Finite Group Databasebeta

← #30755313

Subgroups — #30755313

Order 17152.

406 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 113 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#22560380C211yesmaximal, characteristic
3#22558895C211yesmaximal, characteristic
4#22558889C211yesmaximal, characteristic
5#22558881C211yesmaximal, characteristic
6#22558502C211yesmaximal, characteristic
7C67×#1546C211yesmaximal, characteristic
8C2×#14524043C2211yescharacteristic
9#16863746—21no—
10#16863746—21no—
11#16863746C414yes—
12#16863449C2211yescharacteristic
13C2×#14523978—22no—
14#16863420C2211yescharacteristic
15C2×#14523967—22no—
16C2×C4×C67⋊C8C2211yescharacteristic
17#16863144—22no—
18#16863142—22no—
19#16863141—22no—
20#16863118—21no—
21#16863117—21no—
22C2×C4⋊Q8×C67C2211yescharacteristic
23#431×C67C2211yescharacteristic
24#423×C67C2211yescharacteristic
25#332×C67—22no—
26C22×Dic134—21no—
27C2×#11761612—21no—
28C22×C67⋊C8D412yes—
29#14524043—21no—
30#14524043—21no—
31#14524043—21no—
32#14524043C2×C412yes—
33C2×C4⋊Dic67—21no—
34C2×C4⋊Dic67—21no—
35#14523978—44no—
36C4⋊(C67⋊C8)—21no—
37C4⋊(C67⋊C8)—21no—
38C4⋊(C67⋊C8)D412yes—
39#14523967—24no—
40#14523967—24no—
41C22⋊(C67⋊C8)—21no—
42C4×C67⋊C8—21no—
43C4×C67⋊C8C2×C412yes—
44C4×C67⋊C8D411yescharacteristic
45C4×C67⋊C8D411yescharacteristic
46C22×Q8×C67—21no—
47C2×M16×C67—21no—
48C4⋊Q8×C67—21no—
49C4⋊Q8×C67—21no—
50C4⋊Q8×C67—21no—
51C4⋊Q8×C67C2×C412yes—
52C2×C4⋊C4×C67—21no—
53C2×C4⋊C4×C67—21no—
54C2×C42×C67C2311yescharacteristic
55C4⋊C8×C67—21no—
56C4⋊C8×C67—21no—
57C4⋊C8×C67D412yes—
58C22⋊C8×C67—21no—
59C4×C8×C67—21no—
60C2×Dic134—22no—
61C2×Dic134—41no—
62C2×Dic134—82no—
63C22×Dic67—41no—
64#11761612—42no—
65C2×C67⋊C8—22no—
66C2×C67⋊C8—22no—
67C2×C67⋊C8—22no—
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—41no—
77C2×Q8×C67—42no—
78C2×Q8×C67—42no—
79C22×C4×C67—41no—
80C22×C4×C67C2×D411yescharacteristic
81C22×C4×C67#4111yescharacteristic
82C22×C4×C67#4111yescharacteristic
83M16×C67—42no—
84C2×C8×C67—22no—
85C2×C8×C67—41no—
86C4⋊C4×C67—22no—
87C4⋊C4×C67—22no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C42×C67C22×C411yescharacteristic
91C42×C67C2×D411yescharacteristic
92C42×C67C2×D411yescharacteristic
93C42×C67C2×D411yescharacteristic
94C23×C67#12411yescharacteristic
95Q8×C67—44no—
96Q8×C67—81no—
97Q8×C67—81no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—42no—
101C2×C4×C67—82no—
102C2×C4×C67C2×C22⋊C411yescharacteristic
103C2×C4×C67C4×D411yescharacteristic
104C2×C4×C67C4×D411yescharacteristic
105C2×C4×C67#12111yescharacteristic
106C2×C4×C67#12111yescharacteristic
107C2×C4×C67#12711yescharacteristic
108C2×C4×C67AGL(1,ℤ/8)12yes—
109C2×C4×C67#13712yes—
110C2×Dic67—42no—
111C2×Dic67—82no—
112Dic134—82no—
113Dic134—82no—
114C8×C67—42no—
115C8×C67—81no—
116C67⋊C8—22no—
117C67⋊C8—22no—
118C67⋊C8—42no—
119C67⋊C8—81no—
120C22×C67—21no—
121C22×C67—21no—
122C22×C67#39011yescharacteristic
123C22×C67#48912yes—
124C4×C67—21no—
125C4×C67—21no—
126C4×C67—82no—
127C4×C67#41712yes—
128C4×C67#47112yes—
129C4×C67#47312yes—
130C4×C67#49611yescharacteristic
131C4×C67#49711yescharacteristic
132Dic67—82no—
133#9775—671nomaximal, Sylow
134C2×C67—22no—
135C2×C67#195911yescharacteristic
136C2×C67#197711yescharacteristic
