Finite Group Databasebeta

← #28379631

Subgroups — #28379631

Order 13952.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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#20296093C212yesmaximal, complemented
2#20296006C212yesmaximal, complemented
3#20296005C212yesmaximal, complemented
4D109⋊#99C214yesmaximal, complemented
5C2×#16812757C214yesmaximal, complemented
6C2×#16812744C211yesmaximal, characteristic
7D4×#89137—28no—
8#16812778C2214yescomplemented
9C2×C22⋊#89137C2218yescomplemented
10#16812758C2214yescomplemented
11#16812757C2214yescomplemented
12#16812757C2218yescomplemented
13#16812744—28no—
14C2×#13182029C411yescharacteristic, complemented
15C2×#13182029C2211yescharacteristic
16C2×#13182028C411yescharacteristic, complemented
17C2×#13182028C412yescomplemented
18C2×#13182028C2211yescharacteristic
19C22×C22⋊D109C412yescomplemented
20C22×C22⋊D109C2212yes—
21C2×D4×D109C411yescharacteristic, complemented
22C2×D4×D109C2211yescharacteristic
23C2×D4×D109C2212yes—
24C22×D4×C109C411yescharacteristic, complemented
25C22×#89137—28no—
26C4⋊#89137—24no—
27C22⋊#89137—28no—
28C22⋊#89137C2318yes—
29C4×#89137—24no—
30C23×D109C2×C412yescomplemented
31C23×D109C2312yes—
32#13182029—24no—
33#13182029—24no—
34#13182028—24no—
35#13182028—28no—
36#13182028—24no—
37C2×Dic218C2×C411yescharacteristic, complemented
38C2×Dic218C2×C411yescharacteristic, complemented
39C2×D436C2×C411yescharacteristic
40C2×D436C2311yescharacteristic
41C2×C22⋊D109—28no—
42C2×C22⋊D109—28no—
43C2×C22⋊D109C2×C412yes—
44C2×C22⋊D109C2×C412yescomplemented
45C2×C22⋊D109C2×C412yescomplemented
46C2×C22⋊D109C2×C411yescharacteristic, complemented
47C2×C22⋊D109C2×C412yescomplemented
48C2×C22⋊D109C2312yes—
49C2×C22⋊D109C2311yescharacteristic
50D4×D109—28no—
51D4×D109—24no—
52D4×D109D418yes—
53C22×Dic109C2×C412yescomplemented
54C22×Dic109C2×C412yescomplemented
55C2×C4×D109C2×C411yescharacteristic, complemented
56C2×C4×D109C2×C412yescomplemented
57C2×C4×D109C2311yescharacteristic
58C24×C109C2×C412yescomplemented
59C2×D4×C109—24no—
60C2×D4×C109C2×C411yescharacteristic
61C2×D4×C109C2×C411yescharacteristic, complemented
62C2×D4×C109C2×C412yescomplemented
63C22×C4×C109C2×C411yescharacteristic, complemented
64C23×C109—24no—
65C23×C109—82no—
66C23×C109C22×C411yescharacteristic
67C23×C109C22×C412yes—
68C23×C109C22×C412yes—
69C22×D109—24no—
70C22×D109—82no—
71C22×D109—82no—
72C22×D109C22×C411yescharacteristic
73C22×D109C22×C412yes—
74C22×D109C2×D418yes—
75C22×D109C2411yescharacteristic
76C2×#89137—28no—
77C2×#89137—88no—
78D4×C109—24no—
79D4×C109—22no—
80D4×C109C22⋊C414yes—
81C2×C4×C109—22no—
82C2×C4×C109C22×C411yescharacteristic
83C2×C4×C109C22×C411yescharacteristic
84C22⋊D109—24no—
85C22⋊D109—24no—
86C22⋊D109—24no—
87C22⋊D109—24no—
88C22⋊D109—22no—
89C22⋊D109—22no—
90C22⋊D109C22⋊C418yes—
91C2×Dic109—24no—
92C2×Dic109—24no—
93C2×Dic109C22×C411yescharacteristic
94C2×Dic109C22×C412yes—
95C2×Dic109C22×C411yescharacteristic
96D436—22no—
97D436C22⋊C414yes—
98C4×D109—24no—
99C4×D109—22no—
100C4×D109C2×D414yes—
101Dic218—22no—
102Dic218—22no—
103C22×C109—22no—
104C22×C109—21no—
105C22×C109—42no—
106C22×C109—42no—
107C22×C109—41no—
108C22×C109—84no—
109C22×C109C2×C22⋊C414yes—
110C22×C109C23×C411yescharacteristic
111C2×D109—21no—
112C2×D109—48no—
113C2×D109—42no—
114C2×D109—41no—
115C2×D109—41no—
116C2×D109—84no—
117C2×D109C2×C22⋊C414yes—
118C2×D109C22×D412yes—
119#89137—48no—
120C4×C109—21no—
121C4×C109C2×C22⋊C412yes—
122Dic109—22no—
123Dic109—21no—
124Dic109C2×C22⋊C412yes—
125C2×C109—21no—
126C2×C109—24no—
127C2×C109—42no—
128C2×C109—41no—
129C2×C109C22×C22⋊C411yescharacteristic
130D109—22no—
131D109—24no—
132D109—41no—
133#2883—1091nomaximal, Sylow
134C109#288311yescharacteristic, complemented, Sylow
135C2×H(5,𝔽2)—1091no—
136#518—1092no—
137#410—1092no—
138#410—1092no—
139C2×#99—1094no—
140C2×#99—1094no—
