Finite Group Databasebeta

← #30755419

Subgroups — #30755419

Order 17152.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#16863746C211yesmaximal, characteristic
2#22561199C211yesmaximal, characteristic
3#22561197C211yesmaximal, characteristic
4#22558967C211yesmaximal, characteristic
5#22558893C211yesmaximal, characteristic
6#22558490C211yesmaximal, characteristic
7C67×#1548C211yesmaximal, characteristic
8C2×#14524043C2211yescharacteristic
9#16863746—21no—
10#16863746—21no—
11#16863746—22no—
12#16863420C2211yescharacteristic
13C2×#14523967C414yes—
14C4×#11761612C2211yescharacteristic
15C2×C4×C67⋊C8C2211yescharacteristic
16#16863165—22no—
17#16863164—22no—
18#16863148—22no—
19#16863143—22no—
20#16863118—21no—
21C8×C67⋊C8—21no—
22C2×C4⋊Q8×C67C2211yescharacteristic
23#431×C67C2211yescharacteristic
24#423×C67C2211yescharacteristic
25#333×C67—22no—
26C22×Dic134—21no—
27C2×#11761612—21no—
28C2×#11761612D412yes—
29C22×C67⋊C8C2×C412yes—
30#14524043—21no—
31#14524043—21no—
32#14524043—21no—
33#14524043—21no—
34C2×C4⋊Dic67—21no—
35C2×C4⋊Dic67C2×C412yes—
36C4⋊(C67⋊C8)—21no—
37C4⋊(C67⋊C8)—21no—
38#14523967—24no—
39#14523967—24no—
40#14523954—21no—
41C4×C67⋊C8—21no—
42C4×C67⋊C8D411yescharacteristic
43C4×C67⋊C8D411yescharacteristic
44C4×C67⋊C8D412yes—
45C22×Q8×C67C2×C412yes—
46C2×M16×C67—21no—
47C4⋊Q8×C67—21no—
48C4⋊Q8×C67—21no—
49C4⋊Q8×C67—21no—
50C4⋊Q8×C67—21no—
51C2×C4⋊C4×C67—21no—
52C2×C4⋊C4×C67—21no—
53C2×C42×C67C2311yescharacteristic
54C4⋊C8×C67—21no—
55C4⋊C8×C67—21no—
56C4⋊C8×C67D412yes—
57C22⋊C8×C67—21no—
58C4×C8×C67—21no—
59C2×Dic134—41no—
60C2×Dic134—41no—
61C2×Dic134—82no—
62C22×Dic67—22no—
63#11761612—22no—
64#11761612—42no—
65#11761612Q1614yes—
66C2×C67⋊C8—22no—
67C2×C67⋊C8—42no—
68C2×C67⋊C8—41no—
69C2×C67⋊C8C22⋊C412yes—
70C2×C67⋊C8C22⋊C412yes—
71C4⋊Dic67—22no—
72C4⋊Dic67—22no—
73C4⋊Dic67—41no—
74C4⋊Dic67—41no—
75C2×Q8×C67—22no—
76C2×Q8×C67—22no—
77C2×Q8×C67—44no—
78C22×C4×C67—22no—
79C22×C4×C67C22×C411yescharacteristic
80C22×C4×C67C2×D411yescharacteristic
81C22×C4×C67#4111yescharacteristic
82M16×C67—42no—
83C2×C8×C67—22no—
84C2×C8×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C42×C67C2×D411yescharacteristic
90C42×C67C2×D411yescharacteristic
91C42×C67C2×D411yescharacteristic
92C42×C67#4111yescharacteristic
93C23×C67C4×D411yescharacteristic
94Q8×C67—44no—
95Q8×C67—82no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—42no—
99C2×C4×C67—82no—
100C2×C4×C67C2×C22⋊C411yescharacteristic
101C2×C4×C67C4×D411yescharacteristic
102C2×C4×C67#12111yescharacteristic
103C2×C4×C67#12111yescharacteristic
104C2×C4×C67#12411yescharacteristic
105C2×C4×C67#12711yescharacteristic
106C2×C4×C67C2×Q1612yes—
107C2×C4×C67#13512yes—
108C2×Dic67—42no—
109C2×Dic67—82no—
110Dic134—82no—
111Dic134—82no—
112C8×C67—42no—
113C8×C67—81no—
114C67⋊C8—22no—
115C67⋊C8—22no—
116C67⋊C8—24no—
117C67⋊C8—81no—
118C22×C67—21no—
119C22×C67—21no—
120C22×C67#39011yescharacteristic
121C22×C67C4×Q1612yes—
122C4×C67—21no—
123C4×C67—21no—
124C4×C67—82no—
125C4×C67#41312yes—
126C4×C67#46712yes—
127C4×C67#47112yes—
128C4×C67#49411yescharacteristic
129C4×C67#49511yescharacteristic
130Dic67—82no—
131#9795—671nomaximal, Sylow
132C2×C67—22no—
133C2×C67#196711yescharacteristic