137C2×C67#197911yescharacteristic
138C2×#487—671no—
139#1848—671no—
140#1549—671no—
141#1548—671no—
142#1546—671no—
143#1546D6711yescharacteristic
144#1452—671no—
145C67#977511yescharacteristic, complemented, Sylow
146C2×C4⋊Q8—671no—
147C2×C4⋊Q8C2×D6711yescharacteristic
148#487—674no—
149#487—1341no—
150#487—1341no—
151#431—671no—
152#431C2×D6711yescharacteristic
153C2×#108—1342no—
154#423—671no—
155#423C2×D6711yescharacteristic
156C2×#103—1342no—
157C2×C4×C8—671no—
158#333—1342no—
159#332—22no—
160#332—1342no—
161#332—1342no—
162#322—1341no—
163#322—1341no—
164C22×Q8—21no—
165C22×Q8—1341no—
166C2×M16—21no—
167C2×M16—1341no—
168C22×C8—672no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—672no—
173C4⋊Q8—1341no—
174C4⋊Q8—1341no—
175C4⋊Q8—1341no—
176C4⋊Q8C4×D6712yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—1341no—
180C2×C4⋊C4—1341no—
181C2×C42C22×D6711yescharacteristic
182#108—2684no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—672no—
186C4⋊C8—1341no—
187C4⋊C8—1341no—
188C4⋊C8C22⋊D6712yes—
189#103—1344no—
190#103—1344no—
191C22⋊C8—21no—
192C22⋊C8—1341no—
193C4×C8—21no—
194C4×C8—672no—
195C4×C8—671no—
196C4×C8—671no—
197C4×C8—1341no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—2681no—
203C2×Q8—5362no—
204C2×Q8—1342no—
205C22×C4—41no—
206C22×C4—2681no—
207C22×C4D4×D6711yescharacteristic
208C22×C4#1176162711yescharacteristic
209C22×C4#1176162811yescharacteristic
210M16—42no—
211M16—2682no—
212C2×C8—22no—
213C2×C8—41no—
214C2×C8—2681no—
215C2×C8—672no—
216C2×C8—672no—
217C2×C8—1342no—
218C2×C8—1342no—
219C2×C8—1342no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—2681no—
225C4⋊C4—2681no—
226C4⋊C4—1342no—
227C4⋊C4—1342no—
228C42C2×C4×D6711yescharacteristic
229C42D4×D6711yescharacteristic
230C42D4×D6711yescharacteristic
231C42C2×C22⋊D6711yescharacteristic
232C23#1452402911yescharacteristic
233Q8—44no—
234Q8—81no—
235Q8—81no—
236Q8—5362no—
237Q8—5362no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—82no—
242C2×C4—2682no—
243C2×C4—5362no—
244C2×C4C22⋊C4×D6711yescharacteristic
245C2×C4(C22⋊D67)⋊C411yescharacteristic
246C2×C4C4×C22⋊D6711yescharacteristic
247C2×C4#1452401511yescharacteristic
248C2×C4#1452401611yescharacteristic
249C2×C4#1452403911yescharacteristic
250C2×C4#1452406712yes—
251C2×C4#1452407212yes—
252C8—42no—
253C8—81no—
254C8—2682no—
255C8—5361no—
256C8—1342no—
257C8—1342no—
258C22—21no—
259C22—21no—
260C22#1686329611yescharacteristic
261C22#1686375412yes—
262C4—21no—
263C4—21no—
264C4—82no—
265C4—5362no—
266C4#1686339412yes—
267C4#1686366212yes—
268C4#1686366912yes—
269C4#1686378411yescharacteristic
270C4#1686378711yescharacteristic
271C2—22no—
272C2#2256102411yescharacteristic
273C2#2256111311yescharacteristic
274C2#2256112311yescharacteristic
subgroupG/H
centerC22#16863296
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1546C2
socleC22×C67#390
prime radicalC23×C67#124
cosocleC2×C42×C67C23
hypercenter#1546D67
hypocenterC67#9775
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
145. C67#9775semidirect

Maximal subgroups

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

subgroupindex
C2×#168637462
#225603802
#225588952
#225588892
#225588812
#225585022
C67×#15462
#977567