Finite Group Databasebeta

← #28335707

Subgroups — #28335707

Order 13696.

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

normal subgroups only

This group has 234 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#20293292C211yesmaximal, characteristic
2#20292904C211yesmaximal, characteristic
3#20292834C211yesmaximal, characteristic, complemented
4#20292373C211yesmaximal, characteristic, complemented
5#20292359C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic
7#500×C107C211yesmaximal, characteristic, complemented
8C2×Q8×D107—21no—
9C2×#13133808—21no—
10#16812108—21no—
11#16812087—22no—
12#16812085—21no—
13C4×Dic214C2211yescharacteristic
14C4×D428C2211yescharacteristic
15#16812071—21no—
16#16812042—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18C4⋊(C107⋊C8)C2211yescharacteristic
19#16812033—21no—
20#16812029—21no—
21C4⋊Q8×C107C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23#107×C107—22no—
24C4×C8×C107C2211yescharacteristic
25C2×Dic214—22no—
26C2×Dic214D411yescharacteristic
27Q8×D107—42no—
28Q8×D107—42no—
29C2×D428D411yescharacteristic
30C2×C4×D107—22no—
31C2×C4×D107—21no—
32#13133808—42no—
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C4⋊Dic107D411yescharacteristic
36C4⋊Dic107D411yescharacteristic
37Dic107⋊C4—22no—
38Dic107⋊C4—21no—
39(C2×D107)⋊C4—22no—
40(C2×D107)⋊C4—21no—
41C4×Dic107—21no—
42C2×Q8×C107—21no—
43C2×Q8×C107—21no—
44C2×C8×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107—21no—
47C4⋊C4×C107—21no—
48C4⋊C4×C107D412yes—
49C42×C107C2311yescharacteristic
50C22×D107—21no—
51Q8×C107—41no—
52Q8×C107—41no—
53Q8×C107—41no—
54Q8×C107—41no—
55C2×C4×C107—22no—
56C2×C4×C107—41no—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×C4×C107C2×D411yescharacteristic
60C2×Dic107—21no—
61C2×Dic107—21no—
62C2×Dic107—21no—
63D428—21no—
64D428Q1612yescomplemented
65C4×D107—44no—
66C4×D107—41no—
67C4×D107—41no—
68Dic214—21no—
69Dic214—44no—
70Dic214D812yes—
71C8×C107—22no—
72C107⋊C8—41no—
73C107⋊C8—41no—
74C22×C107#12011yescharacteristic
75C2×D107—22no—
76C2×D107—41no—
77C2×D107—41no—
78C4×C107—21no—
79C4×C107—42no—
80C4×C107—81no—
81C4×C107C2×D811yescharacteristic
82C4×C107C2×Q1611yescharacteristic
83C4×C107AGL(1,ℤ/8)11yescharacteristic
84C4×C107#13711yescharacteristic
85Dic107—22no—
86Dic107—41no—
87Dic107—41no—
88C2×C107#44711yescharacteristic
89C2×C107#45111yescharacteristic
90C2×C107#45611yescharacteristic
91D107—42no—
92#1649—1071nomaximal, Sylow
93C107#164911yescharacteristic, complemented, Sylow
94D4×Q8—1071no—
95#500D10711yescharacteristic, complemented
96#474—1071no—
97#462—1071no—
98#332—1071no—
99Q8⋊C8—1071no—
100D4⋊C8—1071no—
101C22×Q8—2141no—
102C2×Q16—2141no—
103C4⋊Q8—2141no—
104C4⋊Q8C2×D10711yescharacteristic
105C4⋊Q8C2×D10711yescharacteristic
106C22⋊Q8—2141no—
107C22⋊Q8—2142no—
108C4×Q8—1071no—
109C4×D4—1071no—
110C4×D4—2141no—
111#107—22no—
112#107—2141no—
113C4⋊C8—1071no—
114C4⋊C8—1071no—
115#103—2141no—
116#102—2141no—
117C4×C8C2×D10711yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—1071no—
121C2×Q8—4282no—
122C2×Q8—4282no—
123C2×Q8—2142no—
124C2×D4—1071no—
125C22×C4—2142no—
126C22×C4—2141no—
127Q16—4282no—
128C2×C8—21no—
129C2×C8—2141no—
130C2×C8—2141no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1071no—
135C4⋊C4—1071no—
136C4⋊C4—2142no—
137C4⋊C4—2141no—
138C4⋊C4C22⋊D10712yes—
139C22⋊C4—2142no—
140C22⋊C4—2141no—
141C42—2141no—
142C42C22×D10711yescharacteristic
143C23—2141no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—1072no—
149Q8—4284no—
150Q8—2141no—
151D4—1072no—
152D4—2141no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—4284no—
156C2×C4—4281no—
157C2×C4—4281no—
158C2×C4—2141no—
159C2×C4—2141no—
160C2×C4—2141no—
161C2×C4D4×D10711yescharacteristic
162C2×C4D4×D10711yescharacteristic
163C2×C4C2×C22⋊D10711yescharacteristic
164C8—22no—
165C8—4281no—
166C8—4281no—
167C22—4281no—
168C22—4281no—
169C22—2142no—
170C22#1681207711yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—4281no—
175C4—4281no—
176C4—2142no—
177C4D8×D10711yescharacteristic
178C4Q16×D10711yescharacteristic
179C4#1681213311yescharacteristic
180C4#1681214111yescharacteristic
181C2—4282no—
182C2#2029275911yescharacteristic
183C2#2029278111yescharacteristic
184C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#500×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#500D107
hypocenterC107#1649
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #20292834C2semidirect
4. #20292373C2semidirect
5. #20292359C2semidirect
7. #500×C107C2semidirect
64. D428Q16semidirect
93. C107#1649semidirect
95. #500D107semidirect

Maximal subgroups

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

subgroupindex
#202932922
#202929042
#202928342
#202923732
#202923592
#202923562
#500×C1072
#1649107