Finite Group Databasebeta

← #28335454

Subgroups — #28335454

Order 13696.

228 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293253C211yesmaximal, characteristic
2#20292958C212yesmaximal, complemented
3#20292552C211yesmaximal, characteristic
4#20292365C212yesmaximal, complemented
5#323×C107C211yesmaximal, characteristic
6#16812088—21no—
7#16812088C2212yes—
8#16812086—22no—
9#16812086—22no—
10C2×C4⋊Dic107C2211yescharacteristic
11C2×C22⋊Dic107—21no—
12C2×C22⋊Dic107—21no—
13#16812041—22no—
14#16812034—22no—
15C107⋊#95—21no—
16C107⋊#95C2211yescharacteristic
17C2×C22⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19C2×Dic214—21no—
20C2×Dic214D412yes—
21C22×Dic107—21no—
22C22×Dic107—21no—
23C22×Dic107—21no—
24C22×Dic107—21no—
25C4⋊Dic107—21no—
26C4⋊Dic107—22no—
27C4⋊Dic107D412yes—
28Dic107⋊C4—22no—
29Dic107⋊C4—22no—
30C22⋊Dic107—22no—
31C22⋊Dic107—22no—
32C22⋊Dic107—42no—
33C24×C107D411yescharacteristic
34C22×C4×C107D411yescharacteristic
35C22×C4×C107C2311yescharacteristic
36C2×C8×C107—22no—
37C22⋊C4×C107—22no—
38C23×C107—21no—
39C23×C107—21no—
40C23×C107—41no—
41C23×C107—41no—
42C23×C107SD1612yes—
43C23×C107C2×D411yescharacteristic
44C2×C4×C107—21no—
45C2×C4×C107—41no—
46C2×C4×C107—41no—
47C2×C4×C107C2×D412yes—
48C2×Dic107—22no—
49C2×Dic107—22no—
50C2×Dic107—41no—
51C2×Dic107—41no—
52C2×Dic107—41no—
53C2×Dic107—41no—
54C2×Dic107—41no—
55C2×Dic107—41no—
56C2×Dic107—81no—
57C2×Dic107—81no—
58Dic214—42no—
59Dic214—81no—
60C8×C107—42no—
61C22×C107—21no—
62C22×C107—21no—
63C22×C107—22no—
64C22×C107—42no—
65C22×C107—41no—
66C22×C107—41no—
67C22×C107—81no—
68C22×C107#12011yescharacteristic
69C22×C107C2×SD1611yescharacteristic
70C22×C107#13711yescharacteristic
71C4×C107—22no—
72C4×C107—41no—
73Dic107—42no—
74Dic107—42no—
75Dic107—81no—
76Dic107—81no—
77C2×C107—21no—
78C2×C107—21no—
79C2×C107—42no—
80C2×C107#44811yescharacteristic
81C2×C107#45511yescharacteristic
82C2×C107#45811yescharacteristic
83#1608—1071nomaximal, Sylow
84C107#160811yescharacteristic, complemented, Sylow
85#543—1071no—
86#483—1072no—
87#397—1071no—
88#328—1072no—
89#323D10711yescharacteristic
90C22⋊Q8—1072no—
91C22⋊Q8—2142no—
92C22⋊Q8—2142no—
93C22⋊Q8—2141no—
94C2×C4⋊C4—1071no—
95C2×C22⋊C4—2141no—
96C2×C22⋊C4—2141no—
97C2×C22⋊C4C2×D10711yescharacteristic
98#106—2142no—
99#103—2142no—
100C22⋊C8C2×D10712yes—
101#95—1071no—
102#95—2141no—
103C24D42811yescharacteristic
104C2×Q8—1072no—
105C2×Q8—2141no—
106C22×C4—2141no—
107C22×C4—2141no—
108C22×C4—2141no—
109C22×C4—2141no—
110C22×C4D42811yescharacteristic
111C22×C4C22×D10711yescharacteristic
112C2×C8—22no—
113C4⋊C4—1072no—
114C4⋊C4—2142no—
115C4⋊C4—2141no—
116C4⋊C4—2142no—
117C4⋊C4—2142no—
118C22⋊C4—22no—
119C22⋊C4—4282no—
120C22⋊C4—2142no—
121C22⋊C4—2142no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1313380712yes—
127C23C2×D42811yescharacteristic
128Q8—8561no—
129Q8—4282no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—8561no—
134C2×C4—8561no—
135C2×C4—4281no—
136C2×C4—4281no—
137C2×C4—4281no—
138C2×C4—4281no—
139C2×C4—4281no—
140C2×C4—4281no—
141C2×C4—2142no—
142C2×C4—2142no—
143C2×C4D4×D10712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#1681207911yescharacteristic
153C22C2×#1313380711yescharacteristic
154C22#1681214211yescharacteristic
155C4—22no—
156C4—41no—
157C4—8561no—
158C4—8561no—
159C4—4282no—
160C4—4282no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#2029276811yescharacteristic
165C2#2029280711yescharacteristic
166C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#323×C107C2
socleC22×C107#120
prime radicalC24×C107D4
hypercenter#323D107
hypocenterC107#1608
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20292958C2semidirect
4. #20292365C2semidirect
84. C107#1608semidirect

Maximal subgroups

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

subgroupindex
#202932532
#202929582
#202925522
#202923652
#323×C1072
#1608107