Finite Group Databasebeta

← #28374847

Subgroups — #28374847

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#20295681C211yesmaximal, characteristic, complemented
2D4×D436C211yesmaximal, characteristic, complemented
3#20295373C211yesmaximal, characteristic, complemented
4#20295341C212yesmaximal, complemented
5#20295306C211yesmaximal, characteristic, complemented
6#20295261C212yesmaximal, complemented
7#20295221C211yesmaximal, characteristic, complemented
8#20295158C212yesmaximal, complemented
9C8×D436C211yesmaximal, characteristic, complemented
10#102×D109C212yesmaximal, complemented
11C4×D8×C109C211yesmaximal, characteristic, complemented
12C2×#13182029—21no—
13C22×D436—21no—
14C2×D4×D109C2212yes—
15C2×#13182014C2212yescomplemented
16C2×C8×D109C2212yescomplemented
17#16812699C2211yescharacteristic, complemented
18#16812696C2211yescharacteristic
19#16812681—22no—
20#16812678—21no—
21#16812674C2212yes—
22#16812673—21no—
23#16812670C2212yescomplemented
24#16812669—22no—
25C4×D436C2211yescharacteristic
26C4⋊C4×D109C2212yescomplemented
27#16812632C2212yescomplemented
28C4⋊#11724784C2211yescharacteristic, complemented
29#16812621C2212yescomplemented
30#16812621C2212yescomplemented
31#16812620C2212yes—
32#16812619C2212yescomplemented
33#16812611C2212yescomplemented
34C2×D8×C109C2211yescharacteristic
35C4×D4×C109C2211yescharacteristic
36C4×D4×C109C2211yescharacteristic, complemented
37#107×C109C2211yescharacteristic, complemented
38#102×C109C2211yescharacteristic
39#102×C109C2211yescharacteristic, complemented
40C4×C8×C109C2211yescharacteristic, complemented
41C23×D109—22no—
42#13182029—44no—
43C2×Dic218—21no—
44C2×D436—22no—
45C2×D436—21no—
46C2×D436—21no—
47C2×D436—44no—
48C2×D436D411yescharacteristic
49C2×D436C2312yes—
50C2×C22⋊D109—22no—
51C2×C22⋊D109—22no—
52D4×D109—24no—
53D4×D109—42no—
54C2×C4×D109—22no—
55C2×C4×D109C2312yes—
56#13182014—42no—
57C2×#11724784C2312yes—
58C8×D109—42no—
59C4⋊Dic109—21no—
60C4⋊Dic109D411yescharacteristic
61C4⋊Dic109C2312yes—
62#13182008—22no—
63C22⋊Dic109—22no—
64#13182006—22no—
65#13182006D412yes—
66C2×D4×C109C2311yescharacteristic
67C2×D4×C109C2311yescharacteristic
68C22×C4×C109—21no—
69C22×C4×C109—21no—
70D8×C109—22no—
71C2×C8×C109C2311yescharacteristic
72C2×C8×C109C2311yescharacteristic
73C4⋊C4×C109C2311yescharacteristic
74C4⋊C4×C109C2311yescharacteristic
75C22⋊C4×C109—21no—
76C22⋊C4×C109—21no—
77C42×C109C2311yescharacteristic
78C23×C109—21no—
79C23×C109—21no—
80C22×D109—22no—
81C22×D109—44no—
82C22×D109—44no—
83C22×D109—82no—
84C22×D109C2×D412yes—
85D4×C109—21no—
86D4×C109—21no—
87D4×C109—21no—
88D4×C109#4112yescomplemented
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—42no—
92C2×C4×C109—42no—
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2×D411yescharacteristic
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—44no—
97C22⋊D109—44no—
98C2×Dic109—22no—
99C2×Dic109C2×D412yes—
100D436—22no—
101D436—22no—
102D436—44no—
103D436—42no—
104D436—81no—
105D436D814yescomplemented
106C4×D109—22no—
107C4×D109—22no—
108C4×D109—44no—
109Dic218—42no—
110C8×C109—21no—
111C8×C109—21no—
112#11724784—22no—
113C22×C109—22no—
114C22×C109—41no—
115C22×C109—41no—
116C22×C109—41no—
117C22×C109—41no—
118C22×C109—41no—
119C22×C109C22×D411yescharacteristic
120C2×D109—22no—
121C2×D109—22no—
122C2×D109—42no—
123C2×D109—42no—
124C2×D109—42no—
125C2×D109—88no—
126C2×D109C2×D814yes—
127C4×C109—21no—
128C4×C109—41no—
129C4×C109—41no—
130C4×C109C2×D812yes—
131C4×C109C2×#4111yescharacteristic
132C4×C109H(5,𝔽2)11yescharacteristic
133Dic109—22no—
134Dic109—42no—
135C2×C109—42no—
136C2×C109—81no—
137C2×C109#54611yescharacteristic
138C2×C109C22×D811yescharacteristic
139C2×C109#57711yescharacteristic
140D109—24no—
