Finite Group Databasebeta

← #30771838

Subgroups — #30771838

Order 17152.

480 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16863193C212yesmaximal
2#22560205C212yesmaximal, complemented
3#22558336C212yesmaximal, complemented
4C67×#2326C211yesmaximal, characteristic
5#16863420—22no—
6C2×#14523962C2214yes—
7#16863193—24no—
8#16863165—22no—
9C2×#124×C67C2211yescharacteristic
10C2×C4×Q8×C67C2211yescharacteristic
11C2×C4×D4×C67C411yescharacteristic
12#391×C67C2211yescharacteristic
13#390×C67C411yescharacteristic
14#388×C67—22no—
15#385×C67—22no—
16#384×C67C411yescharacteristic
17#376×C67C411yescharacteristic
18C2×#11761612—24no—
19C4⋊(C67⋊C8)—44no—
20#14523962—24no—
21#14523962—24no—
22C22×Q8×C67C2311yescharacteristic
23C22×D4×C67C2×C411yescharacteristic
24C23×C4×C67—21no—
25#124×C67—22no—
26#124×C67D412yes—
27#124×C67Q812yes—
28C4×Q8×C67—42no—
29C4×D4×C67—42no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67C2×C411yescharacteristic
32C2×C4⋊C4×C67C2×C411yescharacteristic
33C2×C4⋊C4×C67D412yes—
34C2×C22⋊C4×C67—22no—
35C2×C22⋊C4×C67—21no—
36C2×C22⋊C4×C67—22no—
37C2×C42×C67C2×C411yescharacteristic
38C2×C42×C67C2×C411yescharacteristic
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67D412yes—
41#95×C67—21no—
42#95×C67—21no—
43#95×C67—22no—
44#11761612—44no—
45#11761612—44no—
46C2×C67⋊C8—44no—
47C24×C67—21no—
48C2×Q8×C67—41no—
49C2×Q8×C67—41no—
50C2×Q8×C67C2×D411yescharacteristic
51C2×Q8×C67C2×Q811yescharacteristic
52C2×Q8×C67#4112yes—
53C2×D4×C67—21no—
54C2×D4×C67—41no—
55C2×D4×C67—41no—
56C2×D4×C67C4⋊C412yes—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—42no—
61C22×C4×C67—82no—
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22⋊C412yes—
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22×C411yescharacteristic
66C22×C4×C67C2×D412yes—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—82no—
71C22⋊C4×C67—24no—
72C22⋊C4×C67—24no—
73C22⋊C4×C67—42no—
74C22⋊C4×C67—82no—
75C42×C67—22no—
76C42×C67—24no—
77C42×C67—21no—
78C42×C67—41no—
79C42×C67C4⋊C412yes—
80C23×C67—22no—
81C23×C67—42no—
82C23×C67—42no—
83C23×C67—41no—
84C23×C67—41no—
85C23×C67C2×C22⋊C411yescharacteristic
86Q8×C67—41no—
87Q8×C67—41no—
88Q8×C67—42no—
89D4×C67—42no—
90D4×C67—42no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—21no—
97C2×C4×C67—22no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—42no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—42no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—84no—
109C2×C4×C67#9912yes—
110C2×C4×C67C2×C4⋊C411yescharacteristic
111C2×C4×C67#11711yescharacteristic
112C2×C4×C67#12112yes—
113C2×C4×C67C22⋊Q812yes—
114C67⋊C8—84no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67—44no—
118C22×C67—42no—
119C22×C67—82no—
120C22×C67—81no—
121C22×C67—81no—
122C22×C67#38011yescharacteristic
123C22×C67C2×#9911yescharacteristic
124C22×C67#41011yescharacteristic
125C4×C67—22no—
126C4×C67—21no—
127C4×C67—21no—
128C4×C67—22no—
129C4×C67—22no—
130C4×C67—42no—
131C4×C67—42no—
132C4×C67—41no—
133C4×C67—81no—
134C4×C67—81no—
135#12803—671nomaximal, Sylow
136C2×C67—21no—
137C2×C67—21no—
138C2×C67—82no—
139C2×C67#179911yescharacteristic
140C2×C67#213311yescharacteristic
141C2×C67#213411yescharacteristic
142#2326D6711yescharacteristic
143C2×#355—672no—
144#1801—672no—
145#1394—672no—
146C67#1280311yescharacteristic, complemented, Sylow
147C2×#124C2×D6711yescharacteristic
148C2×C4×Q8C2×D6711yescharacteristic
149C2×C4×D4Dic6711yescharacteristic
150#423—1342no—
151C2×#101—674no—
152#391C2×D6711yescharacteristic
153#390Dic6711yescharacteristic
154#388—22no—
155#385—22no—
156#384Dic6711yescharacteristic
157#376Dic6711yescharacteristic
158#355—1344no—
159#343—1342no—
160C22×Q8C22×D6711yescharacteristic
161C22×D4C2×Dic6711yescharacteristic
162C23×C4—21no—
163C2×M16—1344no—
164#124—22no—
165#124Dic13412yes—
166#124D26812yes—
167C4×Q8—42no—
168C4×D4—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×Dic6711yescharacteristic
171C2×C4⋊C4C2×Dic6711yescharacteristic
172C2×C4⋊C4C22⋊D6712yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C42C2×Dic6711yescharacteristic
177C2×C42C2×Dic6711yescharacteristic
178C2×C42C2×Dic6711yescharacteristic
179C2×C42C22⋊D6712yes—
180C4⋊C8—2684no—
181#101—1344no—
182#101—1344no—
183#95—21no—
184#95—21no—
185#95—22no—
186C24—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8C2×D26811yescharacteristic
190C2×Q8C2×Dic13411yescharacteristic
191C2×Q8#1176162812yes—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4C4⋊Dic6712yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—82no—
201C22×C4C22⋊Dic6711yescharacteristic
202C22×C4C22⋊Dic6712yes—
203C22×C4C22⋊Dic6711yescharacteristic
204C22×C4C22×Dic6711yescharacteristic
205C22×C4C2×C22⋊D6712yes—
206M16—2684no—
207M16—2684no—
208C2×C8—2684no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C42—24no—
218C42—22no—
219C42—21no—
220C42—41no—
221C42C4⋊Dic6712yes—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23C2×C22⋊Dic6711yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4#1452395812yes—
252C2×C4C2×C4⋊Dic6711yescharacteristic
253C2×C4#1452400211yescharacteristic
254C2×C4#1452401712yes—
255C2×C4#1452402212yes—
256C8—5364no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—82no—
262C22—81no—
263C22—81no—
264C22#1686325611yescharacteristic
265C22C2×#1452395811yescharacteristic
266C22#1686337011yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—81no—
276C4—81no—
277C2—21no—
278C2—21no—
279C2—82no—
280C2#2256019711yescharacteristic
281C2#2256189811yescharacteristic
282C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC22×Q8C22×D67
FittingC67×#2326C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC22×Q8×C67C23
hypercenter#2326D67
hypocenterC67#12803
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
2. #22560205C2semidirect
3. #22558336C2semidirect
146. C67#12803semidirect

Maximal subgroups

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

subgroupindex
C2×#168631932
#225602052
#225583362
C67×#23262
#1280367