Finite Group Databasebeta

← #28336671

Subgroups — #28336671

Order 13696.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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#20292670C212yesmaximal, complemented
2C2×#16812026C211yesmaximal, characteristic
3#20292530C211yesmaximal, characteristic
4#20292386C212yesmaximal, complemented
5#518×C107C211yesmaximal, characteristic
6C2×#13133802C2212yes—
7C2×C4⋊Dic107C2211yescharacteristic
8C2×C22⋊Dic107—22no—
9C2×C4×Dic107C2211yescharacteristic
10#16812042—22no—
11#16812040—22no—
12#16812026C414yescomplemented
13C22×D4×C107C2211yescharacteristic
14C4×D4×C107—22no—
15C4×D4×C107—22no—
16#117×C107C2212yes—
17C2×C22⋊C4×C107—22no—
18C22×Dic107—21no—
19C22×Dic107—21no—
20#13133802—22no—
21#13133802C2×C414yescomplemented
22C2×C107⋊C8—22no—
23C4⋊Dic107—22no—
24C22⋊Dic107—44no—
25C4×Dic107—22no—
26C24×C107D412yes—
27C2×D4×C107—22no—
28C2×D4×C107—22no—
29C2×D4×C107C2×C412yes—
30C2×D4×C107Q812yes—
31C22×C4×C107—22no—
32C22×C4×C107—22no—
33C22×C4×C107C2311yescharacteristic
34C4⋊C4×C107D412yes—
35C4⋊C4×C107D412yes—
36C22⋊C4×C107—24no—
37C22⋊C4×C107—22no—
38C42×C107—22no—
39C23×C107—22no—
40C23×C107—22no—
41C23×C107—42no—
42C23×C107—42no—
43C23×C107C4⋊C414yes—
44C23×C107C2×D411yescharacteristic
45D4×C107—24no—
46D4×C107—24no—
47C2×C4×C107—22no—
48C2×C4×C107—22no—
49C2×C4×C107—82no—
50C2×C4×C107—82no—
51C2×C4×C107C22×C411yescharacteristic
52C2×C4×C107C2×D412yes—
53C2×C4×C107C2×Q811yescharacteristic
54C2×C4×C107#4112yes—
55C2×Dic107—22no—
56C2×Dic107—41no—
57C2×Dic107—41no—
58C2×Dic107—42no—
59C2×Dic107—41no—
60C107⋊C8—24no—
61C22×C107—21no—
62C22×C107—21no—
63C22×C107—24no—
64C22×C107—44no—
65C22×C107—42no—
66C22×C107—42no—
67C22×C107—82no—
68C22×C107C2×C4⋊C411yescharacteristic
69C22×C107#12011yescharacteristic
70C22×C107#12311yescharacteristic
71C4×C107—42no—
72C4×C107—42no—
73C4×C107C4×D412yes—
74C4×C107C22⋊Q812yes—
75Dic107—42no—
76Dic107—81no—
77C2×C107—21no—
78C2×C107—21no—
79C2×C107—44no—
80C2×C107#38511yescharacteristic
81C2×C107#45311yescharacteristic
82C2×C107#45511yescharacteristic
83#1856—1071nomaximal, Sylow
84C107#185611yescharacteristic, complemented, Sylow
85#518D10711yescharacteristic
86#428—1072no—
87C2×#100—1071no—
88#390—1071no—
89#339—1072no—
90C22×D4C2×D10711yescharacteristic
91C2×M16—1072no—
92C4×D4—22no—
93C4×D4—22no—
94#117C2×D10712yes—
95C2×C4⋊C4—1071no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—2142no—
98C2×C42—1071no—
99#107—2142no—
100#106—2142no—
101#100—1074no—
102C24C22⋊D10712yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4Dic21412yes—
106C2×D4C4×D10712yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—2141no—
110C22×C4—2141no—
111C22×C4C22×D10711yescharacteristic
112M16—1074no—
113M16—2142no—
114C2×C8—2142no—
115C4⋊C4—2142no—
116C4⋊C4C22⋊D10712yes—
117C4⋊C4C22⋊D10712yes—
118C22⋊C4—24no—
119C22⋊C4—22no—
120C22⋊C4—4284no—
121C42—22no—
122C42—2142no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23Dic107⋊C414yes—
128C23C2×C22⋊D10711yescharacteristic
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—4282no—
136C2×C4—4281no—
137C2×C4—4281no—
138C2×C4—4281no—
139C2×C4—2142no—
140C2×C4C2×C4×D10711yescharacteristic
141C2×C4C2×C22⋊D10712yes—
142C2×C4C2×Dic21411yescharacteristic
143C2×C4#1313381812yes—
144C8—2144no—
145C22—21no—
146C22—21no—
147C22—24no—
148C22—44no—
149C22—42no—
150C22—42no—
151C22—82no—
152C22C2×Dic107⋊C411yescharacteristic
153C22#1681207811yescharacteristic
154C22#1681209211yescharacteristic
155C4—42no—
156C4—42no—
157C4—8561no—
158C4—4282no—
159C4C4×C22⋊D10712yes—
160C4#1681208812yes—
161C2—21no—
162C2—21no—
163C2—44no—
164C2#2029250011yescharacteristic
165C2#2029279511yescharacteristic
166C2#2029280511yescharacteristic
subgroupG/H
centerC22#16812078
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#518×C107C2
socleC22×C107#120
prime radicalC22×D4×C107C22
cosocleC22×C4×C107C23
hypercenter#518D107
hypocenterC107#1856
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20292670C2semidirect
4. #20292386C2semidirect
12. #16812026C4semidirect
21. #13133802C2×C4semidirect
84. C107#1856semidirect

Maximal subgroups

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

subgroupindex
#202926702
C2×#168120262
#202925302
#202923862
#518×C1072
#1856107