Finite Group Databasebeta

← #28334947

Subgroups — #28334947

Order 13696.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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#20292680C211yesmaximal, characteristic, complemented
2#20292473C211yesmaximal, characteristic, complemented
3#20292364C212yesmaximal, complemented
4#20292361C212yesmaximal
5#529×C107C211yesmaximal, characteristic
6C2×C4×Dic107—21no—
7C4⋊(C107⋊C8)—21no—
8C22⋊(C107⋊C8)C2212yes—
9C4×C107⋊C8—21no—
10C107⋊#95—21no—
11C107⋊#95C2212yes—
12#126×C107—21no—
13#125×C107C411yescharacteristic, complemented
14#124×C107C411yescharacteristic, complemented
15#123×C107—21no—
16C22⋊Q8×C107C2211yescharacteristic
17#121×C107C2211yescharacteristic
18C4×Q8×C107C411yescharacteristic, complemented
19C4×D4×C107—21no—
20C4×D4×C107C411yescharacteristic, complemented
21#117×C107—21no—
22C2×C42×C107C2211yescharacteristic
23C22×Dic107—22no—
24C2×C107⋊C8—22no—
25C4×Dic107—24no—
26C2×Q8×C107C2×C411yescharacteristic
27C2×D4×C107—21no—
28C2×D4×C107C2×C411yescharacteristic
29C22×C4×C107—21no—
30C22×C4×C107D411yescharacteristic
31C22×C4×C107D411yescharacteristic
32C22×C4×C107C2311yescharacteristic
33C4⋊C4×C107—21no—
34C4⋊C4×C107—21no—
35C4⋊C4×C107—21no—
36C4⋊C4×C107—21no—
37C4⋊C4×C107C2×C411yescharacteristic
38C4⋊C4×C107C2×C411yescharacteristic
39C22⋊C4×C107—21no—
40C22⋊C4×C107—21no—
41C22⋊C4×C107—21no—
42C22⋊C4×C107—21no—
43C22⋊C4×C107—21no—
44C42×C107—21no—
45C42×C107C2×C411yescharacteristic
46C42×C107C2×C411yescharacteristic
47C42×C107D411yescharacteristic
48C42×C107D411yescharacteristic
49C23×C107—21no—
50C23×C107C2×D411yescharacteristic
51Q8×C107—41no—
52D4×C107—41no—
53D4×C107—42no—
54C2×C4×C107—22no—
55C2×C4×C107—22no—
56C2×C4×C107—21no—
57C2×C4×C107—21no—
58C2×C4×C107—21no—
59C2×C4×C107—41no—
60C2×C4×C107—42no—
61C2×C4×C107C22⋊C411yescharacteristic
62C2×C4×C107C22⋊C411yescharacteristic
63C2×C4×C107C22⋊C411yescharacteristic
64C2×C4×C107C22⋊C411yescharacteristic
65C2×C4×C107C22×C411yescharacteristic
66C2×C4×C107C2×D411yescharacteristic
67C2×Dic107—24no—
68C2×Dic107—42no—
69C2×Dic107—42no—
70C107⋊C8—42no—
71C22×C107—21no—
72C22×C107—21no—
73C22×C107—41no—
74C22×C107—41no—
75C22×C107—41no—
76C22×C107C2×C22⋊C411yescharacteristic
77C22×C107#13512yes—
78C4×C107—21no—
79C4×C107—21no—
80C4×C107—21no—
81C4×C107—21no—
82C4×C107—21no—
83C4×C107—41no—
84C4×C107—41no—
85C4×C107—41no—
86C4×C107#10412yes—
87Dic107—44no—
88C2×C107—22no—
89C2×C107—81no—
90C2×C107#41011yescharacteristic
91C2×C107#41611yescharacteristic
92C2×C107C2×#10411yescharacteristic
93#1500—1071nomaximal, Sylow
94C107#150011yescharacteristic, complemented, Sylow
95#529D10711yescharacteristic
96#431—1071no—
97#376—1071no—
98#328—1072no—
99#327—1072no—
100#126—21no—
101#125Dic10711yescharacteristic, complemented
102#124Dic10711yescharacteristic, complemented
103#123—21no—
104C22⋊Q8C2×D10711yescharacteristic
105#121C2×D10711yescharacteristic
106C4×Q8Dic10711yescharacteristic, complemented
107C4×D4—21no—
108C4×D4Dic10711yescharacteristic, complemented
109#117—21no—
110C2×C42—2141no—
111C2×C42C2×D10711yescharacteristic
112C4⋊C8—2141no—
113C22⋊C8—1072no—
114C4×C8—2141no—
115#95—1072no—
116#95—2141no—
117C2×Q8C2×Dic10711yescharacteristic
118C2×D4—21no—
119C2×D4C2×Dic10711yescharacteristic
120C22×C4—21no—
121C22×C4—2142no—
122C22×C4C22⋊D10711yescharacteristic
123C22×C4C22⋊D10711yescharacteristic
124C22×C4C22×D10711yescharacteristic
125C2×C8—2142no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4C2×Dic10711yescharacteristic
131C4⋊C4C2×Dic10711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42—21no—
138C42—2144no—
139C42C2×Dic10711yescharacteristic
140C42C2×Dic10711yescharacteristic
141C42C22⋊D10711yescharacteristic
142C42C22⋊D10711yescharacteristic
143C23—21no—
144C23C2×C22⋊D10711yescharacteristic
145Q8—41no—
146D4—41no—
147D4—42no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—4282no—
156C2×C4—4282no—
157C2×C4—2144no—
158C2×C4C22⋊Dic10711yescharacteristic
159C2×C4C22⋊Dic10711yescharacteristic
160C2×C4C22⋊Dic10711yescharacteristic
161C2×C4C22⋊Dic10711yescharacteristic
162C2×C4C22×Dic10711yescharacteristic
163C2×C4C2×C22⋊D10711yescharacteristic
164C8—4282no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×C22⋊Dic10711yescharacteristic
171C22#1681212912yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—4284no—
181C4#1681203612yes—
182C2—22no—
183C2—81no—
184C2#2029260111yescharacteristic
185C2#2029262311yescharacteristic
186C2C2×#1681203611yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#529×C107C2
socleC22×C107C2×C22⋊C4
prime radical#121×C107C22
cosocleC22×C4×C107C23
hypercenter#529D107
hypocenterC107#1500
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20292680C2semidirect
2. #20292473C2semidirect
3. #20292364C2semidirect
13. #125×C107C4semidirect
14. #124×C107C4semidirect
18. C4×Q8×C107C4semidirect
20. C4×D4×C107C4semidirect
94. C107#1500semidirect
101. #125Dic107semidirect
102. #124Dic107semidirect
106. C4×Q8Dic107semidirect
108. C4×D4Dic107semidirect

Maximal subgroups

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

subgroupindex
#202926802
#202924732
#202923642
#202923612
#529×C1072
#1500107