Finite Group Databasebeta

← #28363064

Subgroups — #28363064

Order 13952.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#13182012C211yesmaximal, characteristic
2C2×#16812632C211yesmaximal, characteristic, complemented
3C2×#16812630C211yesmaximal, characteristic, complemented
4#20294883C211yesmaximal, characteristic
5#20294883C211yesmaximal, characteristic
6#20294789C211yesmaximal, characteristic, complemented
7#340×C109C211yesmaximal, characteristic, complemented
8C22×C4×D109C2211yescharacteristic
9C2×#13182012—22no—
10C2×#13182012—22no—
11C2×#13182012C414yescomplemented
12C22×#11724784C2211yescharacteristic
13C2×C4⋊Dic109C2211yescharacteristic
14C2×C4⋊Dic109C2211yescharacteristic
15C2×#13182006—21no—
16C2×#13182006—21no—
17#16812632—22no—
18#16812630—22no—
19#16812606—21no—
20#16812606—21no—
21C22×C8×C109C2211yescharacteristic
22C2×C4⋊C4×C109C2211yescharacteristic
23C2×C4⋊C4×C109C2211yescharacteristic
24C23×D109D411yescharacteristic
25C22×Dic109—21no—
26C22×Dic109—21no—
27C22×Dic109D411yescharacteristic
28C2×C4×D109—21no—
29C2×C4×D109—21no—
30C2×C4×D109—21no—
31C2×C4×D109—21no—
32C2×C4×D109C2×C412yes—
33C2×C4×D109Q812yes—
34#13182012—28no—
35C2×#11724784C2×C412yes—
36C2×#11724784D412yes—
37C2×#11724784D412yes—
38C4⋊Dic109—21no—
39C4⋊Dic109—21no—
40C4⋊Dic109—21no—
41C4⋊Dic109—21no—
42#13182006—42no—
43#13182006—42no—
44C22×C4×C109—21no—
45C22×C4×C109—21no—
46C22×C4×C109C2311yescharacteristic
47C2×C8×C109—21no—
48C2×C8×C109—21no—
49C2×C8×C109C2×C412yes—
50C4⋊C4×C109—21no—
51C4⋊C4×C109—21no—
52C4⋊C4×C109—21no—
53C4⋊C4×C109—21no—
54C23×C109C2×D411yescharacteristic
55C22×D109—21no—
56C22×D109—21no—
57C22×D109—21no—
58C22×D109—21no—
59C22×D109—21no—
60C22×D109—21no—
61C22×D109C4⋊C412yes—
62C2×C4×C109—41no—
63C2×C4×C109—41no—
64C2×C4×C109—41no—
65C2×C4×C109—41no—
66C2×C4×C109—41no—
67C2×C4×C109—41no—
68C2×C4×C109C22×C411yescharacteristic
69C2×C4×C109C2×D411yescharacteristic
70C2×C4×C109C2×D411yescharacteristic
71C2×C4×C109C2×Q811yescharacteristic
72C2×C4×C109#4111yescharacteristic
73C2×C4×C109#4111yescharacteristic
74C2×Dic109—21no—
75C2×Dic109—21no—
76C2×Dic109—41no—
77C2×Dic109—41no—
78C2×Dic109—41no—
79C2×Dic109—41no—
80C2×Dic109—41no—
81C2×Dic109—41no—
82C2×Dic109C4⋊C412yes—
83C4×D109—22no—
84C4×D109—22no—
85C4×D109—22no—
86C4×D109—22no—
87C8×C109—22no—
88#11724784C22⋊C414yescomplemented
89C22×C109C2×C4⋊C411yescharacteristic
90C22×C109#12111yescharacteristic
91C22×C109#12111yescharacteristic
92C22×C109AGL(1,ℤ/8)11yescharacteristic
93C22×C109AGL(1,ℤ/8)11yescharacteristic
94C22×C109#13711yescharacteristic
95C22×C109#13711yescharacteristic
96C2×D109—22no—
97C2×D109—22no—
98C2×D109—22no—
99C2×D109—41no—
100C2×D109—41no—
101C2×D109—41no—
102C2×D109—41no—
103C4×C109—81no—
104C4×C109—81no—
105C4×C109C2×C22⋊C411yescharacteristic
106C4×C109#11711yescharacteristic
107C4×C109C22⋊Q811yescharacteristic
108C4×C109C22⋊Q811yescharacteristic
109Dic109—22no—
110Dic109—81no—
111Dic109—81no—
112C2×C109#38011yescharacteristic
113C2×C109#42811yescharacteristic
114C2×C109#42811yescharacteristic
115C2×C109#46811yescharacteristic
116C2×C109#46811yescharacteristic
117C2×C109#46911yescharacteristic
118C2×C109#47011yescharacteristic
119D109—42no—
