Finite Group Databasebeta

← #28374846

Subgroups — #28374846

Order 13952.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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#20295680C211yesmaximal, characteristic, complemented
2D4×D436C211yesmaximal, characteristic, complemented
3#20295373C211yesmaximal, characteristic, complemented
4#20295340C212yesmaximal, complemented
5#20295307C211yesmaximal, characteristic, complemented
6#20295262C212yesmaximal, complemented
7#20295222C211yesmaximal, characteristic, complemented
8#20295162C212yesmaximal, complemented
9C4×D872C211yesmaximal, characteristic, complemented
10C2×#16812621C212yesmaximal, complemented
11C8×D4×C109C211yesmaximal, characteristic, complemented
12C2×#13182028—21no—
13C22×D436C2212yes—
14C2×D4×D109—21no—
15C2×D872C2211yescharacteristic
16C2×D872C2212yescomplemented
17#16812699C2211yescharacteristic, complemented
18#16812696C2211yescharacteristic
19#16812683—22no—
20#16812679—21no—
21#16812674—21no—
22#16812673C2212yes—
23#16812673C2212yescomplemented
24#16812669—22no—
25C4×D436C2211yescharacteristic
26C4×D436C2211yescharacteristic, complemented
27C2×C4⋊Dic109C2212yescomplemented
28#16812633C2212yescomplemented
29#16812633C2211yescharacteristic, complemented
30#16812621C2211yescharacteristic
31#16812621C2212yes—
32#16812621C2212yescomplemented
33#16812621C2211yescharacteristic, complemented
34#16812621C2212yescomplemented
35#16812621C2212yescomplemented
36C22×C8×C109C2212yescomplemented
37C4×D4×C109C2211yescharacteristic
38C4⋊C8×C109C2211yescharacteristic, complemented
39C22⋊C8×C109C2212yescomplemented
40C4×C8×C109C2211yescharacteristic, complemented
41C23×D109—22no—
42#13182028—44no—
43C2×Dic218—21no—
44C2×D436—24no—
45C2×D436—22no—
46C2×D436—21no—
47C2×D436—21no—
48C2×D436—42no—
49C2×D436C2312yes—
50C2×D436C2311yescharacteristic
51C2×D436C2311yescharacteristic
52C2×C22⋊D109—22no—
53C2×C22⋊D109—22no—
54D4×D109—44no—
55C22×Dic109—22no—
56C2×C4×D109—21no—
57C2×C4×D109—21no—
58D872—22no—
59D872—42no—
60C4⋊Dic109—22no—
61C4⋊Dic109—21no—
62C4⋊Dic109C2311yescharacteristic
63C4⋊Dic109C2312yes—
64C4⋊Dic109C2311yescharacteristic
65#13182006—21no—
66#13182006—21no—
67#13182006—22no—
68#13182006—22no—
69C2×D4×C109D411yescharacteristic
70C22×C4×C109C2312yes—
71C2×C8×C109—42no—
72C2×C8×C109C2311yescharacteristic
73C2×C8×C109C2311yescharacteristic
74C2×C8×C109C2312yes—
75C4⋊C4×C109D411yescharacteristic
76C22⋊C4×C109D412yes—
77C42×C109C2311yescharacteristic
78C23×C109C2×D412yes—
79C22×D109—21no—
80C22×D109—21no—
81C22×D109—22no—
82C22×D109—44no—
83C22×D109—44no—
84C22×D109—82no—
85D4×C109D814yescomplemented
86C2×C4×C109—22no—
87C2×C4×C109—22no—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2×D412yes—
90C2×C4×C109C2×D411yescharacteristic
91C2×C4×C109C2411yescharacteristic
92C22⋊D109—44no—
93C22⋊D109—44no—
94C2×Dic109—21no—
95C2×Dic109—21no—
96C2×Dic109—22no—
97C2×Dic109—44no—
98D436—21no—
99D436—22no—
100D436—21no—
101D436—21no—
102D436—22no—
103D436—44no—
104D436—42no—
105D436—81no—
106D436#4112yescomplemented
107C4×D109—42no—
108C4×D109—42no—
109Dic218—42no—
110C8×C109—21no—
111C8×C109—21no—
112C8×C109—22no—
113C22×C109—22no—
114C22×C109—22no—
115C22×C109C2×D814yes—
116C22×C109C22×D411yescharacteristic
117C2×D109—22no—
118C2×D109—41no—
119C2×D109—41no—
120C2×D109—42no—
121C2×D109—41no—
122C2×D109—41no—
123C2×D109—42no—
124C2×D109—42no—
125C2×D109—41no—
126C2×D109—88no—
127C4×C109—22no—
128C4×C109—21no—
129C4×C109C2×D812yes—
130C4×C109C2×#4111yescharacteristic
131C4×C109H(5,𝔽2)11yescharacteristic
132Dic109—42no—
133Dic109—41no—
134Dic109—41no—
135C2×C109—24no—
136C2×C109#54611yescharacteristic
137C2×C109C22×D811yescharacteristic
138C2×C109#57711yescharacteristic
139D109—42no—
140D109—81no—
141D109—82no—
142#3294—1091nomaximal, Sylow
143C109#329411yescharacteristic, complemented, Sylow
144#547—1091no—
145D42—1091no—
146#486—1091no—
147#480—1092no—
148#474—1091no—
