Finite Group Databasebeta

← #30744493

Subgroups — #30744493

Order 17152.

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560352C211yesmaximal, characteristic, complemented
2#22560352C211yesmaximal, characteristic, complemented
3#22558032C212yesmaximal, complemented
4#22558030C212yesmaximal
5C67×#2860C211yesmaximal, characteristic
6#16863151—21no—
7#16863151—21no—
8#16863151—21no—
9#16863151—21no—
10#16863151C2212yes—
11#16863151C2212yes—
12C2×#126×C67—21no—
13#390×C67C411yescharacteristic
14#390×C67C411yescharacteristic
15#390×C67C2211yescharacteristic
16#389×C67C411yescharacteristic
17#389×C67C411yescharacteristic
18#389×C67C2211yescharacteristic
19#388×C67—21no—
20#388×C67—21no—
21#388×C67—21no—
22C43×C67C2211yescharacteristic
23C22×C67⋊C8—22no—
24C22×C67⋊C8—22no—
25C22×D4×C67C2×C411yescharacteristic
26#126×C67—44no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67C2×C411yescharacteristic
31C2×C4⋊C4×C67C2×C411yescharacteristic
32C2×C4⋊C4×C67C2×C411yescharacteristic
33C2×C22⋊C4×C67—21no—
34C2×C22⋊C4×C67—21no—
35C2×C22⋊C4×C67—21no—
36C2×C22⋊C4×C67—21no—
37C2×C22⋊C4×C67—21no—
38C2×C22⋊C4×C67—21no—
39C2×C42×C67C2×C411yescharacteristic
40C2×C42×C67C2×C411yescharacteristic
41C2×C42×C67D411yescharacteristic
42C2×C42×C67D411yescharacteristic
43C2×C42×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67C2311yescharacteristic
46#95×C67—21no—
47#95×C67—21no—
48#95×C67—21no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8—42no—
55C24×C67—21no—
56C2×D4×C67—22no—
57C2×D4×C67—41no—
58C2×D4×C67—41no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22×C411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69C4⋊C4×C67—22no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—42no—
75C22⋊C4×C67—42no—
76C22⋊C4×C67—42no—
77C22⋊C4×C67—42no—
78C22⋊C4×C67—42no—
79C22⋊C4×C67—42no—
80C22⋊C4×C67—42no—
81C42×C67—22no—
82C42×C67—22no—
83C42×C67—22no—
84C42×C67—41no—
85C42×C67—41no—
86C42×C67—41no—
87C42×C67—41no—
88C23×C67—41no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67—41no—
95C23×C67C2×C22⋊C411yescharacteristic
96D4×C67—82no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67#13112yes—
125C2×C4×C67#13512yes—
126C2×C4×C67AGL(1,ℤ/8)12yes—
127C67⋊C8—82no—
128C67⋊C8—82no—
129C22×C67—81no—
130C22×C67—81no—
131C22×C67—81no—
132C22×C67—81no—
133C22×C67—81no—
134C22×C67—81no—
135C22×C67—81no—
136C22×C67#40811yescharacteristic
137C22×C67#41011yescharacteristic
138C22×C67#41311yescharacteristic
139C22×C67#41611yescharacteristic
140C22×C67#41811yescharacteristic
141C22×C67#42111yescharacteristic
142C22×C67#42111yescharacteristic
143C4×C67—22no—
144C4×C67—22no—
145C4×C67—22no—
146C4×C67—41no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—81no—
151C4×C67—81no—
152C4×C67—81no—
153#7726—671nomaximal, Sylow
154C2×C67—161no—
155C2×C67#147311yescharacteristic
156C2×C67#148211yescharacteristic
157C2×C67#149611yescharacteristic
158C2×C67#151411yescharacteristic
159C2×C67#152111yescharacteristic
160C2×C67#153511yescharacteristic
161C2×C67#155311yescharacteristic
162#2860D6711yescharacteristic
163#1839—671no—
164#1839—671no—
165#1271—672no—
166#1270—672no—
167C67#772611yescharacteristic, complemented, Sylow
168C2×#126—21no—
169#390Dic6711yescharacteristic
170#390Dic6711yescharacteristic
171#390C2×D6711yescharacteristic
172#389Dic6711yescharacteristic
173#389Dic6711yescharacteristic
174#389C2×D6711yescharacteristic
175#388—21no—
176#388—21no—
177#388—21no—
178C43C2×D6711yescharacteristic
179#336—672no—
180#336—672no—
181#336—1341no—
182#336—1341no—
183#336—1341no—
184#336—1341no—
185C22×D4C2×Dic6711yescharacteristic
186C22×C8—1342no—
187C22×C8—1342no—
188#126—44no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C2×Dic6711yescharacteristic
193C2×C4⋊C4C2×Dic6711yescharacteristic
194C2×C4⋊C4C2×Dic6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42C2×Dic6711yescharacteristic
202C2×C42C2×Dic6711yescharacteristic
203C2×C42C22⋊D6711yescharacteristic
204C2×C42C22⋊D6711yescharacteristic
205C2×C42C22⋊D6711yescharacteristic
206C2×C42C22⋊D6711yescharacteristic
207C2×C42C22×D6711yescharacteristic
208#95—21no—
209#95—21no—
210#95—21no—
211C24—21no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4C22⋊Dic6711yescharacteristic
219C22×C4C22⋊Dic6711yescharacteristic
220C22×C4C22⋊Dic6711yescharacteristic
221C22×C4C22⋊Dic6711yescharacteristic
222C22×C4C22×Dic6711yescharacteristic
223C22×C4C2×C22⋊D6711yescharacteristic
224C22×C4C2×C22⋊D6711yescharacteristic
225C2×C8—2682no—
226C2×C8—2682no—
227C2×C8—2682no—
228C2×C8—2682no—
229C2×C8—2682no—
230C2×C8—2682no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊Dic6711yescharacteristic
258D4—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4#1452405012yes—
287C2×C4#1452406312yes—
288C2×C4#1452406912yes—
289C8—5362no—
290C8—5362no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1686336311yescharacteristic
299C22#1686337011yescharacteristic
300C22#1686338011yescharacteristic
301C22#1686339211yescharacteristic
302C22#1686340111yescharacteristic
303C22#1686341311yescharacteristic
304C22#1686341311yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C2—161no—
316C2#2255857811yescharacteristic
317C2#2255861711yescharacteristic
318C2#2255867811yescharacteristic
319C2#2255875611yescharacteristic
320C2#2255878611yescharacteristic
321C2#2255884011yescharacteristic
322C2#2255891411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2860C2
socleC23×C67C2×C22⋊C4
prime radicalC22×D4×C67C2×C4
cosocleC2×C42×C67C23
hypercenter#2860D67
hypocenterC67#7726
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #22560352C2semidirect
2. #22560352C2semidirect
3. #22558032C2semidirect
167. C67#7726semidirect

Maximal subgroups

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

subgroupindex
#225603522
#225580322
#225580302
C67×#28602
#772667