Finite Group Databasebeta

← #28372724

Subgroups — #28372724

Order 13952.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812732C211yesmaximal, characteristic, complemented
2C2×#16812707C211yesmaximal, characteristic, complemented
3#20295607C211yesmaximal, characteristic
4#20295602C211yesmaximal, characteristic, complemented
5#20295285C212yesmaximal, complemented
6#20295273C212yesmaximal, complemented
7#20295268C212yesmaximal, complemented
8#20295256C212yesmaximal, complemented
9#20295020C211yesmaximal, characteristic, complemented
10C2×#16812622C211yesmaximal, characteristic, complemented
11#428×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C2×#13182029C2211yescharacteristic
14C22×Dic218C2211yescharacteristic
15#16812732—24no—
16C2×Dic436C2212yes—
17C2×#13182018C2212yes—
18#16812707—24no—
19C2×#13182012C2212yes—
20C22×#11724784C2211yescharacteristic
21C2×C8×D109C2212yes—
22#16812699—21no—
23#16812687—21no—
24#16812683—22no—
25#16812679—21no—
26#16812679C2212yes—
27#16812679C2212yes—
28#16812679C2212yes—
29#16812674—21no—
30#16812674C2212yes—
31#16812673—22no—
32#16812622C2212yes—
33#16812622C2211yescharacteristic
34#16812622C2212yes—
35#16812622C2211yescharacteristic
36#16812619C2211yescharacteristic
37#16812619C2211yescharacteristic
38C2×M16×C109C2211yescharacteristic
39#117×C109C2211yescharacteristic
40C2×C4⋊C4×C109C2211yescharacteristic
41#107×C109C2212yes—
42#106×C109C2212yes—
43#13182029—22no—
44#13182029—22no—
45#13182029—22no—
46#13182029—22no—
47C2×Dic218—22no—
48C2×Dic218—22no—
49C2×Dic218—21no—
50C2×Dic218D411yescharacteristic
51C2×Dic218C2312yes—
52C2×Dic218C2311yescharacteristic
53C2×D436—22no—
54C2×D436D411yescharacteristic
55C2×D436C2311yescharacteristic
56C2×C22⋊D109—21no—
57C2×C22⋊D109D412yes—
58C22×Dic109—21no—
59C2×C4×D109—21no—
60C2×C4×D109C2312yes—
61Dic436—24no—
62#13182018—24no—
63#13182012—24no—
64C2×#11724784—22no—
65C2×#11724784C2311yescharacteristic
66C2×#11724784C2311yescharacteristic
67C8×D109—24no—
68C4⋊Dic109—22no—
69C4⋊Dic109—22no—
70C4⋊Dic109—22no—
71#13182006—22no—
72#13182006—22no—
73#13182006—22no—
74#13182006—22no—
75C22×C4×C109—21no—
76C22×C4×C109C2311yescharacteristic
77M16×C109D414yescomplemented
78C2×C8×C109C2312yes—
79C4⋊C4×C109—21no—
80C4⋊C4×C109C2311yescharacteristic
81C4⋊C4×C109C2312yes—
82C4⋊C4×C109C2311yescharacteristic
83C22⋊C4×C109—21no—
84C42×C109—21no—
85C23×C109C2×D411yescharacteristic
86C22×D109—21no—
87C22×D109C2×D412yes—
88C2×C4×C109—21no—
89C2×C4×C109—21no—
90C2×C4×C109—22no—
91C2×C4×C109—81no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2×D412yes—
94C2×C4×C109#4112yes—
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—24no—
97C22⋊D109—42no—
98C2×Dic109—21no—
99C2×Dic109—22no—
100C2×Dic109—42no—
101C2×Dic109C2×D412yes—
102D436—22no—
103D436—21no—
104D436—21no—
105D436—82no—
106C4×D109—22no—
107C4×D109—22no—
108C4×D109—42no—
109Dic218—22no—
110Dic218—22no—
111Dic218—21no—
112Dic218—22no—
113Dic218—21no—
114Dic218—42no—
115C8×C109C2×D414yes—
116#11724784—21no—
117#11724784—21no—
118C22×C109—21no—
119C22×C109—21no—
120C22×C109#12112yes—
121C22×C109C22×D411yescharacteristic
122C2×D109—22no—
123C2×D109—22no—
124C2×D109—22no—
125C2×D109—41no—
126C2×D109—41no—
127C2×D109—41no—
128C4×C109—42no—
129C4×C109—41no—
130C4×C109—41no—
131C4×C109#12112yes—
132C4×C109C22×D411yescharacteristic
133C4×C109C2×#4111yescharacteristic
134Dic109—22no—
135Dic109—42no—
136Dic109—41no—
137C2×C109—22no—
138C2×C109C2×#12111yescharacteristic
139C2×C109#57711yescharacteristic
