Finite Group Databasebeta

← #30670554

Subgroups — #30670554

Order 16768.

440 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 137 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#22553241C212yesmaximal, complemented
2#22553239C212yesmaximal, complemented
3#22553214C211yesmaximal, characteristic
4#22553213C211yesmaximal, characteristic
5#22553213C211yesmaximal, characteristic
6#22553211C211yesmaximal, characteristic, complemented
7#22553204C212yesmaximal, complemented
8#22553201C212yesmaximal, complemented
9C4⋊C4×Dic131C211yesmaximal, characteristic, complemented
10C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
11#382×C131C211yesmaximal, characteristic, complemented
12C22×D524C2211yescharacteristic
13C2×C4⋊Dic131C2211yescharacteristic
14C2×C4⋊Dic131C2212yes—
15C2×Dic131⋊C4C2211yescharacteristic
16C2×Dic131⋊C4C2211yescharacteristic
17C2×(C2×D131)⋊C4C2212yes—
18C2×(C2×D131)⋊C4C2212yes—
19C2×(C2×D131)⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2212yes—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×C4×Dic131C2211yescharacteristic
24C2×C4×Dic131C2211yescharacteristic
25C2×C4×Dic131C2212yes—
26C131⋊#95C2211yescharacteristic
27C131⋊#95C2211yescharacteristic
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C2×C4⋊C4×C131C2211yescharacteristic
31C2×C4⋊C4×C131C2211yescharacteristic
32C2×C42×C131C2211yescharacteristic
33#95×C131C2212yes—
34#95×C131C2212yes—
35C23×D131C2312yes—
36C2×D524—22no—
37C2×D524—22no—
38C2×D524—22no—
39C22×Dic131C2311yescharacteristic
40C22×Dic131C2311yescharacteristic
41C22×Dic131C2312yes—
42C22×Dic131C2311yescharacteristic
43C22×Dic131C2311yescharacteristic
44C4⋊Dic131—22no—
45C4⋊Dic131—42no—
46Dic131⋊C4—22no—
47Dic131⋊C4—22no—
48(C2×D131)⋊C4—24no—
49(C2×D131)⋊C4—24no—
50(C2×D131)⋊C4—42no—
51(C2×D131)⋊C4—42no—
52(C2×D131)⋊C4—42no—
53(C2×D131)⋊C4—42no—
54C4×Dic131—22no—
55C4×Dic131—22no—
56C4×Dic131—42no—
57C22×C4×C131C2311yescharacteristic
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2312yes—
60C22×C4×C131C2312yes—
61C22×C4×C131C2311yescharacteristic
62C4⋊C4×C131—22no—
63C4⋊C4×C131—22no—
64C42×C131—22no—
65C22×D131—22no—
66C22×D131—22no—
67C22×D131—22no—
68C22×D131—22no—
69C22×D131—22no—
70C22×D131—22no—
71C22×D131—22no—
72D524—44no—
73C2×Dic131—21no—
74C2×Dic131—21no—
75C2×Dic131—21no—
76C2×Dic131—21no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—21no—
83C2×Dic131—21no—
84C2×Dic131—22no—
85C2×Dic131—21no—
86C2×Dic131#4112yes—
87C2×Dic131#4112yes—
88C23×C131C2411yescharacteristic
89C2×C4×C131—21no—
90C2×C4×C131—21no—
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—22no—
94C2×C4×C131—22no—
95C2×C4×C131—21no—
96C2×C4×C131—21no—
97C2×C4×C131—22no—
98C2×C4×C131—21no—
99C2×C4×C131—21no—
100C2×C4×C131—22no—
101C2×C4×C131#4112yes—
102C2×C4×C131#4112yes—
103C2×C4×C131#4112yes—
104C22×C131C2×#4111yescharacteristic
105C22×C131C2×#4111yescharacteristic
106C22×C131C2×#4111yescharacteristic
107C22×C131C2×#4111yescharacteristic
108C22×C131C2×#4111yescharacteristic
109C22×C131H(5,𝔽2)11yescharacteristic
110C22×C131H(5,𝔽2)11yescharacteristic
111C2×D131—42no—
112C2×D131—42no—
113C2×D131—42no—
114C2×D131—42no—
115C2×D131—42no—
116C2×D131—42no—
117C2×D131—42no—
118C4×C131—41no—
119C4×C131—41no—
120C4×C131—42no—
121C4×C131—42no—
122C4×C131#12614yes—
123Dic131—22no—
124Dic131—22no—
125Dic131—41no—
126Dic131—42no—
127Dic131—41no—
128C2×C131C2×#12611yescharacteristic
129C2×C131#52911yescharacteristic
130C2×C131#54011yescharacteristic
