Finite Group Databasebeta

← #28372544

Subgroups — #28372544

Order 13952.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812725C211yesmaximal, characteristic
2C22×#13182017C211yesmaximal, characteristic, complemented
3#20295602C211yesmaximal, characteristic, complemented
4#20295241C214yesmaximal, complemented
5#20295231C214yesmaximal, complemented
6#20295053C211yesmaximal, characteristic, complemented
7#20295034C211yesmaximal, characteristic, complemented
8#20295020C211yesmaximal, characteristic, complemented
9#518×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11C22×Dic218C2211yescharacteristic
12#16812725—24no—
13C2×#13182017—24no—
14C2×#13182017C2212yes—
15C2×#13182017C2214yes—
16C2×#13182014C2212yes—
17C2×#13182013C2211yescharacteristic
18C22×#11724784C2211yescharacteristic
19#16812699C2212yescomplemented
20#16812687C2212yescomplemented
21#16812683—22no—
22#16812679—22no—
23C4⋊#11724784C2214yescomplemented
24#16812622C2212yescomplemented
25#16812622C2214yescomplemented
26#16812619C2212yescomplemented
27C22×D4×C109C2211yescharacteristic
28C4×D4×C109—22no—
29C4×D4×C109C2214yes—
30#117×C109C2211yescharacteristic
31#117×C109C2211yescharacteristic
32C2×C22⋊C4×C109—22no—
33#13182029—22no—
34#13182029—22no—
35C2×Dic218—22no—
36C2×Dic218—22no—
37C2×Dic218—21no—
38C2×Dic218C2312yes—
39C2×Dic218C2311yescharacteristic
40C2×D436C2311yescharacteristic
41C2×C22⋊D109—21no—
42C22×Dic109—21no—
43C2×C4×D109—21no—
44#13182017—28no—
45#13182017—24no—
46#13182014—24no—
47#13182013—22no—
48C2×#11724784—22no—
49C2×#11724784C2312yes—
50C2×#11724784C2312yes—
51C4⋊Dic109—24no—
52#13182006—24no—
53C24×C109—21no—
54C2×D4×C109—21no—
55C2×D4×C109—22no—
56C2×D4×C109D414yes—
57C2×D4×C109C2312yes—
58C22×C4×C109—24no—
59C22×C4×C109C2311yescharacteristic
60C4⋊C4×C109D412yes—
61C4⋊C4×C109C2312yes—
62C22⋊C4×C109—21no—
63C22⋊C4×C109—24no—
64C22⋊C4×C109D412yes—
65C42×C109—21no—
66C42×C109C2312yes—
67C23×C109—22no—
68C23×C109—24no—
69C23×C109—82no—
70C23×C109C2×D411yescharacteristic
71C22×D109—21no—
72D4×C109—24no—
73D4×C109—22no—
74D4×C109C2×D414yes—
75C2×C4×C109—22no—
76C2×C4×C109—84no—
77C2×C4×C109C2×D414yes—
78C2×C4×C109C2×D411yescharacteristic
79C2×C4×C109C2×D412yes—
80C2×C4×C109#4112yes—
81C2×C4×C109C2411yescharacteristic
82C22⋊D109—42no—
83C2×Dic109—22no—
84C2×Dic109—21no—
85C2×Dic109—42no—
86D436—21no—
87D436—21no—
88C4×D109—42no—
89Dic218—22no—
90Dic218—21no—
91Dic218—22no—
92Dic218—21no—
93Dic218—42no—
94#11724784—22no—
95#11724784—22no—
96C22×C109—22no—
97C22×C109—24no—
98C22×C109—44no—
99C22×C109—84no—
100C22×C109#12112yes—
101C22×C109C22×D411yescharacteristic
102C2×D109—41no—
103C2×D109—42no—
104C4×C109—24no—
105C4×C109—42no—
106C4×C109#12112yes—
107C4×C109C22×D411yescharacteristic
108C4×C109C2×#4111yescharacteristic
109Dic109—42no—
110Dic109—41no—
111C2×C109—22no—
112C2×C109—44no—
113C2×C109C2×#12111yescharacteristic
114C2×C109#57712yes—
115D109—81no—
116#3042—1091nomaximal, Sylow
117C109#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1091no—
