Finite Group Databasebeta

← #28372312

Subgroups — #28372312

Order 13952.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812745C211yesmaximal, characteristic
2C2×#16812729C211yesmaximal, characteristic, complemented
3C22×#13182018C211yesmaximal, characteristic, complemented
4#20295602C211yesmaximal, characteristic, complemented
5#20295191C214yesmaximal, complemented
6#20295178C214yesmaximal, complemented
7#20295030C211yesmaximal, characteristic, complemented
8#20294992C211yesmaximal, characteristic, complemented
9#416×C109C211yesmaximal, characteristic, complemented
10#16812745—24no—
11#16812745—24no—
12C2×#13182030C2212yes—
13C2×#13182029—21no—
14C2×#13182029C2211yescharacteristic
15C22×Dic218C2211yescharacteristic, complemented
16C22×D436—21no—
17C22×D436C2211yescharacteristic
18C2×D4×D109—22no—
19C2×D4×D109C2212yes—
20#16812729—24no—
21C2×#13182019C2212yescomplemented
22C2×#13182018—24no—
23C2×#13182018C2214yescomplemented
24C2×#13182017C2212yescomplemented
25C2×#13182013C2211yescharacteristic, complemented
26#16812699—21no—
27#16812687—21no—
28#16812683—22no—
29#16812679C2214yescomplemented
30#16812619C2214yescomplemented
31#16812611C2214yescomplemented
32C2×#41×C109C2211yescharacteristic
33C22×C8×C109C2211yescharacteristic, complemented
34#117×C109C2211yescharacteristic, complemented
35#103×C109C2212yescomplemented
36#102×C109C2212yescomplemented
37C23×D109—21no—
38C23×D109—22no—
39#13182030—24no—
40#13182030—24no—
41#13182029—28no—
42#13182029—22no—
43#13182029—22no—
44C2×Dic218—22no—
45C2×Dic218—22no—
46C2×Dic218—21no—
47C2×Dic218D411yescharacteristic
48C2×Dic218C2312yes—
49C2×D436—28no—
50C2×D436—22no—
51C2×D436—21no—
52C2×D436—22no—
53C2×D436D411yescharacteristic
54C2×D436D414yes—
55C2×D436C2312yes—
56C2×C22⋊D109—22no—
57C2×C22⋊D109D412yes—
58D4×D109—216no—
59D4×D109—24no—
60D4×D109—24no—
61C22×Dic109—21no—
62C2×C4×D109—22no—
63C2×C4×D109C2312yes—
64#13182019—24no—
65#13182018—28no—
66#13182017—24no—
67#13182013—22no—
68C2×#11724784C2312yes—
69C4⋊Dic109—24no—
70#13182006—24no—
71#41×C109—22no—
72#41×C109D414yes—
73C2×Q8×C109C2311yescharacteristic
74C2×D4×C109—21no—
75C2×D4×C109C2311yescharacteristic
76C22×C4×C109—21no—
77C22×C4×C109C2311yescharacteristic
78C2×C8×C109—22no—
79C2×C8×C109C2312yes—
80C4⋊C4×C109C2312yes—
81C22⋊C4×C109—21no—
82C42×C109—21no—
83C23×C109—21no—
84C23×C109C2×D411yescharacteristic
85C22×D109—22no—
86C22×D109—28no—
87C22×D109—24no—
88C22×D109—84no—
89C22×D109—82no—
90C22×D109C2×D412yes—
91Q8×C109—21no—
92Q8×C109C2×D412yes—
93D4×C109—24no—
94D4×C109—21no—
95D4×C109C2×D412yes—
96C2×C4×C109—21no—
97C2×C4×C109—24no—
98C2×C4×C109—22no—
99C2×C4×C109C2×D411yescharacteristic
100C2×C4×C109C2×D412yes—
101C2×C4×C109C2×D412yes—
102C2×C4×C109C2411yescharacteristic
103C22⋊D109—28no—
104C22⋊D109—24no—
105C2×Dic109—22no—
106C2×Dic109—42no—
107C2×Dic109C2×D412yes—
108D436—28no—
109D436—22no—
110D436—24no—
111D436—22no—
112D436C2×D414yes—
113C4×D109—28no—
114C4×D109—22no—
115C4×D109—22no—
116Dic218—22no—
117Dic218—22no—
118Dic218—22no—
119Dic218—42no—
120C8×C109—22no—
121#11724784—22no—
122C22×C109—22no—
123C22×C109—22no—
124C22×C109—42no—
125C22×C109#12012yes—
126C22×C109C22×D411yescharacteristic
127C2×D109—22no—
128C2×D109—24no—
129C2×D109—24no—
130C2×D109—44no—
131C2×D109—84no—
132C2×D109—88no—
133C4×C109—22no—
134C4×C109—42no—
135C4×C109#12012yes—
136C4×C109C22×D411yescharacteristic
137C4×C109C22×D411yescharacteristic
138Dic109—22no—
139Dic109—42no—
140C2×C109—22no—
141C2×C109—42no—
142C2×C109C2×#12011yescharacteristic
143C2×C109#57712yes—
144D109—42no—
145D109—44no—
146#3009—1091nomaximal, Sylow
147C109#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—1091no—
149C2×#137—1091no—
