Finite Group Databasebeta

← #30670958

Subgroups — #30670958

Order 16768.

604 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16862244C211yesmaximal, characteristic, complemented
2C2×#16862235C211yesmaximal, characteristic
3C2×#16862234C211yesmaximal, characteristic
4C2×#16862208C211yesmaximal, characteristic
5#22553241C214yesmaximal, complemented
6#22553217C211yesmaximal, characteristic, complemented
7#22553171C214yesmaximal, complemented
8C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
9#391×C131C211yesmaximal, characteristic, complemented
10C22×Dic262C2211yescharacteristic
11C22×D524C2211yescharacteristic
12C22×C4×D131C2212yes—
13#16862244—24no—
14#16862235—24no—
15#16862234—24no—
16#16862208—24no—
17C2×Dic131⋊C4C2212yes—
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2214yes—
21C2×(C2×D131)⋊C4C2214yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×C4×Dic131C2211yescharacteristic
24C2×C4×Dic131C2211yescharacteristic
25C131⋊#95C2212yes—
26C131⋊#95C2214yes—
27C22×Q8×C131C2211yescharacteristic
28C2×C4⋊C4×C131C2211yescharacteristic
29C2×C42×C131C2211yescharacteristic
30#95×C131C2214yes—
31C23×D131C2312yes—
32C2×Dic262—22no—
33C2×Dic262—22no—
34C2×Dic262—22no—
35C2×D524—22no—
36C2×D524—22no—
37C2×D524D414yes—
38C22×Dic131C2312yes—
39C22×Dic131C2312yes—
40C22×Dic131C2311yescharacteristic
41C22×Dic131C2311yescharacteristic
42C2×C4×D131—24no—
43C2×C4×D131—28no—
44Dic131⋊C4—24no—
45Dic131⋊C4—24no—
46Dic131⋊C4—24no—
47(C2×D131)⋊C4—28no—
48(C2×D131)⋊C4—28no—
49(C2×D131)⋊C4—24no—
50C4×Dic131—22no—
51C4×Dic131D414yes—
52C2×Q8×C131—22no—
53C2×Q8×C131—22no—
54C2×Q8×C131—22no—
55C22×C4×C131C2312yes—
56C22×C4×C131C2312yes—
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C4⋊C4×C131—22no—
60C42×C131—22no—
61C22×D131—22no—
62C22×D131—24no—
63C22×D131—22no—
64C22×D131—24no—
65C22×D131C2×D414yes—
66Dic262—44no—
67D524—28no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—22no—
71C2×Dic131—22no—
72C2×Dic131—22no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131C2×D412yes—
76C2×Dic131C2×D412yes—
77C2×Dic131#4114yes—
78C4×D131—48no—
79C23×C131C2411yescharacteristic
80Q8×C131—44no—
81C2×C4×C131—22no—
82C2×C4×C131—22no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—22no—
87C2×C4×C131—24no—
88C2×C4×C131—22no—
89C2×C4×C131C2×D412yes—
90C2×C4×C131C2×D412yes—
91C2×C4×C131#4112yes—
92C22×C131C22×D411yescharacteristic
93C22×C131C22×D411yescharacteristic
94C22×C131C2×#4112yes—
95C22×C131C2×#4111yescharacteristic
96C22×C1312−1+412yes—
97C2×D131—24no—
98C2×D131—28no—
99C2×D131—42no—
100C2×D131—42no—
101C2×D131—44no—
102C4×C131—42no—
103C4×C131—42no—
104C4×C131—42no—
105C4×C131#12114yes—
106Dic131—22no—
107Dic131—42no—
108Dic131—42no—
109Dic131#12414yescomplemented
110C2×C131C2×#12111yescharacteristic
111C2×C131C2×#12411yescharacteristic
112C2×C131#53611yescharacteristic
