Finite Group Databasebeta

← #28374842

Subgroups — #28374842

Order 13952.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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#20295679C211yesmaximal, characteristic, complemented
2D4×D436C211yesmaximal, characteristic, complemented
3#20295374C211yesmaximal, characteristic, complemented
4#20295341C212yesmaximal, complemented
5#20295263C212yesmaximal, complemented
6#20295222C211yesmaximal, characteristic, complemented
7#20295161C212yesmaximal, complemented
8C4×#13182014C211yesmaximal, characteristic, complemented
9#20295091C211yesmaximal, characteristic, complemented
10#102×D109C212yesmaximal, complemented
11#474×C109C211yesmaximal, characteristic, complemented
12C2×#13182030—21no—
13C22×D436—21no—
14C2×D4×D109C2212yes—
15C2×D872C2212yescomplemented
16C2×#13182014C2211yescharacteristic
17C2×C8×D109C2212yescomplemented
18#16812696C2211yescharacteristic
19#16812684—22no—
20#16812675—21no—
21#16812674C2212yes—
22#16812674C2212yescomplemented
23#16812673—21no—
24#16812669—22no—
25C4×D436C2211yescharacteristic
26C4×D436C2211yescharacteristic, complemented
27C4⋊C4×D109C2212yescomplemented
28#16812632C2211yescharacteristic, complemented
29#16812621C2212yescomplemented
30#16812620C2211yescharacteristic
31#16812619C2211yescharacteristic, complemented
32#16812619C2212yescomplemented
33#16812619C2212yescomplemented
34#16812611C2212yescomplemented
35C4×#11724784C2211yescharacteristic, complemented
36C4⋊Q8×C109C2211yescharacteristic, complemented
37C4×D4×C109C2211yescharacteristic
38#107×C109C2212yescomplemented
39C4⋊C8×C109C2211yescharacteristic, complemented
40#102×C109C2212yes—
41C23×D109—22no—
42#13182030—44no—
43C2×D436—22no—
44C2×D436—22no—
45C2×D436—21no—
46C2×D436—21no—
47C2×D436—44no—
48C2×D436D411yescharacteristic
49C2×D436C2312yes—
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—22no—
52D4×D109—24no—
53D4×D109—42no—
54C2×C4×D109—22no—
55C2×C4×D109—21no—
56C2×C4×D109C2312yes—
57D872—42no—
58#13182014—22no—
59C2×#11724784C2311yescharacteristic
60C2×#11724784C2311yescharacteristic
61C8×D109—42no—
62C4⋊Dic109D411yescharacteristic
63C4⋊Dic109C2311yescharacteristic
64#13182008—22no—
65#13182006—21no—
66#13182006—22no—
67#13182006—22no—
68#13182006D412yes—
69C2×Q8×C109—21no—
70C2×D4×C109C2311yescharacteristic
71C22×C4×C109—21no—
72C2×C8×C109C2312yes—
73C4⋊C4×C109—21no—
74C4⋊C4×C109C2312yes—
75C4⋊C4×C109C2311yescharacteristic
76C22⋊C4×C109—21no—
77C42×C109C2311yescharacteristic
78C23×C109—21no—
79C22×D109—21no—
80C22×D109—22no—
81C22×D109—44no—
82C22×D109—44no—
83C22×D109—82no—
84C22×D109C2×D412yes—
85Q8×C109—42no—
86D4×C109—21no—
87D4×C109#4112yescomplemented
88C2×C4×C109—21no—
89C2×C4×C109—22no—
90C2×C4×C109—42no—
91C2×C4×C109C2×D411yescharacteristic
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—44no—
95C2×Dic109—21no—
96C2×Dic109C2×D412yes—
97D436—21no—
98D436—22no—
99D436—21no—
100D436—22no—
101D436—44no—
102D436—44no—
103D436—42no—
104D436—81no—
105D436D814yescomplemented
106C4×D109—22no—
107C4×D109—22no—
108C4×D109—44no—
109C4×D109—42no—
110C8×C109—22no—
111#11724784—21no—
112#11724784—21no—
113C22×C109—22no—
114C22×C109—41no—
115C22×C109—41no—
116C22×C109C22×D411yescharacteristic
117C2×D109—22no—
118C2×D109—22no—
119C2×D109—42no—
120C2×D109—41no—
121C2×D109—41no—
122C2×D109—42no—
123C2×D109—42no—
124C2×D109—41no—
125C2×D109—88no—
126C2×D109C2×D814yes—
127C4×C109—21no—
128C4×C109—42no—
129C4×C109—41no—
130C4×C109C2×D812yes—
131C4×C109C2×#4111yescharacteristic
132C4×C109H(5,𝔽2)11yescharacteristic
133Dic109—22no—
134Dic109—41no—
135C2×C109—42no—
136C2×C109#54611yescharacteristic
137C2×C109C22×D811yescharacteristic
138C2×C109#57711yescharacteristic
139D109—24no—
140D109—81no—
141D109—82no—
142#3294—1091nomaximal, Sylow