140C2×C109#57811yescharacteristic
141D109—42no—
142D109—81no—
143#3063—1091nomaximal, Sylow
144C109#306311yescharacteristic, complemented, Sylow
145C2×#137—1091no—
146C2×#131—1091no—
147#537—1091no—
148#536—1091no—
149#470—1092no—
150#468—1092no—
151#467—1092no—
152#465—1092no—
153#428D10911yescharacteristic, complemented
154#416—1091no—
155C2×#103—1091no—
156C2×#41—1091no—
157C2×#41—1091no—
158C22×Q8—1091no—
159#137—2184no—
160C2×Q16—1092no—
161C2×SD16—1092no—
162#131—2184no—
163C2×M16—1092no—
164C2×M16C2×D10911yescharacteristic
165C22×C8—1091no—
166C22×C8—1092no—
167C4⋊Q8—2181no—
168#124—2181no—
169#123—2182no—
170C22⋊Q8—1092no—
171C22⋊Q8—1092no—
172C22⋊Q8—1092no—
173C22⋊Q8—2181no—
174#121—1092no—
175#121—2181no—
176#121—2182no—
177#117C2×D10911yescharacteristic
178C2×C4⋊C4C2×D10911yescharacteristic
179#107C2×D10912yes—
180#106C2×D10912yes—
181#103—1092no—
182#103—1092no—
183#103—1091no—
184#103—1091no—
185#102—1091no—
186#102—1091no—
187#41—2182no—
188#41—2182no—
189#41—2182no—
190#41—2182no—
191C2×Q8—1091no—
192C2×Q8—1092no—
193C2×Q8—1091no—
194C2×Q8—2182no—
195C2×Q8—2182no—
196C2×Q8—2181no—
197C2×D4—1092no—
198C2×D4—1091no—
199C2×D4—1091no—
200C2×D4—2182no—
201C2×D4—2181no—
202C22×C4—21no—
203C22×C4—1092no—
204C22×C4—2181no—
205C22×C4—2181no—
206C22×C4C22×D10911yescharacteristic
207Q16—2184no—
208SD16—2184no—
209M16—2184no—
210M16D43614yescomplemented
211C2×C8—1091no—
212C2×C8—1091no—
213C2×C8—2182no—
214C2×C8—2184no—
215C2×C8C22×D10912yes—
216C4⋊C4—21no—
217C4⋊C4—2182no—
218C4⋊C4—2182no—
219C4⋊C4—2182no—
220C4⋊C4C22×D10911yescharacteristic
221C4⋊C4C22×D10911yescharacteristic
222C4⋊C4C22×D10912yes—
223C22⋊C4—21no—
224C22⋊C4—2182no—
225C22⋊C4—2182no—
226C22⋊C4—2182no—
227C22⋊C4—2182no—
228C42—21no—
229C23—1092no—
230C23—2181no—
231C23D4×D10911yescharacteristic
232Q8—4362no—
233Q8—2182no—
234Q8—2181no—
235Q8—2182no—
236Q8—2181no—
237Q8—2182no—
238D4—8722no—
239D4—4362no—
240D4—2184no—
241D4—2182no—
242D4—2181no—
243D4—2181no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—81no—
248C2×C4—1092no—
249C2×C4—4362no—
250C2×C4—4362no—
251C2×C4—2181no—
252C2×C4—2182no—
253C2×C4—2182no—
254C2×C4—2182no—
255C2×C4D4×D10911yescharacteristic
256C2×C4C2×D43612yes—
257C2×C4#1318203012yes—
258C2×C4C23×D10911yescharacteristic
259C8—2181no—
260C8—2181no—
261C8C2×D43614yes—
262C22—21no—
263C22—21no—
264C22—4361no—
265C22—4361no—
266C22—4361no—
267C22—2182no—
268C22—2182no—
269C22—2182no—
270C22#1681267412yes—
271C22C2×D4×D10911yescharacteristic
272C4—42no—
273C4—41no—
274C4—41no—
275C4—4361no—
276C4—4362no—
277C4—2182no—
278C4#1681267412yes—
279C4C22×D43611yescharacteristic
280C4C2×#1318203011yescharacteristic
281C2—22no—
282C2—8721no—
283C2—4362no—
284C2C2×#1681267411yescharacteristic
285C2#2029583611yescharacteristic
286C2#2029584211yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#428×C109C2
socleC22×C109C22×D4
prime radical#20295607C2
hypercenter#428D109
hypocenterC109#3063
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C2×#16812707C2semidirect
4. #20295602C2semidirect
5. #20295285C2semidirect
6. #20295273C2semidirect
7. #20295268C2semidirect
8. #20295256C2semidirect
9. #20295020C2semidirect
10. C2×#16812622C2semidirect
11. #428×C109C2semidirect
77. M16×C109D4semidirect
144. C109#3063semidirect
153. #428D109semidirect
210. M16D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×#168127072
#202956072
#202956022
#202952852
#202952732
#202952682
#202952562
#202950202
C2×#168126222
#428×C1092
#3063109