Finite Group Databasebeta

← #30666363

Subgroups — #30666363

Order 16768.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#16862234C211yesmaximal, characteristic
2#22553789C211yesmaximal, characteristic, complemented
3#22553318C211yesmaximal, characteristic
4#22553316C211yesmaximal, characteristic
5C2×#16862164C211yesmaximal, characteristic, complemented
6#22553072C211yesmaximal, characteristic, complemented
7#339×C131C211yesmaximal, characteristic, complemented
8C22×Dic262C2211yescharacteristic
9C22×D524C2211yescharacteristic
10C2×#14523497C2211yescharacteristic
11#16862234—22no—
12#16862234C414yescomplemented
13C4×Dic262—22no—
14#16862211—22no—
15#16862205C2211yescharacteristic
16#16862202—22no—
17C2×(C2×D131)⋊C4—22no—
18#16862168—22no—
19#16862166—22no—
20#16862164—22no—
21C2×M16×C131C2211yescharacteristic
22#117×C131C2211yescharacteristic
23C2×C42×C131C2211yescharacteristic
24C23×D131—21no—
25C2×Dic262—21no—
26C2×Dic262—21no—
27C2×Dic262—21no—
28C2×Dic262—21no—
29C2×Dic262C2×C412yes—
30C2×Dic262D412yes—
31C2×D524—21no—
32C2×D524—21no—
33C2×D524—21no—
34C2×D524—21no—
35C2×D524—21no—
36C2×D524C2×C412yes—
37C22×Dic131D412yes—
38#14523497—21no—
39#14523497—21no—
40C2×C131⋊C8—21no—
41C4⋊Dic131D412yes—
42Dic131⋊C4—24no—
43C22⋊Dic131—21no—
44(C2×D131)⋊C4—28no—
45C4×Dic131—24no—
46C4×Dic131—21no—
47C22×C4×C131—21no—
48C22×C4×C131C2311yescharacteristic
49M16×C131—21no—
50M16×C131—21no—
51C2×C8×C131—21no—
52C4⋊C4×C131D412yes—
53C22⋊C4×C131—21no—
54C42×C131—21no—
55C42×C131—21no—
56C42×C131C2×C412yes—
57C22×D131—22no—
58C22×D131—44no—
59C22×D131—41no—
60C22×D131—41no—
61Dic262—22no—
62Dic262—22no—
63Dic262—22no—
64Dic262—22no—
65D524—22no—
66D524—22no—
67D524—41no—
68D524—41no—
69C2×Dic131—22no—
70C2×Dic131—22no—
71C2×Dic131—22no—
72C2×Dic131C22⋊C414yes—
73C131⋊C8—41no—
74C23×C131C2×D411yescharacteristic
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—42no—
78C2×C4×C131C22×C411yescharacteristic
79C2×C4×C131C2×D411yescharacteristic
80C2×C4×C131C2×D411yescharacteristic
81C2×C4×C131C2×D411yescharacteristic
82C2×C4×C131#4111yescharacteristic
83C2×C4×C131#4111yescharacteristic
84C8×C131—41no—
85C22×C131—22no—
86C22×C131C2×C22⋊C411yescharacteristic
87C22×C131#12011yescharacteristic
88C22×C131#12311yescharacteristic
89C2×D131—44no—
90C2×D131—42no—
91C2×D131—82no—
92C2×D131—82no—
93C4×C131—42no—
94C4×C131—42no—
95C4×C131C4×D411yescharacteristic
96C4×C131C4×D411yescharacteristic
97C4×C131#12111yescharacteristic
98C4×C131#12111yescharacteristic
99Dic131—24no—
100Dic131—42no—
101C2×C131—21no—
102C2×C131—21no—
103C2×C131#38611yescharacteristic
104C2×C131#45412yes—
105D131—82no—
106C131#188811yescharacteristic, complemented, Sylow
107#1888—1311nomaximal, Sylow
108C2×#124—1311no—
109#519—1311no—
110#418—1311no—
111#418—1311no—
112C2×#101—1311no—
113#339—1311no—
114#339D13111yescharacteristic, complemented
115C22×Q8—1311no—
116C22×D4—1311no—
117C2×M16—1311no—
118C2×M16C2×D13111yescharacteristic
119#124—2622no—
120#124—1314no—
121C4×Q8—2622no—
122C4×Q8—2622no—
123#117—2622no—
124#117—1311no—
125#117C2×D13111yescharacteristic
126C2×C22⋊C4—2622no—
127C2×C42C2×D13111yescharacteristic
128#102—2622no—
129#102—2622no—
130#101—2622no—
131C24—2621no—
132C2×Q8—2621no—
133C2×Q8—2621no—
134C2×Q8—2621no—
135C2×Q8—2621no—
136C2×Q8—1312no—
137C2×Q8—1312no—
138C2×D4—2621no—
139C2×D4—2621no—
140C2×D4—2621no—
141C2×D4—2621no—
142C2×D4—2621no—
143C2×D4—1312no—
144C22×C4—21no—
145C22×C4—1312no—
146C22×C4C22×D13111yescharacteristic
147M16—21no—
148M16—21no—
149M16—2621no—
150M16—2621no—
151C2×C8—21no—
152C2×C8—2621no—
153C4⋊C4—2624no—
154C4⋊C4—1312no—
155C4⋊C4D52412yes—
156C22⋊C4—21no—
157C22⋊C4—2628no—
158C22⋊C4—2621no—
159C42—21no—
160C42—21no—
161C42—2624no—
162C42—2621no—
163C42C4×D13112yes—
164C23—2622no—
165C23—5244no—
166C23—5241no—
167C23—5241no—
168C23D4×D13111yescharacteristic
169Q8—2622no—
170Q8—2622no—
171Q8—2622no—
172Q8—2622no—
173D4—2622no—
174D4—2622no—
175D4—5241no—
176D4—5241no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—2622no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—1314no—
184C2×C4C2×C4×D13111yescharacteristic
185C2×C4D4×D13111yescharacteristic
186C2×C4D4×D13111yescharacteristic
187C2×C4C2×D52411yescharacteristic
188C2×C4#1452351311yescharacteristic
189C2×C4#1452351411yescharacteristic
190C8—41no—
191C8—5241no—
192C22—22no—
193C22—5244no—
194C22—5242no—
195C22—10482no—
196C22—10482no—
197C22C22⋊C4×D13111yescharacteristic
198C22#1686221611yescharacteristic
199C22#1686223111yescharacteristic
200C4—42no—
201C4—42no—
202C4—2624no—
203C4—5242no—
204C4#1686220811yescharacteristic
205C4C4×D52411yescharacteristic
206C4#1686221911yescharacteristic
207C4#1686222111yescharacteristic
208C2—21no—
209C2—21no—
210C2—10482no—
211C2#2255319011yescharacteristic
212C2#2255348612yes—
subgroupG/H
centerC22#16862216
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#339×C131C2
socleC22×C131#120
prime radicalC22×D524C22
cosocleC22×C4×C131C23
hypercenter#339D131
hypocenterC131#1888
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #22553789C2semidirect
5. C2×#16862164C2semidirect
6. #22553072C2semidirect
7. #339×C131C2semidirect
12. #16862234C4semidirect
106. C131#1888semidirect
114. #339D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622342
#225537892
#225533182
#225533162
C2×#168621642
#225530722
#339×C1312
#1888131