Finite Group Databasebeta

← #28372543

Subgroups — #28372543

Order 13952.

494 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812727C211yesmaximal, characteristic
2C22×#13182016C211yesmaximal, characteristic, complemented
3#20295600C211yesmaximal, characteristic, complemented
4#20295500C211yesmaximal, characteristic, complemented
5#20295244C214yesmaximal, complemented
6#20295230C214yesmaximal, complemented
7#20295054C211yesmaximal, characteristic, complemented
8#20295020C211yesmaximal, characteristic, complemented
9#419×C109C211yesmaximal, characteristic, complemented
10C2×#13182029C2211yescharacteristic
11C22×D436C2211yescharacteristic
12#16812727—24no—
13C2×#13182018C2212yes—
14C2×#13182016—24no—
15C2×#13182016C2214yes—
16C2×D872C2212yes—
17C22×#11724784C2211yescharacteristic
18#16812697C2212yescomplemented
19#16812688C2212yescomplemented
20#16812682—22no—
21#16812675—22no—
22C4×D436—22no—
23#16812661C2214yes—
24#16812658C2211yescharacteristic
25C22⋊C4×D109—22no—
26#16812628C2214yescomplemented
27#16812622C2212yescomplemented
28#16812619C2212yescomplemented
29C22×Q8×C109C2211yescharacteristic
30C2×M16×C109C2211yescharacteristic
31#117×C109C2211yescharacteristic
32#103×C109C2214yescomplemented
33C23×D109—21no—
34#13182029—22no—
35#13182029—22no—
36C2×Dic218C2311yescharacteristic
37C2×D436—21no—
38C2×D436—22no—
39C2×D436D414yes—
40C2×D436C2311yescharacteristic
41C2×D436C2312yes—
42C2×C22⋊D109—21no—
43C2×C4×D109—21no—
44C2×C4×D109—24no—
45#13182018—24no—
46#13182016—28no—
47D872—24no—
48C2×#11724784—22no—
49C2×#11724784C2312yes—
50C4⋊Dic109D412yes—
51#13182008—24no—
52C22⋊Dic109D412yes—
53#13182006—24no—
54#13182006—24no—
55C4×Dic109C2312yes—
56C2×Q8×C109—21no—
57C2×Q8×C109—22no—
58C2×Q8×C109—22no—
59C2×Q8×C109C2312yes—
60C22×C4×C109—21no—
61C22×C4×C109C2311yescharacteristic
62M16×C109—22no—
63C2×C8×C109C2312yes—
64C4⋊C4×C109C2312yes—
65C22⋊C4×C109—21no—
66C42×C109—21no—
67C23×C109C2×D411yescharacteristic
68C22×D109—22no—
69C22×D109—24no—
70C22×D109—21no—
71C22×D109—82no—
72Q8×C109—22no—
73Q8×C109—22no—
74Q8×C109—42no—
75C2×C4×C109—22no—
76C2×C4×C109—22no—
77C2×C4×C109—42no—
78C2×C4×C109C2×D411yescharacteristic
79C2×C4×C109C2×D412yes—
80C2×C4×C109#4112yes—
81C2×C4×C109C2411yescharacteristic
82C22⋊D109—42no—
83C2×Dic109—21no—
84C2×Dic109C2×D414yes—
85D436—24no—
86D436—21no—
87D436—22no—
88D436—21no—
89D436C2×D414yes—
90C4×D109—42no—
91C4×D109—84no—
92Dic218—21no—
93Dic218—21no—
94C8×C109—22no—
95#11724784—22no—
96C22×C109—22no—
97C22×C109#12112yes—
98C22×C109C22×D411yescharacteristic
99C2×D109—24no—
100C2×D109—44no—
101C2×D109—41no—
102C2×D109—42no—
103C2×D109—84no—
104C4×C109—42no—
105C4×C109—42no—
106C4×C109#12112yes—
107C4×C109C22×D411yescharacteristic
108C4×C109C2×#4111yescharacteristic
109Dic109—24no—
110Dic109—41no—
111C2×C109—22no—
112C2×C109C2×#12111yescharacteristic
113C2×C109#57712yes—
114D109—44no—
115D109—81no—
116#3042—1091nomaximal, Sylow
