Finite Group Databasebeta

← #30675490

Subgroups — #30675490

Order 16768.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16862279C211yesmaximal, characteristic
2C22×#14523503C211yesmaximal, characteristic, complemented
3#22553886C211yesmaximal, characteristic
4#22553789C211yesmaximal, characteristic, complemented
5#22553537C214yesmaximal, complemented
6#22553521C214yesmaximal, complemented
7#22553340C211yesmaximal, characteristic, complemented
8#22553306C211yesmaximal, characteristic
9#419×C131C211yesmaximal, characteristic, complemented
10C2×#14523513C2211yescharacteristic
11C22×Dic262C2211yescharacteristic
12#16862279—24no—
13C2×Dic524C2212yes—
14C2×#14523503—24no—
15C2×#14523503C2214yes—
16C2×#14523502C2212yes—
17C22×C131⋊C8C2211yescharacteristic
18#16862244C2212yes—
19#16862235C2212yes—
20#16862229—22no—
21#16862222—22no—
22C4×Dic262—22no—
23#16862211C2214yes—
24#16862205C2211yescharacteristic
25#16862202—22no—
26Dic131⋊C8C2214yes—
27#16862169C2212yes—
28#16862166C2212yes—
29C22×Q8×C131C2211yescharacteristic
30C2×M16×C131C2211yescharacteristic
31#117×C131C2211yescharacteristic
32#103×C131C2214yes—
33#14523513—22no—
34#14523513—22no—
35C2×Dic262—21no—
36C2×Dic262—22no—
37C2×Dic262D414yes—
38C2×Dic262C2311yescharacteristic
39C2×Dic262C2312yes—
40C2×D524C2311yescharacteristic
41C2×C22⋊D131—21no—
42C22×Dic131—21no—
43C2×C4×D131—21no—
44Dic524—24no—
45#14523503—28no—
46#14523502—24no—
47C2×C131⋊C8—22no—
48C2×C131⋊C8C2312yes—
49C4⋊Dic131D412yes—
50Dic131⋊C4—24no—
51Dic131⋊C4—24no—
52C22⋊Dic131D412yes—
53(C2×D131)⋊C4—24no—
54C4×Dic131—24no—
55C4×Dic131C2312yes—
56C2×Q8×C131—21no—
57C2×Q8×C131—22no—
58C2×Q8×C131—22no—
59C2×Q8×C131C2312yes—
60C22×C4×C131—21no—
61C22×C4×C131C2311yescharacteristic
62M16×C131—22no—
63C2×C8×C131C2312yes—
64C4⋊C4×C131C2312yes—
65C22⋊C4×C131—21no—
66C42×C131—21no—
67C22×D131—21no—
68Dic262—24no—
69Dic262—22no—
70Dic262—21no—
71Dic262—21no—
72Dic262C2×D414yes—
73D524—21no—
74D524—21no—
75C22⋊D131—42no—
76C2×Dic131—24no—
77C2×Dic131—22no—
78C2×Dic131—21no—
79C2×Dic131C2×D414yes—
80C4×D131—42no—
81C131⋊C8—22no—
82C23×C131C2×D411yescharacteristic
83Q8×C131—22no—
84Q8×C131—22no—
85Q8×C131—42no—
86C2×C4×C131—22no—
87C2×C4×C131—22no—
88C2×C4×C131—42no—
89C2×C4×C131C2×D411yescharacteristic
90C2×C4×C131C2×D412yes—
91C2×C4×C131#4112yes—
92C2×C4×C131C2411yescharacteristic
93C8×C131—22no—
94C22×C131—22no—
95C22×C131#12112yes—
96C22×C131C22×D411yescharacteristic
97C2×D131—41no—
98C2×D131—42no—
99C4×C131—42no—
100C4×C131—42no—
101C4×C131#12112yes—
102C4×C131C22×D411yescharacteristic
103C4×C131C2×#4111yescharacteristic
104Dic131—24no—
105Dic131—24no—
106Dic131—41no—
107C2×C131—22no—
108C2×C131C2×#12111yescharacteristic
