Finite Group Databasebeta

← #28372611

Subgroups — #28372611

Order 13952.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812718C211yesmaximal, characteristic, complemented
2C22×C8×D109C211yesmaximal, characteristic, complemented
3#20295536C212yesmaximal, complemented
4#20295268C211yesmaximal, characteristic, complemented
5#20295267C211yesmaximal, characteristic, complemented
6#20295259C211yesmaximal, characteristic, complemented
7#20295254C212yesmaximal, complemented
8#20295253C212yesmaximal, complemented
9#20295063C211yesmaximal, characteristic, complemented
10#20295019C212yesmaximal, complemented
11#466×C109C211yesmaximal, characteristic, complemented
12C2×#13182028C2212yes—
13C22×C4×D109C2211yescharacteristic
14#16812718—24no—
15C2×Dic436C2211yescharacteristic, complemented
16C2×#13182017C2212yescomplemented
17C22×#11724784C2211yescharacteristic
18C2×C8×D109—24no—
19C2×C8×D109C2211yescharacteristic
20C2×C8×D109C2211yescharacteristic
21C2×C8×D109C2211yescharacteristic
22C2×C8×D109C2211yescharacteristic
23#16812681—22no—
24#16812679C2212yes—
25#16812678C2212yes—
26#16812670C2212yes—
27#16812667—22no—
28D4×Dic109—22no—
29#16812659—22no—
30#16812657C2212yes—
31#16812633C2211yescharacteristic, complemented
32#16812630C2212yescomplemented
33#16812624C2212yescomplemented
34#16812622C2212yescomplemented
35#16812620C2212yescomplemented
36C2×D8×C109C2211yescharacteristic, complemented
37C22×C8×C109C2211yescharacteristic
38#121×C109C2212yes—
39#107×C109C2211yescharacteristic, complemented
40#102×C109C2212yescomplemented
41C23×D109D411yescharacteristic
42#13182028—24no—
43#13182028—24no—
44C2×Dic218C2312yes—
45C2×C22⋊D109—22no—
46C2×C22⋊D109—22no—
47C22×Dic109—22no—
48C22×Dic109D411yescharacteristic
49C2×C4×D109—22no—
50C2×C4×D109—22no—
51C2×C4×D109D411yescharacteristic
52C2×C4×D109D411yescharacteristic
53C2×C4×D109C2311yescharacteristic
54C2×C4×D109C2311yescharacteristic
55Dic436—22no—
56#13182017—24no—
57C2×#11724784—22no—
58C2×#11724784C2311yescharacteristic
59C2×#11724784C2311yescharacteristic
60C8×D109—22no—
61C8×D109—22no—
62C8×D109—22no—
63C8×D109D414yescomplemented
64C4⋊Dic109—22no—
65C4⋊Dic109C2312yes—
66#13182008—22no—
67C22⋊Dic109—22no—
68C22⋊Dic109—22no—
69#13182006—22no—
70#13182006—22no—
71C4×Dic109—22no—
72C2×D4×C109—22no—
73C2×D4×C109C2312yes—
74C22×C4×C109C2311yescharacteristic
75D8×C109—22no—
76C2×C8×C109—22no—
77C2×C8×C109C2311yescharacteristic
78C2×C8×C109C2311yescharacteristic
79C4⋊C4×C109C2312yes—
80C22⋊C4×C109—22no—
81C23×C109—22no—
82C23×C109C2×D411yescharacteristic
83C22×D109—22no—
84C22×D109—22no—
85C22×D109—22no—
86C22×D109C2×D411yescharacteristic
87C22×D109C2×D411yescharacteristic
88D4×C109—22no—
89D4×C109—22no—
90D4×C109—44no—
91C2×C4×C109—21no—
92C2×C4×C109—21no—
93C2×C4×C109—22no—
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—44no—
97C22⋊D109—44no—
98C2×Dic109—22no—
99C2×Dic109—22no—
100C2×Dic109—22no—
101C2×Dic109—44no—
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D411yescharacteristic
104C4×D109—22no—
105C4×D109—21no—
106C4×D109—21no—
107C4×D109—22no—
108C4×D109C2×D412yes—
109C4×D109#4112yes—
110Dic218—22no—
111Dic218—22no—
112C8×C109—21no—
113C8×C109C2×D412yes—
114#11724784—21no—
115#11724784C2×D412yes—
116C22×C109—21no—
117C22×C109—21no—
118C22×C109—42no—
119C22×C109—42no—
120C22×C109—42no—
121C22×C109#13512yescomplemented
122C22×C109C22×D411yescharacteristic
123C2×D109—28no—
124C2×D109—21no—
125C2×D109—21no—
126C2×D109—21no—
127C2×D109—21no—
128C2×D109#12112yes—
129C2×D109#13512yescomplemented
130C4×C109—21no—
131C4×C109—42no—
132C4×C109C22×D411yescharacteristic
133C4×C109C2×#4111yescharacteristic
134Dic109—21no—
135Dic109—42no—
136Dic109—42no—
137Dic109#12112yes—
138C2×C109—22no—
139C2×C109—82no—
140C2×C109C2×#12111yescharacteristic
141C2×C109C2×#13511yescharacteristic
142C2×C109C2×#13511yescharacteristic
143D109—22no—
144D109—22no—
145#3050—1091nomaximal, Sylow