117C109#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1091no—
119C22×SD16—1091no—
120#536—1091no—
121#518—1091no—
122#463—1094no—
123#461—1094no—
124#424—1091no—
125#419D10911yescharacteristic, complemented
126#416—1091no—
127C2×#41—1091no—
128C22×Q8C2×D10911yescharacteristic
129C22×D4—1091no—
130AGL(1,ℤ/8)—2184no—
131C2×SD16—1094no—
132C2×SD16—1092no—
133C2×SD16—2184no—
134C2×D8—1092no—
135C2×M16C2×D10911yescharacteristic
136C22×C8—1091no—
137C4⋊Q8—1092no—
138#124—1092no—
139#123—2182no—
140C22⋊Q8—2182no—
141C4×D4—1094no—
142C4×D4—2182no—
143#117—1091no—
144#117C2×D10911yescharacteristic
145C2×C22⋊C4—2182no—
146C4⋊C8—1094no—
147#103—1092no—
148#103C2×D10914yescomplemented
149#102—1092no—
150C24—2181no—
151#41—2182no—
152#41—2182no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1091no—
157C2×Q8C22×D10912yes—
158C2×D4—1094no—
159C2×D4—1092no—
160C2×D4—1091no—
161C2×D4—2181no—
162C2×D4—2181no—
163C2×D4—2182no—
164C22×C4—21no—
165C22×C4—2184no—
166C22×C4—2181no—
167C22×C4C22×D10911yescharacteristic
168SD16—2184no—
169SD16—2188no—
170D8—2184no—
171M16—22no—
172C2×C8—1092no—
173C2×C8—2182no—
174C2×C8C22×D10912yes—
175C4⋊C4—1092no—
176C4⋊C4—2184no—
177C4⋊C4C22×D10912yes—
178C22⋊C4—21no—
179C22⋊C4—1092no—
180C22⋊C4—2184no—
181C22⋊C4—2184no—
182C42—21no—
183C42—1092no—
184C23—8722no—
185C23—2184no—
186C23—2181no—
187C23—2182no—
188C23D4×D10911yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2181no—
193Q8—2181no—
194D4—1094no—
195D4—4362no—
196D4—2184no—
197D4—2181no—
198D4—2181no—
199D4—2182no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—8724no—
204C2×C4—1094no—
205C2×C4—4362no—
206C2×C4—2181no—
207C2×C4D4×D10911yescharacteristic
208C2×C4D4×D10912yes—
209C2×C4#1318202912yes—
210C2×C4C23×D10911yescharacteristic
211C8—22no—
212C8—2182no—
213C22—22no—
214C22—8724no—
215C22—4361no—
216C22—4362no—
217C22—4364no—
218C22—2184no—
219C22#1681267212yes—
220C22C2×D4×D10911yescharacteristic
221C4—42no—
222C4—42no—
223C4—4361no—
224C4—2184no—
225C4#1681267212yes—
226C4C2×D4×D10911yescharacteristic
227C4C2×#1318202911yescharacteristic
228C2—22no—
229C2—8721no—
230C2—4364no—
231C2C2×#1681267211yescharacteristic
232C2#2029583512yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#419×C109C2
socleC22×C109C22×D4
prime radicalC2×#16812727C2
hypercenter#419D109
hypocenterC109#3042
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#13182016C2semidirect
3. #20295600C2semidirect
4. #20295500C2semidirect
5. #20295244C2semidirect
6. #20295230C2semidirect
7. #20295054C2semidirect
8. #20295020C2semidirect
9. #419×C109C2semidirect
18. #16812697C22semidirect
19. #16812688C22semidirect
26. #16812628C22semidirect
27. #16812622C22semidirect
28. #16812619C22semidirect
32. #103×C109C22semidirect
117. C109#3042semidirect
125. #419D109semidirect
148. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C22×#131820162
#202956002
#202955002
#202952442
#202952302
#202950542
#202950202
#419×C1092
#3042109