Finite Group Databasebeta

← #28335625

Subgroups — #28335625

Order 13696.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic214C211yesmaximal, characteristic
2#20292977C211yesmaximal, characteristic, complemented
3#20292919C211yesmaximal, characteristic, complemented
4#20292372C211yesmaximal, characteristic, complemented
5#20292357C211yesmaximal, characteristic, complemented
6D4⋊(C107⋊C8)C211yesmaximal, characteristic
7#476×C107C211yesmaximal, characteristic
8C22×Dic214—21no—
9#16812109C2211yescharacteristic
10#16812108—21no—
11#16812088—21no—
12#16812086—22no—
13C4×Dic214C2211yescharacteristic
14D4×Dic107—21no—
15#16812042—21no—
16C4⋊(C107⋊C8)C2211yescharacteristic
17#16812033—22no—
18#16812032—21no—
19C4×C107⋊C8C2211yescharacteristic
20C4⋊Q8×C107C2211yescharacteristic
21C4×D4×C107C2211yescharacteristic
22#106×C107—21no—
23C4⋊C8×C107C2211yescharacteristic
24#102×C107—21no—
25C2×Dic214—22no—
26C2×Dic214—21no—
27C2×Dic214—42no—
28C2×Dic214—42no—
29C2×Dic214D411yescharacteristic
30C22×Dic107—22no—
31C2×C107⋊C8—21no—
32C2×C107⋊C8—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107D412yes—
35C4⋊Dic107D411yescharacteristic
36Dic107⋊C4—22no—
37Dic107⋊C4—21no—
38C22⋊Dic107—22no—
39C4×Dic107—21no—
40C2×Q8×C107—21no—
41C2×D4×C107D411yescharacteristic
42C22×C4×C107—21no—
43C2×C8×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107D411yescharacteristic
47C22⋊C4×C107—21no—
48C42×C107C2311yescharacteristic
49C23×C107—21no—
50Q8×C107—42no—
51D4×C107—21no—
52D4×C107Q1612yes—
53C2×C4×C107—21no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107—41no—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C2×Dic107—21no—
61C2×Dic107—21no—
62C2×Dic107—22no—
63C2×Dic107—44no—
64Dic214—21no—
65Dic214—44no—
66Dic214—41no—
67Dic214—41no—
68Dic214SD1612yes—
69C8×C107—41no—
70C107⋊C8—41no—
71C107⋊C8—41no—
72C22×C107—22no—
73C22×C107—41no—
74C22×C107—41no—
75C22×C107#12011yescharacteristic
76C4×C107—21no—
77C4×C107—41no—
78C4×C107—81no—
79C4×C107C2×SD1611yescharacteristic
80C4×C107C2×Q1611yescharacteristic
81C4×C107AGL(1,ℤ/8)11yescharacteristic
82C4×C107#13711yescharacteristic
83Dic107—22no—
84Dic107—42no—
85Dic107—41no—
86C2×C107—42no—
87C2×C107#45011yescharacteristic
88C2×C107#45111yescharacteristic
89C2×C107#45411yescharacteristic
90#1637—1071nomaximal, Sylow
91C107#163711yescharacteristic, complemented, Sylow
92D4×Q8—1071no—
93#487—1071no—
94#477—1071no—
95#476D10711yescharacteristic
96#332—1071no—
97Q8⋊C8—1071no—
98D4⋊C8—1071no—
99C22×Q8—2141no—
100C4⋊Q8—1071no—
101C4⋊Q8—2141no—
102C4⋊Q8C2×D10711yescharacteristic
103C22⋊Q8—2142no—
104C22⋊Q8—2141no—
105C4×Q8—1071no—
106C4×D4—2141no—
107C4×D4C2×D10711yescharacteristic
108#107—2141no—
109#106—21no—
110C4⋊C8—1071no—
111C4⋊C8C2×D10711yescharacteristic
112#103—2141no—
113#103—2142no—
114#102—21no—
115C4×C8—1071no—
116C2×Q8—21no—
117C2×Q8—1071no—
118C2×Q8—4282no—
119C2×Q8—4282no—
120C2×Q8—2142no—
121C2×Q8—2141no—
122C2×D4C22⋊D10711yescharacteristic
123C22×C4—21no—
124C22×C4—2142no—
125C2×C8—21no—
126C2×C8—2141no—
127C2×C8—2141no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1072no—
131C4⋊C4—1071no—
132C4⋊C4—2142no—
133C4⋊C4—2141no—
134C4⋊C4—2141no—
135C4⋊C4C22⋊D10711yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—2142no—
138C42—2141no—
139C42C22×D10711yescharacteristic
140C23—21no—
141Q8—42no—
142Q8—1072no—
143Q8—4284no—
144Q8—4281no—
145Q8—4281no—
146Q8—2141no—
147D4—21no—
148D4#1313380812yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—4284no—
154C2×C4—2141no—
155C2×C4—2142no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×C22⋊D10711yescharacteristic
160C8—41no—
161C8—4281no—
162C8—4281no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1681207711yescharacteristic
167C4—21no—
168C4—41no—
169C4—81no—
170C4—4282no—
171C4—4281no—
172C4—2142no—
173C4SD16×D10711yescharacteristic
174C4C2×#1313380811yescharacteristic
175C4#1681213611yescharacteristic
176C4#1681213911yescharacteristic
177C2—42no—
178C2#2029277611yescharacteristic
179C2#2029278211yescharacteristic
180C2#2029279811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#476×C107C2
socleC22×C107#120
prime radicalC2×D4×C107D4
hypercenter#476D107
hypocenterC107#1637
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20292977C2semidirect
3. #20292919C2semidirect
4. #20292372C2semidirect
5. #20292357C2semidirect
91. C107#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2142
#202929772
#202929192
#202923722
#202923572
D4⋊(C107⋊C8)2
#476×C1072
#1637107