146C109#305011yescharacteristic, complemented, Sylow
147C2×#135—1091no—
148C23×C8—1091no—
149#525—1092no—
150#467—1091no—
151#467—1091no—
152#466—1091no—
153#466D10911yescharacteristic, complemented
154#465—1092no—
155#465—1092no—
156#427—1091no—
157#416—1092no—
158C2×#41—1092no—
159C23×C4—1091no—
160#135—2184no—
161C2×Q16—1091no—
162C2×SD16—1092no—
163C2×D8C2×D10911yescharacteristic, complemented
164C22×C8—1091no—
165C22×C8—1091no—
166C22×C8—1091no—
167C22×C8—1091no—
168C22×C8—1091no—
169C22×C8—2184no—
170C22×C8C2×D10911yescharacteristic
171#123—2182no—
172C22⋊Q8—1092no—
173C22⋊Q8—1092no—
174#121—1092no—
175#121C2×D10912yes—
176#120—2182no—
177C4×D4—2182no—
178C4×D4—2182no—
179#117—1092no—
180#107—1091no—
181#107C2×D10911yescharacteristic, complemented
182#106—1092no—
183#103—1092no—
184#103—1092no—
185#102—1092no—
186#102C2×D10912yescomplemented
187C24—1091no—
188#41—2184no—
189#41—2184no—
190C2×Q8—1092no—
191C2×D4—22no—
192C2×D4—2182no—
193C2×D4—2182no—
194C2×D4C22×D10912yes—
195C22×C4—1091no—
196C22×C4—1091no—
197C22×C4—1091no—
198C22×C4—1091no—
199C22×C4—1091no—
200C22×C4—2182no—
201C22×C4—2182no—
202C22×C4—2182no—
203C22×C4C22×D10911yescharacteristic
204Q16—2182no—
205SD16—2184no—
206D8—22no—
207C2×C8—22no—
208C2×C8—1091no—
209C2×C8—1094no—
210C2×C8—1091no—
211C2×C8—2182no—
212C2×C8—2182no—
213C2×C8—2182no—
214C2×C8—2182no—
215C2×C8C22×D10911yescharacteristic
216C2×C8C22×D10911yescharacteristic
217C4⋊C4—1092no—
218C4⋊C4—2182no—
219C4⋊C4—2182no—
220C4⋊C4C22×D10912yes—
221C22⋊C4—22no—
222C22⋊C4—2182no—
223C22⋊C4—2182no—
224C22⋊C4—2182no—
225C22⋊C4—2182no—
226C42—2182no—
227C23—22no—
228C23—1091no—
229C23—1091no—
230C23—2182no—
231C23—2182no—
232C23—2182no—
233C23D4×D10911yescharacteristic
234Q8—2182no—
235Q8—2182no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—4364no—
240D4—4364no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—1091no—
245C2×C4—1092no—
246C2×C4—1092no—
247C2×C4—1091no—
248C2×C4—4364no—
249C2×C4—2182no—
250C2×C4—2182no—
251C2×C4—2182no—
252C2×C4—2182no—
253C2×C4—2182no—
254C2×C4—2181no—
255C2×C4—2181no—
256C2×C4D4×D10911yescharacteristic
257C2×C4C23×D10911yescharacteristic
258C8—21no—
259C8—1092no—
260C8—2181no—
261C8D4×D10912yes—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—1092no—
268C22—1092no—
269C22—2188no—
270C22—2181no—
271C22—2181no—
272C22—2181no—
273C22—2181no—
274C22#1681271812yescomplemented
275C22C2×D4×D10911yescharacteristic
276C4—21no—
277C4—42no—
278C4—1092no—
279C4—4362no—
280C4—4362no—
281C4—2181no—
282C4C2×D4×D10911yescharacteristic
283C4#41×D10911yescharacteristic
284C2—22no—
285C2—82no—
286C2—2182no—
287C2—2182no—
288C2#121×D10911yescharacteristic
289C2#135×D10911yescharacteristic
290C2C2×#1681271811yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#466×C109C2
socleC22×C109C22×D4
hypercenter#466D109
hypocenterC109#3050
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16812718C2semidirect
2. C22×C8×D109C2semidirect
3. #20295536C2semidirect
4. #20295268C2semidirect
5. #20295267C2semidirect
6. #20295259C2semidirect
7. #20295254C2semidirect
8. #20295253C2semidirect
9. #20295063C2semidirect
10. #20295019C2semidirect
11. #466×C109C2semidirect
15. C2×Dic436C22semidirect
16. C2×#13182017C22semidirect
31. #16812633C22semidirect
32. #16812630C22semidirect
33. #16812624C22semidirect
34. #16812622C22semidirect
35. #16812620C22semidirect
36. C2×D8×C109C22semidirect
39. #107×C109C22semidirect
40. #102×C109C22semidirect
63. C8×D109D4semidirect
121. C22×C109#135semidirect
129. C2×D109#135semidirect
146. C109#3050semidirect
153. #466D109semidirect
163. C2×D8C2×D109semidirect
181. #107C2×D109semidirect
186. #102C2×D109semidirect
274. C22#16812718semidirect

Maximal subgroups

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

subgroupindex
C2×#168127182
C22×C8×D1092
#202955362
#202952682
#202952672
#202952592
#202952542
#202952532
#202950632
#202950192
#466×C1092
#3050109