Finite Group Databasebeta

← #30765461

Subgroups — #30765461

Order 17152.

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

normal subgroups only

This group has 412 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
1#22558553C211yesmaximal, characteristic, complemented
2#22558533C211yesmaximal, characteristic, complemented
3#22558143C212yesmaximal, complemented
4#22558141C212yesmaximal
5C67×#2545C211yesmaximal, characteristic
6#16863453—21no—
7#16863449—21no—
8#16863122—21no—
9#16863122—21no—
10#16863122C2212yes—
11#16863122C2212yes—
12C2×C4⋊Q8×C67C411yescharacteristic
13C2×#124×C67C411yescharacteristic
14C2×#124×C67C411yescharacteristic
15C2×C4×Q8×C67C2211yescharacteristic
16#396×C67—21no—
17#396×C67—21no—
18#391×C67C2211yescharacteristic
19#390×C67C2211yescharacteristic
20#388×C67—21no—
21#388×C67—21no—
22C4×C4⋊C4×C67C411yescharacteristic
23C4⋊(C67⋊C8)—41no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27#14523954—41no—
28C4×C67⋊C8—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C22×Q8×C67C2×C411yescharacteristic
31C22×D4×C67C2×C411yescharacteristic
32C4⋊Q8×C67—42no—
33#124×C67—41no—
34#124×C67—41no—
35#124×C67—42no—
36C4×Q8×C67—41no—
37C4×Q8×C67—41no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C4⋊C4×C67C2311yescharacteristic
44C2×C22⋊C4×C67—21no—
45C2×C22⋊C4×C67—21no—
46C2×C22⋊C4×C67—21no—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67—21no—
50C2×C42×C67C2×C411yescharacteristic
51C2×C42×C67C2×C411yescharacteristic
52C2×C42×C67C2×C411yescharacteristic
53C2×C42×C67D411yescharacteristic
54C2×C42×C67D411yescharacteristic
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—42no—
60C24×C67—21no—
61C2×Q8×C67—22no—
62C2×Q8×C67—22no—
63C2×Q8×C67—22no—
64C2×Q8×C67—41no—
65C2×Q8×C67—41no—
66C2×Q8×C67—41no—
67C2×D4×C67—22no—
68C2×D4×C67—41no—
69C2×D4×C67—41no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—21no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22⋊C411yescharacteristic
77C22×C4×C67C22×C411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C22×C4×C67C2×D411yescharacteristic
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C4⋊C4×C67—42no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C22⋊C4×C67—42no—
89C22⋊C4×C67—42no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67—41no—
92C22⋊C4×C67—42no—
93C42×C67—21no—
94C42×C67—21no—
95C42×C67—21no—
96C42×C67—21no—
97C42×C67—21no—
98C42×C67—21no—
99C42×C67—41no—
100C42×C67—41no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67—41no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67—41no—
108C23×C67C2×C22⋊C411yescharacteristic
109Q8×C67—44no—
110Q8×C67—82no—
111D4×C67—82no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—22no—
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—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—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67#9911yescharacteristic
136C2×C4×C67#9911yescharacteristic
137C2×C4×C67#10012yes—
138C2×C4×C67#13112yes—
139C67⋊C8—82no—
140C67⋊C8—82no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—81no—
144C22×C67—81no—
145C22×C67—81no—
146C22×C67—81no—
147C22×C67—81no—
148C22×C67—81no—
149C22×C67—81no—
150C22×C67#40811yescharacteristic
151C22×C67C2×#9911yescharacteristic
152C22×C67C2×#10011yescharacteristic
153C4×C67—21no—
154C4×C67—21no—
155C4×C67—22no—
156C4×C67—22no—
157C4×C67—42no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—81no—
163C4×C67—81no—
164#11565—671nomaximal, Sylow
165C2×C67—21no—
166C2×C67—21no—
167C2×C67—161no—
168C2×C67#147111yescharacteristic
169C2×C67#212611yescharacteristic
170C2×C67#213311yescharacteristic
171#2545D6711yescharacteristic
172#1467—671no—
173#1461—671no—
174#1319—672no—
175#1318—672no—
176C67#1156511yescharacteristic, complemented, Sylow
177C2×C4⋊Q8Dic6711yescharacteristic
178C2×#124Dic6711yescharacteristic
179C2×#124Dic6711yescharacteristic
180C2×C4×Q8C2×D6711yescharacteristic
181#432—1341no—
182#431—1341no—
183#396—21no—
184#396—21no—
185#391C2×D6711yescharacteristic
186#390C2×D6711yescharacteristic
187#388—21no—
188#388—21no—
189C4×C4⋊C4Dic6711yescharacteristic
190#324—672no—
191#324—672no—
192#324—1341no—
193#324—1341no—
194C22×Q8C2×Dic6711yescharacteristic
195C22×Q8C2×Dic6711yescharacteristic
196C22×D4C2×Dic6711yescharacteristic
197C4⋊Q8—42no—
198#124—41no—
199#124—41no—
200#124—42no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22⋊D6711yescharacteristic
207C2×C4⋊C4C22⋊D6711yescharacteristic
208C2×C4⋊C4C22×D6711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42C2×Dic6711yescharacteristic
216C2×C42C2×Dic6711yescharacteristic
217C2×C42C2×Dic6711yescharacteristic
218C2×C42C22⋊D6711yescharacteristic
219C2×C42C22⋊D6711yescharacteristic
220C4⋊C8—2681no—
221C4⋊C8—2681no—
222C22⋊C8—1342no—
223C22⋊C8—1342no—
224#97—2681no—
225C4×C8—2681no—
226#95—21no—
227#95—21no—
228#95—21no—
229C24—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C22⋊Dic6711yescharacteristic
243C22×C4C22⋊Dic6711yescharacteristic
244C22×C4C22⋊Dic6711yescharacteristic
245C22×C4C22⋊Dic6711yescharacteristic
246C22×C4C22×Dic6711yescharacteristic
247C22×C4C2×C22⋊D6711yescharacteristic
248C22×C4C2×C22⋊D6711yescharacteristic
249C2×C8—2682no—
250C2×C8—2682no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C22⋊Dic6711yescharacteristic
280Q8—44no—
281Q8—82no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4#1452395811yescharacteristic
307C2×C4#1452395811yescharacteristic
308C2×C4#1452396012yes—
309C2×C4#1452405012yes—
310C8—5362no—
311C8—5362no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#1686336311yescharacteristic
322C22C2×#1452395811yescharacteristic
323C22C2×#1452396011yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—161no—
338C2#2255856911yescharacteristic
339C2#2256187411yescharacteristic
340C2#2256189811yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2545C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2545D67
hypocenterC67#11565
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. #22558553C2semidirect
2. #22558533C2semidirect
3. #22558143C2semidirect
176. C67#11565semidirect

Maximal subgroups

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

subgroupindex
#225585532
#225585332
#225581432
#225581412
C67×#25452
#1156567