149#466—1092no—
150#459—1091no—
151#447—1092no—
152C4×D8—1091no—
153C8×D4D10911yescharacteristic, complemented
154C2×#102—1092no—
155C2×#41—2181no—
156C22×D4—1092no—
157C22×D4—2181no—
158C2×D8—1092no—
159C2×D8—1091no—
160C22×C8C2×D10912yescomplemented
161C4⋊Q8—1091no—
162#127—1091no—
163#123—2182no—
164C22⋊Q8—2181no—
165#121—1092no—
166#121—1092no—
167#121—2181no—
168#120—2182no—
169C4×D4—1091no—
170C4×D4—1091no—
171C4×D4C2×D10911yescharacteristic
172C2×C4⋊C4—1092no—
173#107—1091no—
174#107—1092no—
175C4⋊C8C2×D10911yescharacteristic, complemented
176#102—1091no—
177#102—1092no—
178#102—1092no—
179#102—1092no—
180#102—1091no—
181#102—1092no—
182C22⋊C8C2×D10912yescomplemented
183C4×C8C2×D10911yescharacteristic, complemented
184C24—2182no—
185#41—4364no—
186C2×Q8—2181no—
187C2×D4—1091no—
188C2×D4—1092no—
189C2×D4—1091no—
190C2×D4—4362no—
191C2×D4—4364no—
192C2×D4—2184no—
193C2×D4—2182no—
194C2×D4—2182no—
195C2×D4—2181no—
196C2×D4—2182no—
197C2×D4—2181no—
198C2×D4D43611yescharacteristic
199C22×C4—2182no—
200C22×C4—2181no—
201C22×C4—2181no—
202C22×C4C22×D10912yes—
203D8—4362no—
204D8—2182no—
205C2×C8—42no—
206C2×C8C22×D10911yescharacteristic
207C2×C8C22×D10911yescharacteristic
208C2×C8C22×D10912yes—
209C4⋊C4—1091no—
210C4⋊C4—1092no—
211C4⋊C4—1091no—
212C4⋊C4—2182no—
213C4⋊C4—2181no—
214C4⋊C4D43611yescharacteristic
215C22⋊C4—2181no—
216C22⋊C4—2181no—
217C22⋊C4—2182no—
218C22⋊C4—2182no—
219C22⋊C4D43612yes—
220C42C22×D10911yescharacteristic
221C23—8722no—
222C23—4364no—
223C23—4364no—
224C23—2181no—
225C23—2182no—
226C23—2181no—
227C23C2×D43612yes—
228Q8—4362no—
229D4—8721no—
230D4—1092no—
231D4—4364no—
232D4—4364no—
233D4—4364no—
234D4—4362no—
235D4—2181no—
236D4—2182no—
237D4—2181no—
238D4—2182no—
239D4—2181no—
240D4D87214yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—4364no—
244C2×C4—4362no—
245C2×C4—4362no—
246C2×C4—2181no—
247C2×C4—2181no—
248C2×C4—2182no—
249C2×C4C2×D43611yescharacteristic
250C2×C4C2×D43612yes—
251C2×C4C2×D43611yescharacteristic
252C2×C4C23×D10911yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—8728no—
259C22—4361no—
260C22—4362no—
261C22—4362no—
262C22—4361no—
263C22—4362no—
264C22—4361no—
265C22—4361no—
266C22—4361no—
267C22—2182no—
268C22C2×D87214yes—
269C22C22×D43611yescharacteristic
270C4—21no—
271C4—22no—
272C4—4361no—
273C4—4361no—
274C4—4362no—
275C4C2×D87212yes—
276C4#41×D10911yescharacteristic
277C4#1681274411yescharacteristic
278C2—24no—
279C2—8722no—
280C2—8721no—
281C2—4362no—
282C2#2029567111yescharacteristic
283C2C22×D87211yescharacteristic
284C2#2029583811yescharacteristic
subgroupG/H
centerC22C22×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC8×D4×C109C2
socleC22×C109C22×D4
hypercenterC8×D4D109
hypocenterC109#3294
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20295680C2semidirect
2. D4×D436C2semidirect
3. #20295373C2semidirect
4. #20295340C2semidirect
5. #20295307C2semidirect
6. #20295262C2semidirect
7. #20295222C2semidirect
8. #20295162C2semidirect
9. C4×D872C2semidirect
10. C2×#16812621C2semidirect
11. C8×D4×C109C2semidirect
16. C2×D872C22semidirect
17. #16812699C22semidirect
23. #16812673C22semidirect
26. C4×D436C22semidirect
27. C2×C4⋊Dic109C22semidirect
28. #16812633C22semidirect
29. #16812633C22semidirect
32. #16812621C22semidirect
33. #16812621C22semidirect
34. #16812621C22semidirect
35. #16812621C22semidirect
36. C22×C8×C109C22semidirect
38. C4⋊C8×C109C22semidirect
39. C22⋊C8×C109C22semidirect
40. C4×C8×C109C22semidirect
85. D4×C109D8semidirect
106. D436#41semidirect
143. C109#3294semidirect
153. C8×D4D109semidirect
160. C22×C8C2×D109semidirect
175. C4⋊C8C2×D109semidirect
182. C22⋊C8C2×D109semidirect
183. C4×C8C2×D109semidirect
240. D4D872semidirect

Maximal subgroups

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

subgroupindex
#202956802
D4×D4362
#202953732
#202953402
#202953072
#202952622
#202952222
#202951622
C4×D8722
C2×#168126212
C8×D4×C1092
#3294109