Finite Group Databasebeta

← #28374839

Subgroups — #28374839

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
1D4×D436C211yesmaximal, characteristic, complemented
2#20295374C211yesmaximal, characteristic, complemented
3#20295339C212yesmaximal, complemented
4#20295303C211yesmaximal, characteristic, complemented
5#20295260C212yesmaximal, complemented
6#20295221C211yesmaximal, characteristic, complemented
7#20295159C212yesmaximal, complemented
8C4×#13182014C211yesmaximal, characteristic, complemented
9D4×#11724784C211yesmaximal, characteristic, complemented
10C2×#16812619C212yesmaximal, complemented
11#547×C109C211yesmaximal, characteristic, complemented
12C22×D436C2212yes—
13C2×D4×D109—21no—
14C2×#13182014C2211yescharacteristic
15C2×#13182014C2212yescomplemented
16C22×#11724784C2212yescomplemented
17#16812696C2211yescharacteristic
18#16812674—21no—
19#16812673C2212yes—
20#16812669—22no—
21C4×D436C2211yescharacteristic
22#16812632C2212yescomplemented
23#16812632C2211yescharacteristic, complemented
24C4⋊#11724784C2211yescharacteristic, complemented
25#16812620C2211yescharacteristic, complemented
26#16812619C2211yescharacteristic
27#16812619C2212yes—
28#16812619C2212yescomplemented
29#16812619C2212yescomplemented
30#16812619C2212yescomplemented
31C22⋊#11724784C2212yescomplemented
32C4×#11724784C2211yescharacteristic, complemented
33C2×#41×C109—21no—
34C4⋊Q8×C109C2211yescharacteristic, complemented
35#123×C109—22no—
36C22⋊Q8×C109—21no—
37#121×C109C2212yescomplemented
38C4×D4×C109C2211yescharacteristic
39C4×D4×C109C2211yescharacteristic, complemented
40C2×C4⋊C4×C109C2212yescomplemented
41C23×D109—22no—
42C2×D436—24no—
43C2×D436—22no—
44C2×D436—21no—
45C2×D436—21no—
46C2×D436—42no—
47C2×D436C2312yes—
48C2×D436C2311yescharacteristic
49C2×C22⋊D109—22no—
50D4×D109—44no—
51C2×C4×D109—21no—
52#13182014—22no—
53#13182014—42no—
54C2×#11724784—42no—
55C2×#11724784C2311yescharacteristic
56C2×#11724784C2311yescharacteristic
57C2×#11724784C2312yes—
58C4⋊Dic109C2311yescharacteristic
59#13182006—21no—
60#13182006—22no—
61#41×C109—44no—
62C2×Q8×C109—21no—
63C2×D4×C109—22no—
64C2×D4×C109D411yescharacteristic
65C2×D4×C109C2311yescharacteristic
66C22×C4×C109—22no—
67C22×C4×C109—21no—
68C22×C4×C109C2312yes—
69C4⋊C4×C109—22no—
70C4⋊C4×C109—21no—
71C4⋊C4×C109D411yescharacteristic
72C4⋊C4×C109C2311yescharacteristic
73C4⋊C4×C109C2312yes—
74C22⋊C4×C109—21no—
75C22⋊C4×C109—22no—
76C22⋊C4×C109D412yes—
77C42×C109C2311yescharacteristic
78C23×C109—21no—
79C23×C109C2×D412yes—
80C22×D109—21no—
81C22×D109—22no—
82C22×D109—44no—
83C22×D109—44no—
84C22×D109—82no—
85Q8×C109—42no—
86D4×C109—21no—
87D4×C109—21no—
88D4×C109—44no—
89D4×C109D814yescomplemented
90C2×C4×C109—22no—
91C2×C4×C109—22no—
92C2×C4×C109—21no—
93C2×C4×C109—22no—
94C2×C4×C109—44no—
95C2×C4×C109—42no—
96C2×C4×C109C2×D411yescharacteristic
97C2×C4×C109C2×D412yes—
98C2×C4×C109C2×D411yescharacteristic
99C2×C4×C109C2411yescharacteristic
100C22⋊D109—44no—
101C2×Dic109—21no—
102D436—22no—
103D436—21no—
104D436—22no—
105D436—44no—
106D436—42no—
107D436—81no—
108D436#4112yescomplemented
109C4×D109—42no—
110#11724784—21no—
111#11724784—21no—
112#11724784—22no—
113C22×C109—22no—
114C22×C109—22no—
115C22×C109—41no—
116C22×C109—41no—
117C22×C109—41no—
118C22×C109C2×D814yes—
119C22×C109C22×D411yescharacteristic
120C2×D109—22no—
121C2×D109—41no—
122C2×D109—41no—
123C2×D109—42no—
124C2×D109—42no—
125C2×D109—42no—
126C2×D109—88no—
127C4×C109—22no—
128C4×C109—21no—
129C4×C109—41no—
130C4×C109—42no—
131C4×C109C2×D812yes—
132C4×C109C2×#4111yescharacteristic
133C4×C109H(5,𝔽2)11yescharacteristic
134Dic109—41no—
135C2×C109—24no—
136C2×C109—81no—
137C2×C109#54611yescharacteristic
138C2×C109C22×D811yescharacteristic
139C2×C109#57711yescharacteristic
140D109—42no—
141D109—82no—
142#3294—1091nomaximal, Sylow
143C109#329411yescharacteristic, complemented, Sylow
144#547D10911yescharacteristic, complemented
