Finite Group Databasebeta

← #30766890

Subgroups — #30766890

Order 17152.

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

normal subgroups only

This group has 408 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
1#22565194C211yesmaximal, characteristic
2#22560205C211yesmaximal, characteristic
3#22558216C214yesmaximal
4C67×#1461C211yesmaximal, characteristic
5C2×#14524043C411yescharacteristic
6C2×#14524041—21no—
7#16863420—21no—
8C2×#14523962—21no—
9#16863326C412yes—
10#16863326C2212yes—
11#16863301—21no—
12#16863294—21no—
13#16863294—21no—
14C4×C4⋊Dic67C411yescharacteristic
15#16863165C2212yes—
16C2×C4×Q8×C67C2211yescharacteristic
17#431×C67—21no—
18#324×C67—21no—
19#324×C67C2212yes—
20C22×Dic134—21no—
21C2×#11761612—22no—
22#14524043—41no—
23#14524043—41no—
24#14524041—42no—
25#14524041—42no—
26C2×C4⋊Dic67—21no—
27C2×C4⋊Dic67—21no—
28C2×C4⋊Dic67C2×C412yes—
29C2×Dic67⋊C4—22no—
30C2×Dic67⋊C4—22no—
31C2×C4×Dic67—21no—
32C4⋊(C67⋊C8)—41no—
33C4⋊(C67⋊C8)—41no—
34#14523962—42no—
35C67⋊#95—22no—
36C67⋊#95—22no—
37C67⋊#95C2×C412yes—
38C22×Q8×C67D411yescharacteristic
39C4×Q8×C67—41no—
40C4×Q8×C67—41no—
41C2×C4⋊C4×C67C2×C411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C4⋊C4×C67C2311yescharacteristic
44C2×C42×C67C2×C411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47C4⋊C8×C67—41no—
48C22⋊C8×C67—22no—
49C4×C8×C67—41no—
50C2×Dic134—42no—
51C2×Dic134—42no—
52C2×Dic134—42no—
53C22×Dic67—22no—
54C22×Dic67—22no—
55#11761612—42no—
56#11761612—42no—
57C2×C67⋊C8—42no—
58C4⋊Dic67—24no—
59C4⋊Dic67—41no—
60C4⋊Dic67—41no—
61C4⋊Dic67—42no—
62Dic67⋊C4—44no—
63Dic67⋊C4—42no—
64Dic67⋊C4—42no—
65C4×Dic67—42no—
66C2×Q8×C67—41no—
67C2×Q8×C67—41no—
68C2×Q8×C67D812yes—
69C2×Q8×C67SD1612yes—
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22×C411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C2×C8×C67—42no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—21no—
85C42×C67—21no—
86C42×C67—21no—
87C23×C67C2×C22⋊C411yescharacteristic
88Q8×C67—42no—
89Q8×C67—42no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—22no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—41no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67#9911yescharacteristic
104C2×C4×C67#9911yescharacteristic
105C2×C4×C67#10212yes—
106C2×C4×C67#10212yes—
107C2×C4×C67C2×D811yescharacteristic
108C2×C4×C67C2×SD1611yescharacteristic
109C2×Dic67—24no—
110C2×Dic67—42no—
111C2×Dic67—42no—
112C2×Dic67—42no—
113C2×Dic67—42no—
114C2×Dic67—42no—
115Dic134—84no—
116C8×C67—82no—
117C67⋊C8—82no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67C2×#9911yescharacteristic
121C22×C67C2×#10211yescharacteristic
122C22×C67#42111yescharacteristic
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—22no—
126C4×C67—22no—
127C4×C67—21no—
128C4×C67—41no—
129C4×C67—41no—
130C4×C67—41no—
131C4×C67#35612yes—
132Dic67—44no—
133Dic67—82no—
134#11883—671nomaximal, Sylow
135C2×C67—21no—
136C2×C67—21no—
137C2×C67#151911yescharacteristic
138C2×C67#212311yescharacteristic
139C2×C67C2×#35611yescharacteristic
