Finite Group Databasebeta

← #28337403

Subgroups — #28337403

Order 13696.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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#20293180C211yesmaximal, characteristic
2M16×Dic107C211yesmaximal, characteristic
3C4×Dic107⋊C4C211yesmaximal, characteristic
4#20292388C211yesmaximal, characteristic
5#20292387C211yesmaximal, characteristic
6#20292386C211yesmaximal, characteristic
7#339×C107C211yesmaximal, characteristic
8C2×#13133802C2211yescharacteristic
9#16812108C411yescharacteristic, complemented
10#16812108C411yescharacteristic, complemented
11#16812100C411yescharacteristic, complemented
12#16812100C411yescharacteristic, complemented
13#16812086—21no—
14#16812086—21no—
15C4×Dic214—21no—
16C4×Dic214—21no—
17#16812068C2211yescharacteristic
18C2×Dic107⋊C4—21no—
19C2×C4×Dic107C2211yescharacteristic
20C2×C4×Dic107C2211yescharacteristic
21#16812019—21no—
22C8×Dic107—21no—
23C107⋊#95—21no—
24C2×M16×C107C2211yescharacteristic
25#117×C107C2211yescharacteristic
26C2×C42×C107C2211yescharacteristic
27C2×Dic214—21no—
28C2×Dic214—21no—
29C22×Dic107—21no—
30C22×Dic107D411yescharacteristic
31C22×Dic107D411yescharacteristic
32#13133802—21no—
33#13133802Q812yescomplemented
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107—21no—
37C4⋊Dic107C2×C411yescharacteristic
38C4⋊Dic107C2×C411yescharacteristic
39Dic107⋊C4—24no—
40Dic107⋊C4—21no—
41Dic107⋊C4—21no—
42Dic107⋊C4—21no—
43Dic107⋊C4—21no—
44C22⋊Dic107—21no—
45C4×Dic107—21no—
46C4×Dic107—21no—
47C4×Dic107—21no—
48C4×Dic107C2×C411yescharacteristic
49C4×Dic107C2×C411yescharacteristic
50C4×Dic107D411yescharacteristic
51C4×Dic107D411yescharacteristic
52C22×C4×C107—21no—
53C22×C4×C107C2311yescharacteristic
54M16×C107—21no—
55M16×C107Q812yescomplemented
56C2×C8×C107—21no—
57C4⋊C4×C107C2×C411yescharacteristic
58C4⋊C4×C107C2×C411yescharacteristic
59C22⋊C4×C107—21no—
60C42×C107—21no—
61C42×C107—21no—
62C42×C107—21no—
63C23×C107C2×D411yescharacteristic
64C2×C4×C107—22no—
65C2×C4×C107—21no—
66C2×C4×C107—21no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107C22×C411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×Q811yescharacteristic
72C2×C4×C107C2×Q811yescharacteristic
73C2×C4×C107#4111yescharacteristic
74C2×C4×C107#4111yescharacteristic
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—21no—
78C2×Dic107—21no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107—41no—
82C2×Dic107C22⋊C411yescharacteristic
83C2×Dic107C22⋊C411yescharacteristic
84C2×Dic107C22⋊C411yescharacteristic
85C2×Dic107C22⋊C411yescharacteristic
86Dic214—42no—
87Dic214—42no—
88C8×C107—22no—
89C107⋊C8—22no—
90C22×C107—21no—
91C22×C107—21no—
92C22×C107C2×C22⋊C411yescharacteristic
93C22×C107#12411yescharacteristic
94C22×C107C4⋊Q811yescharacteristic
95C4×C107—42no—
96C4×C107—41no—
97C4×C107—41no—
98C4×C107C4×Q811yescharacteristic
99C4×C107C4×Q811yescharacteristic
100C4×C107C22⋊Q811yescharacteristic
101C4×C107C22⋊Q811yescharacteristic
102Dic107—21no—
103Dic107—21no—
104Dic107—21no—
105Dic107—42no—
106Dic107—41no—
107Dic107—41no—
108Dic107#10412yes—
