Finite Group Databasebeta

← #30766220

Subgroups — #30766220

Order 17152.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14523962C211yesmaximal, characteristic, complemented
2#22558533C211yesmaximal, characteristic, complemented
3#22558239C212yesmaximal, complemented
4#22558212C212yesmaximal
5C67×#2632C211yesmaximal, characteristic
6#16863449—21no—
7C2×#14523962—21no—
8C4×#11761612—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×#124×C67C411yescharacteristic
13C2×C4×Q8×C67C2211yescharacteristic
14#399×C67—21no—
15#395×C67C411yescharacteristic
16#395×C67C2211yescharacteristic
17#394×C67C411yescharacteristic
18#394×C67C2211yescharacteristic
19#393×C67—21no—
20#388×C67—21no—
21#388×C67—21no—
22#383×C67C411yescharacteristic
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—41no—
25#14523962—24no—
26C22⋊(C67⋊C8)—22no—
27#14523954—41no—
28C4×C67⋊C8—41no—
29C4×C67⋊C8—41no—
30C22×Q8×C67C2×C411yescharacteristic
31C22×Q8×C67D411yescharacteristic
32C22×D4×C67C2×C411yescharacteristic
33#124×C67—41no—
34#124×C67—41no—
35C4×Q8×C67—41no—
36C4×Q8×C67—41no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C42×C67C2×C411yescharacteristic
49C2×C42×C67D411yescharacteristic
50C2×C42×C67D411yescharacteristic
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67C2×C411yescharacteristic
56#95×C67C2×C411yescharacteristic
57#11761612—24no—
58#11761612—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C24×C67—21no—
62C2×Q8×C67—22no—
63C2×Q8×C67—22no—
64C2×Q8×C67—41no—
65C2×Q8×C67—41no—
66C2×Q8×C67—81no—
67C2×D4×C67—41no—
68C2×D4×C67—81no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C4⋊C4×C67—41no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—81no—
83C4⋊C4×C67—81no—
84C22⋊C4×C67—42no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—41no—
87C22⋊C4×C67—42no—
88C22⋊C4×C67—42no—
89C22⋊C4×C67—81no—
90C22⋊C4×C67—81no—
91C42×C67—21no—
92C42×C67—21no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67—21no—
96C42×C67—21no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67—41no—
104C23×C67C2×C22⋊C411yescharacteristic
105Q8×C67—44no—
106Q8×C67—82no—
107D4×C67—82no—
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—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
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—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67#9911yescharacteristic
132C2×C4×C67#9911yescharacteristic
133C2×C4×C67#10412yes—
134C2×C4×C67#13512yes—
135C67⋊C8—44no—
136C67⋊C8—82no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—81no—
140C22×C67—81no—
141C22×C67—81no—
142C22×C67—81no—
143C22×C67—81no—
144C22×C67—81no—
145C22×C67—81no—
146C22×C67C2×#9911yescharacteristic
147C22×C67#41611yescharacteristic
148C22×C67C2×#10411yescharacteristic
149C4×C67—21no—
150C4×C67—21no—
151C4×C67—22no—
152C4×C67—22no—
153C4×C67—41no—
154C4×C67—41no—
155C4×C67—41no—
156C4×C67—41no—
157C4×C67—81no—
158C4×C67—81no—
159C4×C67—81no—
160#11738—671nomaximal, Sylow
161C2×C67—21no—
162C2×C67—21no—
163C2×C67—161no—
164C2×C67#150111yescharacteristic
165C2×C67#212211yescharacteristic
166C2×C67#213311yescharacteristic
167#2632D6711yescharacteristic
168C4×#101—671no—
169#1461—671no—
170#1354—672no—
171#1345—672no—
172C67#1173811yescharacteristic, complemented, Sylow
173C2×#124Dic6711yescharacteristic
174C2×C4×Q8C2×D6711yescharacteristic
175#431—1341no—
176C2×#101—1341no—
177C4×M16—1341no—
178#399—21no—
179#395Dic6711yescharacteristic
180#395C2×D6711yescharacteristic
181#394Dic6711yescharacteristic
182#394C2×D6711yescharacteristic
183#393—21no—
184#388—21no—
185#388—21no—
186#383Dic6711yescharacteristic
187#343—672no—
188#324—672no—
189#324—1341no—
190C22×Q8C2×Dic6711yescharacteristic
191C22×Q8C22⋊D6711yescharacteristic
192C22×D4C2×Dic6711yescharacteristic
193C2×M16—1342no—
194#124—41no—
195#124—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C2×Dic6711yescharacteristic
201C2×C4⋊C4C22⋊D6711yescharacteristic
202C2×C4⋊C4C22×D6711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42C2×Dic6711yescharacteristic
210C2×C42C22⋊D6711yescharacteristic
211C2×C42C22⋊D6711yescharacteristic
212C4⋊C8—2681no—
213#101—1344no—
214C22⋊C8—1342no—
215#97—2681no—
216C4×C8—2681no—
217C4×C8—2681no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×Dic6711yescharacteristic
223#95C2×Dic6711yescharacteristic
224C24—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—81no—
230C2×D4—41no—
231C2×D4—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22⋊Dic6711yescharacteristic
238C22×C4C22⋊Dic6711yescharacteristic
239C22×C4C22×Dic6711yescharacteristic
240C22×C4C2×C22⋊D6711yescharacteristic
241C22×C4C2×C22⋊D6711yescharacteristic
242M16—2682no—
243M16—1344no—
244C2×C8—2682no—
245C2×C8—2682no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C22⋊C4—42no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊Dic6711yescharacteristic
272Q8—44no—
273Q8—82no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4#1452395811yescharacteristic
299C2×C4#1452395811yescharacteristic
300C2×C4#1452397012yes—
301C2×C4#1452406312yes—
302C8—2684no—
303C8—5362no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×#1452395811yescharacteristic
314C22#1686339211yescharacteristic
315C22C2×#1452397011yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C2—21no—
328C2—21no—
329C2—161no—
330C2#2255870011yescharacteristic
331C2#2256185911yescharacteristic
332C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2632C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2632D67
hypocenterC67#11738
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. C4×#14523962C2semidirect
2. #22558533C2semidirect
3. #22558239C2semidirect
172. C67#11738semidirect

Maximal subgroups

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

subgroupindex
C4×#145239622
#225585332
#225582392
#225582122
C67×#26322
#1173867