145D42—1091no—
146#486—1091no—
147#480—1092no—
148#474—1091no—
149#466—1092no—
150#459—1091no—
151#447—1092no—
152C4×D8—1091no—
153C8×D4—1091no—
154C2×#102—1092no—
155C2×#41—21no—
156C22×D4—1092no—
157C22×D4—2181no—
158C2×D8—1092no—
159C2×D8—1091no—
160C22×C8—1092no—
161C4⋊Q8C2×D10911yescharacteristic, complemented
162#127—1091no—
163#123—22no—
164C22⋊Q8—21no—
165#121—1092no—
166#121—2181no—
167#121C2×D10912yescomplemented
168#120—2182no—
169C4×D4—1091no—
170C4×D4C2×D10911yescharacteristic
171C4×D4C2×D10911yescharacteristic, complemented
172C2×C4⋊C4C2×D10912yescomplemented
173#107—1091no—
174#107—1092no—
175C4⋊C8—1091no—
176#102—1091no—
177#102—1091no—
178#102—1092no—
179#102—1092no—
180#102—1092no—
181#102—1092no—
182C22⋊C8—1092no—
183C4×C8—1091no—
184C24—2182no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—1091no—
189C2×D4—1092no—
190C2×D4—4362no—
191C2×D4—4364no—
192C2×D4—2184no—
193C2×D4—2182no—
194C2×D4—2181no—
195C2×D4—2182no—
196C2×D4—2181no—
197C2×D4C22⋊D10911yescharacteristic
198C2×D4C22×D10911yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2181no—
202C22×C4C22×D10912yes—
203D8—4362no—
204D8—2182no—
205C2×C8—1091no—
206C2×C8—1091no—
207C2×C8—1092no—
208C2×C8—4362no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1091no—
212C4⋊C4C22⋊D10911yescharacteristic
213C4⋊C4C22×D10911yescharacteristic
214C4⋊C4C22×D10912yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2181no—
218C22⋊C4—2182no—
219C22⋊C4C22⋊D10912yes—
220C42C22×D10911yescharacteristic
221C23—21no—
222C23—8722no—
223C23—4364no—
224C23—4364no—
225C23—2182no—
226C23—2181no—
227C23C2×C22⋊D10912yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—8721no—
233D4—1092no—
234D4—4364no—
235D4—4364no—
236D4—4362no—
237D4—2182no—
238D4—2182no—
239D4—2181no—
240D4#1318201414yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—4362no—
248C2×C4—2181no—
249C2×C4C2×C22⋊D10911yescharacteristic
250C2×C4C2×C22⋊D10912yes—
251C2×C4C2×C22⋊D10911yescharacteristic
252C2×C4C23×D10911yescharacteristic
253C8—2181no—
254C8—2181no—
255C8—2182no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—8728no—
262C22—4362no—
263C22—4362no—
264C22—4362no—
265C22—4361no—
266C22—4361no—
267C22—2182no—
268C22C2×#1318201414yes—
269C22C22×C22⋊D10911yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—4361no—
275C4C2×#1318201412yes—
276C4#41×D10911yescharacteristic
277C4#1681274511yescharacteristic
278C2—24no—
279C2—81no—
280C2—8722no—
281C2—4362no—
282C2#2029566411yescharacteristic
283C2C22×#1318201411yescharacteristic
284C2#2029583411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#547×C109C2
socleC22×C109C22×D4
hypercenter#547D109
hypocenterC109#3294
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D436C2semidirect
2. #20295374C2semidirect
3. #20295339C2semidirect
4. #20295303C2semidirect
5. #20295260C2semidirect
6. #20295221C2semidirect
7. #20295159C2semidirect
8. C4×#13182014C2semidirect
9. D4×#11724784C2semidirect
10. C2×#16812619C2semidirect
11. #547×C109C2semidirect
15. C2×#13182014C22semidirect
16. C22×#11724784C22semidirect
22. #16812632C22semidirect
23. #16812632C22semidirect
24. C4⋊#11724784C22semidirect
25. #16812620C22semidirect
28. #16812619C22semidirect
29. #16812619C22semidirect
30. #16812619C22semidirect
31. C22⋊#11724784C22semidirect
32. C4×#11724784C22semidirect
34. C4⋊Q8×C109C22semidirect
37. #121×C109C22semidirect
39. C4×D4×C109C22semidirect
40. C2×C4⋊C4×C109C22semidirect
89. D4×C109D8semidirect
108. D436#41semidirect
143. C109#3294semidirect
144. #547D109semidirect
161. C4⋊Q8C2×D109semidirect
167. #121C2×D109semidirect
171. C4×D4C2×D109semidirect
172. C2×C4⋊C4C2×D109semidirect
240. D4#13182014semidirect

Maximal subgroups

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

subgroupindex
D4×D4362
#202953742
#202953392
#202953032
#202952602
#202952212
#202951592
C4×#131820142
D4×#117247842
C2×#168126192
#547×C1092
#3294109