Finite Group Databasebeta

← C2×#20293238 #28349828

Subgroups — C2×#20293238 #28349828

Order 13696.

1004 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 485 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#20293238C2116yesmaximal, direct product factor
2C2×#16812104C214yesmaximal
3C2×#16812101C211yesmaximal, characteristic, complemented
4C2×#16812099C214yesmaximal, complemented
5C2×#16812085C212yesmaximal
6C2×Q8×Dic107C212yesmaximal, complemented
7C2×#16812066C211yesmaximal, characteristic
8C2×C4⋊Q8×C107C211yesmaximal, characteristic, complemented
9C22×C4×D107C2211yescharacteristic
10#16812104C22132yes—
11#16812101C2218yescomplemented
12#16812099C22132yescomplemented
13#16812085C22116yes—
14Q8×Dic107C22116yescomplemented
15#16812066C2218yes—
16C2×C4⋊Dic107C2212yes—
17C2×C4⋊Dic107C2214yes—
18C2×Dic107⋊C4C2214yes—
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2214yes—
21C2×(C2×D107)⋊C4C2214yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×C4×Dic107C2214yes—
24C2×C4×Dic107C2211yescharacteristic
25C22×Q8×C107C2212yes—
26C4⋊Q8×C107C2218yescomplemented
27C2×C4⋊C4×C107C2214yes—
28C2×C42×C107C2211yescharacteristic
29C23×D107C2311yescharacteristic
30C22×Dic107C2312yes—
31C22×Dic107C2314yes—
32C22×Dic107C2311yescharacteristic
33C2×C4×D107—24no—
34C2×C4×D107C2314yes—
35C4⋊Dic107C2318yes—
36C4⋊Dic107C23116yes—
37Dic107⋊C4C2318yes—
38Dic107⋊C4C23116yes—
39Dic107⋊C4C23116yes—
40(C2×D107)⋊C4C2318yes—
41(C2×D107)⋊C4C23116yes—
42C4×Dic107C2314yes—
43C4×Dic107C23116yes—
44C2×Q8×C107—28no—
45C2×Q8×C107C2318yes—
46C22×C4×C107C2314yes—
47C22×C4×C107C2312yes—
48C22×C4×C107C2311yescharacteristic
49C4⋊C4×C107C23116yes—
50C42×C107C2314yes—
51C23×C107C2411yescharacteristic
52C22×D107—22no—
53C22×D107—24no—
54C22×D107C2412yes—
55Q8×C107—216no—
56C2×C4×C107—28no—
57C2×C4×C107—24no—
58C2×C4×C107#4114yes—
59C2×C4×C107C2414yes—
60C2×C4×C107C2418yes—
61C2×C4×C107C2412yes—
62C2×Dic107—22no—
63C2×Dic107—24no—
64C2×Dic107—28no—
65C2×Dic107C2414yes—
66C2×Dic107C2418yes—
67C2×Dic107C2412yes—
68C4×D107—28no—
69C22×C107C2×#4112yes—
70C22×C1072−1+414yes—
71C22×C107C2511yescharacteristic
72C2×D107—22no—
73C2×D107—24no—
74C2×D107—44no—
75C4×C107—28no—
76C4×C107—24no—
77C4×C107C2×#4114yes—
78Dic107—22no—
79Dic107—24no—
80Dic107—28no—
81C2×C107#54114yescomplemented
82C2×C107C22×#4111yescharacteristic
83C2×C107C2×2−1+412yes—
84D107—42no—
85C2×#541—1071nomaximal, Sylow
86C107C2×#54111yescharacteristic, complemented, Sylow
87#541—10716no—
88C2×C4⋊Q8D10711yescharacteristic, complemented
89C2×#126—1074no—
90C2×#125—1074no—
91C2×#125—1071no—
92C2×C22⋊Q8—1072no—
93C2×C4×Q8—1072no—
94C2×#117—1071no—
95C22×Q8C2×D10712yes—
96C23×C4—1071no—
97C4⋊Q8C2×D10718yescomplemented
98#126—10732no—
99#125—10732no—
100#125—1078no—
101C22⋊Q8—10716no—
102C4×Q8—10716no—
103#117—1078no—
104C2×C4⋊C4—1072no—
105C2×C4⋊C4—1072no—
106C2×C4⋊C4—1074no—
107C2×C4⋊C4—1074no—
108C2×C4⋊C4—1074no—
109C2×C4⋊C4C2×D10714yes—
110C2×C22⋊C4—1072no—
111C2×C22⋊C4—1074no—
112C2×C42—1071no—
113C2×C42—1074no—
114C2×C42C2×D10711yescharacteristic
115C24—1071no—
116C2×Q8—28no—
117C2×Q8C22×D10718yes—
118C22×C4—1072no—
119C22×C4—1074no—
120C22×C4—1071no—
121C22×C4—1074no—
122C22×C4—2144no—
123C22×C4C22×D10714yes—
124C22×C4C22×D10712yes—
125C22×C4C22×D10711yescharacteristic
126C4⋊C4—1078no—
127C4⋊C4—1078no—
128C4⋊C4—10716no—
129C4⋊C4—10716no—
130C4⋊C4—10716no—
131C4⋊C4C22×D107116yes—
132C22⋊C4—1078no—
133C22⋊C4—10716no—
134C42—1074no—
135C42—10716no—
136C42C22×D10714yes—
137C23—1072no—
138C23—2142no—
139C23—2144no—
140C23C23×D10711yescharacteristic
141Q8—216no—
142C2×C4—28no—
143C2×C4—24no—
144C2×C4—1074no—
145C2×C4—1078no—
146C2×C4—1072no—
147C2×C4—2142no—
148C2×C4—2144no—
149C2×C4—2148no—
150C2×C4—2148no—
151C2×C4#1313381714yes—
152C2×C4C23×D10718yes—
153C2×C4C23×D10714yes—
154C2×C4C23×D10712yes—
155C22—4284no—
156C22—2142no—
157C22—2144no—
158C22C2×#1313381712yes—
159C22#1681215714yes—
160C22C24×D10711yescharacteristic
161C4—28no—
162C4—24no—
163C4—2142no—
164C4—2144no—
165C4—2148no—
166C4C2×#1313381714yes—
167C2—4282no—
168C2#2029323814yesdirect product factor
169C2C22×#1313381711yescharacteristic
170C2C2×#1681215712yes—
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×C4⋊Q8×C107C2
socleC23×C107C24
prime radicalC23×D107C23
hypercenterC2×C4⋊Q8D107
hypocenterC107C2×#541
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20293238C2direct (N × H)
3. C2×#16812101C2semidirect
4. C2×#16812099C2semidirect
6. C2×Q8×Dic107C2semidirect
8. C2×C4⋊Q8×C107C2semidirect
11. #16812101C22semidirect
12. #16812099C22semidirect
14. Q8×Dic107C22semidirect
26. C4⋊Q8×C107C22semidirect
81. C2×C107#541semidirect
86. C107C2×#541semidirect
88. C2×C4⋊Q8D107semidirect
97. C4⋊Q8C2×D107semidirect
168. C2#20293238direct (N × H)

Maximal subgroups

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

subgroupindex
#202932382
C2×#168121042
C2×#168121012
C2×#168120992
C2×#168120852
C2×Q8×Dic1072
C2×#168120662
C2×C4⋊Q8×C1072
C2×#541107