143C109#329411yescharacteristic, complemented, Sylow
144#547—1091no—
145D42—1091no—
146#486—1091no—
147#480—1092no—
148#474D10911yescharacteristic, complemented
149#466—1092no—
150#459—1091no—
151#447—1092no—
152C4×D8—1091no—
153C8×D4—1091no—
154C2×#102—1092no—
155C2×#41—2181no—
156C22×D4—1092no—
157C22×D4—2181no—
158C2×D8—1091no—
159C2×D8—1092no—
160C22×C8—1092no—
161C4⋊Q8C2×D10911yescharacteristic, complemented
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#107C2×D10912yescomplemented
175C4⋊C8C2×D10911yescharacteristic, complemented
176#102—1091no—
177#102—1092no—
178#102—1092no—
179#102—1091no—
180#102—1092no—
181#102C2×D10912yes—
182C22⋊C8—1092no—
183C4×C8—1091no—
184C24—2182no—
185#41—4364no—
186C2×Q8—21no—
187C2×D4—1091no—
188C2×D4—1092no—
189C2×D4—1091no—
190C2×D4—4364no—
191C2×D4—4362no—
192C2×D4—2182no—
193C2×D4—2182no—
194C2×D4—2182no—
195C2×D4—2184no—
196C2×D4—2181no—
197C2×D4—2181no—
198C2×D4C22×D10911yescharacteristic
199C22×C4—21no—
200C22×C4—1092no—
201C22×C4—2182no—
202C22×C4—2181no—
203D8—4362no—
204D8—2182no—
205C2×C8—1091no—
206C2×C8—1091no—
207C2×C8—4362no—
208C2×C8C22×D10912yes—
209C4⋊C4—21no—
210C4⋊C4—1091no—
211C4⋊C4—1091no—
212C4⋊C4—2182no—
213C4⋊C4C22×D10912yes—
214C4⋊C4C22×D10911yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1092no—
217C22⋊C4—2182no—
218C22⋊C4—2182no—
219C22⋊C4—2181no—
220C42C22×D10911yescharacteristic
221C23—21no—
222C23—8722no—
223C23—1092no—
224C23—4364no—
225C23—4364no—
226C23—2181no—
227C23—2182no—
228Q8—42no—
229D4—21no—
230D4—8721no—
231D4—1094no—
232D4—4364no—
233D4—4364no—
234D4—4364no—
235D4—4362no—
236D4—2181no—
237D4—2182no—
238D4—2181no—
239D4—2182no—
240D4#1318203012yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1092no—
245C2×C4—4364no—
246C2×C4—4362no—
247C2×C4—2181no—
248C2×C4—2182no—
249C2×C4—2182no—
250C2×C4D4×D10911yescharacteristic
251C2×C4D4×D10911yescharacteristic
252C2×C4C23×D10911yescharacteristic
253C8—22no—
254C8—2181no—
255C8—2181no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—8728no—
260C22—1094no—
261C22—4361no—
262C22—4362no—
263C22—4362no—
264C22—4361no—
265C22—4362no—
266C22—4361no—
267C22—2182no—
268C22—2182no—
269C22C2×D4×D10911yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—4361no—
274C4—2182no—
275C4D8×D10912yes—
276C4C2×#1318203011yescharacteristic
277C4#1681274511yescharacteristic
278C2—42no—
279C2—8722no—
280C2—8721no—
281C2—2184no—
282C2#2029566711yescharacteristic
283C2C2×D8×D10911yescharacteristic
284C2#2029583511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#474×C109C2
socleC22×C109C22×D4
hypercenter#474D109
hypocenterC109#3294
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20295679C2semidirect
2. D4×D436C2semidirect
3. #20295374C2semidirect
4. #20295341C2semidirect
5. #20295263C2semidirect
6. #20295222C2semidirect
7. #20295161C2semidirect
8. C4×#13182014C2semidirect
9. #20295091C2semidirect
10. #102×D109C2semidirect
11. #474×C109C2semidirect
15. C2×D872C22semidirect
17. C2×C8×D109C22semidirect
22. #16812674C22semidirect
26. C4×D436C22semidirect
27. C4⋊C4×D109C22semidirect
28. #16812632C22semidirect
29. #16812621C22semidirect
31. #16812619C22semidirect
32. #16812619C22semidirect
33. #16812619C22semidirect
34. #16812611C22semidirect
35. C4×#11724784C22semidirect
36. C4⋊Q8×C109C22semidirect
38. #107×C109C22semidirect
39. C4⋊C8×C109C22semidirect
87. D4×C109#41semidirect
105. D436D8semidirect
143. C109#3294semidirect
148. #474D109semidirect
161. C4⋊Q8C2×D109semidirect
174. #107C2×D109semidirect
175. C4⋊C8C2×D109semidirect
240. D4#13182030semidirect

Maximal subgroups

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

subgroupindex
#202956792
D4×D4362
#202953742
#202953412
#202952632
#202952222
#202951612
C4×#131820142
#202950912
#102×D1092
#474×C1092
#3294109