Finite Group Databasebeta

← #28335624

Subgroups — #28335624

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
1Q8×D428C211yesmaximal, characteristic
2#20292977C211yesmaximal, characteristic, complemented
3#20292914C211yesmaximal, characteristic
4#20292374C211yesmaximal, characteristic, complemented
5Q8⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic
7#477×C107C211yesmaximal, characteristic, complemented
8C2×Q8×D107—21no—
9#16812109—21no—
10#16812109C2211yescharacteristic
11#16812089—22no—
12#16812089—21no—
13C4×D428—21no—
14C4×D428C2211yescharacteristic
15#16812040—21no—
16C4⋊(C107⋊C8)C2211yescharacteristic
17#16812033—22no—
18#16812029—21no—
19C4×C107⋊C8C2211yescharacteristic
20C4⋊Q8×C107C2211yescharacteristic
21C4×Q8×C107C2211yescharacteristic
22#107×C107—21no—
23C4⋊C8×C107C2211yescharacteristic
24#103×C107—21no—
25C2×Dic214—22no—
26C2×Dic214—21no—
27Q8×D107—42no—
28Q8×D107—42no—
29C2×D428D411yescharacteristic
30C2×C4×D107—22no—
31C2×C4×D107—21no—
32C2×C107⋊C8—21no—
33C2×C107⋊C8—21no—
34C4⋊Dic107—22no—
35C4⋊Dic107—21no—
36C4⋊Dic107D412yes—
37C4⋊Dic107D411yescharacteristic
38(C2×D107)⋊C4—22no—
39(C2×D107)⋊C4—21no—
40C2×Q8×C107—21no—
41C2×Q8×C107D411yescharacteristic
42C2×C8×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107D411yescharacteristic
47C42×C107—21no—
48C42×C107C2311yescharacteristic
49C22×D107—21no—
50Q8×C107—21no—
51Q8×C107—42no—
52Q8×C107SD1612yes—
53C2×C4×C107—21no—
54C2×C4×C107—21no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×Dic107—22no—
60C2×Dic107—21no—
61D428—21no—
62D428Q1612yes—
63C4×D107—44no—
64C4×D107—41no—
65C4×D107—41no—
66Dic214—44no—
67Dic214—41no—
68Dic214—41no—
69C8×C107—41no—
70C107⋊C8—41no—
71C107⋊C8—41no—
72C22×C107#12011yescharacteristic
73C2×D107—22no—
74C2×D107—41no—
75C2×D107—41no—
76C4×C107—22no—
77C4×C107—21no—
78C4×C107—41no—
79C4×C107—81no—
80C4×C107C2×SD1611yescharacteristic
81C4×C107C2×Q1611yescharacteristic
82C4×C107AGL(1,ℤ/8)11yescharacteristic
83C4×C107#13711yescharacteristic
84Dic107—41no—
85Dic107—42no—
86C2×C107#45011yescharacteristic
87C2×C107#45111yescharacteristic
88C2×C107#45411yescharacteristic
89D107—42no—
90#1637—1071nomaximal, Sylow
91C107#163711yescharacteristic, complemented, Sylow
92D4×Q8—1071no—
93#487—1071no—
94#477D10711yescharacteristic, complemented
95#476—1071no—
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×Q8C2×D10711yescharacteristic
106C4×D4—1071no—
107C4×D4—2141no—
108#107—21no—
109#106—2141no—
110C4⋊C8—1071no—
111C4⋊C8C2×D10711yescharacteristic
112#103—21no—
113#103—2142no—
114#102—2141no—
115C4×C8—1071no—
116C2×Q8—21no—
117C2×Q8—4282no—
118C2×Q8—4282no—
119C2×Q8—2142no—
120C2×Q8—2141no—
121C2×Q8C22⋊D10711yescharacteristic
122C2×D4—1071no—
123C22×C4—2142no—
124C22×C4—2141no—
125C2×C8—21no—
126C2×C8—2141no—
127C2×C8—2141no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1072no—
132C4⋊C4—1071no—
133C4⋊C4—2142no—
134C4⋊C4—2141no—
135C4⋊C4C22⋊D10711yescharacteristic
136C22⋊C4—2142no—
137C22⋊C4—2141no—
138C42—21no—
139C42C22×D10711yescharacteristic
140C23—2141no—
141Q8—21no—
142Q8—42no—
143Q8—4284no—
144Q8—4281no—
145Q8—4281no—
146Q8#1313380512yes—
147D4—1072no—
148D4—2141no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—4284no—
153C2×C4—4281no—
154C2×C4—4281no—
155C2×C4—2142no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×C22⋊D10711yescharacteristic
160C8—41no—
161C8—4281no—
162C8—4281no—
163C22—4281no—
164C22—4281no—
165C22—2142no—
166C22#1681207711yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—81no—
171C4—4282no—
172C4—4281no—
173C4C2×#1313380511yescharacteristic
174C4Q16×D10711yescharacteristic
175C4#1681213411yescharacteristic
176C4#1681214111yescharacteristic
177C2—4282no—
178C2#2029277711yescharacteristic
179C2#2029278111yescharacteristic
180C2#2029279811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#477×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#477D107
hypocenterC107#1637
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #20292977C2semidirect
4. #20292374C2semidirect
5. Q8⋊(C107⋊C8)C2semidirect
7. #477×C107C2semidirect
91. C107#1637semidirect
94. #477D107semidirect

Maximal subgroups

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

subgroupindex
Q8×D4282
#202929772
#202929142
#202923742
Q8⋊(C107⋊C8)2
#202923542
#477×C1072
#1637107