Finite Group Databasebeta

← #30753201

Subgroups — #30753201

Order 17152.

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

normal subgroups only

This group has 506 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#22565194C211yesmaximal, characteristic
2#22560205C211yesmaximal, characteristic
3#22558880C212yesmaximal
4#22558111C212yesmaximal
5C67×#2982C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7C2×#14524041—21no—
8#16863420C2212yes—
9C2×#14523962—21no—
10#16863326C414yes—
11#16863301—21no—
12#16863294—22no—
13C4×C4⋊Dic67C2211yescharacteristic
14#16863165—21no—
15#16863141—24no—
16C2×C4×Q8×C67C2211yescharacteristic
17C8×Q8×C67—24no—
18#431×C67—22no—
19#431×C67C2212yes—
20C22×Dic134—21no—
21C2×#11761612—22no—
22#14524043—22no—
23#14524043—22no—
24#14524041—42no—
25#14524041—42no—
26C2×C4⋊Dic67—21no—
27C2×C4⋊Dic67—21no—
28C2×C4⋊Dic67C2×C412yes—
29C2×Dic67⋊C4—24no—
30C2×C4×Dic67—21no—
31C4⋊(C67⋊C8)—22no—
32C4⋊(C67⋊C8)—22no—
33#14523962—42no—
34C67⋊#95—24no—
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—
48C2×Dic134—42no—
49C2×Dic134—41no—
50C2×Dic134—41no—
51C2×Dic134—42no—
52C22×Dic67—22no—
53C22×Dic67—22no—
54#11761612—42no—
55#11761612—42no—
56C2×C67⋊C8—42no—
57C4⋊Dic67—24no—
58C4⋊Dic67—24no—
59C4⋊Dic67—41no—
60C4⋊Dic67—41no—
61Dic67⋊C4—48no—
62C4×Dic67—41no—
63C4×Dic67—41no—
64C2×Q8×C67—22no—
65C2×Q8×C67—22no—
66C2×Q8×C67D812yes—
67C2×Q8×C67SD1612yes—
68C22×C4×C67C22⋊C412yes—
69C22×C4×C67C22⋊C412yes—
70C22×C4×C67C22×C411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C2×C8×C67—24no—
74C4⋊C4×C67—24no—
75C4⋊C4×C67—22no—
76C42×C67—22no—
77C42×C67—22no—
78C42×C67C2×D412yes—
79C42×C67#4112yes—
80C23×C67C2×C22⋊C411yescharacteristic
81Q8×C67—24no—
82Q8×C67—24no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—22no—
87C2×C4×C67—41no—
88C2×C4×C67—42no—
89C2×C4×C67#10214yes—
90C2×C4×C67C4×D412yes—
91C2×C4×C67#12011yescharacteristic
92C2×C4×C67#12112yes—
93C2×C4×C67#12311yescharacteristic
94C2×C4×C67C2×D811yescharacteristic
95C2×C4×C67C2×SD1611yescharacteristic
96C2×C4×C67#13512yes—
97C2×Dic67—24no—
98C2×Dic67—42no—
99C2×Dic67—42no—
100C2×Dic67—42no—
101C2×Dic67—42no—
102C2×Dic67—42no—
103Dic134—82no—
104Dic134—82no—
105C8×C67—44no—
106C67⋊C8—82no—
107C22×C67—21no—
108C22×C67—21no—
109C22×C67#38611yescharacteristic
110C22×C67C2×#10211yescharacteristic
111C22×C67#41611yescharacteristic
112C4×C67—21no—
113C4×C67—24no—
114C4×C67—22no—
115C4×C67—42no—
116C4×C67#45612yes—
117C4×C67#46512yes—
118C4×C67#46612yes—
119Dic67—44no—
120Dic67—82no—
121#9394—671nomaximal, Sylow
122C2×C67—21no—
123C2×C67—21no—
124C2×C67#188611yescharacteristic
125C2×C67#189311yescharacteristic
126C2×C67#191511yescharacteristic
127#2982D6711yescharacteristic
128#2546—671no—
129#1801—671no—
130#1546—672no—
131#1303—672no—
