Finite Group Databasebeta

← #28346840

Subgroups — #28346840

Order 13696.

380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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#20293183C211yesmaximal, characteristic
2C4⋊Q8×D107C211yesmaximal, characteristic
3#20292997C211yesmaximal, characteristic, complemented
4#20292996C211yesmaximal, characteristic, complemented
5#20292995C211yesmaximal, characteristic, complemented
6#20292976C214yesmaximal, complemented
7#20292634C214yesmaximal
8#20292579C211yesmaximal, characteristic
9#491×C107C211yesmaximal, characteristic, complemented
10C2×Q8×D107C2212yes—
11C2×#13133801C2212yes—
12#16812109C2211yescharacteristic
13#16812108—22no—
14#16812108C2212yes—
15#16812107C2212yes—
16#16812101C2211yescharacteristic
17#16812087—22no—
18#16812074—22no—
19#16812067C2212yes—
20C4⋊C4×D107—22no—
21C42×D107C2211yescharacteristic
22#16812034C2214yes—
23#16812033C2214yes—
24#16812032C2214yes—
25#16812020C2211yescharacteristic
26C8×Dic107C2212yes—
27C4⋊Q8×C107C2211yescharacteristic
28#125×C107C2211yescharacteristic
29#103×C107C2214yes—
30#97×C107C2211yescharacteristic
31C2×Dic214—22no—
32C2×Dic214—22no—
33C2×Dic214C2312yes—
34Q8×D107—28no—
35C2×C4×D107—22no—
36C2×C4×D107D412yes—
37C2×C4×D107C2311yescharacteristic
38#13133801—24no—
39C2×C107⋊C8C2312yes—
40C4⋊Dic107—22no—
41C4⋊Dic107C2312yes—
42Dic107⋊C4—22no—
43Dic107⋊C4—24no—
44Dic107⋊C4—22no—
45(C2×D107)⋊C4—22no—
46C4×Dic107—22no—
47C4×Dic107D412yes—
48C4×Dic107C2311yescharacteristic
49C2×Q8×C107C2312yes—
50C2×C8×C107C2312yes—
51C4⋊C4×C107—22no—
52C4⋊C4×C107—22no—
53C4⋊C4×C107C2312yes—
54C42×C107C2311yescharacteristic
55C22×D107C2×D411yescharacteristic
56Q8×C107—24no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107C2×D412yes—
60C2×C4×C107C2411yescharacteristic
61C2×Dic107—22no—
62C2×Dic107—22no—
63C2×Dic107C2×D411yescharacteristic
64C2×Dic107C2×D412yes—
65C4×D107—24no—
66C4×D107—44no—
67C4×D107#4114yes—
68Dic214—24no—
69Dic214—44no—
70Dic214—44no—
71C8×C107—22no—
72C107⋊C8—22no—
73C22×C107C22×D411yescharacteristic
74C2×D107—22no—
75C2×D107#12412yes—
76C4×C107—22no—
77C4×C107—42no—
78C4×C107—42no—
79C4×C107C2×#4112yes—
80Dic107—42no—
81Dic107—42no—
82Dic107#12412yes—
83Dic107#13714yes—
84C2×C107C2×#12411yescharacteristic
85C2×C107C2×#13712yes—
86D107—22no—
87#3139—1071nomaximal, Sylow
88C107#313911yescharacteristic, complemented, Sylow
89#533—1071no—
90C2×C4⋊Q8—1071no—
91#491—1071no—
92#491—1071no—
93#491—1071no—
94#491D10711yescharacteristic, complemented
95#487—1074no—
96#419—1074no—
97C4×M16—1071no—
98C22×Q8—1072no—
99C2×M16—1072no—
100C4⋊Q8—1072no—
101C4⋊Q8—1072no—
102C4⋊Q8—1071no—
103C4⋊Q8—2142no—
104C4⋊Q8C2×D10711yescharacteristic
105#125—1071no—
106#125C2×D10711yescharacteristic
107C22⋊Q8—2142no—
108C4×Q8—2142no—
109#117—1072no—
110C2×C4⋊C4—2142no—
111C2×C42—1071no—
112#103—1074no—
113#103—1074no—
114#103—1074no—
115#103C2×D10714yes—
116#97—1071no—
117#97C2×D10711yescharacteristic
118C4×C8—1072no—
119C2×Q8—1072no—
120C2×Q8—2142no—
121C2×Q8—2142no—
122C2×Q8—2148no—
123C2×Q8C22×D10712yes—
124C22×C4—1071no—
125C22×C4—1072no—
126C22×C4—2142no—
127M16—2144no—
128C2×C8—1072no—
129C2×C8C22×D10712yes—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—1072no—
133C4⋊C4—2144no—
134C4⋊C4—2142no—
135C4⋊C4—2142no—
136C4⋊C4—2142no—
137C4⋊C4C22×D10712yes—
138C22⋊C4—2142no—
139C42—1072no—
140C42—1071no—
141C42—2142no—
142C42C22×D10711yescharacteristic
143C23—1071no—
144Q8—24no—
145Q8—4284no—
146Q8—4284no—
147Q8—2144no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—1071no—
151C2×C4—1072no—
152C2×C4—1074no—
153C2×C4—4284no—
154C2×C4—2144no—
155C2×C4—2142no—
156C2×C4—2142no—
157C2×C4D4×D10712yes—
158C2×C4C23×D10711yescharacteristic
159C8—22no—
160C8—2142no—
161C22—1072no—
162C22—2142no—
163C22C2×D4×D10711yescharacteristic
164C4—22no—
165C4—42no—
166C4—42no—
167C4—1072no—
168C4—1074no—
169C4—4282no—
170C4—4282no—
171C4#41×D10712yes—
172C2—2142no—
173C2#124×D10711yescharacteristic
174C2#137×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#491×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#491D107
hypocenterC107#3139
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
3. #20292997C2semidirect
4. #20292996C2semidirect
5. #20292995C2semidirect
6. #20292976C2semidirect
9. #491×C107C2semidirect
88. C107#3139semidirect
94. #491D107semidirect

Maximal subgroups

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

subgroupindex
#202931832
C4⋊Q8×D1072
#202929972
#202929962
#202929952
#202929762
#202926342
#202925792
#491×C1072
#3139107