Finite Group Databasebeta

← #28335163

Subgroups — #28335163

Order 13696.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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#20293175C211yesmaximal, characteristic
2#20292683C211yesmaximal, characteristic, complemented
3#20292679C211yesmaximal, characteristic, complemented
4#20292374C211yesmaximal, characteristic, complemented
5#20292372C211yesmaximal, characteristic, complemented
6#20292371C211yesmaximal, characteristic, complemented
7#331×C107C211yesmaximal, characteristic, complemented
8C2×#13133817—21no—
9#16812109C2211yescharacteristic
10#16812096—21no—
11#16812080—21no—
12#16812080C412yes—
13D4×Dic107—21no—
14D4×Dic107C412yes—
15C42×D107C2211yescharacteristic
16C4⋊(C107⋊C8)C2211yescharacteristic
17C4⋊(C107⋊C8)C2211yescharacteristic
18(C2×D107)⋊C8—21no—
19(C2×D107)⋊C8—21no—
20#16812019—21no—
21C8×Dic107—21no—
22#127×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25#13133817—42no—
26#13133817—42no—
27C2×Dic214—21no—
28C2×C22⋊D107—22no—
29C22×Dic107—22no—
30C2×C4×D107C2×C411yescharacteristic
31C2×C4×D107D411yescharacteristic
32C2×C4×D107D411yescharacteristic
33C2×C107⋊C8—21no—
34C2×C107⋊C8—21no—
35C4⋊Dic107—21no—
36C4⋊Dic107C2×C412yes—
37C22⋊Dic107—22no—
38C22⋊Dic107—22no—
39C4×Dic107C2×C411yescharacteristic
40C4×Dic107D411yescharacteristic
41C4×Dic107D411yescharacteristic
42C2×D4×C107—21no—
43C2×D4×C107—21no—
44C2×D4×C107C2×C412yes—
45C2×C8×C107—21no—
46C2×C8×C107—21no—
47C42×C107C2311yescharacteristic
48C23×C107—22no—
49C22×D107C22⋊C411yescharacteristic
50D4×C107—42no—
51D4×C107—41no—
52D4×C107—41no—
53D4×C107—41no—
54D4×C107—41no—
55C2×C4×C107C22×C411yescharacteristic
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C22⋊D107—44no—
59C2×Dic107—22no—
60C2×Dic107—44no—
61C2×Dic107C22⋊C411yescharacteristic
62C2×Dic107C22⋊C411yescharacteristic
63C2×Dic107C22⋊C411yescharacteristic
64C4×D107—21no—
65C4×D107—21no—
66C4×D107—21no—
67C4×D107—21no—
68C4×D107—41no—
69Dic214—41no—
70Dic214—41no—
71C8×C107—41no—
72C8×C107—41no—
73C107⋊C8—41no—
74C107⋊C8—41no—
75C22×C107—42no—
76C22×C107—42no—
77C22×C107—42no—
78C22×C107C2×C22⋊C411yescharacteristic
79C2×D107—21no—
80C2×D107—21no—
81C2×D107—21no—
82C4×C107—21no—
83C4×C107#13511yescharacteristic
84C4×C107#13511yescharacteristic
85C4×C107AGL(1,ℤ/8)11yescharacteristic
86C4×C107AGL(1,ℤ/8)11yescharacteristic
87Dic107—21no—
88Dic107—21no—
89Dic107—21no—
90Dic107—42no—
91C2×C107—82no—
92C2×C107#41311yescharacteristic
93C2×C107#41611yescharacteristic
94C2×C107#41811yescharacteristic
95D107—41no—
96#1553—1071nomaximal, Sylow
97C107#155311yescharacteristic, complemented, Sylow
98#532—1071no—
99#432—1071no—
100#431—1071no—
101#332—1071no—
102#332—1071no—
103#331—1071no—
104#331D10711yescharacteristic, complemented
105C2×#41—2141no—
106C4⋊Q8—1071no—
107#127C2×D10711yescharacteristic
108#124—2141no—
109#121—1072no—
110#121—2141no—
111C4×D4—1072no—
112C4×D4—2141no—
113C2×C42—1071no—
114C4⋊C8—1071no—
115C4⋊C8—1071no—
116C4⋊C8C2×D10711yescharacteristic
117C4⋊C8C2×D10711yescharacteristic
118C22⋊C8—2141no—
119C22⋊C8—2141no—
120#97—2141no—
121C4×C8—2141no—
122#41—4282no—
123#41—4282no—
124C2×Q8—2141no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—2142no—
128C2×D4C4×D10712yes—
129C22×C4—1071no—
130C22×C4—1071no—
131C22×C4—1071no—
132C22×C4—2142no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2141no—
136C2×C8—2141no—
137C4⋊C4—1072no—
138C4⋊C4—2141no—
139C22⋊C4—2142no—
140C22⋊C4—2142no—
141C42—1071no—
142C42—1071no—
143C42—1071no—
144C42C22×D10711yescharacteristic
145C23—22no—
146C23—1071no—
147Q8—4281no—
148Q8—4281no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—4284no—
155C2×C4—1071no—
156C2×C4—1071no—
157C2×C4—1071no—
158C2×C4—4281no—
159C2×C4—4284no—
160C2×C4—2142no—
161C2×C4—2141no—
162C2×C4—2141no—
163C2×C4—2141no—
164C2×C4—2141no—
165C2×C4C2×C4×D10711yescharacteristic
166C2×C4D4×D10711yescharacteristic
167C2×C4D4×D10711yescharacteristic
168C8—41no—
169C8—41no—
170C8—4281no—
171C8—4281no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—2141no—
176C22—2141no—
177C22—2141no—
178C22C22⋊C4×D10711yescharacteristic
179C4—21no—
180C4—4282no—
181C4—2141no—
182C4—2141no—
183C4—2141no—
184C4#1681212811yescharacteristic
185C4#1681213011yescharacteristic
186C4#1681213311yescharacteristic
187C4#1681213611yescharacteristic
188C2—82no—
189C2—4281no—
190C2#2029260811yescharacteristic
191C2#2029262011yescharacteristic
192C2#2029263011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D107
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
Fitting#331×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293175C2
cosocleC42×C107C23
hypercenter#331D107
hypocenterC107#1553
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20292683C2semidirect
3. #20292679C2semidirect
4. #20292374C2semidirect
5. #20292372C2semidirect
6. #20292371C2semidirect
7. #331×C107C2semidirect
97. C107#1553semidirect
104. #331D107semidirect

Maximal subgroups

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

subgroupindex
#202931752
#202926832
#202926792
#202923742
#202923722
#202923712
#331×C1072
#1553107