Finite Group Databasebeta

← #28378423

Subgroups — #28378423

Order 13952.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#20296048C211yesmaximal, characteristic, complemented
2#20295962C211yesmaximal, characteristic, complemented
3#20295878C214yesmaximal, complemented
4#20295577C211yesmaximal, characteristic
5C2×C4×#89137—21no—
6C4⋊#11724786—21no—
7#16812756C2212yes—
8#16812754—21no—
9(C2×#89137)⋊C4—21no—
10(C2×#89137)⋊C4C2212yes—
11C2×#13182030—21no—
12#16812696C411yescharacteristic, complemented
13#16812688—21no—
14#16812674—21no—
15#16812674C2212yes—
16#16812661—21no—
17#16812661C412yescomplemented
18C42×D109C2211yescharacteristic
19C4⋊Q8×C109C411yescharacteristic, complemented
20C22×#89137—22no—
21C2×#11724786—22no—
22C4×#89137—24no—
23#13182030—44no—
24C2×D436—22no—
25C2×D436—21no—
26C2×D436—21no—
27C2×D436C2×C412yes—
28C2×C4×D109—22no—
29C2×C4×D109D411yescharacteristic
30C2×C4×D109D411yescharacteristic
31C2×C4×D109C2311yescharacteristic
32#13182006—22no—
33#13182006—22no—
34C4×Dic109C2×C411yescharacteristic
35C4×Dic109D411yescharacteristic
36C4×Dic109D411yescharacteristic
37C2×Q8×C109—21no—
38C4⋊C4×C109—21no—
39C4⋊C4×C109C2×C412yes—
40C42×C109C2×C411yescharacteristic
41C22×D109—22no—
42C22×D109C2×D411yescharacteristic
43C2×#89137—24no—
44C2×#89137—42no—
45C2×#89137—42no—
46Q8×C109—42no—
47C2×C4×C109—22no—
48C2×C4×C109C22⋊C411yescharacteristic
49C2×C4×C109C22⋊C411yescharacteristic
50C2×C4×C109C22×C411yescharacteristic
51C2×Dic109C22⋊C411yescharacteristic
52C2×Dic109C22⋊C411yescharacteristic
53C2×Dic109C2×D411yescharacteristic
54D436—42no—
55D436—42no—
56D436—44no—
57D436—42no—
58C4×D109—22no—
59C4×D109—22no—
60C4×D109—41no—
61C4×D109—44no—
62#11724786—42no—
63C22×C109C2×C22⋊C411yescharacteristic
64C2×D109—21no—
65C2×D109—21no—
66C2×D109—42no—
67C2×D109—42no—
68C2×D109—42no—
69C2×D109AGL(1,ℤ/8)12yes—
70#89137—44no—
71C4×C109—21no—
72C4×C109—21no—
73C4×C109—42no—
74C4×C109#10412yes—
75Dic109—21no—
76Dic109—21no—
77Dic109—21no—
78C2×C109#41011yescharacteristic
79C2×C109#41811yescharacteristic
80C2×C109C2×#10411yescharacteristic
81D109—22no—
82D109—82no—
83#1505—1091nomaximal, Sylow
84C109#150511yescharacteristic, complemented, Sylow
85#532—1091no—
86#432—1091no—
87#376—1091no—
88#327—1094no—
89C2×#41—2181no—
90C4⋊Q8#8913711yescharacteristic, complemented
91#127—1091no—
92#124—2181no—
93#121—1092no—
94#121—2181no—
95C4×D4—1092no—
96C4×D4—2181no—
97C2×C42—1091no—
98C2×C42—2181no—
99C4⋊C8—2181no—
100C22⋊C8—1092no—
101#97—2181no—
102#95—1092no—
103#95—2181no—
104#41—4364no—
105C2×Q8—21no—
106C2×D4—1092no—
107C2×D4—2182no—
108C2×D4—2181no—
109C2×D4—2181no—
110C22×C4—1091no—
111C22×C4—1091no—
112C22×C4—1091no—
113C22×C4—2182no—
114C22×C4—2182no—
115C2×C8—2182no—
116C4⋊C4—21no—
117C4⋊C4C2×#8913712yes—
118C22⋊C4—2182no—
119C22⋊C4—2182no—
120C42—1091no—
121C42—1091no—
122C42—1091no—
123C42—2184no—
124C42C2×#8913711yescharacteristic
125C23—1091no—
126C23—2182no—
127Q8—42no—
128D4—4362no—
129D4—4362no—
130D4—4364no—
131D4—4362no—
132C2×C4—22no—
133C2×C4—1091no—
134C2×C4—1091no—
135C2×C4—1091no—
136C2×C4—4361no—
137C2×C4—4364no—
138C2×C4—4362no—
139C2×C4—4362no—
140C2×C4—2182no—
141C2×C4—2184no—
142C2×C4—2182no—
143C2×C4C22⋊#8913711yescharacteristic
144C2×C4C22⋊#8913711yescharacteristic
145C2×C4C22×#8913711yescharacteristic
146C8—4362no—
147C22—1092no—
148C22—4362no—
149C22—4362no—
150C22—4362no—
151C22—2181no—
152C22—2181no—
153C22C2×C22⋊#8913711yescharacteristic
154C4—21no—
155C4—21no—
156C4—42no—
157C4—4364no—
158C4—2181no—
159C4—2181no—
160C4—2181no—
161C4#1681276412yes—
162C2—8722no—
163C2—2182no—
164C2#2029600711yescharacteristic
165C2#2029601811yescharacteristic
166C2C2×#1681276411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC4⋊Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radical#20295577C2
cosocleC2×C4×D109C23
hypercenterC4⋊Q8#89137
hypocenterC109#1505
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20296048C2semidirect
2. #20295962C2semidirect
3. #20295878C2semidirect
12. #16812696C4semidirect
17. #16812661C4semidirect
19. C4⋊Q8×C109C4semidirect
84. C109#1505semidirect
90. C4⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202960482
#202959622
#202958782
#202955772
#1505109