Finite Group Databasebeta

← #28376164

Subgroups — #28376164

Order 13952.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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
2#135×D109C211yesmaximal, characteristic, complemented
3#20295469C211yesmaximal, characteristic, complemented
4#20295466C211yesmaximal, characteristic, complemented
5#20295465C212yesmaximal, complemented
6#20295462C211yesmaximal, characteristic, complemented
7#20295460C212yesmaximal, complemented
8#20295457C212yesmaximal, complemented
9#20295455C212yesmaximal, complemented
10M32×D109C211yesmaximal, characteristic, complemented
11#509×C109C211yesmaximal, characteristic, complemented
12C2×#13182028—21no—
13#41×D109—21no—
14#16812722C2211yescharacteristic, complemented
15#16812721C2211yescharacteristic
16#16812720—21no—
17#16812719—21no—
18#16812718—21no—
19#16812718—21no—
20#16812718C2212yes—
21#16812718C2211yescharacteristic, complemented
22#16812718C2212yescomplemented
23C2×Dic436C2211yescharacteristic, complemented
24Q16×D109C2211yescharacteristic, complemented
25C2×#13182017—21no—
26SD16×D109—21no—
27D8×D109C2211yescharacteristic, complemented
28C2×C8×D109C2211yescharacteristic
29Dic872C2212yescomplemented
30#16812645C2212yescomplemented
31#16812644C2212yescomplemented
32#16812643C2212yescomplemented
33#16812643C2212yescomplemented
34#16812640C2212yescomplemented
35#16812639C2211yescharacteristic, complemented
36#16812638C2212yescomplemented
37C16×D109C2212yescomplemented
38#135×C109C2211yescharacteristic
39C2×D8×C109C2211yescharacteristic, complemented
40SD32×C109C2212yescomplemented
41D16×C109C2212yescomplemented
42M32×C109C2211yescharacteristic, complemented
43#13182030—21no—
44#13182029—21no—
45#13182029—41no—
46#13182028—22no—
47#13182028—22no—
48#13182028—21no—
49#13182028—41no—
50#13182028—41no—
51#13182028—41no—
52C2×Dic218—21no—
53Q8×D109—21no—
54C2×C22⋊D109—41no—
55D4×D109—21no—
56D4×D109—41no—
57C22×Dic109—41no—
58C2×C4×D109—41no—
59C2×C4×D109D411yescharacteristic
60Dic436—21no—
61Dic436C2311yescharacteristic
62Dic436C2312yes—
63#13182019—21no—
64#13182018—21no—
65#13182017—21no—
66#13182017—22no—
67#13182017—22no—
68#13182016—21no—
69D872C2311yescharacteristic
70#13182014—21no—
71C2×#11724784D411yescharacteristic
72C8×D109D411yescharacteristic
73C8×D109D411yescharacteristic
74C8×D109C2311yescharacteristic
75C8×D109C2311yescharacteristic
76#13182004C2312yes—
77#41×C109—21no—
78C2×D4×C109—21no—
79Q16×C109C2311yescharacteristic
80SD16×C109—21no—
81D8×C109—21no—
82D8×C109C2311yescharacteristic
83D8×C109C2312yes—
84C2×C8×C109C2311yescharacteristic
85C16×C109C2312yes—
86C23×C109—41no—
87C22×D109—41no—
88C22×D109D811yescharacteristic
89Q8×C109—21no—
90D4×C109—22no—
91D4×C109—21no—
92D4×C109—41no—
93D4×C109—41no—
94C2×C4×C109—41no—
95C2×C4×C109C2×D411yescharacteristic
96C22⋊D109—41no—
97C22⋊D109—41no—
98C22⋊D109—42no—
99C22⋊D109—42no—
100C22⋊D109—41no—
101C2×Dic109—41no—
102C2×Dic109—41no—
103C2×Dic109—42no—
104C2×Dic109—42no—
105C2×Dic109D811yescharacteristic
106D436—21no—
107D436—41no—
108C4×D109—41no—
109C4×D109—41no—
110C4×D109—41no—
111C4×D109D811yescharacteristic
112C4×D109D811yescharacteristic
113C4×D109C2×D411yescharacteristic
114C4×D109C2×D411yescharacteristic
115Dic218—22no—
116Dic218—21no—
117Dic218—41no—
118Dic218—41no—
119C8×C109C2×D411yescharacteristic
120C8×C109C2411yescharacteristic
121#11724784C2×D411yescharacteristic
122#11724784C2×D411yescharacteristic
123C22×C109—42no—
124C22×C109—41no—
125C22×C109—82no—
126C22×C109C2×D811yescharacteristic
127C2×D109—41no—
128C2×D109—41no—
129C2×D109—82no—
130C2×D109C2×D811yescharacteristic
131C2×D109C2×D811yescharacteristic
132C4×C109—41no—
133C4×C109C2×D811yescharacteristic
134C4×C109C22×D411yescharacteristic
135Dic109—41no—
136Dic109—42no—
137Dic109C2×D811yescharacteristic
138Dic109C2×D811yescharacteristic
139C2×C109—21no—
140C2×C109—81no—
141C2×C109—82no—
142C2×C109C22×D811yescharacteristic
143D109—21no—
144D109—21no—
145D109—81no—
146#3414—1091nomaximal, Sylow
