Finite Group Databasebeta

← #30666159

Subgroups — #30666159

Order 16768.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16862204C211yesmaximal, characteristic
2C2×#16862179C211yesmaximal, characteristic, complemented
3C2×(C2×D131)⋊C8C211yesmaximal, characteristic
4#22553169C211yesmaximal, characteristic
5#22553075C211yesmaximal, characteristic, complemented
6#22553074C211yesmaximal, characteristic, complemented
7C2×#106×C131C211yesmaximal, characteristic, complemented
8C22×C4×D131C2211yescharacteristic
9C22×C131⋊C8C2211yescharacteristic
10#16862204—22no—
11#16862204—22no—
12C2×C4⋊Dic131C2211yescharacteristic
13C2×C4⋊Dic131C2211yescharacteristic
14C2×(C2×D131)⋊C4—21no—
15C2×(C2×D131)⋊C4—21no—
16C2×C4×Dic131—21no—
17#16862179—22no—
18(C2×D131)⋊C8C414yescomplemented
19C131⋊#95—21no—
20C22×C8×C131C2211yescharacteristic
21C2×C4⋊C4×C131C2211yescharacteristic
22C2×C4⋊C4×C131C2211yescharacteristic
23#106×C131—22no—
24C23×D131D411yescharacteristic
25C22×Dic131—21no—
26C22×Dic131—21no—
27C22×Dic131D411yescharacteristic
28C2×C4×D131—21no—
29C2×C4×D131—21no—
30C2×C4×D131—21no—
31C2×C4×D131—21no—
32C2×C4×D131C2×C412yes—
33C2×C4×D131Q812yes—
34C2×C131⋊C8—21no—
35C2×C131⋊C8—21no—
36C2×C131⋊C8C2×C412yes—
37C4⋊Dic131—21no—
38C4⋊Dic131—21no—
39C4⋊Dic131—21no—
40C4⋊Dic131D412yes—
41(C2×D131)⋊C4—24no—
42(C2×D131)⋊C4—42no—
43C4×Dic131—24no—
44C22×C4×C131—21no—
45C22×C4×C131—21no—
46C22×C4×C131C2311yescharacteristic
47C2×C8×C131—21no—
48C2×C8×C131—21no—
49C2×C8×C131C2×C412yes—
50C4⋊C4×C131—21no—
51C4⋊C4×C131—21no—
52C4⋊C4×C131—21no—
53C4⋊C4×C131D412yes—
54C22×D131—21no—
55C22×D131—21no—
56C22×D131—21no—
57C22×D131—21no—
58C22×D131—21no—
59C22×D131—21no—
60C22×D131C4⋊C412yes—
61C2×Dic131—21no—
62C2×Dic131—21no—
63C2×Dic131—22no—
64C2×Dic131—41no—
65C2×Dic131—41no—
66C2×Dic131—41no—
67C2×Dic131—41no—
68C2×Dic131—41no—
69C2×Dic131C4⋊C412yes—
70C4×D131—22no—
71C4×D131—22no—
72C4×D131—22no—
73C4×D131—22no—
74C131⋊C8—22no—
75C23×C131C2×D411yescharacteristic
76C2×C4×C131—22no—
77C2×C4×C131—41no—
78C2×C4×C131—41no—
79C2×C4×C131—41no—
80C2×C4×C131—41no—
81C2×C4×C131—41no—
82C2×C4×C131C22×C411yescharacteristic
83C2×C4×C131C2×D411yescharacteristic
84C2×C4×C131C2×D411yescharacteristic
85C2×C4×C131C2×Q811yescharacteristic
86C2×C4×C131#4111yescharacteristic
87C2×C4×C131#4111yescharacteristic
88C8×C131—22no—
89C22×C131C2×C4⋊C411yescharacteristic
90C22×C131#12011yescharacteristic
91C22×C131#12311yescharacteristic
92C22×C131#13511yescharacteristic
93C22×C131#13511yescharacteristic
94C22×C131AGL(1,ℤ/8)11yescharacteristic
95C22×C131#13711yescharacteristic
96C2×D131—22no—
97C2×D131—22no—
98C2×D131—22no—
99C2×D131—41no—
100C2×D131—41no—
101C2×D131—41no—
102C2×D131—41no—
103C4×C131—42no—
104C4×C131—81no—
105C4×C131C4×D411yescharacteristic
106C4×C131C4×D411yescharacteristic
107C4×C131C22⋊Q811yescharacteristic
108C4×C131C22⋊Q811yescharacteristic
109Dic131—22no—
110Dic131—42no—
111Dic131—81no—
112C2×C131#38511yescharacteristic
113C2×C131#42711yescharacteristic
114C2×C131#42811yescharacteristic
115C2×C131#44911yescharacteristic
116C2×C131#45211yescharacteristic
117C2×C131#48211yescharacteristic
118C2×C131#48511yescharacteristic
