Finite Group Databasebeta

← #30666379

Subgroups — #30666379

Order 16768.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#22553819C211yesmaximal, characteristic
2#22553296C211yesmaximal, characteristic, complemented
3#22553275C212yesmaximal, complemented
4#22553083C212yesmaximal, complemented
5#568×C131C211yesmaximal, characteristic, complemented
6C22×C4×D131C2211yescharacteristic
7C2×#14523497C2212yes—
8#16862227C412yes—
9#16862222—21no—
10#16862217—21no—
11#16862214C412yes—
12C4×C22⋊D131—22no—
13#16862205C2211yescharacteristic
14#16862165—21no—
15#16862163—21no—
16(C2×D131)⋊C8—24no—
17C2×#41×C131C2211yescharacteristic
18#131×C131—24no—
19C2×M16×C131—22no—
20C2×M16×C131C2212yes—
21C23×D131C2×C411yescharacteristic
22C2×C22⋊D131—22no—
23C22×Dic131C2×C411yescharacteristic
24C2×C4×D131—21no—
25C2×C4×D131—22no—
26C2×C4×D131—21no—
27C2×C4×D131—22no—
28#14523497—22no—
29#14523497—22no—
30C2×C131⋊C8—22no—
31C4⋊Dic131—21no—
32Dic131⋊C4—22no—
33C22⋊Dic131—22no—
34C22⋊Dic131C2×C412yes—
35(C2×D131)⋊C4—22no—
36C4×Dic131—21no—
37#41×C131—22no—
38#41×C131—22no—
39C2×Q8×C131D411yescharacteristic
40C2×D4×C131C2×C412yes—
41C2×D4×C131D411yescharacteristic
42C22×C4×C131D412yes—
43C22×C4×C131C2311yescharacteristic
44M16×C131—24no—
45M16×C131—22no—
46M16×C131—22no—
47C2×C8×C131—24no—
48C2×C8×C131D414yescomplemented
49C22×D131—21no—
50C22×D131—22no—
51C22×D131—42no—
52C22⋊D131—44no—
53C2×Dic131—22no—
54C2×Dic131—21no—
55C2×Dic131—22no—
56C4×D131—41no—
57C4×D131—42no—
58C4×D131—41no—
59C131⋊C8—42no—
60C23×C131C22⋊C412yes—
61C23×C131C22×C411yescharacteristic
62Q8×C131—22no—
63D4×C131—24no—
64D4×C131—22no—
65C2×C4×C131—22no—
66C2×C4×C131—22no—
67C2×C4×C131C22⋊C412yes—
68C2×C4×C131C2×D412yes—
69C2×C4×C131C2×D411yescharacteristic
70C2×C4×C131C2×D411yescharacteristic
71C2×C4×C131#4112yes—
72C8×C131—24no—
73C22×C131—22no—
74C22×C131—41no—
75C22×C131—42no—
76C22×C131C2×C22⋊C411yescharacteristic
77C22×C131C4×D412yes—
78C2×D131—22no—
79C2×D131—24no—
80C2×D131—41no—
81C2×D131—42no—
82C2×D131—41no—
83C4×C131—22no—
84C4×C131#12011yescharacteristic
85C4×C131#12112yes—
86C4×C131#12311yescharacteristic
87Dic131—42no—
88Dic131—41no—
89C2×C131—21no—
90C2×C131—22no—
91C2×C131—42no—
92C2×C131#38611yescharacteristic
93D131—42no—
94C131#189111yescharacteristic, complemented, Sylow
95#1891—1311nomaximal, Sylow
96#568D13111yescharacteristic, complemented
97#525—1311no—
98#413—1311no—
99#407—1312no—
100#344—1312no—
101C2×#41C2×D13111yescharacteristic
102C23×C4—1311no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1312no—
106C2×M16C2×D13112yes—
107#123—1312no—
108C22⋊Q8—2621no—
109#121—2621no—
110#120—1312no—
111C4×D4—2622no—
112#117—1311no—
113#101—2621no—
114#100—2621no—
115C22⋊C8—2624no—
116C24—1311no—
117#41—22no—
118#41—22no—
119C2×Q8D52411yescharacteristic
120C2×D4—2622no—
121C2×D4C4×D13112yes—
122C2×D4D52411yescharacteristic
123C22×C4—2622no—
124C22×C4—2621no—
125C22×C4—2621no—
126C22×C4—2622no—
127C22×C4—1311no—
128C22×C4C22⋊D13112yes—
129C22×C4C22×D13111yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2622no—
134M16—2622no—
135C2×C8—24no—
136C2×C8—2622no—
137C2×C8C22⋊D13114yescomplemented
138C4⋊C4—2622no—
139C4⋊C4—2621no—
140C22⋊C4—2622no—
141C22⋊C4—2622no—
142C22⋊C4—1312no—
143C42—2621no—
144C23—2621no—
145C23—2622no—
146C23—5242no—
147C23(C2×D131)⋊C412yes—
148C23C2×C4×D13111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—5244no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—2622no—
156C2×C4—2621no—
157C2×C4—2622no—
158C2×C4—5241no—
159C2×C4—5242no—
160C2×C4—5241no—
161C2×C4(C2×D131)⋊C412yes—
162C2×C4C2×C22⋊D13112yes—
163C2×C4C2×C22⋊D13111yescharacteristic
164C2×C4C2×D52411yescharacteristic
165C2×C4#1452351312yes—
166C8—24no—
167C8—5242no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—2622no—
172C22—2624no—
173C22—5241no—
174C22—5242no—
175C22—5241no—
176C22C2×(C2×D131)⋊C411yescharacteristic
177C22C4×C22⋊D13112yes—
178C4—22no—
179C4—5242no—
180C4—5241no—
181C4#1686221511yescharacteristic
182C4#1686222012yes—
183C4#1686222911yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—5242no—
188C2#2255319111yescharacteristic
subgroupG/H
centerC4#16862215
commutatorC23×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#568×C131C2
socleC2×C131#386
prime radical#22553819C2
cosocleC22×C4×C131C23
hypercenter#568D131
hypocenterC131#1891
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22553296C2semidirect
3. #22553275C2semidirect
4. #22553083C2semidirect
5. #568×C131C2semidirect
48. C2×C8×C131D4semidirect
94. C131#1891semidirect
96. #568D131semidirect
137. C2×C8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225538192
#225532962
#225532752
#225530832
#568×C1312
#1891131