Finite Group Databasebeta

← #30664985

Subgroups — #30664985

Order 16768.

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

normal subgroups only

This group has 236 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#22553938C211yesmaximal, characteristic
2#22553647C212yesmaximal, complemented
3#22553048C212yesmaximal, complemented
4#22553036C211yesmaximal, characteristic
5#399×C131C211yesmaximal, characteristic
6#16862225—21no—
7#16862225C2212yes—
8#16862223—22no—
9#16862223—22no—
10C2×C22⋊Dic131—21no—
11#16862179—22no—
12#16862169—22no—
13C22⋊(C131⋊C8)C2212yes—
14C2×C4⋊C4×C131—21no—
15C2×C4⋊C4×C131C2211yescharacteristic
16C2×C22⋊C4×C131—21no—
17C2×C22⋊C4×C131C2211yescharacteristic
18#95×C131C2211yescharacteristic
19C2×Dic262—21no—
20C2×Dic262D412yes—
21C22×Dic131—21no—
22C22×Dic131—21no—
23C2×C131⋊C8—22no—
24C4⋊Dic131—21no—
25C4⋊Dic131D412yes—
26Dic131⋊C4—22no—
27Dic131⋊C4—22no—
28C22⋊Dic131—22no—
29C22⋊Dic131—22no—
30C24×C131D411yescharacteristic
31C22×C4×C131—21no—
32C22×C4×C131—21no—
33C22×C4×C131D411yescharacteristic
34C22×C4×C131C2311yescharacteristic
35C4⋊C4×C131—22no—
36C4⋊C4×C131—42no—
37C22⋊C4×C131—22no—
38C22⋊C4×C131—42no—
39Dic262—42no—
40Dic262—81no—
41C2×Dic131—22no—
42C2×Dic131—22no—
43C2×Dic131—81no—
44C2×Dic131—81no—
45C131⋊C8—42no—
46C23×C131—21no—
47C23×C131—21no—
48C23×C131—41no—
49C23×C131—41no—
50C23×C131Q1612yes—
51C23×C131C2×D411yescharacteristic
52C2×C4×C131—22no—
53C2×C4×C131—21no—
54C2×C4×C131—41no—
55C2×C4×C131—41no—
56C2×C4×C131—41no—
57C2×C4×C131—41no—
58C2×C4×C131—41no—
59C2×C4×C131—41no—
60C2×C4×C131—41no—
61C2×C4×C131C2×D412yes—
62C22×C131—21no—
63C22×C131—21no—
64C22×C131—22no—
65C22×C131—42no—
66C22×C131—41no—
67C22×C131—41no—
68C22×C131—81no—
69C22×C131#12011yescharacteristic
70C22×C131C2×Q1611yescharacteristic
71C22×C131#13711yescharacteristic
72C4×C131—22no—
73C4×C131—42no—
74C4×C131—41no—
75C4×C131—81no—
76Dic131—42no—
77Dic131—42no—
78C2×C131—21no—
79C2×C131—21no—
80C2×C131—42no—
81C2×C131#45111yescharacteristic
82C2×C131#45311yescharacteristic
83C2×C131#45811yescharacteristic
84C131#161411yescharacteristic, complemented, Sylow
85#1614—1311nomaximal, Sylow
86#543—1311no—
87#484—1312no—
88#399D13111yescharacteristic
89#328—1312no—
90#323—1311no—
91C22⋊Q8—2622no—
92C22⋊Q8—2622no—
93C22⋊Q8—2621no—
94C22⋊Q8—1312no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×D13111yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—2621no—
99C2×C22⋊C4C2×D13111yescharacteristic
100#107—2622no—
101#103—2622no—
102C22⋊C8—1312no—
103#95C2×D13111yescharacteristic
104C24C22⋊D13111yescharacteristic
105C2×Q8—2621no—
106C2×Q8—1312no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—2621no—
110C22×C4—2621no—
111C22×C4C22⋊D13111yescharacteristic
112C22×C4C22×D13111yescharacteristic
113C2×C8—2622no—
114C4⋊C4—22no—
115C4⋊C4—42no—
116C4⋊C4—2622no—
117C4⋊C4—2621no—
118C4⋊C4—2622no—
119C4⋊C4—1312no—
120C22⋊C4—22no—
121C22⋊C4—42no—
122C22⋊C4—2622no—
123C22⋊C4—2622no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23#1452350312yes—
129C23C2×C22⋊D13111yescharacteristic
130Q8—5242no—
131Q8—10481no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—2622no—
142C2×C4—2622no—
143C2×C4—10481no—
144C2×C4—10481no—
145C2×C4D4×D13112yes—
146C8—5242no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1686221411yescharacteristic
155C22C2×#1452350311yescharacteristic
156C22#1686227611yescharacteristic
157C4—22no—
158C4—42no—
159C4—41no—
160C4—81no—
161C4—5242no—
162C4—5242no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2255346711yescharacteristic
167C2#2255347911yescharacteristic
168C2#2255350111yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#399×C131C2
socleC22×C131#120
prime radicalC24×C131D4
hypercenter#399D131
hypocenterC131#1614
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #22553647C2semidirect
3. #22553048C2semidirect
84. C131#1614semidirect

Maximal subgroups

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

subgroupindex
#225539382
#225536472
#225530482
#225530362
#399×C1312
#1614131