Finite Group Databasebeta

← #30752498

Subgroups — #30752498

Order 17152.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#22571157C211yesmaximal, characteristic
2#22569149C211yesmaximal, characteristic
3#22558894C211yesmaximal, characteristic
4#22558772C211yesmaximal, characteristic
5#22558067C211yesmaximal, characteristic
6#22558040C211yesmaximal, characteristic
7C67×#1993C211yesmaximal, characteristic
8#16863953C2211yescharacteristic
9#16863775C412yescomplemented
10#16863746C412yescomplemented
11#16863603—22no—
12#16863552—22no—
13#16863524—22no—
14#16863524—22no—
15#16863449C2211yescharacteristic
16#16863449C2211yescharacteristic
17#16863422—21no—
18#16863420C2211yescharacteristic
19#16863351—21no—
20#16863142—22no—
21#16863132—22no—
22C2×C4⋊Q8×C67C2211yescharacteristic
23C2×#97×C67C2211yescharacteristic
24#382×C67C2211yescharacteristic
25#340×C67—21no—
26#340×C67—21no—
27C2×#11761618—24no—
28C2×#11761612—21no—
29#14524043C2×C411yescharacteristic
30#14524043C2×C411yescharacteristic
31#14524035—21no—
32#14524022—21no—
33#14524022D412yes—
34C4×Dic134—21no—
35C4×Dic134D412yes—
36#14524002—21no—
37C4⋊(C67⋊C8)—21no—
38C4⋊(C67⋊C8)—21no—
39C4⋊(C67⋊C8)—21no—
40C4⋊(C67⋊C8)D412yes—
41Dic67⋊C8—24no—
42#14523967—24no—
43C22⋊(C67⋊C8)—21no—
44C22⋊(C67⋊C8)—21no—
45#14523953—22no—
46C4×C67⋊C8—21no—
47C4×C67⋊C8C2×C412yes—
48C8×Dic67—22no—
49C22×Q8×C67—21no—
50C22×C8×C67D411yescharacteristic
51C22×C8×C67D411yescharacteristic
52C4⋊Q8×C67—21no—
53C4⋊Q8×C67—21no—
54C4⋊Q8×C67—21no—
55C4⋊Q8×C67C2×C412yes—
56C2×C4⋊C4×C67—21no—
57C2×C4⋊C4×C67—21no—
58C2×C4⋊C4×C67—21no—
59C2×C42×C67C2311yescharacteristic
60#97×C67—21no—
61#97×C67—21no—
62#95×C67—21no—
63#95×C67—21no—
64C2×Dic134—21no—
65C2×Dic134C22⋊C412yes—
66#11761618—84no—
67#11761612—41no—
68#11761612—41no—
69C2×C67⋊C8—22no—
70C2×C67⋊C8—22no—
71C2×C67⋊C8—41no—
72C4⋊Dic67—21no—
73C4⋊Dic67—21no—
74C4⋊Dic67—21no—
75C4⋊Dic67C22⋊C412yes—
76Dic67⋊C4—22no—
77Dic67⋊C4—22no—
78C22⋊Dic67—22no—
79C4×Dic67—22no—
80C2×Q8×C67—22no—
81C2×Q8×C67—41no—
82C2×Q8×C67—82no—
83C22×C4×C67—41no—
84C22×C4×C67—41no—
85C22×C4×C67C2×D411yescharacteristic
86C22×C4×C67C2×D411yescharacteristic
87C22×C4×C67#4111yescharacteristic
88C2×C8×C67—21no—
89C2×C8×C67—21no—
90C2×C8×C67—21no—
91C2×C8×C67—21no—
92C2×C8×C67—21no—
93C2×C8×C67Q1612yescomplemented
94C4⋊C4×C67—22no—
95C4⋊C4×C67—22no—
96C4⋊C4×C67—41no—
97C4⋊C4×C67—41no—
98C4⋊C4×C67—41no—
99C4⋊C4×C67—41no—
100C42×C67C22×C411yescharacteristic
101C42×C67C2×D411yescharacteristic
102C42×C67C2×D411yescharacteristic
103C42×C67#4111yescharacteristic
104C23×C67#12111yescharacteristic
105Q8×C67—44no—
106Q8×C67—82no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—42no—
110C2×C4×C67—82no—
111C2×C4×C67—81no—
112C2×C4×C67—81no—
113C2×C4×C67—81no—
114C2×C4×C67C2×C22⋊C411yescharacteristic
115C2×C4×C67C4×D411yescharacteristic
116C2×C4×C67C4×D411yescharacteristic
117C2×C4×C67#12011yescharacteristic
118C2×C4×C67#12111yescharacteristic
119C2×C4×C67#12311yescharacteristic
120C2×C4×C67C2×Q1611yescharacteristic
121C2×C4×C67#13511yescharacteristic
122C2×C4×C67#13711yescharacteristic
123C2×C4×C67#13711yescharacteristic
124C2×Dic67—22no—
125C2×Dic67—22no—
126Dic134—42no—
127Dic134—81no—
128C8×C67—22no—
129C8×C67—41no—
130C67⋊C8—42no—
131C67⋊C8—42no—
132C67⋊C8—81no—
133C22×C67—21no—
134C22×C67—21no—
135C22×C67#38611yescharacteristic
136C22×C67#46211yescharacteristic
137C22×C67#46411yescharacteristic
138C4×C67—21no—
139C4×C67—21no—
140C4×C67—21no—
141C4×C67—21no—
142C4×C67—82no—
143C4×C67—161no—
144C4×C67C4×Q1611yescharacteristic
145C4×C67#44111yescharacteristic
146C4×C67#45111yescharacteristic
147C4×C67#45211yescharacteristic
148Dic67—42no—
149Dic67—42no—
150#9290—671nomaximal, Sylow