131C2×C131#55311yescharacteristic
132C2×C131#55511yescharacteristic
133C2×C131#55911yescharacteristic
134C2×C131#55911yescharacteristic
135D131—82no—
136C131#251311yescharacteristic, complemented, Sylow
137#2513—1311nomaximal, Sylow
138#397—1312no—
139#397—1312no—
140#390—1311no—
141#390—1311no—
142#390—1311no—
143#389—1311no—
144#388—1312no—
145#388—1312no—
146#382D13111yescharacteristic, complemented
147C4×C4⋊C4—1311no—
148C4×C4⋊C4—1311no—
149C22×D4—1311no—
150C2×C4⋊C4—1311no—
151C2×C4⋊C4—1311no—
152C2×C4⋊C4—1311no—
153C2×C4⋊C4—1312no—
154C2×C4⋊C4C2×D13111yescharacteristic
155C2×C4⋊C4C2×D13111yescharacteristic
156C2×C22⋊C4—1312no—
157C2×C22⋊C4—1312no—
158C2×C22⋊C4—1312no—
159C2×C22⋊C4—1312no—
160C2×C22⋊C4—1312no—
161C2×C22⋊C4—1312no—
162C2×C42—1311no—
163C2×C42—1311no—
164C2×C42—1312no—
165C2×C42C2×D13111yescharacteristic
166#95—1311no—
167#95—1312no—
168#95—1311no—
169#95—1312no—
170#95C2×D13112yes—
171#95C2×D13112yes—
172C24—1312no—
173C2×D4—2622no—
174C2×D4—2622no—
175C2×D4—2622no—
176C22×C4—1311no—
177C22×C4—1311no—
178C22×C4—1312no—
179C22×C4—1311no—
180C22×C4—1311no—
181C22×C4C22×D13112yes—
182C22×C4C22×D13111yescharacteristic
183C22×C4C22×D13111yescharacteristic
184C22×C4C22×D13112yes—
185C22×C4C22×D13111yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—2622no—
189C4⋊C4—2622no—
190C4⋊C4—2622no—
191C4⋊C4—5242no—
192C22⋊C4—2624no—
193C22⋊C4—2624no—
194C22⋊C4—5242no—
195C22⋊C4—5242no—
196C22⋊C4—5242no—
197C22⋊C4—5242no—
198C42—22no—
199C42—2622no—
200C42—2622no—
201C42—5242no—
202C23—2622no—
203C23—2622no—
204C23—2622no—
205C23—2622no—
206C23—2622no—
207C23—2622no—
208C23—2622no—
209C23C23×D13111yescharacteristic
210D4—5244no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—2621no—
224C2×C4—2621no—
225C2×C4—2621no—
226C2×C4—2621no—
227C2×C4—2621no—
228C2×C4—2621no—
229C2×C4—2621no—
230C2×C4—2622no—
231C2×C4—2622no—
232C2×C4—2621no—
233C2×C4—2621no—
234C2×C4—2622no—
235C2×C4—2621no—
236C2×C4—1312no—
237C2×C4—1312no—
238C2×C4#1452351212yes—
239C2×C4#1452351312yes—
240C2×C4#1452351412yes—
241C22—5242no—
242C22—5242no—
243C22—5242no—
244C22—5242no—
245C22—5242no—
246C22—5242no—
247C22—5242no—
248C22#41×D13111yescharacteristic
249C22#41×D13111yescharacteristic
250C22C2×#1452351211yescharacteristic
251C22C2×#1452351311yescharacteristic
252C22C2×#1452351411yescharacteristic
253C22#1686229211yescharacteristic
254C22#1686229211yescharacteristic
255C4—41no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—2622no—
260C4—2622no—
261C4—5241no—
262C4—5242no—
263C4—5241no—
264C4#1686224114yes—
265C2—10482no—
266C2C2×#1686224111yescharacteristic
267C2#2255384711yescharacteristic
268C2#2255391811yescharacteristic
269C2#2255400611yescharacteristic
270C2#2255402511yescharacteristic
271C2#2255404411yescharacteristic
272C2#2255404411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#382×C131C2
prime radicalC22×D524C22
hypercenter#382D131
hypocenterC131#2513
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #22553241C2semidirect
2. #22553239C2semidirect
6. #22553211C2semidirect
7. #22553204C2semidirect
8. #22553201C2semidirect
9. C4⋊C4×Dic131C2semidirect
10. C4×Dic131⋊C4C2semidirect
11. #382×C131C2semidirect
136. C131#2513semidirect
146. #382D131semidirect

Maximal subgroups

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

subgroupindex
#225532412
#225532392
#225532142
#225532132
#225532112
#225532042
#225532012
C4⋊C4×Dic1312
C4×Dic131⋊C42
#382×C1312
#2513131