147C109#341411yescharacteristic, complemented, Sylow
148C2×#135—1091no—
149C2×#135—1091no—
150#510—1091no—
151#510—1092no—
152#510—1091no—
153#509—1091no—
154#509—1092no—
155#509D10911yescharacteristic, complemented
156#508—1092no—
157#508—1092no—
158C2×M32—1091no—
159C2×#41—2181no—
160C2×#41—2181no—
161#135—1091no—
162#135—1092no—
163#135—1092no—
164#135—1091no—
165#135—1091no—
166#135—2181no—
167#135—2181no—
168#135—2181no—
169#135—2181no—
170#135C2×D10911yescharacteristic
171C2×Q16—1091no—
172C2×Q16—1091no—
173C2×SD16—2181no—
174C2×SD16—2181no—
175C2×D8—1091no—
176C2×D8C2×D10911yescharacteristic, complemented
177C22×C8—1091no—
178Q32—1092no—
179Q32—1092no—
180SD32—1092no—
181SD32—1092no—
182SD32—1092no—
183SD32C2×D10912yescomplemented
184D16—1092no—
185D16C2×D10912yescomplemented
186M32—1091no—
187M32—1092no—
188M32C2×D10911yescharacteristic, complemented
189C2×C16—1092no—
190#41—21no—
191#41—4361no—
192#41—4361no—
193#41—4361no—
194#41—4361no—
195#41—2182no—
196#41—2182no—
197#41—2181no—
198#41—2181no—
199#41—2181no—
200C2×Q8—2181no—
201C2×Q8—2181no—
202C2×D4—21no—
203C2×D4—4361no—
204C2×D4—4361no—
205C2×D4—2181no—
206C22×C4—1091no—
207C22×C4—4361no—
208C22×C4—4361no—
209Q16—1091no—
210Q16—1092no—
211Q16—2181no—
212Q16—2181no—
213Q16C22×D10911yescharacteristic
214SD16—21no—
215SD16—2182no—
216SD16—2182no—
217SD16—2181no—
218SD16—2181no—
219SD16—2181no—
220D8—21no—
221D8—1091no—
222D8—2181no—
223D8C22×D10912yes—
224D8C22×D10911yescharacteristic
225C2×C8—1091no—
226C2×C8—1091no—
227C2×C8—1091no—
228C2×C8—1091no—
229C2×C8—1091no—
230C2×C8C22×D10911yescharacteristic
231C16—1092no—
232C16C22×D10912yes—
233C23—41no—
234C23—1091no—
235C23—4361no—
236Q8—21no—
237Q8—4361no—
238Q8—4361no—
239Q8—2182no—
240Q8—2181no—
241D4—22no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—4361no—
246D4—4361no—
247D4—4361no—
248D4—4362no—
249D4—4362no—
250D4—4361no—
251D4—2181no—
252C2×C4—41no—
253C2×C4—1091no—
254C2×C4—1091no—
255C2×C4—1091no—
256C2×C4—1091no—
257C2×C4—1091no—
258C2×C4—4361no—
259C2×C4—4361no—
260C2×C4—4361no—
261C2×C4—4362no—
262C2×C4—4362no—
263C2×C4—4361no—
264C2×C4—4361no—
265C2×C4D4×D10911yescharacteristic
266C8—1091no—
267C8—1091no—
268C8D4×D10911yescharacteristic
269C8C23×D10911yescharacteristic
270C22—41no—
271C22—42no—
272C22—82no—
273C22—8722no—
274C22—1091no—
275C22—1091no—
276C22—4361no—
277C22—4361no—
278C22D8×D10911yescharacteristic
279C4—41no—
280C4—1091no—
281C4—1091no—
282C4—4361no—
283C4—4362no—
284C4D8×D10911yescharacteristic
285C4C2×D4×D10911yescharacteristic
286C2—21no—
287C2—82no—
288C2—81no—
289C2—8721no—
290C2—2181no—
291C2—2181no—
292C2C2×D8×D10911yescharacteristic
subgroupG/H
centerC2C2×D8×D109
commutator, cosocleC8×C109C24
FrattiniC8C23×D109
Fitting#509×C109C2
socleC2×C109C22×D8
hypercenter#509D109
hypocenterC109#3414
LCSG ▷ C8×C109 ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. C2×#16812718C2semidirect
2. #135×D109C2semidirect
3. #20295469C2semidirect
4. #20295466C2semidirect
5. #20295465C2semidirect
6. #20295462C2semidirect
7. #20295460C2semidirect
8. #20295457C2semidirect
9. #20295455C2semidirect
10. M32×D109C2semidirect
11. #509×C109C2semidirect
14. #16812722C22semidirect
21. #16812718C22semidirect
22. #16812718C22semidirect
23. C2×Dic436C22semidirect
24. Q16×D109C22semidirect
27. D8×D109C22semidirect
29. Dic872C22semidirect
30. #16812645C22semidirect
31. #16812644C22semidirect
32. #16812643C22semidirect
33. #16812643C22semidirect
34. #16812640C22semidirect
35. #16812639C22semidirect
36. #16812638C22semidirect
37. C16×D109C22semidirect
39. C2×D8×C109C22semidirect
40. SD32×C109C22semidirect
41. D16×C109C22semidirect
42. M32×C109C22semidirect
147. C109#3414semidirect
155. #509D109semidirect
176. C2×D8C2×D109semidirect
183. SD32C2×D109semidirect
185. D16C2×D109semidirect
188. M32C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127182
#135×D1092
#202954692
#202954662
#202954652
#202954622
#202954602
#202954572
#202954552
M32×D1092
#509×C1092
#3414109