Finite Group Databasebeta

← #30765319

Subgroups — #30765319

Order 17152.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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#22558553C212yesmaximal, complemented
2#22558155C212yesmaximal, complemented
3#22558141C212yesmaximal
4C67×#2326C211yesmaximal, characteristic
5#16863453—22no—
6#16863122—22no—
7#16863122C2214yes—
8C2×#124×C67C411yescharacteristic
9C2×C4×Q8×C67C2211yescharacteristic
10C2×C4×D4×C67C411yescharacteristic
11#391×C67C411yescharacteristic
12#390×C67C2211yescharacteristic
13#388×C67—22no—
14#385×C67—22no—
15#384×C67C2211yescharacteristic
16#376×C67C411yescharacteristic
17C4⋊(C67⋊C8)—42no—
18C22⋊(C67⋊C8)—24no—
19#14523954—42no—
20C22×Q8×C67C2×C411yescharacteristic
21C22×D4×C67C2×C411yescharacteristic
22C23×C4×C67—21no—
23#124×C67—22no—
24#124×C67—22no—
25C4×Q8×C67—42no—
26C4×D4×C67—42no—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67C2×C411yescharacteristic
29C2×C4⋊C4×C67D412yes—
30C2×C4⋊C4×C67C2311yescharacteristic
31C2×C22⋊C4×C67—22no—
32C2×C22⋊C4×C67—21no—
33C2×C22⋊C4×C67—22no—
34C2×C42×C67C2×C411yescharacteristic
35C2×C42×C67C2×C411yescharacteristic
36C2×C42×C67C2×C411yescharacteristic
37C2×C42×C67D412yes—
38#95×C67—21no—
39#95×C67—21no—
40#95×C67—22no—
41C2×C67⋊C8—44no—
42C24×C67—21no—
43C2×Q8×C67—21no—
44C2×Q8×C67—21no—
45C2×Q8×C67—41no—
46C2×Q8×C67—41no—
47C2×D4×C67—21no—
48C2×D4×C67—41no—
49C2×D4×C67—41no—
50C2×D4×C67M1612yes—
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—42no—
55C22×C4×C67—82no—
56C22×C4×C67C22⋊C412yes—
57C22×C4×C67C22⋊C411yescharacteristic
58C22×C4×C67C22⋊C411yescharacteristic
59C22×C4×C67C22×C411yescharacteristic
60C22×C4×C67C2×D412yes—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—82no—
65C22⋊C4×C67—28no—
66C22⋊C4×C67—42no—
67C22⋊C4×C67—82no—
68C42×C67—22no—
69C42×C67—22no—
70C42×C67—22no—
71C42×C67—21no—
72C42×C67—41no—
73C42×C67M1612yes—
74C23×C67—22no—
75C23×C67—42no—
76C23×C67—42no—
77C23×C67—41no—
78C23×C67—41no—
79C23×C67C2×C22⋊C411yescharacteristic
80Q8×C67—42no—
81Q8×C67—42no—
82D4×C67—42no—
83D4×C67—42no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—22no—
87C2×C4×C67—22no—
88C2×C4×C67—21no—
89C2×C4×C67—22no—
90C2×C4×C67—22no—
91C2×C4×C67—22no—
92C2×C4×C67—41no—
93C2×C4×C67—41no—
94C2×C4×C67—41no—
95C2×C4×C67—42no—
96C2×C4×C67—41no—
97C2×C4×C67—41no—
98C2×C4×C67—42no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—84no—
103C2×C4×C67#9912yes—
104C2×C4×C67C2×M1611yescharacteristic
105C2×C4×C67C2×M1611yescharacteristic
106C67⋊C8—84no—
107C22×C67—21no—
108C22×C67—21no—
109C22×C67—44no—
110C22×C67—42no—
111C22×C67—82no—
112C22×C67—81no—
113C22×C67—81no—
114C22×C67#40711yescharacteristic
115C22×C67C2×#9911yescharacteristic
116C22×C67#41311yescharacteristic
117C4×C67—22no—
118C4×C67—21no—
119C4×C67—21no—
120C4×C67—42no—
121C4×C67—42no—
122C4×C67—41no—
123C4×C67—42no—
124C4×C67—81no—
125C4×C67—81no—
126#11534—671nomaximal, Sylow
127C2×C67—21no—
128C2×C67—21no—
129C2×C67—82no—
130C2×C67#146511yescharacteristic
131C2×C67#212811yescharacteristic
132C2×C67#213311yescharacteristic
133#2326D6711yescharacteristic
134#1467—672no—
135#1325—672no—
136#1318—672no—
137C67#1153411yescharacteristic, complemented, Sylow
138C2×#124Dic6711yescharacteristic
139C2×C4×Q8C2×D6711yescharacteristic
140C2×C4×D4Dic6711yescharacteristic
141#432—1342no—
142#391Dic6711yescharacteristic
143#390C2×D6711yescharacteristic
144#388—22no—
145#385—22no—
146#384C2×D6711yescharacteristic
147#376Dic6711yescharacteristic
148#324—674no—
149#324—1342no—
150C22×Q8C2×Dic6711yescharacteristic
151C22×D4C2×Dic6711yescharacteristic
152C23×C4—21no—
153#124—22no—
154#124—22no—
155C4×Q8—42no—
156C4×D4—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic6711yescharacteristic
159C2×C4⋊C4C22⋊D6712yes—
160C2×C4⋊C4C22×D6711yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42C2×Dic6711yescharacteristic
165C2×C42C2×Dic6711yescharacteristic
166C2×C42C2×Dic6711yescharacteristic
167C2×C42C22⋊D6712yes—
168C4⋊C8—2682no—
169C22⋊C8—1344no—
170#97—2682no—
171#95—21no—
172#95—22no—
173#95—21no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1176161212yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic6712yes—
189C22×C4C22⋊Dic6711yescharacteristic
190C22×C4C22⋊Dic6711yescharacteristic
191C22×C4C22×Dic6711yescharacteristic
192C22×C4C2×C22⋊D6712yes—
193C2×C8—2684no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C22⋊C4—28no—
199C22⋊C4—42no—
200C22⋊C4—82no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42#1176161212yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊Dic6711yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—84no—
236C2×C4#1452395812yes—
237C2×C4C2×#1176161211yescharacteristic
238C2×C4C2×#1176161211yescharacteristic
239C8—5364no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#1686336011yescharacteristic
248C22C2×#1452395811yescharacteristic
249C22#1686338011yescharacteristic
250C4—21no—
251C4—21no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2255854711yescharacteristic
263C2#2256188111yescharacteristic
264C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2326C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2326D67
hypocenterC67#11534
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #22558553C2semidirect
2. #22558155C2semidirect
137. C67#11534semidirect

Maximal subgroups

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

subgroupindex
#225585532
#225581552
#225581412
C67×#23262
#1153467