Finite Group Databasebeta

← #28378461

Subgroups — #28378461

Order 13952.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#20296044C211yesmaximal, characteristic, complemented
2#20295970C211yesmaximal, characteristic, complemented
3#20295877C214yesmaximal, complemented
4#20295574C211yesmaximal, characteristic
5C2×C4⋊#89137—21no—
6#16812766—21no—
7#16812756C2212yes—
8C4×#11724786—21no—
9(C2×#89137)⋊C4—21no—
10(C2×#89137)⋊C4C2212yes—
11C2×#13182028—21no—
12#16812699C411yescharacteristic, complemented
13#16812686—21no—
14#16812670—21no—
15#16812670C2212yes—
16D4×Dic109—21no—
17D4×Dic109C412yescomplemented
18C42×D109C2211yescharacteristic
19#127×C109C411yescharacteristic, complemented
20C22×#89137—22no—
21C2×#11724786—22no—
22C4⋊#89137—42no—
23#13182028—42no—
24#13182028—42no—
25C2×Dic218—21no—
26C2×C22⋊D109—22no—
27C22×Dic109—22no—
28C2×C4×D109D411yescharacteristic
29C2×C4×D109D411yescharacteristic
30C2×C4×D109C2311yescharacteristic
31C4⋊Dic109—21no—
32C4⋊Dic109C2×C412yes—
33C22⋊Dic109—22no—
34C22⋊Dic109—22no—
35C4×Dic109C2×C411yescharacteristic
36C4×Dic109D411yescharacteristic
37C4×Dic109D411yescharacteristic
38C2×D4×C109—21no—
39C2×D4×C109—21no—
40C2×D4×C109C2×C412yes—
41C42×C109C2×C411yescharacteristic
42C23×C109—22no—
43C22×D109C2×D411yescharacteristic
44C2×#89137—42no—
45C2×#89137—42no—
46C2×#89137—42no—
47D4×C109—42no—
48D4×C109—42no—
49D4×C109—42no—
50C2×C4×C109C22⋊C411yescharacteristic
51C2×C4×C109C22⋊C411yescharacteristic
52C2×C4×C109C22×C411yescharacteristic
53C22⋊D109—42no—
54C22⋊D109—42no—
55C2×Dic109—22no—
56C2×Dic109—44no—
57C2×Dic109C22⋊C411yescharacteristic
58C2×Dic109C22⋊C411yescharacteristic
59C2×Dic109C2×D411yescharacteristic
60C4×D109—41no—
61C4×D109—41no—
62C4×D109D812yes—
63C4×D109SD1612yes—
64Dic218—42no—
65#11724786—42no—
66C22×C109—42no—
67C22×C109—42no—
68C22×C109—42no—
69C22×C109C2×C22⋊C411yescharacteristic
70C2×D109—21no—
71C2×D109—21no—
72C2×D109C2×D811yescharacteristic
73C2×D109C2×SD1611yescharacteristic
74#89137—82no—
75C4×C109—21no—
76C4×C109—21no—
77C4×C109#10212yes—
78Dic109—21no—
79Dic109—21no—
80Dic109—42no—
81Dic109#10212yes—
82C2×C109—82no—
83C2×C109#41011yescharacteristic
84C2×C109C2×#10211yescharacteristic
85C2×C109#42111yescharacteristic
86D109—21no—
87D109—21no—
88#1517—1091nomaximal, Sylow
89C109#151711yescharacteristic, complemented, Sylow
90#532—1091no—
91#431—1091no—
92#382—1091no—
93#327—1094no—
94C2×#41—2181no—
95C4⋊Q8—1091no—
96#127#8913711yescharacteristic, complemented
97#124—2181no—
98#121—1092no—
99#121—2181no—
100C4×D4—1092no—
101C4×D4—2181no—
102C2×C4⋊C4—2181no—
103C2×C42—1091no—
104C4⋊C8—2181no—
105C22⋊C8—1092no—
106C4×C8—2181no—
107#95—1092no—
108#95—2181no—
109#41—4362no—
110#41—4362no—
111C2×Q8—2181no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—2182no—
115C2×D4C2×#8913712yes—
116C22×C4—1091no—
117C22×C4—1091no—
118C22×C4—1091no—
119C22×C4—2182no—
120C22×C4—2182no—
121C2×C8—2182no—
122C4⋊C4—1092no—
123C4⋊C4—4362no—
124C4⋊C4—2181no—
125C22⋊C4—2182no—
126C22⋊C4—2182no—
127C42—1091no—
128C42—1091no—
129C42—1091no—
130C42C2×#8913711yescharacteristic
131C23—22no—
132C23—1091no—
133Q8—4362no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—4362no—
138D4—4362no—
139C2×C4—1091no—
140C2×C4—1092no—
141C2×C4—1092no—
142C2×C4—1091no—
143C2×C4—1091no—
144C2×C4—4361no—
145C2×C4—4361no—
146C2×C4—4362no—
147C2×C4—4362no—
148C2×C4—4362no—
149C2×C4—4364no—
150C2×C4—2182no—
151C2×C4C22⋊#8913711yescharacteristic
152C2×C4C22⋊#8913711yescharacteristic
153C2×C4C22×#8913711yescharacteristic
154C8—4362no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22—1091no—
159C22—1091no—
160C22—2181no—
161C22—2181no—
162C22C2×C22⋊#8913711yescharacteristic
163C4—21no—
164C4—21no—
165C4—8722no—
166C4—1092no—
167C4—4362no—
168C4—2181no—
169C4—2181no—
170C4#1681276112yes—
171C2—82no—
172C2—2181no—
173C2—2181no—
174C2#2029600611yescharacteristic
175C2C2×#1681276111yescharacteristic
176C2#2029602411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#127×C109C4
socleC22×C109C2×C22⋊C4
prime radical#20295574C2
cosocleC2×C4×D109C23
hypercenter#127#89137
hypocenterC109#1517
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20296044C2semidirect
2. #20295970C2semidirect
3. #20295877C2semidirect
12. #16812699C4semidirect
17. D4×Dic109C4semidirect
19. #127×C109C4semidirect
89. C109#1517semidirect
96. #127#89137semidirect

Maximal subgroups

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

subgroupindex
#202960442
#202959702
#202958772
#202955742
#1517109