Finite Group Databasebeta

← #28338414

Subgroups — #28338414

Order 13696.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293427C211yesmaximal, characteristic
2#20293070C212yesmaximal, complemented
3#20292885C211yesmaximal, characteristic, complemented
4#20292438C212yesmaximal, complemented
5#349×C107C211yesmaximal, characteristic, complemented
6C2×#13133818—21no—
7#16812142—22no—
8#16812137—22no—
9#16812131—21no—
10#16812131C2212yes—
11C2×Dic428C2211yescharacteristic, complemented
12#16812088—21no—
13Dic856—22no—
14#16812054—22no—
15#16812041C2211yescharacteristic, complemented
16#16812034—21no—
17C2×M16×C107C2211yescharacteristic
18M32×C107C2212yescomplemented
19#13133818—22no—
20#13133818—22no—
21#13133818—41no—
22#13133818—41no—
23C2×Dic214—21no—
24C2×Dic214—21no—
25C2×D428—21no—
26C2×C22⋊D107—41no—
27C2×C4×D107—41no—
28Dic428—21no—
29Dic428—22no—
30Dic428D412yes—
31#13133807—22no—
32#13133807—22no—
33D856—21no—
34D856D412yes—
35C4⋊Dic107—21no—
36Dic107⋊C4—41no—
37C22⋊Dic107—41no—
38C22×C4×C107D411yescharacteristic
39M16×C107—22no—
40C2×C8×C107D411yescharacteristic
41C2×C8×C107C2311yescharacteristic
42C16×C107—22no—
43C23×C107D811yescharacteristic
44C22×D107—41no—
45C2×C4×C107—21no—
46C2×C4×C107—21no—
47C2×C4×C107D811yescharacteristic
48C2×C4×C107C2×D411yescharacteristic
49C22⋊D107—42no—
50C22⋊D107—42no—
51C2×Dic107—41no—
52C2×Dic107—41no—
53C2×Dic107—41no—
54D428—22no—
55D428—41no—
56C4×D107—42no—
57C4×D107—42no—
58Dic214—22no—
59Dic214—41no—
60Dic214—41no—
61Dic214—41no—
62C8×C107—21no—
63C8×C107C2×D412yes—
64C22×C107—21no—
65C22×C107—41no—
66C22×C107C2×D811yescharacteristic
67C2×D107—42no—
68C2×D107—82no—
69C4×C107—21no—
70C4×C107#12011yescharacteristic
71C4×C107AGL(1,ℤ/8)11yescharacteristic
72Dic107—42no—
73Dic107—81no—
74Dic107—81no—
75C2×C107—21no—
76C2×C107—41no—
77C2×C107#44711yescharacteristic
78D107—82no—
79#2191—1071nomaximal, Sylow
80C107#219111yescharacteristic, complemented, Sylow
81#575—1071no—
82#510—1072no—
83#470—1071no—
84#360—1072no—
85#349D10711yescharacteristic, complemented
86C2×#41—2141no—
87#137—2142no—
88AGL(1,ℤ/8)—2142no—
89#135—1072no—
90#135—2141no—
91C2×Q16—1071no—
92C2×M16C2×D10711yescharacteristic
93C22⋊Q8—2141no—
94Q32—2142no—
95SD32—2142no—
96M32C2×D10712yescomplemented
97#106—1071no—
98#103—2141no—
99#41—4281no—
100#41—4281no—
101#41—2142no—
102#41—2142no—
103C2×Q8—2141no—
104C2×Q8—2141no—
105C2×D4—4281no—
106C2×D4—2141no—
107C22×C4—4281no—
108C22×C4D42811yescharacteristic
109Q16—1072no—
110Q16—2141no—
111Q16—2142no—
112SD16—2142no—
113SD16—2142no—
114D8—1072no—
115D8—2141no—
116M16—22no—
117C2×C8D42811yescharacteristic
118C2×C8C22×D10711yescharacteristic
119C4⋊C4—4281no—
120C4⋊C4—2141no—
121C22⋊C4—4281no—
122C16—22no—
123C23—4281no—
124C23D85611yescharacteristic
125Q8—4281no—
126Q8—4281no—
127Q8—4281no—
128Q8—2142no—
129D4—4281no—
130D4—4282no—
131D4—4282no—
132D4—2142no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—4282no—
136C2×C4—4281no—
137C2×C4—4281no—
138C2×C4—4281no—
139C2×C4—4282no—
140C2×C4D85611yescharacteristic
141C2×C4C2×D42811yescharacteristic
142C8—21no—
143C8D4×D10712yes—
144C22—21no—
145C22—41no—
146C22—8562no—
147C22—4282no—
148C22C2×D85611yescharacteristic
149C4—21no—
150C4—8561no—
151C4—8561no—
152C4—4282no—
153C4#1681207911yescharacteristic
154C4#1681213711yescharacteristic
155C2—21no—
156C2—41no—
157C2—8562no—
158C2#2029276311yescharacteristic
subgroupG/H
centerC2#20292763
commutator, cosocleC2×C8×C107C23
FrattiniC2×C8C22×D107
Fitting#349×C107C2
socleC2×C107#447
prime radical#20293427C2
hypercenter#349D107
hypocenterC107#2191
LCSG ▷ C2×C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #20293070C2semidirect
3. #20292885C2semidirect
4. #20292438C2semidirect
5. #349×C107C2semidirect
11. C2×Dic428C22semidirect
15. #16812041C22semidirect
18. M32×C107C22semidirect
80. C107#2191semidirect
85. #349D107semidirect
96. M32C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934272
#202930702
#202928852
#202924382
#349×C1072
#2191107