Finite Group Databasebeta

← #28378412

Subgroups — #28378412

Order 13952.

284 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 70 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#20296045C211yesmaximal, characteristic, complemented
2C4×C4⋊#89137C211yesmaximal, characteristic, complemented
3#20295877C214yesmaximal, complemented
4#20295574C211yesmaximal, characteristic
5C2×C4⋊#89137—21no—
6C2×C4×#89137—21no—
7C4⋊#11724786—21no—
8#16812756C2212yes—
9C4×#11724786—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—24no—
23C4×#89137—42no—
24#13182028—42no—
25#13182028—42no—
26C2×Dic218—21no—
27C2×C22⋊D109—22no—
28C22×Dic109—22no—
29C2×C4×D109D411yescharacteristic
30C2×C4×D109D411yescharacteristic
31C2×C4×D109C2311yescharacteristic
32C4⋊Dic109—21no—
33C4⋊Dic109C2×C412yes—
34C22⋊Dic109—22no—
35C22⋊Dic109—22no—
36C4×Dic109C2×C411yescharacteristic
37C4×Dic109D411yescharacteristic
38C4×Dic109D411yescharacteristic
39C2×D4×C109—21no—
40C2×D4×C109—21no—
41C2×D4×C109C2×C412yes—
42C42×C109C2×C411yescharacteristic
43C23×C109—22no—
44C22×D109C2×D411yescharacteristic
45C2×#89137—24no—
46C2×#89137—42no—
47C2×#89137—42no—
48D4×C109—42no—
49D4×C109—42no—
50D4×C109—42no—
51C2×C4×C109C22⋊C411yescharacteristic
52C2×C4×C109C22⋊C411yescharacteristic
53C2×C4×C109C22×C411yescharacteristic
54C22⋊D109—42no—
55C22⋊D109—42no—
56C2×Dic109—22no—
57C2×Dic109—44no—
58C2×Dic109C22⋊C411yescharacteristic
59C2×Dic109C22⋊C411yescharacteristic
60C2×Dic109C2×D411yescharacteristic
61C4×D109—22no—
62C4×D109—41no—
63C4×D109D812yes—
64C4×D109SD1612yes—
65Dic218—42no—
66#11724786—42no—
67C22×C109—42no—
68C22×C109—42no—
69C22×C109—42no—
70C22×C109C2×C22⋊C411yescharacteristic
71C2×D109—21no—
72C2×D109—21no—
73C2×D109C2×D811yescharacteristic
74C2×D109C2×SD1611yescharacteristic
75#89137—44no—
76C4×C109—21no—
77C4×C109#10212yes—
78C4×C109#10412yes—
79Dic109—21no—
80Dic109—21no—
81Dic109—42no—
82Dic109#10212yes—
83C2×C109—82no—
84C2×C109#41011yescharacteristic
85C2×C109C2×#10211yescharacteristic
86C2×C109C2×#10411yescharacteristic
87D109—21no—
88D109—21no—
89#1502—1091nomaximal, Sylow
90C109#150211yescharacteristic, complemented, Sylow
91#532—1091no—
92#431—1091no—
93C4×C4⋊C4—1091no—
94#327—1094no—
95C2×#41—2181no—
96C4⋊Q8—1091no—
97#127#8913711yescharacteristic, complemented
98#124—2181no—
99#121—1092no—
100#121—2181no—
101C4×D4—1092no—
102C4×D4—2181no—
103C2×C4⋊C4—2181no—
104C2×C42—1091no—
105C2×C42—2181no—
106C4⋊C8—2181no—
107C22⋊C8—1092no—
108C4×C8—2181no—
109#95—1092no—
110#41—4362no—
111#41—4362no—
112C2×Q8—2181no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—2182no—
116C2×D4C2×#8913712yes—
117C22×C4—1091no—
118C22×C4—1091no—
119C22×C4—1091no—
120C22×C4—2182no—
121C22×C4—2182no—
122C2×C8—2182no—
123C4⋊C4—1092no—
124C4⋊C4—2184no—
125C4⋊C4—2181no—
126C22⋊C4—2182no—
127C22⋊C4—2182no—
128C42—1091no—
129C42—1091no—
130C42—1091no—
131C42—4362no—
132C42C2×#8913711yescharacteristic
133C23—22no—
134C23—1091no—
135Q8—4362no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—4362no—
140D4—4362no—
141C2×C4—1091no—
142C2×C4—1092no—
143C2×C4—1092no—
144C2×C4—1091no—
145C2×C4—1091no—
146C2×C4—4361no—
147C2×C4—4364no—
148C2×C4—4362no—
149C2×C4—4362no—
150C2×C4—2184no—
151C2×C4—2182no—
152C2×C4—2182no—
153C2×C4C22⋊#8913711yescharacteristic
154C2×C4C22⋊#8913711yescharacteristic
155C2×C4C22×#8913711yescharacteristic
156C8—4362no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—1091no—
161C22—1091no—
162C22—2181no—
163C22—2181no—
164C22C2×C22⋊#8913711yescharacteristic
165C4—21no—
166C4—1092no—
167C4—4364no—
168C4—4362no—
169C4—2181no—
170C4—2181no—
171C4#1681276112yes—
172C4#1681276312yes—
173C2—82no—
174C2—2181no—
175C2—2181no—
176C2#2029600611yescharacteristic
177C2C2×#1681276111yescharacteristic
178C2C2×#1681276311yescharacteristic
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#1502
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. #20296045C2semidirect
2. C4×C4⋊#89137C2semidirect
3. #20295877C2semidirect
12. #16812699C4semidirect
17. D4×Dic109C4semidirect
19. #127×C109C4semidirect
90. C109#1502semidirect
97. #127#89137semidirect

Maximal subgroups

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

subgroupindex
#202960452
C4×C4⋊#891372
#202958772
#202955742
#1502109