Finite Group Databasebeta

← #28335032

Subgroups — #28335032

Order 13696.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16812088C211yesmaximal, characteristic
2C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic
3#20292363C214yesmaximal, complemented
4#380×C107C211yesmaximal, characteristic
5C22×Dic214C411yescharacteristic, complemented
6C22×C107⋊C8—21no—
7#16812088—21no—
8#16812088—21no—
9#16812088C412yescomplemented
10#16812088C2212yes—
11C2×C4⋊Dic107C411yescharacteristic, complemented
12C2×Dic107⋊C4—21no—
13C2×C22⋊Dic107—21no—
14C22⋊(C107⋊C8)—21no—
15C22⋊(C107⋊C8)C2212yes—
16C23×C4×C107C2211yescharacteristic
17C2×C4⋊C4×C107—21no—
18C2×C22⋊C4×C107—21no—
19#95×C107C2212yes—
20C2×Dic214—21no—
21C2×Dic214—42no—
22C2×Dic214C2×C412yes—
23C22×Dic107—21no—
24C22×Dic107—21no—
25C2×C107⋊C8—22no—
26C2×C107⋊C8—42no—
27C4⋊Dic107—21no—
28C4⋊Dic107C2×C412yes—
29Dic107⋊C4—22no—
30Dic107⋊C4—22no—
31C22⋊Dic107—22no—
32C22⋊Dic107—22no—
33C24×C107D411yescharacteristic
34C22×C4×C107—22no—
35C22×C4×C107—42no—
36C22×C4×C107C2×C411yescharacteristic
37C22×C4×C107C2×C411yescharacteristic
38C22×C4×C107D411yescharacteristic
39C22×C4×C107D411yescharacteristic
40C22×C4×C107D411yescharacteristic
41C22×C4×C107C2311yescharacteristic
42C4⋊C4×C107—42no—
43C22⋊C4×C107—42no—
44C23×C107—41no—
45C23×C107—41no—
46C23×C107C22⋊C411yescharacteristic
47C23×C107C22⋊C411yescharacteristic
48C23×C107SD1612yes—
49C23×C107Q1612yes—
50C23×C107C2×D411yescharacteristic
51C2×C4×C107—22no—
52C2×C4×C107—22no—
53C2×C4×C107—22no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107—41no—
57C2×C4×C107—42no—
58C2×C4×C107—42no—
59C2×C4×C107—42no—
60C2×C4×C107C22⋊C411yescharacteristic
61C2×C4×C107C22⋊C411yescharacteristic
62C2×C4×C107C22×C411yescharacteristic
63C2×C4×C107C2×D411yescharacteristic
64C2×Dic107—22no—
65C2×Dic107—22no—
66C2×Dic107—42no—
67C2×Dic107—42no—
68Dic214—42no—
69Dic214—42no—
70C107⋊C8—42no—
71C22×C107—21no—
72C22×C107—21no—
73C22×C107—21no—
74C22×C107—21no—
75C22×C107—21no—
76C22×C107—21no—
77C22×C107—42no—
78C22×C107—42no—
79C22×C107#10312yes—
80C22×C107#10312yes—
81C22×C107C2×C22⋊C411yescharacteristic
82C22×C107C2×SD1611yescharacteristic
83C22×C107C2×Q1611yescharacteristic
84C4×C107—21no—
85C4×C107—21no—
86C4×C107—21no—
87C4×C107—82no—
88C4×C107#9912yes—
89Dic107—42no—
90Dic107—42no—
91C2×C107—21no—
92C2×C107—22no—
93C2×C107—22no—
94C2×C107—21no—
95C2×C107C2×#9911yescharacteristic
96C2×C107C2×#10311yescharacteristic
97C2×C107#42111yescharacteristic
98#1520—1071nomaximal, Sylow
99C107#152011yescharacteristic, complemented, Sylow
100C2×C22⋊Q8—1071no—
101C2×C22⋊C8—1071no—
102#380D10711yescharacteristic
103#328—1074no—
104C22×Q8—1071no—
105C23×C4C2×D10711yescharacteristic
106C22×C8—2141no—
107C22⋊Q8—1072no—
108C22⋊Q8—1072no—
109C22⋊Q8—2141no—
110C22⋊Q8—2141no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—1071no—
113C2×C4⋊C4—2141no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—2141no—
116C22⋊C8—1072no—
117C22⋊C8—2141no—
118#95C2×D10712yes—
119C24C22⋊D10711yescharacteristic
120C2×Q8—1072no—
121C2×Q8—4282no—
122C2×Q8—2141no—
123C22×C4—22no—
124C22×C4—42no—
125C22×C4—2141no—
126C22×C4—2141no—
127C22×C4C4×D10711yescharacteristic
128C22×C4C4×D10711yescharacteristic
129C22×C4D42811yescharacteristic
130C22×C4D42811yescharacteristic
131C22×C4C22⋊D10711yescharacteristic
132C22×C4C22×D10711yescharacteristic
133C2×C8—4282no—
134C2×C8—2142no—
135C4⋊C4—42no—
136C4⋊C4—1072no—
137C4⋊C4—2141no—
138C4⋊C4—2142no—
139C4⋊C4—2142no—
140C22⋊C4—42no—
141C22⋊C4—2142no—
142C22⋊C4—2142no—
143C23—41no—
144C23—41no—
145C23(C2×D107)⋊C411yescharacteristic
146C23(C2×D107)⋊C411yescharacteristic
147C23#1313380612yes—
148C23#1313380812yes—
149C23C2×C22⋊D10711yescharacteristic
150Q8—4282no—
151Q8—4282no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—4282no—
162C2×C4—4282no—
163C2×C4—2142no—
164C2×C4—2142no—
165C2×C4(C2×D107)⋊C411yescharacteristic
166C2×C4(C2×D107)⋊C411yescharacteristic
167C2×C4C2×C4×D10711yescharacteristic
168C2×C4C2×D42811yescharacteristic
169C8—4282no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#1681203212yes—
179C22#1681203212yes—
180C22C2×(C2×D107)⋊C411yescharacteristic
181C22C2×#1313380611yescharacteristic
182C22C2×#1313380811yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—82no—
187C4—4282no—
188C4—4282no—
189C4#1681202312yes—
190C2—21no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2C2×#1681202311yescharacteristic
195C2C2×#1681203211yescharacteristic
196C2#2029264511yescharacteristic
subgroupG/H
centerC22C2×(C2×D107)⋊C4
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#380×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC24×C107D4
cosocleC22×C4×C107C23
hypercenter#380D107
hypocenterC107#1520
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20292363C2semidirect
5. C22×Dic214C4semidirect
9. #16812088C4semidirect
11. C2×C4⋊Dic107C4semidirect
99. C107#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#168120882
C2×C22⋊(C107⋊C8)2
#202923632
#380×C1072
#1520107