Finite Group Databasebeta

← #28373641

Subgroups — #28373641

Order 13952.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#20295581C211yesmaximal, characteristic, complemented
2#20295555C211yesmaximal, characteristic
3#20295432C211yesmaximal, characteristic, complemented
4#20295421C211yesmaximal, characteristic, complemented
5#20295395C211yesmaximal, characteristic, complemented
6#20295283C212yesmaximal, complemented
7#20295271C212yesmaximal
8#20295118C212yesmaximal, complemented
9#20295117C212yesmaximal, complemented
10#20294978C211yesmaximal, characteristic, complemented
11#489×C109C211yesmaximal, characteristic, complemented
12C2×#13182019C2212yes—
13C2×#13182017C2212yes—
14C2×#13182012C2212yes—
15#16812699C2211yescharacteristic
16#16812692—21no—
17#16812691C2211yescharacteristic
18#16812690—21no—
19#16812681—21no—
20#16812679C2212yes—
21#16812675—21no—
22#16812675C2211yescharacteristic
23#16812670C2211yescharacteristic
24Q8×Dic109—21no—
25Q8×Dic109C2211yescharacteristic
26#16812664C2212yes—
27D4×Dic109C2211yescharacteristic
28#16812659—21no—
29#16812657—21no—
30#16812655—21no—
31C42×D109C2211yescharacteristic
32#16812632C2212yes—
33#16812630C2212yes—
34#16812624C2212yes—
35#16812624C2212yes—
36#16812610C2211yescharacteristic
37C8×Dic109C2212yes—
38C4⋊Q8×C109C2211yescharacteristic
39#124×C109C2211yescharacteristic
40#103×C109C2212yes—
41#102×C109C2212yes—
42#97×C109C2211yescharacteristic
43C2×Dic218C2312yes—
44C2×C22⋊D109—21no—
45C22×Dic109—21no—
46C2×C4×D109D411yescharacteristic
47C2×C4×D109D411yescharacteristic
48C2×C4×D109C2311yescharacteristic
49#13182019—24no—
50#13182017—24no—
51#13182012—24no—
52C2×#11724784C2312yes—
53C4⋊Dic109—21no—
54C4⋊Dic109—21no—
55C4⋊Dic109—22no—
56C4⋊Dic109—21no—
57C4⋊Dic109C2311yescharacteristic
58C4⋊Dic109C2311yescharacteristic
59#13182008—21no—
60#13182008—21no—
61#13182008—21no—
62#13182008—22no—
63C22⋊Dic109—21no—
64C22⋊Dic109—21no—
65#13182006—22no—
66#13182006—21no—
67C4×Dic109—21no—
68C4×Dic109—22no—
69C4×Dic109D411yescharacteristic
70C4×Dic109D411yescharacteristic
71C4×Dic109C2311yescharacteristic
72C2×Q8×C109—21no—
73C2×Q8×C109C2311yescharacteristic
74C2×D4×C109C2311yescharacteristic
75C2×C8×C109C2312yes—
76C4⋊C4×C109—21no—
77C4⋊C4×C109C2312yes—
78C22⋊C4×C109—21no—
79C22⋊C4×C109—21no—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C22×D109C2×D411yescharacteristic
83Q8×C109—22no—
84Q8×C109—42no—
85D4×C109—22no—
86C2×C4×C109—21no—
87C2×C4×C109—22no—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2×D411yescharacteristic
90C2×C4×C109C2411yescharacteristic
91C22⋊D109—42no—
92C2×Dic109—22no—
93C2×Dic109—21no—
94C2×Dic109—21no—
95C2×Dic109—42no—
96C2×Dic109C2×D411yescharacteristic
97C2×Dic109C2×D411yescharacteristic
98C2×Dic109C2×D411yescharacteristic
99C4×D109—22no—
100C4×D109—22no—
101C4×D109—41no—
102Dic218—22no—
103Dic218—22no—
104C8×C109—22no—
105#11724784#4114yes—
106C22×C109—41no—
107C22×C109—41no—
108C22×C109—41no—
109C22×C109C22×D411yescharacteristic
110C2×D109—21no—
111C2×D109—21no—
112C2×D109—21no—
113C4×C109—21no—
114C4×C109—21no—
115C4×C109—41no—
116C4×C109—42no—
117C4×C109C2×#4112yes—
118Dic109—21no—
119Dic109—21no—
120Dic109—42no—
121Dic109—41no—
122Dic109—41no—
123Dic109#13712yes—
124C2×C109—81no—
