Finite Group Databasebeta

← #30753221

Subgroups — #30753221

Order 17152.

574 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 135 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#22565208C211yesmaximal, characteristic
2#22560205C211yesmaximal, characteristic, complemented
3#22558876C212yesmaximal
4#22558111C212yesmaximal, complemented
5C67×#2982C211yesmaximal, characteristic, complemented
6C2×#14524040C2211yescharacteristic
7C2×#14524032—21no—
8#16863420C2212yes—
9C2×#14523962—21no—
10#16863313C414yes—
11#16863298—22no—
12#16863297—21no—
13C4×C4⋊Dic67C2211yescharacteristic
14#16863165—21no—
15#16863139—24no—
16C2×C4×Q8×C67C2211yescharacteristic
17C8×Q8×C67—24no—
18#431×C67—22no—
19#431×C67C2212yes—
20C22×D268—21no—
21C22×D268—21no—
22C22×D268C2×C412yes—
23C2×#11761612—22no—
24#14524040—22no—
25#14524040—22no—
26#14524032—42no—
27#14524032—42no—
28C2×C4⋊Dic67—21no—
29C2×(C2×D67)⋊C4—24no—
30C2×(C2×D67)⋊C4—24no—
31C2×C4×Dic67—21no—
32C4⋊(C67⋊C8)—22no—
33C4⋊(C67⋊C8)—22no—
34#14523962—42no—
35C67⋊#95C2×C412yes—
36C22×Q8×C67D411yescharacteristic
37C4×Q8×C67—22no—
38C4×Q8×C67—22no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C42×C67D412yes—
42C2×C42×C67C2311yescharacteristic
43C4⋊C8×C67—24no—
44C4⋊C8×C67D414yes—
45C22⋊C8×C67—22no—
46C4×C8×C67—24no—
47C4×C8×C67—22no—
48C23×D67—22no—
49C2×D268—24no—
50C2×D268—44no—
51C2×D268—41no—
52C2×D268—42no—
53C2×D268—41no—
54C2×D268—41no—
55C2×D268—41no—
56C2×D268—42no—
57C2×D268—82no—
58C22×Dic67—22no—
59#11761612—42no—
60#11761612—42no—
61C2×C67⋊C8—42no—
62C4⋊Dic67—42no—
63(C2×D67)⋊C4—48no—
64(C2×D67)⋊C4—84no—
65C4×Dic67—41no—
66C4×Dic67—41no—
67C2×Q8×C67—22no—
68C2×Q8×C67—22no—
69C2×Q8×C67D812yes—
70C2×Q8×C67SD1612yes—
71C22×C4×C67C22⋊C412yes—
72C22×C4×C67C22⋊C412yes—
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C2×C8×C67—24no—
77C4⋊C4×C67—24no—
78C4⋊C4×C67—22no—
79C42×C67—22no—
80C42×C67—22no—
81C42×C67C2×D412yes—
82C42×C67#4112yes—
83C23×C67C2×C22⋊C411yescharacteristic
84C22×D67—44no—
85C22×D67—44no—
86C22×D67—42no—
87C22×D67—42no—
88C22×D67—42no—
89Q8×C67—24no—
90Q8×C67—24no—
91C2×C4×C67—22no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—22no—
95C2×C4×C67—41no—
96C2×C4×C67—42no—
97C2×C4×C67#10214yes—
98C2×C4×C67C4×D412yes—
99C2×C4×C67#12011yescharacteristic
100C2×C4×C67#12112yes—
101C2×C4×C67#12311yescharacteristic
102C2×C4×C67C2×D811yescharacteristic
103C2×C4×C67C2×SD1611yescharacteristic
104C2×C4×C67#13512yes—
105C2×Dic67—42no—
106C2×Dic67—42no—
107C2×Dic67—42no—
108D268—84no—
109D268—82no—
110D268—82no—
111D268—82no—
112D268—82no—
113C8×C67—44no—
114C67⋊C8—82no—
115C22×C67—21no—
116C22×C67—21no—
117C22×C67#38611yescharacteristic
118C22×C67C2×#10211yescharacteristic
119C22×C67#41611yescharacteristic
120C2×D67—84no—
121C2×D67—84no—
122C2×D67—82no—
123C2×D67—82no—
124C2×D67—82no—
125C4×C67—21no—
126C4×C67—24no—
127C4×C67—22no—
128C4×C67—42no—
129C4×C67#45412yes—
130C4×C67#46512yes—
131C4×C67#46612yes—
132Dic67—82no—
133#9398—671nomaximal, Sylow
134C2×C67—21no—
135C2×C67—21no—
136C2×C67#188811yescharacteristic
137C2×C67#189311yescharacteristic
138C2×C67#191411yescharacteristic
139D67—162no—
140#2982D6711yescharacteristic, complemented
141#2548—671no—
