Finite Group Databasebeta

← #30766873

Subgroups — #30766873

Order 17152.

440 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 103 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#22560205C211yesmaximal, characteristic, complemented
2#22558533C211yesmaximal, characteristic, complemented
3#22558215C212yesmaximal, complemented
4#22558212C212yesmaximal
5C67×#2536C211yesmaximal, characteristic
6#16863449—21no—
7#16863420—21no—
8C2×#14523962—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×C22⋊Q8×C67—21no—
13C2×C4×Q8×C67C2211yescharacteristic
14C2×C4×D4×C67C411yescharacteristic
15#398×C67C411yescharacteristic
16#398×C67C2211yescharacteristic
17#394×C67C411yescharacteristic
18#394×C67C2211yescharacteristic
19#388×C67—21no—
20#388×C67—21no—
21#382×C67C411yescharacteristic
22#380×C67—21no—
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—41no—
25C4⋊(C67⋊C8)—41no—
26C4⋊(C67⋊C8)—41no—
27#14523962—42no—
28C22⋊(C67⋊C8)—22no—
29C4×C67⋊C8—41no—
30C22×Q8×C67D411yescharacteristic
31C22×D4×C67C2×C411yescharacteristic
32C23×C4×C67—21no—
33C22⋊Q8×C67—42no—
34C22⋊Q8×C67—42no—
35C4×Q8×C67—41no—
36C4×Q8×C67—41no—
37C4×D4×C67—41no—
38C4×D4×C67—41no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C4⋊C4×C67C2311yescharacteristic
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67—21no—
50C2×C22⋊C4×C67—21no—
51C2×C42×C67C2×C411yescharacteristic
52C2×C42×C67D411yescharacteristic
53C2×C42×C67D411yescharacteristic
54#95×C67—21no—
55#95×C67—21no—
56#95×C67—21no—
57#95×C67C2×C411yescharacteristic
58#95×C67C2×C411yescharacteristic
59#11761612—42no—
60#11761612—42no—
61C2×C67⋊C8—42no—
62C2×C67⋊C8—42no—
63C24×C67—21no—
64C2×Q8×C67—41no—
65C2×Q8×C67—41no—
66C2×Q8×C67D812yes—
67C2×Q8×C67SD1612yes—
68C2×D4×C67—41no—
69C2×D4×C67—42no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67—42no—
74C22×C4×C67—42no—
75C22×C4×C67—42no—
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22⋊C411yescharacteristic
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C2×D411yescharacteristic
82C22×C4×C67C2×D411yescharacteristic
83C4⋊C4×C67—41no—
84C4⋊C4×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—42no—
88C4⋊C4×C67—42no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C22⋊C4×C67—42no—
92C22⋊C4×C67—42no—
93C22⋊C4×C67—41no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—42no—
96C22⋊C4×C67—41no—
97C22⋊C4×C67—41no—
98C42×C67—21no—
99C42×C67—21no—
100C42×C67—21no—
101C42×C67—21no—
102C42×C67—21no—
103C42×C67—21no—
104C23×C67—22no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67—41no—
109C23×C67—41no—
110C23×C67—41no—
111C23×C67C2×C22⋊C411yescharacteristic
112Q8×C67—42no—
113Q8×C67—42no—
114D4×C67—42no—
115D4×C67—42no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
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—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—84no—
139C2×C4×C67#9911yescharacteristic
140C2×C4×C67#9911yescharacteristic
141C2×C4×C67#10212yes—
142C2×C4×C67#10212yes—
143C2×C4×C67C2×D811yescharacteristic
144C2×C4×C67C2×SD1611yescharacteristic
145C67⋊C8—82no—
146C67⋊C8—82no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—42no—
150C22×C67—42no—
151C22×C67—42no—
152C22×C67—81no—
153C22×C67—81no—
154C22×C67—81no—
155C22×C67—81no—
156C22×C67C2×#9911yescharacteristic
157C22×C67C2×#10211yescharacteristic
158C22×C67#42111yescharacteristic
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—21no—
162C4×C67—22no—
163C4×C67—22no—
164C4×C67—21no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67—81no—
169C4×C67—81no—
170C4×C67—81no—
171#11879—671nomaximal, Sylow
172C2×C67—21no—
173C2×C67—21no—
174C2×C67—82no—
175C2×C67#151911yescharacteristic
176C2×C67#212011yescharacteristic
177C2×C67#213311yescharacteristic
178#2536D6711yescharacteristic
179#1801—671no—
180#1461—671no—
181#1346—672no—
182#1345—672no—
183C67#1187911yescharacteristic, complemented, Sylow
184C2×C22⋊Q8—21no—
185C2×C4×Q8C2×D6711yescharacteristic
186C2×C4×D4Dic6711yescharacteristic
187#431—1341no—
188#423—1341no—
189C2×#101—1341no—
190#398Dic6711yescharacteristic
191#398C2×D6711yescharacteristic
192#394Dic6711yescharacteristic
193#394C2×D6711yescharacteristic
194#388—21no—
195#388—21no—
196#382Dic6711yescharacteristic
197#380—21no—
198#343—672no—
199#324—672no—
200#324—1341no—
201C22×Q8C22⋊D6711yescharacteristic
202C22×D4C2×Dic6711yescharacteristic
203C23×C4—21no—
204C2×M16—1342no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic6711yescharacteristic
215C2×C4⋊C4C2×Dic6711yescharacteristic
216C2×C4⋊C4C22⋊D6711yescharacteristic
217C2×C4⋊C4C22×D6711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×Dic6711yescharacteristic
224C2×C42C22⋊D6711yescharacteristic
225C2×C42C22⋊D6711yescharacteristic
226C4⋊C8—2681no—
227C4⋊C8—2681no—
228C4⋊C8—2681no—
229#101—2682no—
230C22⋊C8—1342no—
231C4×C8—2681no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C2×Dic6711yescharacteristic
236#95C2×Dic6711yescharacteristic
237C24—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#1176161312yes—
241C2×Q8#1176161612yes—
242C2×D4—41no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊Dic6711yescharacteristic
251C22×C4C22⋊Dic6711yescharacteristic
252C22×C4C22⋊Dic6711yescharacteristic
253C22×C4C22⋊Dic6711yescharacteristic
254C22×C4C22×Dic6711yescharacteristic
255C22×C4C2×C22⋊D6711yescharacteristic
256C22×C4C2×C22⋊D6711yescharacteristic
257M16—2682no—
258M16—2682no—
259C2×C8—2682no—
260C2×C8—2682no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic6711yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—42no—
293D4—42no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4#1452395811yescharacteristic
318C2×C4#1452395811yescharacteristic
319C2×C4#1452396412yes—
320C2×C4#1452396412yes—
321C2×C4C2×#1176161311yescharacteristic
322C2×C4C2×#1176161611yescharacteristic
323C8—5362no—
324C8—5362no—
325C22—21no—
326C22—21no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1452395811yescharacteristic
335C22C2×#1452396411yescharacteristic
336C22#1686341311yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—82no—
352C2#2255877711yescharacteristic
353C2#2256185011yescharacteristic
354C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2536C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2536D67
hypocenterC67#11879
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #22560205C2semidirect
2. #22558533C2semidirect
3. #22558215C2semidirect
183. C67#11879semidirect

Maximal subgroups

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

subgroupindex
#225602052
#225585332
#225582152
#225582122
C67×#25362
#1187967