211yescharacteristic
28C2×C4⋊C4×C109C2214yes—
29C2×C42×C109C2211yescharacteristic
30C23×D109C2311yescharacteristic
31C22×Dic109C2312yes—
32C22×Dic109C2312yes—
33C22×Dic109C2312yes—
34C22×Dic109C2311yescharacteristic
35C2×C4×D109—24no—
36C2×C4×D109—44no—
37C2×C4×D109C2×C414yes—
38C2×C4×D109D414yes—
39C2×C4×D109Q814yes—
40C4⋊Dic109—22no—
41C4⋊Dic109—22no—
42#13182008—24no—
43#13182008—44no—
44#13182008C2×C418yescomplemented
45#13182006—24no—
46#13182006C2×C418yes—
47C4×Dic109—24no—
48C4×Dic109C2×C414yescomplemented
49C22×C4×C109C2312yes—
50C22×C4×C109C2312yes—
51C22×C4×C109C2312yes—
52C22×C4×C109C2311yescharacteristic
53C4⋊C4×C109—22no—
54C4⋊C4×C109—22no—
55C4⋊C4×C109—44no—
56C42×C109C2×C414yescomplemented
57C23×C109C2411yescharacteristic
58C22×D109—22no—
59C22×D109—22no—
60C22×D109C22×C412yes—
61C22×D109C2×D412yes—
62C22×D109C2×Q812yes—
63C2×C4×C109—22no—
64C2×C4×C109—22no—
65C2×C4×C109—22no—
66C2×C4×C109—22no—
67C2×C4×C109—24no—
68C2×C4×C109—24no—
69C2×C4×C109C22×C414yes—
70C2×C4×C109C22×C412yes—
71C2×C4×C109C2×D412yes—
72C2×C4×C109C2×Q812yes—
73C2×Dic109—22no—
74C2×Dic109—22no—
75C2×Dic109—22no—
76C2×Dic109—22no—
77C2×Dic109—24no—
78C2×Dic109—24no—
79C2×Dic109C22×C414yes—
80C2×Dic109C22×C412yes—
81C2×Dic109C2×D412yes—
82C2×Dic109C2×Q812yes—
83C4×D109—48no—
84C4×D109C4⋊C4116yescomplemented
85C22×C109C23×C411yescharacteristic
86C22×C109C22×D411yescharacteristic
87C22×C109C22×Q811yescharacteristic
88C22×C109H(5,𝔽2)12yes—
89C22×C1092−1+412yes—
90C2×D109—24no—
91C2×D109—22no—
92C2×D109—22no—
93C2×D109—24no—
94C2×D109C2×C4⋊C414yes—
95C4×C109—24no—
96C4×C109—42no—
97C4×C109—42no—
98C4×C109C2×C4⋊C414yes—
99Dic109—24no—
100Dic109—42no—
101Dic109—42no—
102Dic109C2×C4⋊C414yes—
103C2×C109C22×C4⋊C411yescharacteristic
104C2×C109#52012yes—
105C2×C109#53511yescharacteristic
106C2×C109#53611yescharacteristic
107C2×C109#54412yes—
108D109—24no—
109#2317—1091nomaximal, Sylow
110C109#231711yescharacteristic, complemented, Sylow
111C2×#117—1091no—
112C2×#117—1091no—
113C22×C4⋊C4—1091no—
114#389—1094no—
115#385—1094no—
116#384—1091no—
117#384D10911yescharacteristic, complemented
118#382—1091no—
119#382—1091no—
120C23×C4—1091no—
121C23×C4—1092no—
122#117—1098no—
123#117—2184no—
124C2×C4⋊C4—1092no—
125C2×C4⋊C4—1094no—
126C2×C4⋊C4—1092no—
127C2×C4⋊C4—1091no—
128C2×C4⋊C4—1091no—
129C2×C4⋊C4—2184no—
130C2×C4⋊C4C2×D10911yescharacteristic
131C2×C4⋊C4C2×D10911yescharacteristic
132C2×C4⋊C4C2×D10914yes—
133C2×C22⋊C4—1092no—
134C2×C22⋊C4—1092no—
135C2×C42—1091no—
136C2×C42—1092no—
137C2×C42C2×D10911yescharacteristic
138#95—1094no—
139#95—1094no—
140C24—1091no—
141C22×C4—1091no—
142C22×C4—1094no—
143C22×C4—1094no—
144C22×C4—1094no—
145C22×C4—1092no—
146C22×C4—1092no—
147C22×C4—1092no—
148C22×C4—4364no—
149C22×C4—2184no—
150C22×C4C22×D10912yes—
151C22×C4C22×D10912yes—
152C22×C4C22×D10912yes—
153C22×C4C22×D10911yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—1098no—
158C4⋊C4—4364no—
159C4⋊C4—2184no—
160C4⋊C4—2182no—
161C4⋊C4—2182no—
162C22⋊C4—1098no—
163C22⋊C4—2184no—
164C42—1094no—
165C42—2184no—
166C42C4×D10914yescomplemented
167C23—1092no—
168C23—1092no—
169C23—1092no—
170C23—2182no—
171C23—2182no—
172C23C23×D10911yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—1092no—
180C2×C4—1092no—
181C2×C4—1092no—
182C2×C4—10916no—
183C2×C4—1094no—
184C2×C4—4368no—
185C2×C4—2182no—
186C2×C4—2182no—
187C2×C4—2184no—
188C2×C4—2184no—
189C2×C4—2182no—
190C2×C4—2182no—
191C2×C4C2×C4×D10914yes—
192C2×C4C2×C4×D10912yes—
193C2×C4D4×D10912yes—
194C2×C4Q8×D10912yes—
195C22—1094no—
196C22—2184no—
197C22—2182no—
198C22—2182no—
199C22—2184no—
200C22C22×C4×D10911yescharacteristic
201C22C2×D4×D10911yescharacteristic
202C22C2×Q8×D10911yescharacteristic
203C22#1681274412yes—
204C22#1681274712yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—1094no—
209C4—4362no—
210C4—4362no—
211C4—2184no—
212C4C4⋊C4×D10914yes—
213C2—2184no—
214C2C2×C4⋊C4×D10911yescharacteristic
215C2#2029550712yes—
216C2#2029558911yescharacteristic
217C2#2029560111yescharacteristic
218C2#2029565312yes—
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#384×C109C2
socle, cosocleC23×C109C24
prime radicalC23×D109C23
hypercenter#384D109
hypocenterC109#2317
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #20294921C2semidirect
6. #20294895C2semidirect
7. #20294892C2semidirect
8. #20294891C2semidirect
9. #384×C109C2semidirect
13. #16812656C4semidirect
44. #13182008C2×C4semidirect
48. C4×Dic109C2×C4semidirect
56. C42×C109C2×C4semidirect
84. C4×D109C4⋊C4semidirect
110. C109#2317semidirect
117. #384D109semidirect
166. C42C4×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126572
C2×#168126562
C2×C4⋊C4×D1092
#202949212
#202948972
#202948952
#202948922
#202948912
#384×C1092
#2317109