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×D8D10911yescharacteristic, complemented
153C8×D4—1091no—
154C2×#102—1092no—
155C2×#41—2181no—
156C22×D4—1092no—
157C22×D4—2181no—
158C2×D8—1092no—
159C2×D8C2×D10911yescharacteristic
160C22×C8—1092no—
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×D4C2×D10911yescharacteristic
171C4×D4C2×D10911yescharacteristic, complemented
172C2×C4⋊C4—1092no—
173#107—1092no—
174#107C2×D10911yescharacteristic, complemented
175C4⋊C8—1091no—
176#102—1092no—
177#102—1092no—
178#102—1092no—
179#102—1092no—
180#102C2×D10911yescharacteristic
181#102C2×D10911yescharacteristic, complemented
182C22⋊C8—1092no—
183C4×C8C2×D10911yescharacteristic, complemented
184C24—2182no—
185#41—4364no—
186C2×Q8—2181no—
187C2×D4—1091no—
188C2×D4—1092no—
189C2×D4—4364no—
190C2×D4—4362no—
191C2×D4—2182no—
192C2×D4—2182no—
193C2×D4—2182no—
194C2×D4—2184no—
195C2×D4—2181no—
196C2×D4—2181no—
197C2×D4C22×D10911yescharacteristic
198C2×D4C22×D10911yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1092no—
202C22×C4—2182no—
203D8—22no—
204D8—4362no—
205C2×C8—1092no—
206C2×C8—4362no—
207C2×C8C22×D10911yescharacteristic
208C2×C8C22×D10911yescharacteristic
209C4⋊C4—1091no—
210C4⋊C4—1092no—
211C4⋊C4—2182no—
212C4⋊C4—2181no—
213C4⋊C4C22×D10911yescharacteristic
214C4⋊C4C22×D10911yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1092no—
218C22⋊C4—2182no—
219C22⋊C4—2182no—
220C42C22×D10911yescharacteristic
221C23—21no—
222C23—21no—
223C23—8722no—
224C23—1092no—
225C23—4364no—
226C23—4364no—
227C23—2182no—
228Q8—4362no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—8721no—
233D4—1094no—
234D4—4364no—
235D4—4364no—
236D4—4364no—
237D4—4362no—
238D4—2182no—
239D4—2182no—
240D4#1318202912yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—1092no—
246C2×C4—4364no—
247C2×C4—2182no—
248C2×C4—2182no—
249C2×C4—2182no—
250C2×C4D4×D10911yescharacteristic
251C2×C4D4×D10911yescharacteristic
252C2×C4C23×D10911yescharacteristic
253C8—21no—
254C8—21no—
255C8—2182no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—8728no—
263C22—1094no—
264C22—4362no—
265C22—4362no—
266C22—4362no—
267C22—2182no—
268C22—2182no—
269C22C2×D4×D10911yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—4362no—
274C4—2182no—
275C4D8×D10912yes—
276C4C2×#1318202911yescharacteristic
277C4#1681274511yescharacteristic
278C2—42no—
279C2—81no—
280C2—8722no—
281C2—2184no—
282C2#2029567211yescharacteristic
283C2C2×D8×D10911yescharacteristic
284C2#2029583311yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC4×D8×C109C2
socleC22×C109C22×D4
hypercenterC4×D8D109
hypocenterC109#3294
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20295681C2semidirect
2. D4×D436C2semidirect
3. #20295373C2semidirect
4. #20295341C2semidirect
5. #20295306C2semidirect
6. #20295261C2semidirect
7. #20295221C2semidirect
8. #20295158C2semidirect
9. C8×D436C2semidirect
10. #102×D109C2semidirect
11. C4×D8×C109C2semidirect
15. C2×#13182014C22semidirect
16. C2×C8×D109C22semidirect
17. #16812699C22semidirect
23. #16812670C22semidirect
26. C4⋊C4×D109C22semidirect
27. #16812632C22semidirect
28. C4⋊#11724784C22semidirect
29. #16812621C22semidirect
30. #16812621C22semidirect
32. #16812619C22semidirect
33. #16812611C22semidirect
36. C4×D4×C109C22semidirect
37. #107×C109C22semidirect
39. #102×C109C22semidirect
40. C4×C8×C109C22semidirect
88. D4×C109#41semidirect
105. D436D8semidirect
143. C109#3294semidirect
152. C4×D8D109semidirect
171. C4×D4C2×D109semidirect
174. #107C2×D109semidirect
181. #102C2×D109semidirect
183. C4×C8C2×D109semidirect
240. D4#13182029semidirect

Maximal subgroups

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

subgroupindex
#202956812
D4×D4362
#202953732
#202953412
#202953062
#202952612
#202952212
#202951582
C8×D4362
#102×D1092
C4×D8×C1092
#3294109