Finite Group Databasebeta

← #30760159

Subgroups — #30760159

Order 17152.

364 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 99 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#22570751C211yesmaximal, characteristic
2#22560985C211yesmaximal, characteristic
3#22558897C211yesmaximal, characteristic
4#22558894C211yesmaximal, characteristic
5#22558611C211yesmaximal, characteristic
6#22558059C211yesmaximal, characteristic
7C2×#333×C67C211yesmaximal, characteristic
8#16863953C2211yescharacteristic
9#16863802—22no—
10#16863799—21no—
11#16863792—21no—
12#16863449C2211yescharacteristic
13C2×#14523979—21no—
14#16863436—21no—
15#16863436—21no—
16#16863420C2211yescharacteristic
17C4×#11761612C2211yescharacteristic
18#16863161—21no—
19#16863144—21no—
20#16863143—21no—
21#16863142—22no—
22#16863126—22no—
23C2×C4⋊Q8×C67C2211yescharacteristic
24C2×C4⋊C8×C67C2211yescharacteristic
25C2×C4⋊C8×C67C2211yescharacteristic
26#333×C67—22no—
27C2×#11761612—21no—
28C2×#11761612D411yescharacteristic
29C2×#11761612D411yescharacteristic
30#14524043D411yescharacteristic
31#14524043D411yescharacteristic
32#14524035—21no—
33#14524022—21no—
34#14524022—21no—
35C4×Dic134—21no—
36C4×Dic134—21no—
37#14524002—21no—
38#14523979—42no—
39#14523976—22no—
40#14523976—22no—
41#14523974—22no—
42#14523974—22no—
43C4⋊(C67⋊C8)—21no—
44C4⋊(C67⋊C8)—21no—
45C4⋊(C67⋊C8)—21no—
46C22⋊(C67⋊C8)—21no—
47#14523954—21no—
48C4×C67⋊C8—21no—
49C4×C67⋊C8—21no—
50C22×Q8×C67—21no—
51C22×C8×C67—21no—
52C22×C8×C67—21no—
53C4⋊Q8×C67—21no—
54C4⋊Q8×C67—21no—
55C4⋊Q8×C67—21no—
56C4⋊Q8×C67D412yes—
57C2×C4⋊C4×C67—21no—
58C2×C4⋊C4×C67—21no—
59C2×C42×C67C2311yescharacteristic
60C4⋊C8×C67—21no—
61C4⋊C8×C67—21no—
62C4⋊C8×C67—21no—
63C4⋊C8×C67D412yes—
64C2×Dic134—21no—
65C2×Dic134—21no—
66#11761612—21no—
67#11761612—21no—
68#11761612—41no—
69#11761612—41no—
70#11761612SD1612yes—
71#11761612Q1612yes—
72C2×C67⋊C8—21no—
73C2×C67⋊C8—21no—
74C2×C67⋊C8—41no—
75C2×C67⋊C8—41no—
76C4⋊Dic67—21no—
77C4⋊Dic67—21no—
78C4⋊Dic67—21no—
79C4⋊Dic67—21no—
80Dic67⋊C4—41no—
81Dic67⋊C4—41no—
82C22⋊Dic67—41no—
83C4×Dic67—41no—
84C2×Q8×C67—22no—
85C2×Q8×C67—41no—
86C2×Q8×C67—82no—
87C22×C4×C67—41no—
88C22×C4×C67C2×D411yescharacteristic
89C22×C4×C67#4111yescharacteristic
90C22×C4×C67#4111yescharacteristic
91C2×C8×C67—22no—
92C2×C8×C67—41no—
93C2×C8×C67—41no—
94C2×C8×C67—41no—
95C2×C8×C67—81no—
96C4⋊C4×C67—22no—
97C4⋊C4×C67—22no—
98C4⋊C4×C67—41no—
99C4⋊C4×C67—41no—
100C42×C67C2×D411yescharacteristic
101C42×C67C2×D411yescharacteristic
102C42×C67C2×D411yescharacteristic
103C42×C67#4111yescharacteristic
104C23×C67#12411yescharacteristic
105Q8×C67—82no—
106Q8×C67—82no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—41no—
110C2×C4×C67—42no—
111C2×C4×C67—82no—
112C2×C4×C67#12011yescharacteristic
113C2×C4×C67#12111yescharacteristic
114C2×C4×C67#12111yescharacteristic
115C2×C4×C67#12111yescharacteristic
116C2×C4×C67#12311yescharacteristic
117C2×C4×C67#12311yescharacteristic
118C2×C4×C67C2×SD1611yescharacteristic
119C2×C4×C67C2×Q1611yescharacteristic
120C2×C4×C67#13711yescharacteristic
121C2×C4×C67#13711yescharacteristic
122C2×Dic67—41no—
123C2×Dic67—41no—
124Dic134—41no—
125Dic134—41no—
126Dic134—81no—
127C8×C67—42no—
128C8×C67—81no—
129C67⋊C8—22no—
130C67⋊C8—22no—
131C67⋊C8—81no—
132C67⋊C8—81no—
133C22×C67—21no—
134C22×C67—21no—
135C22×C67#39411yescharacteristic
136C22×C67#48711yescharacteristic
137C22×C67#49111yescharacteristic
138C4×C67—21no—
139C4×C67—21no—
140C4×C67—21no—
141C4×C67—82no—
142C4×C67#44811yescharacteristic
143C4×C67#45111yescharacteristic
144C4×C67#46111yescharacteristic
145C4×C67#46211yescharacteristic
146C4×C67#47111yescharacteristic
147C4×C67#47311yescharacteristic
148Dic67—81no—
