Finite Group Databasebeta

← #30751842

Subgroups — #30751842

Order 17152.

504 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 127 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#22568773C211yesmaximal, characteristic
2#22566030C211yesmaximal, characteristic
3#22558893C212yesmaximal, complemented
4#22558068C212yesmaximal, complemented
5C67×#1458C211yesmaximal, characteristic
6C22×#11761612—21no—
7C2×#14524043C2211yescharacteristic
8#16863423—24no—
9#16863420—21no—
10#16863420C2212yes—
11#16863335C412yes—
12#16863332C412yes—
13#16863316—21no—
14#16863308—21no—
15#16863294—21no—
16#16863262C2211yescharacteristic
17#16863256—21no—
18#16863143—24no—
19C2×#117×C67C2211yescharacteristic
20#431×C67C2212yes—
21#324×C67—21no—
22#323×C67—21no—
23C22×Dic134—21no—
24C2×#11761612—22no—
25C2×#11761612—22no—
26C2×#11761612—42no—
27C2×#11761612—42no—
28C22×C67⋊C8—22no—
29#14524043—22no—
30#14524043—22no—
31C2×C4⋊Dic67—21no—
32C2×C4⋊Dic67—21no—
33C2×C4⋊Dic67C2×C412yes—
34C2×Dic67⋊C4—22no—
35C2×C22⋊Dic67—22no—
36C4⋊(C67⋊C8)—24no—
37C4⋊(C67⋊C8)—22no—
38C4⋊(C67⋊C8)D414yes—
39C67⋊#95—22no—
40C67⋊#95—22no—
41C67⋊#95—21no—
42C67⋊#95C2×C412yes—
43C23×C4×C67D411yescharacteristic
44#117×C67—22no—
45#117×C67—22no—
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C4⋊C4×C67D411yescharacteristic
48C2×C22⋊C4×C67C2×C411yescharacteristic
49C2×C22⋊C4×C67D411yescharacteristic
50C2×C42×C67D411yescharacteristic
51C2×C42×C67C2311yescharacteristic
52C4⋊C8×C67—22no—
53C22⋊C8×C67—22no—
54C4×C8×C67—22no—
55C2×Dic134—42no—
56C2×Dic134—82no—
57C22×Dic67—22no—
58C22×Dic67—22no—
59#11761612—44no—
60#11761612—42no—
61#11761612—42no—
62C2×C67⋊C8—24no—
63C2×C67⋊C8—44no—
64C4⋊Dic67—24no—
65C4⋊Dic67—41no—
66C4⋊Dic67—41no—
67C4⋊Dic67—41no—
68C4⋊Dic67—41no—
69Dic67⋊C4—82no—
70C22⋊Dic67—82no—
71C24×C67C22⋊C411yescharacteristic
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67SD1612yes—
80C22×C4×C67Q1612yes—
81C22×C4×C67C22×C411yescharacteristic
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C2×C8×C67—42no—
85C4⋊C4×C67—22no—
86C4⋊C4×C67—22no—
87C22⋊C4×C67—22no—
88C22⋊C4×C67—22no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67C2×D412yes—
92C42×C67#4112yes—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67#10312yes—
98C23×C67C2×C22⋊C411yescharacteristic
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—22no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67#10312yes—
112C2×C4×C67C4×D412yes—
113C2×C4×C67#12011yescharacteristic
114C2×C4×C67#12112yes—
115C2×C4×C67#12311yescharacteristic
116C2×C4×C67C2×SD1611yescharacteristic
117C2×C4×C67C2×Q1611yescharacteristic
118C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C67#13711yescharacteristic
120C2×Dic67—42no—
121C2×Dic67—42no—
122C2×Dic67—42no—
123C2×Dic67—42no—
124C2×Dic67—42no—
125C2×Dic67—42no—
126Dic134—84no—
127C8×C67—82no—
128C67⋊C8—44no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—22no—
132C22×C67—42no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—81no—
136C22×C67#38611yescharacteristic
137C22×C67C2×#10311yescharacteristic
138C22×C67#41711yescharacteristic
139C4×C67—22no—
140C4×C67—22no—
141C4×C67—22no—
142C4×C67—41no—
143C4×C67—41no—
144C4×C67#45011yescharacteristic
145C4×C67#45111yescharacteristic
146C4×C67#48311yescharacteristic
147C4×C67#48411yescharacteristic
148Dic67—82no—
149Dic67—82no—
150#9193—671nomaximal, Sylow
151C2×C67—21no—
152C2×C67—21no—
153C2×C67—42no—
154C2×C67#187211yescharacteristic
155C2×C67#191011yescharacteristic
156C2×C67#191511yescharacteristic