125C2×C109#53211yescharacteristic
126C2×C109C2×#13711yescharacteristic
127C2×C109#57511yescharacteristic
128D109—41no—
129#3172—1091nomaximal, Sylow
130C109#317211yescharacteristic, complemented, Sylow
131#533—1091no—
132#529—1091no—
133#501—1091no—
134#497—1091no—
135#491—1091no—
136#489D10911yescharacteristic, complemented
137#470—1092no—
138#468—1092no—
139C4×Q16—1092no—
140C4×SD16—1092no—
141C4×M16—1091no—
142C2×Q16—1092no—
143C2×SD16—1092no—
144C2×M16—1092no—
145C4⋊Q8—1091no—
146C4⋊Q8C2×D10911yescharacteristic
147#126—2181no—
148#125—1091no—
149#125—2181no—
150#124C2×D10911yescharacteristic
151#123—2181no—
152C22⋊Q8—1091no—
153C22⋊Q8—1092no—
154C22⋊Q8—2181no—
155#121—1091no—
156C4×Q8—1091no—
157C4×Q8—1092no—
158C4×Q8—2181no—
159C4×D4—1091no—
160C4×D4—2181no—
161#117—2181no—
162#117—2181no—
163C2×C42—1091no—
164#107—1092no—
165#106—1092no—
166#103—1092no—
167#103—1092no—
168#103C2×D10912yes—
169#102C2×D10912yes—
170#97—1091no—
171#97C2×D10911yescharacteristic
172C4×C8—1092no—
173C2×Q8—21no—
174C2×Q8—1092no—
175C2×Q8C22×D10911yescharacteristic
176C2×D4—2181no—
177C2×D4C22×D10911yescharacteristic
178C22×C4—1091no—
179C22×C4—1091no—
180C22×C4—1091no—
181C22×C4—2181no—
182Q16—2184no—
183SD16—2184no—
184M16—2184no—
185C2×C8—1092no—
186C2×C8C22×D10912yes—
187C4⋊C4—21no—
188C4⋊C4—1091no—
189C4⋊C4—1091no—
190C4⋊C4—2181no—
191C4⋊C4—2182no—
192C4⋊C4—2181no—
193C4⋊C4—2182no—
194C4⋊C4—2181no—
195C4⋊C4—2181no—
196C4⋊C4—2181no—
197C4⋊C4—2181no—
198C4⋊C4C22×D10912yes—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—2182no—
202C22⋊C4—2181no—
203C22⋊C4—2181no—
204C22⋊C4—2181no—
205C42—1091no—
206C42—1091no—
207C42—1091no—
208C42—2181no—
209C42—2182no—
210C42C22×D10911yescharacteristic
211C23—21no—
212C23—1091no—
213Q8—22no—
214Q8—42no—
215Q8—2182no—
216Q8—2182no—
217D4—22no—
218D4—4362no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—1091no—
222C2×C4—1091no—
223C2×C4—1091no—
224C2×C4—4361no—
225C2×C4—4362no—
226C2×C4—2182no—
227C2×C4—2181no—
228C2×C4—2182no—
229C2×C4—2181no—
230C2×C4—2182no—
231C2×C4D4×D10911yescharacteristic
232C2×C4D4×D10911yescharacteristic
233C2×C4C23×D10911yescharacteristic
234C8—22no—
235C8—1094no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—2181no—
240C22—2181no—
241C22—2181no—
242C22C2×D4×D10911yescharacteristic
243C4—21no—
244C4—21no—
245C4—42no—
246C4—41no—
247C4—1092no—
248C4—4361no—
249C4—4361no—
250C4—4362no—
251C4—2181no—
252C4—2181no—
253C4#41×D10912yes—
254C2—81no—
255C2—4361no—
256C2#2029557511yescharacteristic
257C2#137×D10911yescharacteristic
258C2#2029582211yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#489×C109C2
socleC22×C109C22×D4
prime radical#16812670C22
hypercenter#489D109
hypocenterC109#3172
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #20295581C2semidirect
3. #20295432C2semidirect
4. #20295421C2semidirect
5. #20295395C2semidirect
6. #20295283C2semidirect
8. #20295118C2semidirect
9. #20295117C2semidirect
10. #20294978C2semidirect
11. #489×C109C2semidirect
130. C109#3172semidirect
136. #489D109semidirect

Maximal subgroups

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

subgroupindex
#202955812
#202955552
#202954322
#202954212
#202953952
#202952832
#202952712
#202951182
#202951172
#202949782
#489×C1092
#3172109