151C2×C67—21no—
152C2×C67—21no—
153C2×C67#187811yescharacteristic
154C2×C67#190211yescharacteristic
155C2×C67#191011yescharacteristic
156#3209—671no—
157#2961—671no—
158#1993D6711yescharacteristic
159#1549—671no—
160#1518—671no—
161#1286—671no—
162#1275—671no—
163C67#929011yescharacteristic, complemented, Sylow
164#533—671no—
165C2×C4⋊Q8C2×D6711yescharacteristic
166#494—672no—
167#487—672no—
168#462—1342no—
169#451—1342no—
170#446—1342no—
171#446—1342no—
172#431—671no—
173#431—671no—
174#424—1341no—
175#423—671no—
176#405—1341no—
177C2×#97C2×D6711yescharacteristic
178#382C2×D6711yescharacteristic
179#340—21no—
180#340—21no—
181#332—1342no—
182#328—1342no—
183C22×Q8—21no—
184C2×Q16—1344no—
185C2×M16—1341no—
186C22×C8C22⋊D6711yescharacteristic
187C22×C8C22⋊D6711yescharacteristic
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—671no—
192C4⋊Q8—671no—
193C4⋊Q8C4×D6712yes—
194#125—1341no—
195C22⋊Q8—672no—
196C22⋊Q8—1341no—
197C4×Q8—672no—
198C4×Q8—1341no—
199#117—1341no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C42C22×D6711yescharacteristic
204C4⋊C8—672no—
205C4⋊C8—1344no—
206C4⋊C8—1341no—
207C4⋊C8—1341no—
208C4⋊C8—1341no—
209#103—1344no—
210C22⋊C8—1341no—
211C22⋊C8—1341no—
212#97—21no—
213#97—21no—
214#97—1342no—
215C4×C8—672no—
216C4×C8—1342no—
217C4×C8—1341no—
218#95—21no—
219#95—21no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—672no—
224C2×Q8—1341no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4D4×D6711yescharacteristic
228C22×C4C2×C22⋊D6711yescharacteristic
229C22×C4#1176162711yescharacteristic
230Q16—5364no—
231M16—2681no—
232M16—2681no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—2681no—
239C2×C8—1342no—
240C2×C8—1342no—
241C2×C8#1176161812yescomplemented
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—672no—
249C4⋊C4—1341no—
250C4⋊C4—1341no—
251C4⋊C4—1341no—
252C4⋊C4—1342no—
253C4⋊C4—1342no—
254C22⋊C4—1342no—
255C42—1342no—
256C42C2×C4×D6711yescharacteristic
257C42D4×D6711yescharacteristic
258C42D4×D6711yescharacteristic
259C42#1176162811yescharacteristic
260C23#1452401411yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—2682no—
264Q8—5361no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—1342no—
273C2×C4—1342no—
274C2×C4C22⋊C4×D6711yescharacteristic
275C2×C4(C22⋊D67)⋊C411yescharacteristic
276C2×C4C4×C22⋊D6711yescharacteristic
277C2×C4#1452401111yescharacteristic
278C2×C4#1452401611yescharacteristic
279C2×C4#1452402511yescharacteristic
280C2×C4C2×#1176161811yescharacteristic
281C2×C4#1452406311yescharacteristic
282C2×C4#1452407311yescharacteristic
283C2×C4#1452407411yescharacteristic
284C8—22no—
285C8—41no—
286C8—2682no—
287C8—2682no—
288C8—5361no—
289C22—21no—
290C22—21no—
291C22#1686327211yescharacteristic
292C22#1686360311yescharacteristic
293C22#1686361511yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—2682no—
300C4—2682no—
301C4—161no—
302C4C4×#1176161811yescharacteristic
303C4#1686350111yescharacteristic
304C4#1686355111yescharacteristic
305C4#1686355611yescharacteristic
306C2—21no—
307C2—21no—
308C2#2256055311yescharacteristic
309C2#2256067911yescharacteristic
310C2#2256073311yescharacteristic
subgroupG/H
centerC22#16863272
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1993C2
socleC22×C67#386
prime radicalC23×C67#121
cosocleC2×C42×C67C23
hypercenter#1993D67
hypocenterC67#9290
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
9. #16863775C4semidirect
10. #16863746C4semidirect
93. C2×C8×C67Q16semidirect
163. C67#9290semidirect
241. C2×C8#11761618semidirect

Maximal subgroups

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

subgroupindex
#225711572
#225691492
#225588942
#225587722
#225580672
#225580402
C67×#19932
#929067