140#2546—671no—
141#1801—671no—
142#1461D6711yescharacteristic
143#1346—674no—
144C67#1188311yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—671no—
146C2×C4⋊Q8—1341no—
147C2×C4×Q8C2×D6711yescharacteristic
148#431—21no—
149#423—1341no—
150C2×#101—1341no—
151#398—672no—
152#398—672no—
153#391—1341no—
154#389—1341no—
155#389—1341no—
156C4×C4⋊C4—671no—
157#343—672no—
158#324—21no—
159#324C2×D6712yes—
160C22×Q8—1341no—
161C22×Q8D26811yescharacteristic
162C2×M16—1342no—
163C4⋊Q8—2681no—
164C4⋊Q8—2681no—
165C4⋊Q8—2682no—
166C4⋊Q8—2682no—
167C4×Q8—41no—
168C4×Q8—41no—
169C2×C4⋊C4—672no—
170C2×C4⋊C4—1342no—
171C2×C4⋊C4—1341no—
172C2×C4⋊C4—1342no—
173C2×C4⋊C4—1341no—
174C2×C4⋊C4C4×D6711yescharacteristic
175C2×C4⋊C4C22⋊D6711yescharacteristic
176C2×C4⋊C4C22×D6711yescharacteristic
177C2×C42—1341no—
178C2×C42C4×D6711yescharacteristic
179C2×C42D26811yescharacteristic
180C2×C42C22⋊D6711yescharacteristic
181C4⋊C8—41no—
182C4⋊C8—2681no—
183C4⋊C8—2681no—
184#101—2682no—
185C22⋊C8—22no—
186C4×C8—41no—
187#95—672no—
188#95—1342no—
189#95—1342no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—2682no—
193C2×Q8—2682no—
194C2×Q8—2682no—
195C2×Q8D53612yes—
196C2×Q8#1176161712yes—
197C22×C4—1342no—
198C22×C4—1342no—
199C22×C4(C2×D67)⋊C411yescharacteristic
200C22×C4(C2×D67)⋊C411yescharacteristic
201C22×C4(C2×D67)⋊C411yescharacteristic
202C22×C4(C2×D67)⋊C411yescharacteristic
203C22×C4C2×C4×D6711yescharacteristic
204C22×C4C2×C22⋊D6711yescharacteristic
205C22×C4C2×D26811yescharacteristic
206M16—2682no—
207M16—2682no—
208C2×C8—42no—
209C2×C8—2682no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—2682no—
214C4⋊C4—2681no—
215C4⋊C4—2681no—
216C4⋊C4—2684no—
217C4⋊C4—2682no—
218C4⋊C4—2682no—
219C4⋊C4—1344no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—2682no—
227C23C2×(C2×D67)⋊C411yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—5364no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—2682no—
245C2×C4—2682no—
246C2×C4—2682no—
247C2×C4—2682no—
248C2×C4—2682no—
249C2×C4—1344no—
250C2×C4#1452395711yescharacteristic
251C2×C4#1452395711yescharacteristic
252C2×C4#1452396512yes—
253C2×C4#1452396512yes—
254C2×C4C2×D53611yescharacteristic
255C2×C4C2×#1176161711yescharacteristic
256C8—82no—
257C8—5362no—
258C22—21no—
259C22—21no—
260C22C2×#1452395711yescharacteristic
261C22C2×#1452396511yescharacteristic
262C22#1686341211yescharacteristic
263C4—21no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—2684no—
272C4—5362no—
273C4#1686319412yes—
274C2—21no—
275C2—21no—
276C2#2255877811yescharacteristic
277C2#2256186211yescharacteristic
278C2C2×#1686319411yescharacteristic
subgroupG/H
centerC22C2×#14523957
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#1461C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#1461D67
hypocenterC67#11883
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
144. C67#11883semidirect

Maximal subgroups

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

subgroupindex
#225651942
#225602052
#225582162
C67×#14612
#1188367