120#1809—1091nomaximal, Sylow
121C109#180911yescharacteristic, complemented, Sylow
122C22×M16—1091no—
123C2×#107—1091no—
124C2×#106—1091no—
125#380—1091no—
126#380—1091no—
127#340—1091no—
128#340D10911yescharacteristic, complemented
129C23×C4—1091no—
130C2×M16—1094no—
131C2×M16—2182no—
132C2×M16—2182no—
133C22×C8—1091no—
134C22×C8C2×D10911yescharacteristic
135C2×C4⋊C4—1091no—
136C2×C4⋊C4—1091no—
137C2×C4⋊C4C2×D10911yescharacteristic
138C2×C4⋊C4C2×D10911yescharacteristic
139C2×C22⋊C4—2181no—
140C2×C22⋊C4—2181no—
141#107—2182no—
142#106—2182no—
143#95—2181no—
144#95—2181no—
145C24—1091no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1091no—
149C22×C4—1092no—
150C22×C4—1092no—
151C22×C4—2181no—
152C22×C4—2181no—
153C22×C4—2181no—
154C22×C4—2181no—
155C22×C4—2181no—
156C22×C4—2181no—
157C22×C4C22×D10911yescharacteristic
158M16—2188no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—1092no—
162C2×C8—1092no—
163C2×C8—1092no—
164C2×C8C4×D10912yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—2181no—
170C4⋊C4—2181no—
171C4⋊C4—2181no—
172C4⋊C4—2181no—
173C22⋊C4—4362no—
174C22⋊C4—4362no—
175C23—1092no—
176C23—2181no—
177C23—2181no—
178C23—2181no—
179C23—2181no—
180C23—2181no—
181C23—2181no—
182C23D4×D10911yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1092no—
190C2×C4—4361no—
191C2×C4—4361no—
192C2×C4—4361no—
193C2×C4—4361no—
194C2×C4—4361no—
195C2×C4—4361no—
196C2×C4—2181no—
197C2×C4—2181no—
198C2×C4—2182no—
199C2×C4—2182no—
200C2×C4—2182no—
201C2×C4—2182no—
202C2×C4C2×C4×D10911yescharacteristic
203C2×C4D4×D10911yescharacteristic
204C2×C4D4×D10911yescharacteristic
205C2×C4Q8×D10911yescharacteristic
206C2×C4#1318202911yescharacteristic
207C2×C4#1318202911yescharacteristic
208C8—22no—
209C8—1094no—
210C22—4361no—
211C22—4361no—
212C22—4361no—
213C22—4361no—
214C22—2182no—
215C22—2182no—
216C22—2182no—
217C22C4⋊C4×D10911yescharacteristic
218C22#1681267211yescharacteristic
219C22#1681267211yescharacteristic
220C22#1681272311yescharacteristic
221C22#1681272611yescharacteristic
222C22#1681272811yescharacteristic
223C22#1681273111yescharacteristic
224C4—81no—
225C4—81no—
226C4—8721no—
227C4—8721no—
228C4—2182no—
229C4C22⋊C4×D10911yescharacteristic
230C4#1681265611yescharacteristic
231C4#1681267711yescharacteristic
232C4#1681267711yescharacteristic
233C2—4362no—
234C2#2029488511yescharacteristic
235C2#2029506511yescharacteristic
236C2#2029506711yescharacteristic
237C2#2029527111yescharacteristic
238C2#2029527511yescharacteristic
239C2#2029527811yescharacteristic
240C2#2029528311yescharacteristic
subgroupG/H
centerC23D4×D109
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#340×C109C2
socleC23×C109C2×D4
prime radicalC23×D109D4
cosocleC22×C4×C109C23
hypercenter#340D109
hypocenterC109#1809
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#16812632C2semidirect
3. C2×#16812630C2semidirect
6. #20294789C2semidirect
7. #340×C109C2semidirect
11. C2×#13182012C4semidirect
88. #11724784C22⋊C4semidirect
121. C109#1809semidirect
128. #340D109semidirect

Maximal subgroups

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

subgroupindex
C22×#131820122
C2×#168126322
C2×#168126302
#202948832
#202947892
#340×C1092
#1809109