Finite Group Databasebeta

← #28335783

Subgroups — #28335783

Order 13696.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293371C211yesmaximal, characteristic
2#20292995C211yesmaximal, characteristic, complemented
3#20292989C211yesmaximal, characteristic, complemented
4#20292368C211yesmaximal, characteristic, complemented
5#20292367C211yesmaximal, characteristic, complemented
6#20292366C211yesmaximal, characteristic, complemented
7#489×C107C211yesmaximal, characteristic, complemented
8#16812101C2211yescharacteristic
9#16812097C2211yescharacteristic
10#16812093—21no—
11#16812091—21no—
12#16812086—21no—
13#16812085—21no—
14#16812082—21no—
15#16812081—21no—
16#16812033—21no—
17#16812033—21no—
18#16812032—21no—
19#16812029—21no—
20#16812020C2211yescharacteristic
21#16812020C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23#124×C107C2211yescharacteristic
24#103×C107—21no—
25#102×C107—21no—
26#97×C107C2211yescharacteristic
27C2×Dic214D411yescharacteristic
28C2×D428D411yescharacteristic
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C22×Dic107—21no—
32C2×C4×D107—21no—
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107D411yescharacteristic
37C4⋊Dic107D411yescharacteristic
38Dic107⋊C4—21no—
39Dic107⋊C4—21no—
40Dic107⋊C4—21no—
41C22⋊Dic107—21no—
42(C2×D107)⋊C4—21no—
43(C2×D107)⋊C4—21no—
44(C2×D107)⋊C4—21no—
45C2×Q8×C107—21no—
46C2×Q8×C107D411yescharacteristic
47C2×D4×C107D411yescharacteristic
48C2×C8×C107—21no—
49C4⋊C4×C107—21no—
50C4⋊C4×C107—21no—
51C22⋊C4×C107—21no—
52C22⋊C4×C107—21no—
53C42×C107C2311yescharacteristic
54C23×C107—21no—
55C22×D107—21no—
56Q8×C107—41no—
57Q8×C107—42no—
58D4×C107—41no—
59C2×C4×C107—21no—
60C2×C4×C107—41no—
61C2×C4×C107C2×D411yescharacteristic
62C2×C4×C107C2×D411yescharacteristic
63C2×C4×C107C2×D411yescharacteristic
64C22⋊D107—81no—
65C22⋊D107—81no—
66C2×Dic107—21no—
67C2×Dic107—21no—
68C2×Dic107—21no—
69C2×Dic107—81no—
70D428—41no—
71C4×D107—81no—
72Dic214—41no—
73C8×C107—41no—
74C107⋊C8—41no—
75C107⋊C8—41no—
76C22×C107—41no—
77C22×C107—41no—
78C22×C107—41no—
79C22×C107#12011yescharacteristic
80C2×D107—41no—
81C2×D107—41no—
82C2×D107—41no—
83C4×C107—21no—
84C4×C107—21no—
85C4×C107—21no—
86C4×C107—41no—
87C4×C107—81no—
88Dic107—41no—
89Dic107—41no—
90Dic107—41no—
91C2×C107—81no—
92C2×C107#45411yescharacteristic
93C2×C107#45511yescharacteristic
94C2×C107#45511yescharacteristic
95D107—81no—
96#1662—1071nomaximal, Sylow
97C107#166211yescharacteristic, complemented, Sylow
98#562—1071no—
99#491—1071no—
100#489—1071no—
101#489D10711yescharacteristic, complemented
102#330—1071no—
103#329—1071no—
104#329—1071no—
105C4⋊Q8C2×D10711yescharacteristic
106#125—1071no—
107#124—1071no—
108#124C2×D10711yescharacteristic
109#123—2141no—
110#123—2141no—
111C22⋊Q8—2141no—
112C22⋊Q8—2141no—
113#121—2141no—
114#121—2141no—
115#103—21no—
116#103—2141no—
117#103—2141no—
118#103—2141no—
119#102—21no—
120#102—2141no—
121#97—1071no—
122#97—1071no—
123#97C2×D10711yescharacteristic
124C2×Q8—21no—
125C2×Q8—1071no—
126C2×Q8C22⋊D10711yescharacteristic
127C2×D4—1071no—
128C2×D4—2141no—
129C2×D4—2141no—
130C2×D4C22⋊D10711yescharacteristic
131C22×C4—2141no—
132C22×C4—2141no—
133C2×C8—21no—
134C2×C8—2141no—
135C2×C8—2141no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1071no—
139C4⋊C4—1071no—
140C4⋊C4—2141no—
141C4⋊C4—2141no—
142C4⋊C4—2141no—
143C4⋊C4—2141no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—2141no—
147C22⋊C4—2141no—
148C22⋊C4—2141no—
149C22⋊C4—2141no—
150C42C22×D10711yescharacteristic
151C23—21no—
152C23—2141no—
153Q8—41no—
154Q8—42no—
155Q8—4281no—
156D4—41no—
157D4—8561no—
158D4—8561no—
159D4—4281no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—8561no—
163C2×C4—8561no—
164C2×C4—2141no—
165C2×C4—2141no—
166C2×C4—2141no—
167C2×C4D4×D10711yescharacteristic
168C2×C4D4×D10711yescharacteristic
169C2×C4C2×C22⋊D10711yescharacteristic
170C8—41no—
171C8—4281no—
172C8—4281no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—4281no—
177C22—4281no—
178C22—4281no—
179C22#1681207711yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—81no—
185C4—4281no—
186C4—4281no—
187C4—4281no—
188C2—81no—
189C2—8561no—
190C2#2029279811yescharacteristic
191C2#2029280311yescharacteristic
192C2#2029280411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#489×C107C2
socleC22×C107#120
prime radical#20293371C2
hypercenter#489D107
hypocenterC107#1662
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #20292995C2semidirect
3. #20292989C2semidirect
4. #20292368C2semidirect
5. #20292367C2semidirect
6. #20292366C2semidirect
7. #489×C107C2semidirect
97. C107#1662semidirect
101. #489D107semidirect

Maximal subgroups

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

subgroupindex
#202933712
#202929952
#202929892
#202923682
#202923672
#202923662
#489×C1072
#1662107