134C2×C67#196911yescharacteristic
135C2×C67#197711yescharacteristic
136C2×#487—671no—
137#1997—671no—
138#1997—671no—
139#1567—671no—
140#1549—671no—
141#1548D6711yescharacteristic
142C8×M16—671no—
143C67#979511yescharacteristic, complemented, Sylow
144C2×C4⋊Q8—671no—
145C2×C4⋊Q8C2×D6711yescharacteristic
146#487—1341no—
147#487—1341no—
148#487—1342no—
149#431C2×D6711yescharacteristic
150#423—671no—
151#423C2×D6711yescharacteristic
152C2×#103—674no—
153C4×M16—671no—
154C2×C4×C8—671no—
155#343—1342no—
156#343—1342no—
157#335—1342no—
158#333—22no—
159#332—1342no—
160#322—1341no—
161C82—1341no—
162C22×Q8—1341no—
163C22×Q8C4×D6712yes—
164C2×M16—21no—
165C2×M16—672no—
166C2×M16—1341no—
167C22×C8—672no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—1341no—
173C4⋊Q8—1341no—
174C4⋊Q8—1341no—
175C4⋊Q8—1341no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—672no—
179C2×C4⋊C4—1341no—
180C2×C42C22×D6711yescharacteristic
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—1341no—
184C4⋊C8—1341no—
185C4⋊C8C22⋊D6712yes—
186#103—1344no—
187#103—1344no—
188C22⋊C8—21no—
189#97—1341no—
190C4×C8—21no—
191C4×C8—671no—
192C4×C8—671no—
193C4×C8—672no—
194C4×C8—1341no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8—2681no—
199C2×Q8—2681no—
200C2×Q8—5362no—
201C22×C4—22no—
202C22×C4—1342no—
203C22×C4C2×C4×D6711yescharacteristic
204C22×C4D4×D6711yescharacteristic
205C22×C4#1176162711yescharacteristic
206M16—42no—
207M16—2682no—
208M16—674no—
209M16—1342no—
210C2×C8—22no—
211C2×C8—41no—
212C2×C8—2681no—
213C2×C8—2682no—
214C2×C8—672no—
215C2×C8—672no—
216C2×C8—1342no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—2681no—
222C4⋊C4—2681no—
223C4⋊C4—1342no—
224C4⋊C4—1342no—
225C42D4×D6711yescharacteristic
226C42D4×D6711yescharacteristic
227C42C2×C22⋊D6711yescharacteristic
228C42#1176162811yescharacteristic
229C23(C22⋊D67)⋊C411yescharacteristic
230Q8—44no—
231Q8—82no—
232Q8—5362no—
233Q8—5362no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—2682no—
239C2×C4—5362no—
240C2×C4C22⋊C4×D6711yescharacteristic
241C2×C4C4×C22⋊D6711yescharacteristic
242C2×C4#1452401511yescharacteristic
243C2×C4#1452401611yescharacteristic
244C2×C4#1452402911yescharacteristic
245C2×C4#1452403911yescharacteristic
246C2×C4Q16×D6712yes—
247C2×C4#1452406412yes—
248C8—42no—
249C8—81no—
250C8—5361no—
251C8—1342no—
252C8—1342no—
253C8—1344no—
254C22—21no—
255C22—21no—
256C22#1686329611yescharacteristic
257C22#1686348812yes—
258C4—21no—
259C4—21no—
260C4—82no—
261C4—5362no—
262C4#1686337712yes—
263C4#1686363612yes—
264C4#1686366112yes—
265C4#1686377411yescharacteristic
266C4#1686377711yescharacteristic
267C2—22no—
268C2#2256105511yescharacteristic
269C2#2256106711yescharacteristic
270C2#2256111111yescharacteristic
subgroupG/H
centerC22#16863296
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1548C2
socleC22×C67#390
prime radicalC23×C67C4×D4
cosocleC2×C42×C67C23
hypercenter#1548D67
hypocenterC67#9795
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
143. C67#9795semidirect

Maximal subgroups

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

subgroupindex
C2×#168637462
#225611992
#225611972
#225589672
#225588932
#225584902
C67×#15482
#979567