Finite Group Databasebeta

← #28336940

Subgroups — #28336940

Order 13696.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16812137C211yesmaximal, characteristic
2#20293393C211yesmaximal, characteristic, complemented
3#20292645C211yesmaximal, characteristic, complemented
4#20292631C211yesmaximal, characteristic, complemented
5C2×#16812025C211yesmaximal, characteristic, complemented
6C107⋊#338C211yesmaximal, characteristic, complemented
7#344×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×D428C2211yescharacteristic
10#16812137—22no—
11#16812137—22no—
12C2×#13133807C412yescomplemented
13C2×D856C412yescomplemented
14#16812117—22no—
15#16812115—22no—
16C2×#13133802C2211yescharacteristic
17C2×#13133802C2211yescharacteristic
18M16×D107—22no—
19#16812037—22no—
20#16812029—22no—
21#16812025—22no—
22C2×M16×C107C2211yescharacteristic
23C2×M16×C107C2211yescharacteristic
24#117×C107C2211yescharacteristic
25C23×D107—21no—
26#13133818—21no—
27#13133818—21no—
28#13133818D412yes—
29#13133818D412yes—
30C2×Dic214C2×C411yescharacteristic
31C2×D428—21no—
32C2×D428—21no—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—21no—
36C2×D428C2×C412yes—
37C2×D428C2×C411yescharacteristic
38C2×C22⋊D107—21no—
39C2×C4×D107—21no—
40#13133807—24no—
41D856—24no—
42#13133802—21no—
43#13133802—21no—
44#13133802—21no—
45#13133802—21no—
46#13133801—24no—
47C2×C107⋊C8—21no—
48C2×C107⋊C8D412yes—
49C8×D107—24no—
50C22×C4×C107C2311yescharacteristic
51M16×C107—21no—
52M16×C107—21no—
53M16×C107—21no—
54M16×C107D412yes—
55C2×C8×C107—21no—
56C2×C8×C107C2×C412yes—
57C4⋊C4×C107—21no—
58C22⋊C4×C107—21no—
59C42×C107—21no—
60C23×C107#4111yescharacteristic
61C22×D107—21no—
62C22×D107—22no—
63C22×D107—41no—
64C22×D107—41no—
65C22×D107—82no—
66C2×C4×C107—41no—
67C2×C4×C107C22×C411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107#4111yescharacteristic
73C22⋊D107—22no—
74C22⋊D107—22no—
75C2×Dic107—21no—
76D428—22no—
77D428—21no—
78D428—22no—
79D428—41no—
80D428—41no—
81D428C22⋊C412yes—
82C4×D107—22no—
83C4×D107—22no—
84Dic214—21no—
85Dic214C22⋊C412yes—
86C8×C107—22no—
87C8×C107—22no—
88C107⋊C8—22no—
89C107⋊C8—41no—
90C22×C107—41no—
91C22×C107C4×D411yescharacteristic
92C22×C107#12111yescharacteristic
93C22×C107#12311yescharacteristic
94C2×D107—22no—
95C2×D107—42no—
96C2×D107—81no—
97C2×D107—82no—
98C2×D107—82no—
99C2×D107—82no—
100C4×C107—81no—
101C4×C107C2×C22⋊C411yescharacteristic
102C4×C107C4×D411yescharacteristic
103C4×C107#12011yescharacteristic
104C4×C107#12111yescharacteristic
105Dic107—22no—
106C2×C107—21no—
107C2×C107—21no—
108C2×C107—21no—
109C2×C107#38611yescharacteristic
110D107—42no—
111D107—82no—
112#1903—1071nomaximal, Sylow
113C107#190311yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)—1071no—
115#568—1071no—
116#421—1071no—
117#418—1071no—
118C2×#100—1071no—
119#344D10711yescharacteristic, complemented
120#338—1071no—
121C2×#41—1071no—
122C22×D4—1071no—
123AGL(1,ℤ/8)—2142no—
124AGL(1,ℤ/8)—2142no—
125C2×SD16—1072no—
126C2×D8—1072no—
127#131—2142no—
128#131—2142no—
129C2×M16—1071no—
130C2×M16—1071no—
131C2×M16—2142no—
132C2×M16C2×D10711yescharacteristic
133C2×M16C2×D10711yescharacteristic
134#117C2×D10711yescharacteristic
135#104—2142no—
136#102—2142no—
137#100—2142no—
138C24—2141no—
139#41—1072no—
140#41—1072no—
141#41—2141no—
142#41—2141no—
143C2×Q8—1071no—
144C2×D4—1071no—
145C2×D4—1072no—
146C2×D4—2141no—
147C2×D4—2141no—
148C2×D4—2141no—
149C2×D4—2141no—
150C2×D4—2141no—
151C2×D4—2141no—
152C22×C4—2141no—
153C22×C4C22×D10711yescharacteristic
154SD16—2144no—
155D8—2144no—
156M16—21no—
157M16—21no—
158M16—21no—
159M16—2144no—
160M16—2141no—
161M16—2141no—
162M16—2141no—
163M16—2141no—
164M16D42812yes—
165C2×C8—21no—
166C2×C8—1072no—
167C2×C8—2144no—
168C2×C8—2141no—
169C2×C8C4×D10712yes—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—8562no—
174C23—4281no—
175C23—4281no—
176C23—2142no—
177C23—2141no—
178C23#1313381811yescharacteristic
179Q8—1072no—
180Q8—2141no—
181D4—1072no—
182D4—4281no—
183D4—4281no—
184D4—2142no—
185D4—2142no—
186D4—2142no—
187D4—2142no—
188D4—2141no—
189C2×C4—41no—
190C2×C4—2141no—
191C2×C4—2142no—
192C2×C4—2142no—
193C2×C4C2×C4×D10711yescharacteristic
194C2×C4D4×D10711yescharacteristic
195C2×C4D4×D10711yescharacteristic
196C2×C4D4×D10711yescharacteristic
197C2×C4C2×D42811yescharacteristic
198C2×C4#1313381911yescharacteristic
199C8—22no—
200C8—22no—
201C8—4281no—
202C8—2142no—
203C22—41no—
204C22—8562no—
205C22—8562no—
206C22—8562no—
207C22—8561no—
208C22—4282no—
209C22—2142no—
210C22C4×D42811yescharacteristic
211C22#1681208211yescharacteristic
212C22#1681209411yescharacteristic
213C4—81no—
214C4—2142no—
215C4C22⋊C4×D10711yescharacteristic
216C4#1681207111yescharacteristic
217C4#1681207911yescharacteristic
218C4#1681208411yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—8562no—
223C2—4282no—
224C2#2029250511yescharacteristic
subgroupG/H
centerC2#20292505
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#344×C107C2
socleC2×C107#386
prime radicalC2×#16812137C2
cosocleC22×C4×C107C23
hypercenter#344D107
hypocenterC107#1903
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20292645C2semidirect
4. #20292631C2semidirect
5. C2×#16812025C2semidirect
6. C107⋊#338C2semidirect
7. #344×C107C2semidirect
12. C2×#13133807C4semidirect
13. C2×D856C4semidirect
113. C107#1903semidirect
119. #344D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
#202933932
#202926452
#202926312
C2×#168120252
C107⋊#3382
#344×C1072
#1903107