Finite Group Databasebeta

← #28372719

Subgroups — #28372719

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×#16812730C211yesmaximal, characteristic, complemented
2C2×#16812706C211yesmaximal, characteristic, complemented
3#20295604C211yesmaximal, characteristic
4#20295282C212yesmaximal, complemented
5#20295274C212yesmaximal, complemented
6#20295265C212yesmaximal, complemented
7#20295257C212yesmaximal, complemented
8#20295069C211yesmaximal, characteristic, complemented
9#20295020C211yesmaximal, characteristic, complemented
10C2×#16812623C211yesmaximal, characteristic, complemented
11#536×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13#16812730—24no—
14C2×#13182019C2212yes—
15C2×#13182016C2212yes—
16#16812706—24no—
17C2×#13182013C2212yes—
18C2×#13182013C2211yescharacteristic
19C22×#11724784C2212yes—
20C22×#11724784C2211yescharacteristic
21#16812678C2212yes—
22#16812675—21no—
23#16812673C2212yes—
24#16812671—22no—
25#16812670—21no—
26C2×C4⋊Dic109C2211yescharacteristic
27#16812632C2212yes—
28#16812630C2212yes—
29#16812623C2212yes—
30#16812623C2212yes—
31#16812622C2211yescharacteristic
32#16812622C2211yescharacteristic
33#16812619C2211yescharacteristic
34#16812619C2211yescharacteristic
35C2×#41×C109C2211yescharacteristic
36C22×Q8×C109C2211yescharacteristic
37C4⋊Q8×C109—21no—
38#124×C109—21no—
39#123×C109—22no—
40C22⋊Q8×C109C2212yes—
41C22⋊Q8×C109C2212yes—
42#117×C109C2211yescharacteristic
43#13182029—22no—
44#13182029—22no—
45C2×Dic218C2311yescharacteristic
46C2×D436C2311yescharacteristic
47C2×C22⋊D109—21no—
48C2×C22⋊D109—22no—
49C22×Dic109—21no—
50C2×C4×D109—21no—
51#13182019—24no—
52#13182016—24no—
53#13182013—24no—
54#13182013D414yescomplemented
55C2×#11724784—22no—
56C2×#11724784—24no—
57C2×#11724784C2311yescharacteristic
58C2×#11724784C2311yescharacteristic
59C2×#11724784C2312yes—
60C4⋊Dic109—21no—
61C4⋊Dic109C2312yes—
62#13182008—22no—
63C22⋊Dic109—22no—
64#13182006—22no—
65#41×C109—22no—
66#41×C109—22no—
67C2×Q8×C109—21no—
68C2×Q8×C109—22no—
69C2×Q8×C109—22no—
70C2×Q8×C109D411yescharacteristic
71C2×Q8×C109C2312yes—
72C2×D4×C109D412yes—
73C2×D4×C109D411yescharacteristic
74C22×C4×C109—21no—
75C22×C4×C109C2312yes—
76C22×C4×C109C2311yescharacteristic
77C4⋊C4×C109—22no—
78C4⋊C4×C109—22no—
79C4⋊C4×C109C2311yescharacteristic
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C22⋊C4×C109—22no—
83C22⋊C4×C109—22no—
84C42×C109—21no—
85C23×C109C2×D412yes—
86C23×C109C2×D411yescharacteristic
87C22×D109—21no—
88Q8×C109—22no—
89Q8×C109—22no—
90Q8×C109—22no—
91Q8×C109—42no—
92D4×C109—24no—
93D4×C109—22no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—21no—
97C2×C4×C109—21no—
98C2×C4×C109—22no—
99C2×C4×C109—42no—
100C2×C4×C109C2×D412yes—
101C2×C4×C109C2×D412yes—
102C2×C4×C109C2×D411yescharacteristic
103C2×C4×C109#4112yes—
104C2×C4×C109C2411yescharacteristic
105C22⋊D109—42no—
106C22⋊D109—82no—
107C2×Dic109—22no—
108C2×Dic109—21no—
109C2×Dic109—81no—
110D436—21no—
111D436—21no—
112C4×D109—42no—
113Dic218—21no—
114Dic218—21no—
115#11724784—21no—
116#11724784—21no—
117#11724784C2×D414yes—
118C22×C109—21no—
119C22×C109—21no—
120C22×C109—22no—
121C22×C109—22no—
122C22×C109—22no—
123C22×C109#12112yes—
124C22×C109C22×D411yescharacteristic
125C2×D109—41no—
126C2×D109—41no—
127C2×D109—41no—
128C4×C109—22no—
129C4×C109—42no—
130C4×C109—41no—
131C4×C109—41no—
132C4×C109#12112yes—
133C4×C109C22×D411yescharacteristic
134C4×C109C2×#4111yescharacteristic
135Dic109—41no—
136Dic109—42no—
137C2×C109—22no—
