Finite Group Databasebeta

← #30766980

Subgroups — #30766980

Order 17152.

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

normal subgroups only

This group has 568 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#22560201C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558239C214yesmaximal, complemented
4C67×#2409C211yesmaximal, characteristic
5#16863420—21no—
6C2×#14523960—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C23×Q8×C67C411yescharacteristic
12C2×C22⋊Q8×C67—21no—
13C2×C4×D4×C67C2211yescharacteristic
14#395×C67C2212yes—
15#393×C67C412yes—
16#391×C67—21no—
17#386×C67—21no—
18#385×C67—21no—
19#380×C67C411yescharacteristic
20C2×#11761612—22no—
21C22×C67⋊C8—41no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523960—42no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22×Q8×C67—21no—
28C22×Q8×C67—41no—
29C22×Q8×C67—41no—
30C22×Q8×C67—44no—
31C22×Q8×C67C2×C412yes—
32C22×D4×C67D411yescharacteristic
33C23×C4×C67—21no—
34C23×C4×C67C2×C411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C22⋊Q8×C67—44no—
37C4×D4×C67—41no—
38C4×D4×C67—41no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—22no—
48#95×C67—22no—
49#95×C67C2×C412yes—
50#11761612—42no—
51#11761612—42no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—42no—
54C2×C67⋊C8—82no—
55C24×C67C22⋊C411yescharacteristic
56C24×C67C22⋊C411yescharacteristic
57C2×Q8×C67—24no—
58C2×Q8×C67—44no—
59C2×Q8×C67—42no—
60C2×Q8×C67—42no—
61C2×Q8×C67—44no—
62C2×Q8×C67—48no—
63C2×Q8×C67—44no—
64C2×Q8×C67—41no—
65C2×Q8×C67—41no—
66C2×D4×C67—41no—
67C2×D4×C67—41no—
68C2×D4×C67SD1612yes—
69C2×D4×C67Q1612yes—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—22no—
76C22×C4×C67—44no—
77C22×C4×C67—42no—
78C22×C4×C67—42no—
79C22×C4×C67—42no—
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C4⋊C4×C67—41no—
86C4⋊C4×C67—42no—
87C4⋊C4×C67—44no—
88C22⋊C4×C67—41no—
89C22⋊C4×C67—41no—
90C22⋊C4×C67—44no—
91C22⋊C4×C67—42no—
92C42×C67—21no—
93C42×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67—21no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67#10312yes—
103C23×C67#10312yes—
104C23×C67C2×C22⋊C411yescharacteristic
105Q8×C67—44no—
106Q8×C67—44no—
107Q8×C67—44no—
108Q8×C67—44no—
109D4×C67—42no—
110D4×C67—42no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—22no—
114C2×C4×C67—24no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—48no—
121C2×C4×C67—42no—
122C2×C4×C67—42no—
123C2×C4×C67—42no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67—42no—
130C2×C4×C67—81no—
131C2×C4×C67#9911yescharacteristic
132C2×C4×C67#9911yescharacteristic
133C2×C4×C67C2×SD1611yescharacteristic
134C2×C4×C67C2×Q1611yescharacteristic
135C67⋊C8—82no—
136C67⋊C8—82no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—22no—
140C22×C67—22no—
141C22×C67—42no—
142C22×C67—42no—
143C22×C67—42no—
144C22×C67—41no—
145C22×C67—41no—
146C22×C67—41no—
147C22×C67—41no—
148C22×C67—81no—
149C22×C67C2×#9911yescharacteristic
150C22×C67C2×#10311yescharacteristic
151C22×C67#42111yescharacteristic
152C4×C67—21no—
153C4×C67—21no—
154C4×C67—21no—
155C4×C67—41no—
156C4×C67—44no—
157C4×C67—41no—
158C4×C67—41no—
159C4×C67—82no—
160C4×C67AΣL(1,𝔽16)12yes—
161#11904—671nomaximal, Sylow
162C2×C67—21no—
163C2×C67—21no—
164C2×C67—42no—
165C2×C67—42no—
166C2×C67#152011yescharacteristic
167C2×C67C2×AΣL(1,𝔽16)11yescharacteristic
168C2×C67#213511yescharacteristic
169#2409D6711yescharacteristic
170#1800—671no—
171#1459—671no—
172#1354—674no—
173C67#1190411yescharacteristic, complemented, Sylow
174C23×Q8Dic6711yescharacteristic
175C2×C22⋊Q8—21no—
176C2×C4×D4C2×D6711yescharacteristic
177#423—1341no—
178C2×#100—1341no—
179C2×C22⋊C8—1341no—
180#395C2×D6712yes—
181#393Dic6712yes—
182#391—21no—
183#386—21no—
184#385—21no—
185#380Dic6711yescharacteristic
186#343—672no—
187#324—672no—
188#323—1341no—
189C22×Q8—21no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—44no—
193C22×Q8C2×Dic6712yes—
194C22×D4C22⋊D6711yescharacteristic
195C23×C4—21no—
196C23×C4C2×Dic6711yescharacteristic
197C23×C4C22⋊D6711yescharacteristic
198C2×M16—1342no—
199C22×C8—2681no—
200C22⋊Q8—44no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C22×D6711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4C2×Dic6711yescharacteristic
209C2×C22⋊C4C22⋊D6711yescharacteristic
210C2×C42C22⋊D6711yescharacteristic
211C4⋊C8—2681no—
212C4⋊C8—2681no—
213#100—2682no—
214C22⋊C8—2681no—
215C22⋊C8—1342no—
216#95—22no—
217#95—22no—
218#95C2×Dic6712yes—
219C24C22⋊Dic6711yescharacteristic
220C24C22⋊Dic6711yescharacteristic
221C2×Q8—24no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1176161512yes—
233C2×D4#1176161812yes—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊Dic6711yescharacteristic
245C22×C4C22⋊Dic6711yescharacteristic
246C22×C4C22×Dic6711yescharacteristic
247C22×C4C2×C22⋊D6711yescharacteristic
248C22×C4C2×C22⋊D6711yescharacteristic
249M16—2682no—
250M16—2682no—
251C2×C8—2682no—
252C2×C8—2682no—
253C2×C8—5362no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C42—21no—
262C42—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1452396712yes—
272C23#1452396712yes—
273C23C2×C22⋊Dic6711yescharacteristic
274Q8—44no—
275Q8—44no—
276Q8—44no—
277Q8—44no—
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4#1452395811yescharacteristic
301C2×C4#1452395811yescharacteristic
302C2×C4C2×#1176161511yescharacteristic
303C2×C4C2×#1176161811yescharacteristic
304C8—5362no—
305C8—5362no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1452395811yescharacteristic
319C22C2×#1452396711yescharacteristic
320C22#1686341311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—44no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4#1686318312yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2255878111yescharacteristic
335C2C2×#1686318311yescharacteristic
336C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2409C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C4⋊C4×C67C23
hypercenter#2409D67
hypocenterC67#11904
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #22558239C2semidirect
173. C67#11904semidirect

Maximal subgroups

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

subgroupindex
#225602012
#225585272
#225582392
C67×#24092
#1190467