141H(5,𝔽2)—2188no—
142C2×#41—1091no—
143C2×#41—1091no—
144C2×#41—1092no—
145C2×#41—1091no—
146C2×#41—1091no—
147C22×D4—1091no—
148C22×D4—1092no—
149C22×D4—1092no—
150C22×D4—1092no—
151C22×D4—1091no—
152C22×D4#8913711yescharacteristic, complemented
153C4×D4—2188no—
154#117—1094no—
155C2×C22⋊C4—1098no—
156#99—1094no—
157#99—1098no—
158#99—1094no—
159C24—1092no—
160C24—1092no—
161C24C2×#8913712yescomplemented
162#41—2184no—
163#41—2184no—
164#41—2184no—
165#41—2188no—
166#41—2184no—
167C2×Q8—1091no—
168C2×Q8—1091no—
169C2×D4—24no—
170C2×D4—1092no—
171C2×D4—1098no—
172C2×D4—1091no—
173C2×D4—1092no—
174C2×D4—1092no—
175C2×D4—1092no—
176C2×D4—1092no—
177C2×D4—1091no—
178C2×D4—1091no—
179C2×D4—1091no—
180C2×D4—2184no—
181C2×D4—2188no—
182C2×D4—2188no—
183C2×D4—2188no—
184C2×D4C2×#8913711yescharacteristic
185C2×D4C2×#8913712yescomplemented
186C2×D4C2×#8913711yescharacteristic, complemented
187C22×C4—1091no—
188C22×C4—1092no—
189C22×C4—1092no—
190C22×C4—1092no—
191C22×C4—1091no—
192C22×C4—2188no—
193C22×C4C2×#8913711yescharacteristic, complemented
194C4⋊C4—2184no—
195C22⋊C4—1098no—
196C22⋊C4—2188no—
197C42—2184no—
198C23—24no—
199C23—82no—
200C23—8722no—
201C23—8722no—
202C23—1098no—
203C23—1091no—
204C23—1092no—
205C23—1091no—
206C23—2184no—
207C23C22×#8913712yes—
208C23C22×#8913711yescharacteristic
209C23C22×#8913712yes—
210Q8—2182no—
211Q8—2182no—
212D4—24no—
213D4—22no—
214D4—1098no—
215D4—1094no—
216D4—2182no—
217D4—2182no—
218D4—2182no—
219D4—2184no—
220D4—2184no—
221D4—2184no—
222D4—2184no—
223D4C22⋊#8913714yes—
224C2×C4—22no—
225C2×C4—8728no—
226C2×C4—1091no—
227C2×C4—1094no—
228C2×C4—1092no—
229C2×C4—1091no—
230C2×C4—2184no—
231C2×C4—2184no—
232C2×C4—2188no—
233C2×C4—2184no—
234C2×C4—2182no—
235C2×C4C22×#8913711yescharacteristic
236C2×C4C22×#8913711yescharacteristic
237C22—21no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—84no—
243C22—8724no—
244C22—1092no—
245C22—1094no—
246C22—4361no—
247C22—4368no—
248C22—4361no—
249C22—4362no—
250C22—2181no—
251C22C2×C22⋊#8913714yes—
252C22C23×#8913711yescharacteristic
253C4—21no—
254C4—1092no—
255C4—4368no—
256C4—2182no—
257C4—2181no—
258C4C2×C22⋊#8913712yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—42no—
263C2—4361no—
264C2—2182no—
265C2—2184no—
266C2C22×C22⋊#8913711yescharacteristic
subgroupG/H
centerC2C22×C22⋊#89137
commutatorC22×C109C23×C4
FrattiniC22C23×#89137
FittingC22×D4×C109C4
socleC2×C109C22×C22⋊C4
prime radicalC2×#16812744C2
cosocleC22×D109C24
hypercenterC22×D4#89137
hypocenterC109#2883
LCSG ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×D4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20296093C2semidirect
2. #20296006C2semidirect
3. #20296005C2semidirect
4. D109⋊#99C2semidirect
5. C2×#16812757C2semidirect
8. #16812778C22semidirect
9. C2×C22⋊#89137C22semidirect
10. #16812758C22semidirect
11. #16812757C22semidirect
12. #16812757C22semidirect
14. C2×#13182029C4semidirect
16. C2×#13182028C4semidirect
17. C2×#13182028C4semidirect
19. C22×C22⋊D109C4semidirect
21. C2×D4×D109C4semidirect
24. C22×D4×C109C4semidirect
30. C23×D109C2×C4semidirect
37. C2×Dic218C2×C4semidirect
38. C2×Dic218C2×C4semidirect
44. C2×C22⋊D109C2×C4semidirect
45. C2×C22⋊D109C2×C4semidirect
46. C2×C22⋊D109C2×C4semidirect
47. C2×C22⋊D109C2×C4semidirect
53. C22×Dic109C2×C4semidirect
54. C22×Dic109C2×C4semidirect
55. C2×C4×D109C2×C4semidirect
56. C2×C4×D109C2×C4semidirect
58. C24×C109C2×C4semidirect
61. C2×D4×C109C2×C4semidirect
62. C2×D4×C109C2×C4semidirect
63. C22×C4×C109C2×C4semidirect
134. C109#2883semidirect
152. C22×D4#89137semidirect
161. C24C2×#89137semidirect
185. C2×D4C2×#89137semidirect
186. C2×D4C2×#89137semidirect
193. C22×C4C2×#89137semidirect

Maximal subgroups

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

subgroupindex
#202960932
#202960062
#202960052
D109⋊#992
C2×#168127572
C2×#168127442
#2883109