Finite Group Databasebeta

← #30671123

Subgroups — #30671123

Order 16768.

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

normal subgroups only

This group has 532 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
1C2×#16862236C211yesmaximal, characteristic, complemented
2C2×#16862208C211yesmaximal, characteristic
3#22553250C211yesmaximal, characteristic, complemented
4#22553230C211yesmaximal, characteristic
5#22553229C211yesmaximal, characteristic
6#22553204C212yesmaximal, complemented
7#22553201C212yesmaximal, complemented
8#22553171C212yesmaximal, complemented
9#22553171C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#400×C131C211yesmaximal, characteristic, complemented
12C22×D524C2211yescharacteristic
13C22×C4×D131C2212yes—
14#16862236—24no—
15#16862208—24no—
16C2×C4⋊Dic131C2211yescharacteristic
17C2×Dic131⋊C4C2212yes—
18C2×Dic131⋊C4C2211yescharacteristic
19C2×Dic131⋊C4C2211yescharacteristic
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×C4×Dic131C2212yes—
28C2×C4×Dic131C2211yescharacteristic
29C131⋊#95C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2211yescharacteristic
34#95×C131C2211yescharacteristic
35#95×C131C2212yes—
36#95×C131C2212yes—
37C23×D131C2312yes—
38C2×D524—24no—
39C2×D524—22no—
40C22×Dic131C2312yes—
41C22×Dic131C2311yescharacteristic
42C22×Dic131C2311yescharacteristic
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C2×C4×D131—24no—
46C2×C4×D131—24no—
47C2×C4×D131—24no—
48C4⋊Dic131—22no—
49Dic131⋊C4—22no—
50Dic131⋊C4—22no—
51Dic131⋊C4—22no—
52Dic131⋊C4—24no—
53Dic131⋊C4—22no—
54(C2×D131)⋊C4—24no—
55(C2×D131)⋊C4—24no—
56(C2×D131)⋊C4—24no—
57(C2×D131)⋊C4—42no—
58(C2×D131)⋊C4—42no—
59C4×Dic131—42no—
60C4×Dic131D414yes—
61C22×C4×C131C2312yes—
62C22×C4×C131C2312yes—
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2311yescharacteristic
66C4⋊C4×C131—22no—
67C4⋊C4×C131—22no—
68C22×D131—22no—
69C22×D131—22no—
70C22×D131—22no—
71C22×D131—22no—
72C22×D131—22no—
73C22×D131—22no—
74C22×D131#4114yes—
75D524—28no—
76C2×Dic131—21no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—21no—
85C2×Dic131—21no—
86C2×Dic131C2×D412yes—
87C2×Dic131C2×D412yes—
88C2×Dic131#4114yes—
89C4×D131—48no—
90C23×C131C2411yescharacteristic
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—21no—
94C2×C4×C131—21no—
95C2×C4×C131—22no—
96C2×C4×C131—22no—
97C2×C4×C131—22no—
98C2×C4×C131—22no—
99C2×C4×C131—22no—
100C2×C4×C131—22no—
101C2×C4×C131—21no—
102C2×C4×C131—21no—
103C2×C4×C131C2×D412yes—
104C22×C131C22×D411yescharacteristic
105C22×C131C2×#4112yes—
106C22×C131C2×#4112yes—
107C22×C131H(5,𝔽2)11yescharacteristic
108C22×C1312−1+411yescharacteristic
109C2×D131—24no—
110C2×D131—24no—
111C2×D131—24no—
112C2×D131—42no—
113C2×D131—42no—
114C2×D131—42no—
115C2×D131—42no—
116C4×C131—22no—
117C4×C131—41no—
118C4×C131—42no—
119C4×C131—41no—
120C4×C131—42no—
121Dic131—22no—
122Dic131—41no—
123Dic131—41no—
124Dic131—42no—
125Dic131#12414yes—
126C2×C131#52511yescharacteristic
127C2×C131C2×#12411yescharacteristic
128C2×C131#53711yescharacteristic
129C2×C131#55212yes—