109C2×C131#57812yes—
110D131—81no—
111C131#304411yescharacteristic, complemented, Sylow
112#3044—1311nomaximal, Sylow
113C2×#137—1311no—
114C22×Q16—1311no—
115#536—1311no—
116#519—1311no—
117#464—1314no—
118#462—1314no—
119#424—1311no—
120#419D13111yescharacteristic, complemented
121#416—1311no—
122C2×#41—1311no—
123C22×Q8—1311no—
124C22×Q8C2×D13111yescharacteristic
125#137—2624no—
126C2×Q16—2624no—
127C2×Q16—1314no—
128C2×Q16—1312no—
129C2×SD16—1312no—
130C2×M16C2×D13111yescharacteristic
131C22×C8—1311no—
132C4⋊Q8—1312no—
133#124—1312no—
134#123—2622no—
135C22⋊Q8—2622no—
136C4×Q8—2622no—
137C4×Q8—1314no—
138#117—2622no—
139#117—1311no—
140#117C2×D13111yescharacteristic
141C4⋊C8—1314no—
142#103—1312no—
143#103C2×D13114yes—
144#102—1312no—
145#41—2622no—
146#41—2622no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—2621no—
151C2×Q8—2622no—
152C2×Q8—1314no—
153C2×Q8—1311no—
154C2×Q8—1312no—
155C2×Q8C22×D13112yes—
156C2×D4—2621no—
157C2×D4—1311no—
158C22×C4—21no—
159C22×C4—2621no—
160C22×C4—2621no—
161C22×C4C22×D13111yescharacteristic
162Q16—2624no—
163Q16—2628no—
164SD16—2624no—
165M16—22no—
166C2×C8—2622no—
167C2×C8—1312no—
168C2×C8C22×D13112yes—
169C4⋊C4—2624no—
170C4⋊C4—2624no—
171C4⋊C4—1312no—
172C4⋊C4C22×D13112yes—
173C22⋊C4—21no—
174C22⋊C4—2624no—
175C22⋊C4—1312no—
176C42—21no—
177C42—2624no—
178C42—1312no—
179C23—2621no—
180C23D4×D13111yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—42no—
184Q8—2624no—
185Q8—2622no—
186Q8—2621no—
187Q8—2621no—
188Q8—1314no—
189D4—2621no—
190D4—2621no—
191D4—5242no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—2624no—
196C2×C4—2622no—
197C2×C4—2621no—
198C2×C4—5242no—
199C2×C4—1314no—
200C2×C4D4×D13111yescharacteristic
201C2×C4D4×D13112yes—
202C2×C4#1452351312yes—
203C2×C4C23×D13111yescharacteristic
204C8—22no—
205C8—2622no—
206C22—22no—
207C22—5242no—
208C22—5241no—
209C22#1686221912yes—
210C22C2×D4×D13111yescharacteristic
211C4—42no—
212C4—42no—
213C4—2624no—
214C4—2624no—
215C4—5241no—
216C4#1686221912yes—
217C4C2×D4×D13111yescharacteristic
218C4C2×#1452351311yescharacteristic
219C2—22no—
220C2—10481no—
221C2C2×#1686221911yescharacteristic
222C2#2255412812yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#419×C131C2
socleC22×C131C22×D4
prime radicalC2×#14523513C22
hypercenter#419D131
hypocenterC131#3044
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#14523503C2semidirect
4. #22553789C2semidirect
5. #22553537C2semidirect
6. #22553521C2semidirect
7. #22553340C2semidirect
9. #419×C131C2semidirect
111. C131#3044semidirect
120. #419D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622792
C22×#145235032
#225538862
#225537892
#225535372
#225535212
#225533402
#225533062
#419×C1312
#3044131