Finite Group Databasebeta

← #28378453

Subgroups — #28378453

Order 13952.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#20296049C211yesmaximal, characteristic, complemented
2#20295971C211yesmaximal, characteristic, complemented
3#20295878C214yesmaximal, complemented
4#20295577C211yesmaximal, characteristic
5#16812766—21no—
6#16812756C2212yes—
7#16812754—21no—
8(C2×#89137)⋊C4—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—
22#13182030—44no—
23C2×D436—22no—
24C2×D436—21no—
25C2×D436—21no—
26C2×D436C2×C412yes—
27C2×C4×D109—22no—
28C2×C4×D109D411yescharacteristic
29C2×C4×D109D411yescharacteristic
30C2×C4×D109C2311yescharacteristic
31#13182006—22no—
32#13182006—22no—
33C4×Dic109C2×C411yescharacteristic
34C4×Dic109D411yescharacteristic
35C4×Dic109D411yescharacteristic
36C2×Q8×C109—21no—
37C4⋊C4×C109—21no—
38C4⋊C4×C109C2×C412yes—
39C42×C109C2×C411yescharacteristic
40C22×D109—22no—
41C22×D109C2×D411yescharacteristic
42C2×#89137—42no—
43C2×#89137—42no—
44C2×#89137—42no—
45Q8×C109—42no—
46C2×C4×C109—22no—
47C2×C4×C109C22⋊C411yescharacteristic
48C2×C4×C109C22⋊C411yescharacteristic
49C2×C4×C109C22×C411yescharacteristic
50C2×Dic109C22⋊C411yescharacteristic
51C2×Dic109C22⋊C411yescharacteristic
52C2×Dic109C2×D411yescharacteristic
53D436—42no—
54D436—42no—
55D436—42no—
56D436—44no—
57C4×D109—22no—
58C4×D109—41no—
59C4×D109—41no—
60C4×D109—44no—
61#11724786—42no—
62C22×C109C2×C22⋊C411yescharacteristic
63C2×D109—21no—
64C2×D109—21no—
65C2×D109—42no—
66C2×D109—42no—
67C2×D109—42no—
68C2×D109AGL(1,ℤ/8)12yes—
69#89137—82no—
70C4×C109—21no—
71C4×C109—21no—
72C4×C109—21no—
73C4×C109—42no—
74Dic109—21no—
75Dic109—21no—
76Dic109—21no—
77C2×C109#41011yescharacteristic
78C2×C109#41811yescharacteristic
79C2×C109#42111yescharacteristic
80D109—22no—
81D109—82no—
82#1514—1091nomaximal, Sylow
83C109#151411yescharacteristic, complemented, Sylow
84#532—1091no—
85#432—1091no—
86#383—1091no—
87#327—1094no—
88C2×#41—2181no—
89C4⋊Q8#8913711yescharacteristic, complemented
90#127—1091no—
91#124—2181no—
92#121—1092no—
93#121—2181no—
94C4×D4—1092no—
95C4×D4—2181no—
96C2×C42—1091no—
97C4⋊C8—2181no—
98C22⋊C8—1092no—
99#97—2181no—
100#95—1092no—
101#95—2181no—
102#95—2181no—
103#41—4364no—
104C2×Q8—21no—
105C2×D4—1092no—
106C2×D4—2182no—
107C2×D4—2181no—
108C2×D4—2181no—
109C22×C4—1091no—
110C22×C4—1091no—
111C22×C4—1091no—
112C22×C4—2182no—
113C22×C4—2182no—
114C2×C8—2182no—
115C4⋊C4—21no—
116C4⋊C4C2×#8913712yes—
117C22⋊C4—2182no—
118C22⋊C4—2182no—
119C42—1091no—
120C42—1091no—
121C42—1091no—
122C42C2×#8913711yescharacteristic
123C23—1091no—
124C23—2182no—
125Q8—42no—
126D4—4362no—
127D4—4362no—
128D4—4362no—
129D4—4364no—
130C2×C4—22no—
131C2×C4—1091no—
132C2×C4—1091no—
133C2×C4—1091no—
134C2×C4—4361no—
135C2×C4—4361no—
136C2×C4—4364no—
137C2×C4—4362no—
138C2×C4—4362no—
139C2×C4—4362no—
140C2×C4—2182no—
141C2×C4C22⋊#8913711yescharacteristic
142C2×C4C22⋊#8913711yescharacteristic
143C2×C4C22×#8913711yescharacteristic
144C8—4362no—
145C22—1092no—
146C22—4362no—
147C22—4362no—
148C22—4362no—
149C22—2181no—
150C22—2181no—
151C22C2×C22⋊#8913711yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—8722no—
157C4—2181no—
158C4—2181no—
159C4—2181no—
160C2—8722no—
161C2—2182no—
162C2#2029600711yescharacteristic
163C2#2029601811yescharacteristic
164C2#2029602511yescharacteristic
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#1514
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. #20296049C2semidirect
2. #20295971C2semidirect
3. #20295878C2semidirect
12. #16812696C4semidirect
17. #16812661C4semidirect
19. C4⋊Q8×C109C4semidirect
83. C109#1514semidirect
89. C4⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202960492
#202959712
#202958782
#202955772
#1514109