130C2×C131#55312yes—
131D131—44no—
132C131#256211yescharacteristic, complemented, Sylow
133#2562—1311nomaximal, Sylow
134C2×#125—1311no—
135C2×C4×D4—1311no—
136#400—1311no—
137#400D13111yescharacteristic, complemented
138#394—1311no—
139#394—1311no—
140#388—1312no—
141#388—1312no—
142#380—1312no—
143#380—1312no—
144C4×C4⋊C4—1311no—
145C22×D4—1311no—
146C23×C4—1312no—
147#125—2624no—
148C4×D4—2624no—
149C2×C4⋊C4—1311no—
150C2×C4⋊C4—1311no—
151C2×C4⋊C4—1311no—
152C2×C4⋊C4—1311no—
153C2×C4⋊C4—1311no—
154C2×C4⋊C4—1312no—
155C2×C4⋊C4C2×D13111yescharacteristic
156C2×C4⋊C4C2×D13111yescharacteristic
157C2×C22⋊C4—1312no—
158C2×C22⋊C4—1312no—
159C2×C22⋊C4—1312no—
160C2×C22⋊C4—1312no—
161C2×C22⋊C4—1312no—
162C2×C42—1312no—
163C2×C42—1311no—
164#95—1311no—
165#95—1312no—
166#95—1312no—
167#95C2×D13111yescharacteristic
168#95C2×D13112yes—
169#95C2×D13112yes—
170C24—1312no—
171C2×D4—2624no—
172C2×D4—2622no—
173C22×C4—2624no—
174C22×C4—2624no—
175C22×C4—2624no—
176C22×C4—1312no—
177C22×C4—1311no—
178C22×C4—1311no—
179C22×C4—1311no—
180C22×C4—1311no—
181C22×C4C22×D13112yes—
182C22×C4C22×D13112yes—
183C22×C4C22×D13111yescharacteristic
184C22×C4C22×D13111yescharacteristic
185C22×C4C22×D13111yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—2624no—
189C4⋊C4—2622no—
190C4⋊C4—2622no—
191C4⋊C4—2622no—
192C4⋊C4—2622no—
193C4⋊C4—2622no—
194C22⋊C4—2624no—
195C22⋊C4—2624no—
196C22⋊C4—2624no—
197C22⋊C4—5242no—
198C22⋊C4—5242no—
199C42—5242no—
200C42—1314no—
201C23—2622no—
202C23—2622no—
203C23—2622no—
204C23—2622no—
205C23—2622no—
206C23—2622no—
207C23—1314no—
208C23C23×D13111yescharacteristic
209D4—2628no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—2621no—
223C2×C4—2621no—
224C2×C4—2621no—
225C2×C4—2621no—
226C2×C4—2622no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—2622no—
230C2×C4—2621no—
231C2×C4—2621no—
232C2×C4—5248no—
233C2×C4—1312no—
234C2×C4—1312no—
235C2×C4—1314no—
236C2×C4D4×D13112yes—
237C22—2624no—
238C22—2624no—
239C22—2624no—
240C22—5242no—
241C22—5242no—
242C22—5242no—
243C22—5242no—
244C22C2×D4×D13111yescharacteristic
245C22#41×D13112yes—
246C22#41×D13112yes—
247C22#1686229211yescharacteristic
248C22#1686229411yescharacteristic
249C4—22no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—42no—
254C4—2622no—
255C4—5241no—
256C4—5242no—
257C4—5241no—
258C4—1314no—
259C2—5244no—
260C2#2255382011yescharacteristic
261C2#124×D13111yescharacteristic
262C2#2255389211yescharacteristic
263C2#2255399412yes—
264C2#2255401312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#400×C131C2
prime radicalC22×D524C22
hypercenter#400D131
hypocenterC131#2562
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16862236C2semidirect
3. #22553250C2semidirect
6. #22553204C2semidirect
7. #22553201C2semidirect
8. #22553171C2semidirect
9. #22553171C2semidirect
10. #22553163C2semidirect
11. #400×C131C2semidirect
132. C131#2562semidirect
137. #400D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622362
C2×#168622082
#225532502
#225532302
#225532292
#225532042
#225532012
#225531712
#225531632
#400×C1312
#2562131