119D131—42no—
120C131#185411yescharacteristic, complemented, Sylow
121#1854—1311nomaximal, Sylow
122C2×#117—1311no—
123C2×#107—1311no—
124C2×#106D13111yescharacteristic, complemented
125C2×C22⋊C8—1311no—
126#380—1311no—
127#340—1311no—
128#340—1311no—
129C23×C4—1311no—
130C22×C8—1311no—
131C22×C8C2×D13111yescharacteristic
132#117—2622no—
133#117—2622no—
134C2×C4⋊C4—1311no—
135C2×C4⋊C4—1311no—
136C2×C4⋊C4C2×D13111yescharacteristic
137C2×C4⋊C4C2×D13111yescharacteristic
138C2×C22⋊C4—2621no—
139C2×C22⋊C4—2621no—
140C2×C42—2621no—
141#107—2622no—
142#106—22no—
143C22⋊C8—1314no—
144#95—2621no—
145C24—1311no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—2621no—
149C22×C4—2621no—
150C22×C4—2621no—
151C22×C4—2621no—
152C22×C4—2621no—
153C22×C4—2621no—
154C22×C4—1311no—
155C22×C4—1312no—
156C22×C4—1312no—
157C22×C4C22×D13111yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—2621no—
161C2×C8—2621no—
162C2×C8—1312no—
163C2×C8C4×D13112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—2621no—
168C4⋊C4—2621no—
169C4⋊C4—2621no—
170C4⋊C4—1312no—
171C4⋊C4C22⋊D13112yes—
172C22⋊C4—2624no—
173C22⋊C4—5242no—
174C42—2624no—
175C23—2621no—
176C23—2621no—
177C23—2621no—
178C23—2621no—
179C23—2621no—
180C23—2621no—
181C23—1312no—
182C23D4×D13111yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2621no—
190C2×C4—2621no—
191C2×C4—2622no—
192C2×C4—2622no—
193C2×C4—2622no—
194C2×C4—2622no—
195C2×C4—2622no—
196C2×C4—5241no—
197C2×C4—5241no—
198C2×C4—5241no—
199C2×C4—5241no—
200C2×C4—5241no—
201C2×C4—1312no—
202C2×C4C2×C4×D13111yescharacteristic
203C2×C4D4×D13111yescharacteristic
204C2×C4C2×C22⋊D13111yescharacteristic
205C2×C4Q8×D13111yescharacteristic
206C2×C4#1452351311yescharacteristic
207C2×C4#1452351411yescharacteristic
208C8—22no—
209C8—2622no—
210C22—2622no—
211C22—2622no—
212C22—2622no—
213C22—5241no—
214C22—5241no—
215C22—5241no—
216C22—5241no—
217C22C4⋊C4×D13111yescharacteristic
218C22#1686221411yescharacteristic
219C22#1686223111yescharacteristic
220C22#1686226711yescharacteristic
221C22#1686226711yescharacteristic
222C22#1686227311yescharacteristic
223C22#1686227511yescharacteristic
224C4—42no—
225C4—2622no—
226C4—81no—
227C4—5242no—
228C4—10481no—
229C4C4×C22⋊D13111yescharacteristic
230C4#1686220811yescharacteristic
231C4#1686222211yescharacteristic
232C4#1686222411yescharacteristic
233C2—5242no—
234C2#2255318311yescharacteristic
235C2#2255334711yescharacteristic
236C2#2255335111yescharacteristic
237C2#2255345611yescharacteristic
238C2#2255347111yescharacteristic
239C2#2255363811yescharacteristic
240C2#2255365611yescharacteristic
subgroupG/H
centerC23D4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
FittingC2×#106×C131C2
socleC23×C131C2×D4
prime radicalC23×D131D4
cosocleC22×C4×C131C23
hypercenterC2×#106D131
hypocenterC131#1854
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
2. C2×#16862179C2semidirect
5. #22553075C2semidirect
6. #22553074C2semidirect
7. C2×#106×C131C2semidirect
18. (C2×D131)⋊C8C4semidirect
120. C131#1854semidirect
124. C2×#106D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622042
C2×#168621792
C2×(C2×D131)⋊C82
#225531692
#225530752
#225530742
C2×#106×C1312
#1854131