113C2×C131#54712yes—
114C2×C131#55612yes—
115D131—44no—
116C131#254711yescharacteristic, complemented, Sylow
117#2547—1311nomaximal, Sylow
118C2×C4⋊Q8—1311no—
119C2×#124—1311no—
120C2×#124—1311no—
121C2×C4×D4—1311no—
122#397—1314no—
123#391—1311no—
124#391D13111yescharacteristic, complemented
125#380—1314no—
126C4×C4⋊C4—1311no—
127C22×Q8—1311no—
128C22×Q8C2×D13111yescharacteristic
129C22×D4—1311no—
130C23×C4—1312no—
131C4⋊Q8—2624no—
132#124—2624no—
133#124—2624no—
134C4×D4—2624no—
135C2×C4⋊C4—1312no—
136C2×C4⋊C4—1312no—
137C2×C4⋊C4—1312no—
138C2×C4⋊C4C2×D13111yescharacteristic
139C2×C22⋊C4—1312no—
140C2×C22⋊C4—1314no—
141C2×C22⋊C4—1314no—
142C2×C42—1311no—
143C2×C42—1311no—
144C2×C42C2×D13111yescharacteristic
145#95—1312no—
146#95—1314no—
147#95C2×D13114yes—
148C24—1312no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—2622no—
153C2×Q8—2622no—
154C2×Q8—2622no—
155C2×D4—2622no—
156C2×D4—2622no—
157C2×D4—1314no—
158C22×C4—2624no—
159C22×C4—2628no—
160C22×C4—1312no—
161C22×C4—1312no—
162C22×C4—1311no—
163C22×C4—1311no—
164C22×C4C22×D13112yes—
165C22×C4C22×D13112yes—
166C22×C4C22×D13112yes—
167C22×C4C22×D13111yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—2624no—
170C4⋊C4—2624no—
171C4⋊C4—2624no—
172C22⋊C4—2628no—
173C22⋊C4—2628no—
174C22⋊C4—2624no—
175C42—22no—
176C42—2622no—
177C42—1314no—
178C23—2622no—
179C23—2624no—
180C23—2622no—
181C23—2624no—
182C23—1314no—
183C23C23×D13111yescharacteristic
184Q8—44no—
185Q8—5244no—
186D4—2628no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—2622no—
196C2×C4—2622no—
197C2×C4—2622no—
198C2×C4—2622no—
199C2×C4—2622no—
200C2×C4—2622no—
201C2×C4—2622no—
202C2×C4—5248no—
203C2×C4—1312no—
204C2×C4—1312no—
205C2×C4—1314no—
206C2×C4D4×D13112yes—
207C2×C4D4×D13112yes—
208C2×C4#1452351312yes—
209C22—2628no—
210C22—2624no—
211C22—5242no—
212C22—5244no—
213C22—5242no—
214C22C2×D4×D13111yescharacteristic
215C22C2×D4×D13111yescharacteristic
216C22#41×D13112yes—
217C22C2×#1452351311yescharacteristic
218C22#1686229412yes—
219C4—42no—
220C4—42no—
221C4—42no—
222C4—2622no—
223C4—5242no—
224C4—5242no—
225C4—1314no—
226C4#1686221914yes—
227C2—5244no—
228C2C2×#1686221911yescharacteristic
229C2#124×D13111yescharacteristic
230C2#2255388711yescharacteristic
231C2#2255396412yes—
232C2#2255403412yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#391×C131C2
prime radicalC22×D524C22
hypercenter#391D131
hypocenterC131#2547
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16862244C2semidirect
5. #22553241C2semidirect
6. #22553217C2semidirect
7. #22553171C2semidirect
8. C4×Dic131⋊C4C2semidirect
9. #391×C131C2semidirect
109. Dic131#124semidirect
116. C131#2547semidirect
124. #391D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622442
C2×#168622352
C2×#168622342
C2×#168622082
#225532412
#225532172
#225531712
C4×Dic131⋊C42
#391×C1312
#2547131