Finite Group Databasebeta

← #28362194

Subgroups — #28362194

Order 13952.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#20295691C211yesmaximal, characteristic
2#20295318C211yesmaximal, characteristic, complemented
3#20295313C211yesmaximal, characteristic
4#20294772C211yesmaximal, characteristic, complemented
5#20294758C211yesmaximal, characteristic, complemented
6#20294755C211yesmaximal, characteristic
7#487×C109C211yesmaximal, characteristic, complemented
8C2×Q8×D109—21no—
9#16812698—21no—
10#16812677—22no—
11#16812675—21no—
12C4×Dic218C2211yescharacteristic
13C4×D436C2211yescharacteristic
14#16812661—21no—
15#16812632—21no—
16#16812630—21no—
17C4⋊#11724784C2211yescharacteristic
18C4⋊#11724784C2211yescharacteristic
19#16812622—21no—
20#16812619—21no—
21C4⋊Q8×C109C2211yescharacteristic
22C4⋊Q8×C109C2211yescharacteristic
23#103×C109—22no—
24C4×C8×C109C2211yescharacteristic
25C2×Dic218—22no—
26C2×Dic218D411yescharacteristic
27Q8×D109—42no—
28Q8×D109—42no—
29C2×D436D411yescharacteristic
30C2×C4×D109—22no—
31C2×C4×D109—21no—
32C2×#11724784—21no—
33C2×#11724784—21no—
34C4⋊Dic109D411yescharacteristic
35C4⋊Dic109D411yescharacteristic
36#13182008—22no—
37#13182008—21no—
38#13182006—22no—
39#13182006—21no—
40C4×Dic109—21no—
41C2×Q8×C109—21no—
42C2×Q8×C109—21no—
43C2×C8×C109—21no—
44C4⋊C4×C109—21no—
45C4⋊C4×C109—21no—
46C4⋊C4×C109—21no—
47C4⋊C4×C109D412yes—
48C42×C109C2311yescharacteristic
49C22×D109—21no—
50Q8×C109—42no—
51Q8×C109—41no—
52Q8×C109—41no—
53C2×C4×C109—22no—
54C2×C4×C109—41no—
55C2×C4×C109C2×D411yescharacteristic
56C2×C4×C109C2×D411yescharacteristic
57C2×C4×C109C2×D411yescharacteristic
58C2×Dic109—21no—
59C2×Dic109—21no—
60C2×Dic109—21no—
61D436—21no—
62D436Q1612yes—
63C4×D109—44no—
64C4×D109—41no—
65C4×D109—41no—
66Dic218—21no—
67Dic218—44no—
68Dic218SD1612yes—
69C8×C109—41no—
70#11724784—41no—
71#11724784—41no—
72C22×C109#12011yescharacteristic
73C2×D109—22no—
74C2×D109—41no—
75C2×D109—41no—
76C4×C109—21no—
77C4×C109—42no—
78C4×C109—81no—
79C4×C109C2×SD1611yescharacteristic
80C4×C109C2×Q1611yescharacteristic
81C4×C109AGL(1,ℤ/8)11yescharacteristic
82C4×C109#13711yescharacteristic
83Dic109—22no—
84Dic109—41no—
85Dic109—41no—
86C2×C109#45011yescharacteristic
87C2×C109#45111yescharacteristic
88C2×C109#45411yescharacteristic
89D109—42no—
90#1637—1091nomaximal, Sylow
91C109#163711yescharacteristic, complemented, Sylow
92D4×Q8—1091no—
93#487D10911yescharacteristic, complemented
94#477—1091no—
95#476—1091no—
96#332—1091no—
97Q8⋊C8—1091no—
98D4⋊C8—1091no—
99C22×Q8—2181no—
100C4⋊Q8—2181no—
101C4⋊Q8C2×D10911yescharacteristic
102C4⋊Q8C2×D10911yescharacteristic
103C22⋊Q8—2181no—
104C22⋊Q8—2182no—
105C4×Q8—1091no—
106C4×D4—1091no—
107C4×D4—2181no—
108#107—2181no—
109#106—2181no—
110C4⋊C8—1091no—
111C4⋊C8—1091no—
112#103—22no—
113#103—2181no—
114#102—2181no—
115C4×C8C2×D10911yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—1091no—
119C2×Q8—4362no—
120C2×Q8—4362no—
121C2×Q8—2182no—
122C2×D4—1091no—
123C22×C4—2182no—
124C22×C4—2181no—
125C2×C8—21no—
126C2×C8—2181no—
127C2×C8—2181no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1091no—
132C4⋊C4—1091no—
133C4⋊C4—2182no—
134C4⋊C4—2181no—
135C4⋊C4C22⋊D10912yes—
136C22⋊C4—2182no—
137C22⋊C4—2181no—
138C42—2181no—
139C42C22×D10911yescharacteristic
140C23—2181no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—1092no—
145Q8—4364no—
146Q8—2181no—
147D4—1092no—
148D4—2181no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—4364no—
152C2×C4—4361no—
153C2×C4—4361no—
154C2×C4—2181no—
155C2×C4—2181no—
156C2×C4—2181no—
157C2×C4D4×D10911yescharacteristic
158C2×C4D4×D10911yescharacteristic
159C2×C4C2×C22⋊D10911yescharacteristic
160C8—41no—
161C8—4361no—
162C8—4361no—
163C22—4361no—
164C22—4361no—
165C22—2182no—
166C22#1681266711yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—4361no—
171C4—4361no—
172C4—2182no—
173C4SD16×D10911yescharacteristic
174C4Q16×D10911yescharacteristic
175C4#1681272611yescharacteristic
176C4#1681273111yescharacteristic
177C2—4362no—
178C2#2029517511yescharacteristic
179C2#2029518011yescharacteristic
180C2#2029519811yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#487×C109C2
socleC22×C109#120
prime radicalC2×D436D4
hypercenter#487D109
hypocenterC109#1637
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20295318C2semidirect
4. #20294772C2semidirect
5. #20294758C2semidirect
7. #487×C109C2semidirect
91. C109#1637semidirect
93. #487D109semidirect

Maximal subgroups

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

subgroupindex
#202956912
#202953182
#202953132
#202947722
#202947582
#202947552
#487×C1092
#1637109