142#1801—671no—
143#1545—672no—
144#1303—672no—
145C67#939811yescharacteristic, complemented, Sylow
146C2×#127—671no—
147C2×#124—1341no—
148C2×C4×Q8C2×D6711yescharacteristic
149C8×Q8—24no—
150#431—22no—
151#431C2×D6712yes—
152#423—672no—
153C2×#101—1341no—
154#394—674no—
155#390—1341no—
156#390—1342no—
157C4×C4⋊C4—671no—
158#343—1341no—
159#331—1344no—
160C22×Q8D26811yescharacteristic
161C22×D4—672no—
162C22×D4—1341no—
163C22×D4—1341no—
164C2×M16—1342no—
165#127—1342no—
166#127—1342no—
167#124—2682no—
168#124—2682no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—1341no—
172C2×C4⋊C4C4×D6712yes—
173C2×C4⋊C4D26811yescharacteristic
174C2×C22⋊C4—1344no—
175C2×C22⋊C4—1344no—
176C2×C42—1341no—
177C2×C42C22⋊D6712yes—
178C2×C42C22×D6711yescharacteristic
179C4⋊C8—24no—
180C4⋊C8—1342no—
181C4⋊C8—1342no—
182C4⋊C8C22⋊D6714yes—
183#101—2682no—
184C22⋊C8—22no—
185C4×C8—24no—
186C4×C8—22no—
187#95—672no—
188C24—1342no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8D53612yes—
192C2×Q8#1176161712yes—
193C2×D4—2684no—
194C2×D4—2681no—
195C2×D4—2682no—
196C2×D4—2681no—
197C2×D4—2681no—
198C2×D4—2681no—
199C2×D4—2682no—
200C2×D4—5362no—
201C2×D4—1344no—
202C22×C4—1342no—
203C22×C4(C2×D67)⋊C412yes—
204C22×C4(C2×D67)⋊C412yes—
205C22×C4C2×C4×D6711yescharacteristic
206C22×C4C2×C22⋊D6711yescharacteristic
207C22×C4C2×D26811yescharacteristic
208M16—2682no—
209M16—2682no—
210C2×C8—24no—
211C2×C8—2682no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—2682no—
215C22⋊C4—2688no—
216C22⋊C4—5364no—
217C42—22no—
218C42—22no—
219C42—2681no—
220C42—2681no—
221C42C2×C22⋊D6712yes—
222C42#1176162812yes—
223C23—2684no—
224C23—2684no—
225C23—2682no—
226C23—2682no—
227C23—2682no—
228C23C2×(C2×D67)⋊C411yescharacteristic
229Q8—24no—
230Q8—24no—
231D4—5364no—
232D4—5362no—
233D4—5362no—
234D4—5362no—
235D4—5362no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—2682no—
243C2×C4—2682no—
244C2×C4—2682no—
245C2×C4#1452396514yes—
246C2×C4C4×C22⋊D6712yes—
247C2×C4#1452401211yescharacteristic
248C2×C4#1452401712yes—
249C2×C4#1452402611yescharacteristic
250C2×C4C2×D53611yescharacteristic
251C2×C4C2×#1176161711yescharacteristic
252C2×C4#1452406512yes—
253C8—44no—
254C8—5362no—
255C22—21no—
256C22—21no—
257C22—5364no—
258C22—5364no—
259C22—5362no—
260C22—5362no—
261C22—5362no—
262C22#1686327511yescharacteristic
263C22C2×#1452396511yescharacteristic
264C22#1686339311yescharacteristic
265C4—21no—
266C4—24no—
267C4—22no—
268C4—42no—
269C4—5362no—
270C4#1686356912yes—
271C4#1686362512yes—
272C4#1686363212yes—
273C2—21no—
274C2—21no—
275C2—10722no—
276C2#2256061211yescharacteristic
277C2#2256063711yescharacteristic
278C2#2256075611yescharacteristic
subgroupG/H
centerC22#16863275
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2982C2
socleC22×C67#386
prime radicalC2×#14524040C22
cosocleC2×C42×C67C23
hypercenter#2982D67
hypocenterC67#9398
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #22560205C2semidirect
4. #22558111C2semidirect
5. C67×#2982C2semidirect
140. #2982D67semidirect
145. C67#9398semidirect

Maximal subgroups

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

subgroupindex
#225652082
#225602052
#225588762
#225581112
C67×#29822
#939867