150C22×SD16—1091no—
151#536—1091no—
152#452—1094no—
153#450—1094no—
154#418—1091no—
155#416D10911yescharacteristic, complemented
156#408—1091no—
157H(5,𝔽2)—2184no—
158H(5,𝔽2)—2184no—
159C2×#41—1091no—
160C2×#41—1092no—
161C2×#41—2181no—
162C2×#41C2×D10911yescharacteristic
163C22×Q8—1091no—
164C22×D4—1092no—
165C22×D4—1091no—
166C22×D4—2181no—
167C22×D4—2182no—
168#137—2184no—
169C2×Q16—1092no—
170C2×SD16—1092no—
171C2×SD16—1094no—
172C2×SD16—2184no—
173C2×M16—1091no—
174C22×C8C2×D10911yescharacteristic, complemented
175C4⋊Q8—2181no—
176#124—2181no—
177#123—2182no—
178C22⋊Q8—1094no—
179#117C2×D10911yescharacteristic, complemented
180#103C2×D10912yescomplemented
181#102—1094no—
182#102C2×D10912yescomplemented
183C22⋊C8—1094no—
184C24—2182no—
185C24—2181no—
186#41—22no—
187#41—2188no—
188#41—2184no—
189#41—2184no—
190#41—2182no—
191#41—2182no—
192#41D43614yes—
193C2×Q8—1091no—
194C2×Q8—1092no—
195C2×Q8—2182no—
196C2×Q8—2182no—
197C2×Q8—2181no—
198C2×Q8C22×D10911yescharacteristic
199C2×D4—21no—
200C2×D4—1092no—
201C2×D4—1094no—
202C2×D4—1091no—
203C2×D4—1092no—
204C2×D4—2182no—
205C2×D4—2188no—
206C2×D4—2182no—
207C2×D4—2184no—
208C2×D4—21816no—
209C2×D4—2184no—
210C2×D4—2181no—
211C2×D4—2182no—
212C2×D4C22×D10911yescharacteristic
213C22×C4—21no—
214C22×C4—1092no—
215C22×C4—2182no—
216C22×C4—2181no—
217C22×C4C22×D10911yescharacteristic
218Q16—2184no—
219SD16—2188no—
220SD16—2184no—
221M16—2182no—
222C2×C8—22no—
223C2×C8—1092no—
224C2×C8C22×D10912yes—
225C4⋊C4—2184no—
226C4⋊C4C22×D10912yes—
227C22⋊C4—21no—
228C22⋊C4—2184no—
229C42—21no—
230C23—21no—
231C23—8722no—
232C23—8724no—
233C23—1092no—
234C23—2188no—
235C23—2184no—
236C23—2182no—
237C23D4×D10911yescharacteristic
238Q8—21no—
239Q8—4362no—
240Q8—2182no—
241Q8—2182no—
242Q8—2182no—
243Q8C2×D43612yes—
244D4—24no—
245D4—21no—
246D4—1094no—
247D4—2188no—
248D4—2188no—
249D4—2184no—
250D4—2184no—
251D4—2182no—
252D4—2182no—
253D4C2×D43612yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—1092no—
258C2×C4—4362no—
259C2×C4—2188no—
260C2×C4—2182no—
261C2×C4—2182no—
262C2×C4—2182no—
263C2×C4D4×D10911yescharacteristic
264C2×C4D4×D10912yes—
265C2×C4C2×D43612yes—
266C2×C4C23×D10911yescharacteristic
267C8—22no—
268C8—2182no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—8724no—
273C22—8728no—
274C22—4364no—
275C22—2184no—
276C22—2182no—
277C22—2184no—
278C22#1681266912yes—
279C22C2×D4×D10911yescharacteristic
280C4—22no—
281C4—42no—
282C4—4362no—
283C4—2182no—
284C4#1681266912yes—
285C4C2×D4×D10911yescharacteristic
286C4C22×D43611yescharacteristic
287C2—22no—
288C2—42no—
289C2—4364no—
290C2—4362no—
291C2C2×#1681266911yescharacteristic
292C2#2029583312yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#416×C109C2
socleC22×C109C22×D4
prime radicalC2×#16812745C2
hypercenter#416D109
hypocenterC109#3009
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#16812729C2semidirect
3. C22×#13182018C2semidirect
4. #20295602C2semidirect
5. #20295191C2semidirect
6. #20295178C2semidirect
7. #20295030C2semidirect
8. #20294992C2semidirect
9. #416×C109C2semidirect
15. C22×Dic218C22semidirect
21. C2×#13182019C22semidirect
23. C2×#13182018C22semidirect
24. C2×#13182017C22semidirect
25. C2×#13182013C22semidirect
29. #16812679C22semidirect
30. #16812619C22semidirect
31. #16812611C22semidirect
33. C22×C8×C109C22semidirect
34. #117×C109C22semidirect
35. #103×C109C22semidirect
36. #102×C109C22semidirect
147. C109#3009semidirect
155. #416D109semidirect
174. C22×C8C2×D109semidirect
179. #117C2×D109semidirect
180. #103C2×D109semidirect
182. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127452
C2×#168127292
C22×#131820182
#202956022
#202951912
#202951782
#202950302
#202949922
#416×C1092
#3009109