149Dic67—81no—
150#10555—671nomaximal, Sylow
151C2×C67—21no—
152C2×C67—21no—
153C2×C67#204311yescharacteristic
154C2×C67#204811yescharacteristic
155C2×C67#205211yescharacteristic
156#3165—671no—
157#1950—671no—
158#1550—671no—
159#1549—671no—
160C2×#333D6711yescharacteristic
161#1481—671no—
162#1283—671no—
163C67#1055511yescharacteristic, complemented, Sylow
164#533—671no—
165C2×C4⋊Q8C2×D6711yescharacteristic
166#501—1342no—
167#500—1341no—
168#498—1341no—
169#431—671no—
170C2×#108—1341no—
171#428—1341no—
172#428—1341no—
173#423—671no—
174C2×C4⋊C8C2×D6711yescharacteristic
175C2×C4⋊C8C2×D6711yescharacteristic
176C4×M16—671no—
177#341—1341no—
178#333—22no—
179#333—1341no—
180#332—1341no—
181#332—1342no—
182Q8⋊C8—1342no—
183C22×Q8—21no—
184C2×M16—671no—
185C2×M16—671no—
186C2×M16—1341no—
187C22×C8—21no—
188C22×C8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—671no—
193C4⋊Q8—671no—
194C4⋊Q8D26812yes—
195#125—1341no—
196C22⋊Q8—1341no—
197C22⋊Q8—1341no—
198C4×Q8—1341no—
199C4×Q8—1341no—
200#117—1341no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C42C22×D6711yescharacteristic
204#108—2682no—
205#107—1342no—
206#107—1342no—
207#106—1342no—
208#106—1342no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—1341no—
213C4⋊C8—1341no—
214C4⋊C8—1341no—
215C4⋊C8C22⋊D6712yes—
216C22⋊C8—1341no—
217#97—1341no—
218C4×C8—1341no—
219C4×C8—1341no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—1341no—
224C2×Q8—1341no—
225C22×C4—41no—
226C22×C4D4×D6711yescharacteristic
227C22×C4#1176162711yescharacteristic
228C22×C4#1176162811yescharacteristic
229M16—2681no—
230M16—2681no—
231M16—672no—
232M16—672no—
233M16—1341no—
234M16—1341no—
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—81no—
240C2×C8—2681no—
241C2×C8—2681no—
242C2×C8—1341no—
243C2×C8—1341no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—2681no—
249C4⋊C4—2681no—
250C4⋊C4—1341no—
251C4⋊C4—1341no—
252C4⋊C4—1341no—
253C4⋊C4—1341no—
254C22⋊C4—2681no—
255C42—2681no—
256C42D4×D6711yescharacteristic
257C42C2×C22⋊D6711yescharacteristic
258C42C2×D26811yescharacteristic
259C42#1176162711yescharacteristic
260C23#1452402911yescharacteristic
261Q8—82no—
262Q8—82no—
263Q8—2681no—
264Q8—2681no—
265Q8—5361no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—2681no—
272C2×C4—2681no—
273C2×C4#1452401311yescharacteristic
274C2×C4#1452401411yescharacteristic
275C2×C4#1452401511yescharacteristic
276C2×C4#1452401711yescharacteristic
277C2×C4#1452402511yescharacteristic
278C2×C4#1452402711yescharacteristic
279C2×C4SD16×D6711yescharacteristic
280C2×C4Q16×D6711yescharacteristic
281C2×C4#1452407211yescharacteristic
282C2×C4#1452407511yescharacteristic
283C8—42no—
284C8—81no—
285C8—5361no—
286C8—5361no—
287C8—1342no—
288C8—1342no—
289C22—21no—
290C22—21no—
291C22#1686331211yescharacteristic
292C22#1686374511yescharacteristic
293C22#1686376311yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—82no—
298C4—5361no—
299C4—5361no—
300C4#1686353611yescharacteristic
301C4#1686355311yescharacteristic
302C4#1686359911yescharacteristic
303C4#1686360411yescharacteristic
304C4#1686365611yescharacteristic
305C4#1686366811yescharacteristic
306C2—21no—
307C2—21no—
308C2#2256145411yescharacteristic
309C2#2256148111yescharacteristic
310C2#2256150911yescharacteristic
subgroupG/H
centerC22#16863312
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC2×#333×C67C2
socleC22×C67#394
prime radicalC23×C67#124
hypercenterC2×#333D67
hypocenterC67#10555
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
163. C67#10555semidirect

Maximal subgroups

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

subgroupindex
#225707512
#225609852
#225588972
#225588942
#225586112
#225580592
C2×#333×C672
#1055567