138C2×C109—42no—
139C2×C109C2×#12111yescharacteristic
140C2×C109#57711yescharacteristic
141C2×C109#57811yescharacteristic
142D109—81no—
143#3063—1091nomaximal, Sylow
144C109#306311yescharacteristic, complemented, Sylow
145C2×#137—1091no—
146C2×#131—1091no—
147#537—1091no—
148#536D10911yescharacteristic, complemented
149#470—1092no—
150#468—1092no—
151#467—1092no—
152#465—1092no—
153#428—1091no—
154#416—1091no—
155C2×#103—1091no—
156C2×#41—1091no—
157C2×#41C2×D10911yescharacteristic
158C22×Q8C2×D10911yescharacteristic
159#137—2184no—
160C2×Q16—1092no—
161C2×SD16—1092no—
162#131—2184no—
163C2×M16—1091no—
164C2×M16—1092no—
165C22×C8—1091no—
166C22×C8—1092no—
167C4⋊Q8—21no—
168#124—21no—
169#123—22no—
170C22⋊Q8—1092no—
171C22⋊Q8—2181no—
172C22⋊Q8C2×D10912yes—
173C22⋊Q8C2×D10912yes—
174#121—1092no—
175#121—2181no—
176#121—2182no—
177#117C2×D10911yescharacteristic
178C2×C4⋊C4—1091no—
179#107—1092no—
180#106—1092no—
181#103—1092no—
182#103—1092no—
183#103—1091no—
184#103—1091no—
185#102—1091no—
186#102—1091no—
187#41—22no—
188#41—22no—
189#41—2182no—
190#41—2182no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1091no—
195C2×Q8C22⋊D10911yescharacteristic
196C2×Q8C22×D10912yes—
197C2×D4—1091no—
198C2×D4—2182no—
199C2×D4—2181no—
200C2×D4C22⋊D10912yes—
201C2×D4C22⋊D10911yescharacteristic
202C22×C4—21no—
203C22×C4—2181no—
204C22×C4—2181no—
205C22×C4C22×D10912yes—
206C22×C4C22×D10911yescharacteristic
207Q16—2184no—
208SD16—2184no—
209M16—1094no—
210M16—2184no—
211C2×C8—1092no—
212C2×C8—1091no—
213C2×C8—1091no—
214C2×C8—2184no—
215C2×C8—2182no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—1092no—
219C4⋊C4—2182no—
220C4⋊C4—2181no—
221C4⋊C4C22×D10911yescharacteristic
222C4⋊C4C22×D10911yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—2182no—
227C22⋊C4—2182no—
228C42—21no—
229C23—2181no—
230C23C2×C22⋊D10912yes—
231C23C2×C22⋊D10911yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—2181no—
237Q8—2181no—
238D4—24no—
239D4—22no—
240D4—8722no—
241D4—4362no—
242D4—2181no—
243D4—2181no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—8721no—
251C2×C4—4362no—
252C2×C4—2182no—
253C2×C4—2181no—
254C2×C4D4×D10912yes—
255C2×C4C2×C22⋊D10912yes—
256C2×C4C2×C22⋊D10911yescharacteristic
257C2×C4#1318202812yes—
258C2×C4C23×D10911yescharacteristic
259C8—1094no—
260C8—2181no—
261C8—2181no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—4361no—
268C22—4361no—
269C22—4361no—
270C22#1681267112yes—
271C22C22×C22⋊D10911yescharacteristic
272C4—22no—
273C4—42no—
274C4—41no—
275C4—41no—
276C4—4361no—
277C4—4362no—
278C4#1681267112yes—
279C4C2×D4×D10911yescharacteristic
280C4C2×#1318202811yescharacteristic
281C2—22no—
282C2—42no—
283C2—8721no—
284C2C2×#1681267111yescharacteristic
285C2#2029583411yescharacteristic
286C2#2029584111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#536×C109C2
socleC22×C109C22×D4
prime radical#20295604C2
hypercenter#536D109
hypocenterC109#3063
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16812730C2semidirect
2. C2×#16812706C2semidirect
4. #20295282C2semidirect
5. #20295274C2semidirect
6. #20295265C2semidirect
7. #20295257C2semidirect
8. #20295069C2semidirect
9. #20295020C2semidirect
10. C2×#16812623C2semidirect
11. #536×C109C2semidirect
54. #13182013D4semidirect
144. C109#3063semidirect
148. #536D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
C2×#168127062
#202956042
#202952822
#202952742
#202952652
#202952572
#202950692
#202950202
C2×#168126232
#536×C1092
#3063109