157#2905—671no—
158#2628—671no—
159#1549—672no—
160#1458D6711yescharacteristic
161#1286—672no—
162C67#919311yescharacteristic, complemented, Sylow
163C22×M16—1341no—
164C2×C4⋊Q8—671no—
165C2×#117C2×D6711yescharacteristic
166#431C2×D6712yes—
167#424—1344no—
168#423—672no—
169#423—1341no—
170#400—672no—
171#399—672no—
172#395—1341no—
173#393—1341no—
174#389—1341no—
175#382—671no—
176#380—1341no—
177#332—1344no—
178#324—21no—
179#323—21no—
180C22×Q8—1341no—
181C23×C4D26811yescharacteristic
182C2×M16—2682no—
183C2×M16—2682no—
184C2×M16—1342no—
185C2×M16—1342no—
186C22×C8—1342no—
187C4⋊Q8—1342no—
188C4⋊Q8—1342no—
189#117—22no—
190#117—22no—
191C2×C4⋊C4—672no—
192C2×C4⋊C4—1341no—
193C2×C4⋊C4—1341no—
194C2×C4⋊C4—1342no—
195C2×C4⋊C4C4×D6711yescharacteristic
196C2×C4⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4—1342no—
198C2×C22⋊C4C4×D6711yescharacteristic
199C2×C22⋊C4C22⋊D6711yescharacteristic
200C2×C42D26811yescharacteristic
201C2×C42C22×D6711yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—674no—
204C4⋊C8—1344no—
205C4⋊C8—1342no—
206C22⋊C8—22no—
207C4×C8—22no—
208#95—672no—
209#95—1342no—
210#95—1342no—
211#95—1341no—
212C24(C2×D67)⋊C411yescharacteristic
213C2×Q8—2682no—
214C2×Q8—5362no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—1342no—
220C22×C4—1342no—
221C22×C4(C2×D67)⋊C411yescharacteristic
222C22×C4(C2×D67)⋊C411yescharacteristic
223C22×C4(C2×D67)⋊C411yescharacteristic
224C22×C4#1176161712yes—
225C22×C4Dic26812yes—
226C22×C4C2×C4×D6711yescharacteristic
227C22×C4C2×C22⋊D6711yescharacteristic
228C22×C4C2×D26811yescharacteristic
229M16—2684no—
230M16—2682no—
231M16—2682no—
232C2×C8—42no—
233C2×C8—2684no—
234C2×C8—1344no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—2681no—
238C4⋊C4—2681no—
239C4⋊C4—2681no—
240C4⋊C4—2681no—
241C4⋊C4—5362no—
242C4⋊C4—1344no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—5362no—
246C42—21no—
247C42—21no—
248C42D4×D6712yes—
249C42#1176162712yes—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23#1452396812yes—
255C23C2×(C2×D67)⋊C411yescharacteristic
256Q8—5364no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—2682no—
270C2×C4—2682no—
271C2×C4—2682no—
272C2×C4—2682no—
273C2×C4—2682no—
274C2×C4—2682no—
275C2×C4#1452396812yes—
276C2×C4(C22⋊D67)⋊C412yes—
277C2×C4#1452401311yescharacteristic
278C2×C4#1452401512yes—
279C2×C4#1452402711yescharacteristic
280C2×C4C2×#1176161711yescharacteristic
281C2×C4C2×Dic26811yescharacteristic
282C2×C4#1452407111yescharacteristic
283C2×C4#1452407611yescharacteristic
284C8—82no—
285C8—2684no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22#1686327311yescharacteristic
294C22C2×#1452396811yescharacteristic
295C22#1686339811yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—5362no—
302C4—5362no—
303C4#1686354911yescharacteristic
304C4#1686355411yescharacteristic
305C4#1686372711yescharacteristic
306C4#1686373211yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2#2256051411yescharacteristic
311C2#2256073411yescharacteristic
312C2#2256075911yescharacteristic
subgroupG/H
centerC22#16863273
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1458C2
socleC22×C67#386
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#1458D67
hypocenterC67#9193
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #22558893C2semidirect
4. #22558068C2semidirect
162. C67#9193semidirect

Maximal subgroups

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

subgroupindex
#225687732
#225660302
#225588932
#225580682
C67×#14582
#919367