119C22×SD16—1091no—
120#536—1091no—
121#518D10911yescharacteristic, complemented
122#463—1094no—
123#461—1094no—
124#424—1091no—
125#419—1091no—
126#416—1091no—
127C2×#41—1091no—
128C22×Q8—1091no—
129C22×D4C2×D10911yescharacteristic
130AGL(1,ℤ/8)—2184no—
131C2×SD16—1094no—
132C2×SD16—1092no—
133C2×SD16—2184no—
134C2×D8—1092no—
135C2×M16—1091no—
136C22×C8—1091no—
137C4⋊Q8—1092no—
138#124—1092no—
139#123—2182no—
140C22⋊Q8—2182no—
141C4×D4—22no—
142C4×D4C2×D10914yes—
143#117C2×D10911yescharacteristic
144#117C2×D10911yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—1094no—
147#103—1092no—
148#103—1094no—
149#102—1092no—
150C24—21no—
151#41—2182no—
152#41—2182no—
153C2×Q8—1092no—
154C2×Q8—1091no—
155C2×Q8—2182no—
156C2×Q8—2182no—
157C2×Q8—2181no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—1091no—
161C2×D4—2181no—
162C2×D4D43614yes—
163C2×D4C22×D10912yes—
164C22×C4—24no—
165C22×C4—2181no—
166C22×C4—2181no—
167C22×C4C22×D10911yescharacteristic
168SD16—2188no—
169SD16—2184no—
170D8—2184no—
171M16—2182no—
172C2×C8—1092no—
173C2×C8—1092no—
174C2×C8—2182no—
175C4⋊C4—2184no—
176C4⋊C4C22⋊D10912yes—
177C4⋊C4C22×D10912yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—2184no—
181C22⋊C4C22⋊D10912yes—
182C42—21no—
183C42C22×D10912yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2181no—
188C23C2×C22⋊D10911yescharacteristic
189Q8—4362no—
190Q8—2182no—
191Q8—2181no—
192Q8—2182no—
193Q8—2181no—
194D4—24no—
195D4—22no—
196D4—4362no—
197D4—2181no—
198D4—2181no—
199D4C2×D43614yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—4362no—
203C2×C4—4362no—
204C2×C4—2182no—
205C2×C4—2181no—
206C2×C4C2×C22⋊D10914yes—
207C2×C4C2×C22⋊D10911yescharacteristic
208C2×C4C2×D43612yes—
209C2×C4#1318202912yes—
210C2×C4C23×D10911yescharacteristic
211C8—2182no—
212C8—2182no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—4361no—
218C22—4362no—
219C22#1681267312yes—
220C22C22×C22⋊D10911yescharacteristic
221C4—24no—
222C4—42no—
223C4—4361no—
224C4—4362no—
225C4#1681267312yes—
226C4C22×D43611yescharacteristic
227C4C2×#1318202911yescharacteristic
228C2—22no—
229C2—44no—
230C2—8721no—
231C2C2×#1681267311yescharacteristic
232C2#2029583712yes—
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#518×C109C2
socleC22×C109C22×D4
prime radicalC2×#16812725C2
hypercenter#518D109
hypocenterC109#3042
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#13182017C2semidirect
3. #20295602C2semidirect
4. #20295241C2semidirect
5. #20295231C2semidirect
6. #20295053C2semidirect
7. #20295034C2semidirect
8. #20295020C2semidirect
9. #518×C109C2semidirect
19. #16812699C22semidirect
20. #16812687C22semidirect
23. C4⋊#11724784C22semidirect
24. #16812622C22semidirect
25. #16812622C22semidirect
26. #16812619C22semidirect
117. C109#3042semidirect
121. #518D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127252
C22×#131820172
#202956022
#202952412
#202952312
#202950532
#202950342
#202950202
#518×C1092
#3042109