109C2×C107—21no—
110C2×C107—21no—
111C2×C107#39111yescharacteristic
112C2×C107C2×#10411yescharacteristic
113C2×C107#42111yescharacteristic
114#1986—1071nomaximal, Sylow
115C107#198611yescharacteristic, complemented, Sylow
116#533—1071no—
117C4×M16—1071no—
118C4×C4⋊C4—1071no—
119#339—1071no—
120#339—1071no—
121#339—1071no—
122#339D10711yescharacteristic
123C2×M16—1071no—
124C2×M16C2×D10711yescharacteristic
125C4⋊Q8—1071no—
126C4⋊Q8—1071no—
127#125—1071no—
128#125—1071no—
129C22⋊Q8—2141no—
130C22⋊Q8—2141no—
131C4×Q8—2141no—
132C4×Q8—2141no—
133#117—1071no—
134#117C2×D10711yescharacteristic
135C2×C4⋊C4—2141no—
136C2×C42—1071no—
137C2×C42—1071no—
138C2×C42C2×D10711yescharacteristic
139#97—2141no—
140C4×C8—2141no—
141#95—2141no—
142C2×Q8—2141no—
143C2×Q8—2141no—
144C22×C4—21no—
145C22×C4—1071no—
146C22×C4—1071no—
147C22×C4—2141no—
148C22×C4C22×D10711yescharacteristic
149M16—21no—
150M16—1072no—
151M16—2141no—
152M16Dic21412yescomplemented
153C2×C8—21no—
154C2×C8—2141no—
155C4⋊C4—1071no—
156C4⋊C4—1071no—
157C4⋊C4—2141no—
158C4⋊C4—2141no—
159C4⋊C4—2141no—
160C4⋊C4—2141no—
161C4⋊C4—2141no—
162C4⋊C4—2141no—
163C4⋊C4—2144no—
164C4⋊C4C4×D10711yescharacteristic
165C4⋊C4C4×D10711yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—2141no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—1071no—
172C42—1071no—
173C42—1071no—
174C42—1071no—
175C42—2141no—
176C42—2141no—
177C42—2141no—
178C23D4×D10711yescharacteristic
179Q8—4282no—
180Q8—4282no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1071no—
187C2×C4—1071no—
188C2×C4—1071no—
189C2×C4—1071no—
190C2×C4—4281no—
191C2×C4—4281no—
192C2×C4—4281no—
193C2×C4—2142no—
194C2×C4—2142no—
195C2×C4—2141no—
196C2×C4—2141no—
197C2×C4C2×C4×D10711yescharacteristic
198C2×C4D4×D10711yescharacteristic
199C2×C4Q8×D10711yescharacteristic
200C2×C4C2×Dic21411yescharacteristic
201C2×C4#1313381711yescharacteristic
202C2×C4#1313381811yescharacteristic
203C8—22no—
204C8—2142no—
205C22—21no—
206C22—21no—
207C22C22⋊C4×D10711yescharacteristic
208C22#1681209511yescharacteristic
209C22#1681210811yescharacteristic
210C4—42no—
211C4—41no—
212C4—41no—
213C4—1072no—
214C4—4282no—
215C4—4281no—
216C4—4281no—
217C4—2141no—
218C4—2141no—
219C4—2141no—
220C4#1681207411yescharacteristic
221C4C4×Dic21411yescharacteristic
222C4#1681208611yescharacteristic
223C4#1681208711yescharacteristic
224C2—21no—
225C2—21no—
226C2#2029253111yescharacteristic
227C2#104×D10711yescharacteristic
228C2#2029264211yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#339×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC23×C107C2×D4
cosocleC22×C4×C107C23
hypercenter#339D107
hypocenterC107#1986
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #16812108C4semidirect
10. #16812108C4semidirect
11. #16812100C4semidirect
12. #16812100C4semidirect
33. #13133802Q8semidirect
55. M16×C107Q8semidirect
115. C107#1986semidirect
152. M16Dic214semidirect

Maximal subgroups

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

subgroupindex
#202931802
M16×Dic1072
C4×Dic107⋊C42
#202923882
#202923872
#202923862
#339×C1072
#1986107