132C67#939411yescharacteristic, complemented, Sylow
133C2×C4⋊Q8—671no—
134C2×C4⋊Q8—1341no—
135C2×C4×Q8C2×D6711yescharacteristic
136C8×Q8—24no—
137#431—22no—
138#431C2×D6712yes—
139#423—672no—
140C2×#101—1341no—
141#398—674no—
142#391—1341no—
143#389—1342no—
144C4×C4⋊C4—671no—
145#343—1341no—
146#332—1344no—
147C22×Q8—1341no—
148C22×Q8D26811yescharacteristic
149C2×M16—1342no—
150C4⋊Q8—2682no—
151C4⋊Q8—2682no—
152C4⋊Q8—1342no—
153C4⋊Q8—1342no—
154C4×Q8—22no—
155C4×Q8—22no—
156C2×C4⋊C4—672no—
157C2×C4⋊C4—1344no—
158C2×C4⋊C4—1341no—
159C2×C4⋊C4—1341no—
160C2×C4⋊C4C4×D6712yes—
161C2×C4⋊C4D26811yescharacteristic
162C2×C42—1341no—
163C2×C42C22⋊D6712yes—
164C2×C42C22×D6711yescharacteristic
165C4⋊C8—24no—
166C4⋊C8—1342no—
167C4⋊C8—1342no—
168C4⋊C8C22⋊D6714yes—
169#101—2682no—
170C22⋊C8—22no—
171C4×C8—24no—
172C4×C8—22no—
173#95—672no—
174#95—1344no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—2682no—
178C2×Q8—2681no—
179C2×Q8—2681no—
180C2×Q8—2682no—
181C2×Q8D53612yes—
182C2×Q8#1176161712yes—
183C22×C4—1342no—
184C22×C4—1342no—
185C22×C4(C2×D67)⋊C412yes—
186C22×C4(C2×D67)⋊C412yes—
187C22×C4C2×C4×D6711yescharacteristic
188C22×C4C2×C22⋊D6711yescharacteristic
189C22×C4C2×D26811yescharacteristic
190M16—2682no—
191M16—2682no—
192C2×C8—24no—
193C2×C8—2682no—
194C4⋊C4—24no—
195C4⋊C4—22no—
196C4⋊C4—2688no—
197C4⋊C4—2681no—
198C4⋊C4—2681no—
199C4⋊C4—1344no—
200C4⋊C4—1344no—
201C42—22no—
202C42—22no—
203C42—2681no—
204C42—2681no—
205C42C2×C22⋊D6712yes—
206C42#1176162812yes—
207C23C2×(C2×D67)⋊C411yescharacteristic
208Q8—24no—
209Q8—24no—
210Q8—5362no—
211Q8—5362no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—2682no—
219C2×C4—2682no—
220C2×C4—2682no—
221C2×C4—2682no—
222C2×C4—2682no—
223C2×C4—1344no—
224C2×C4#1452396514yes—
225C2×C4C4×C22⋊D6712yes—
226C2×C4#1452401211yescharacteristic
227C2×C4#1452401712yes—
228C2×C4#1452402611yescharacteristic
229C2×C4C2×D53611yescharacteristic
230C2×C4C2×#1176161711yescharacteristic
231C2×C4#1452406512yes—
232C8—44no—
233C8—5362no—
234C22—21no—
235C22—21no—
236C22#1686327511yescharacteristic
237C22C2×#1452396511yescharacteristic
238C22#1686339311yescharacteristic
239C4—21no—
240C4—24no—
241C4—22no—
242C4—42no—
243C4—2684no—
244C4—5362no—
245C4#1686357912yes—
246C4#1686362512yes—
247C4#1686363212yes—
248C2—21no—
249C2—21no—
250C2#2256060211yescharacteristic
251C2#2256063711yescharacteristic
252C2#2256076111yescharacteristic
subgroupG/H
centerC22#16863275
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2982C2
socleC22×C67#386
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2982D67
hypocenterC67#9394
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
132. C67#9394semidirect

Maximal subgroups

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

subgroupindex
#225651942
#225602052
#225588802
#225581112
C67×#29822
#939467