Finite Group Databasebeta

← #30753239

Subgroups — #30753239

Order 17152.

486 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 115 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#22569294C211yesmaximal, characteristic
2#22565194C211yesmaximal, characteristic
3#22558882C212yesmaximal
4#22558110C212yesmaximal
5C67×#1801C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7C2×#14524041—21no—
8#16863527—24no—
9#16863453—22no—
10#16863449C2212yes—
11#16863326C414yes—
12#16863301—21no—
13#16863294—22no—
14C4×C4⋊Dic67C2211yescharacteristic
15#16863143—24no—
16C2×C4×Q8×C67C2211yescharacteristic
17#423×C67C2212yes—
18C2×#101×C67—21no—
19#343×C67—21no—
20C22×Dic134—21no—
21#14524043—22no—
22#14524043—22no—
23#14524041—44no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—21no—
26C2×C4⋊Dic67C2×C412yes—
27C2×Dic67⋊C4—24no—
28C2×C4×Dic67—21no—
29C4⋊(C67⋊C8)—24no—
30C4⋊(C67⋊C8)D414yes—
31C22⋊(C67⋊C8)—22no—
32#14523954—24no—
33C4×C67⋊C8—22no—
34C67⋊#95—24no—
35C67⋊#95C2×C412yes—
36C22×Q8×C67D411yescharacteristic
37C2×M16×C67—22no—
38C4×Q8×C67—22no—
39C4×Q8×C67—22no—
40C2×C4⋊C4×C67C2×C412yes—
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C42×C67D412yes—
43C2×C42×C67C2311yescharacteristic
44C4⋊C8×C67—22no—
45C4⋊C8×C67—22no—
46#101×C67—42no—
47C2×Dic134—42no—
48C2×Dic134—42no—
49C2×Dic134—41no—
50C2×Dic134—41no—
51C22×Dic67—22no—
52C22×Dic67—22no—
53C2×C67⋊C8—24no—
54C4⋊Dic67—24no—
55C4⋊Dic67—24no—
56C4⋊Dic67—41no—
57C4⋊Dic67—41no—
58Dic67⋊C4—48no—
59C4×Dic67—42no—
60C2×Q8×C67—22no—
61C2×Q8×C67—21no—
62C2×Q8×C67—22no—
63C2×Q8×C67—21no—
64C22×C4×C67C22⋊C412yes—
65C22×C4×C67C22⋊C412yes—
66C22×C4×C67C22×C411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68C22×C4×C67C2×D411yescharacteristic
69M16×C67—42no—
70M16×C67—42no—
71C2×C8×C67—42no—
72C4⋊C4×C67—24no—
73C4⋊C4×C67—22no—
74C42×C67—22no—
75C42×C67—22no—
76C42×C67C2×D412yes—
77C42×C67#4112yes—
78C23×C67C2×C22⋊C411yescharacteristic
79Q8×C67—42no—
80Q8×C67—42no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—41no—
87C2×C4×C67—42no—
88C2×C4×C67C4×D412yes—
89C2×C4×C67#12011yescharacteristic
90C2×C4×C67#12112yes—
91C2×C4×C67#12311yescharacteristic
92C2×C4×C67AGL(1,ℤ/8)12yes—
93C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
94C2×C4×C67#13711yescharacteristic
95C2×Dic67—24no—
96C2×Dic67—42no—
97C2×Dic67—42no—
98C2×Dic67—42no—
99C2×Dic67—42no—
100C2×Dic67—42no—
101Dic134—82no—
102Dic134—82no—
103C8×C67—82no—
104C67⋊C8—44no—
105C22×C67—21no—
106C22×C67—21no—
107C22×C67#38611yescharacteristic
108C22×C67#41711yescharacteristic
109C22×C67#41811yescharacteristic
110C4×C67—21no—
111C4×C67—22no—
112C4×C67—42no—
113C4×C67—42no—
114C4×C67#45612yes—
115C4×C67#46812yes—
116C4×C67#46912yes—
117Dic67—44no—
118Dic67—82no—
119#9402—671nomaximal, Sylow
120C2×C67—21no—
121C2×C67—21no—
122C2×C67#188611yescharacteristic
123C2×C67#189511yescharacteristic
124C2×C67#191511yescharacteristic
125#2983—671no—
126#2546—671no—
127#1801D6711yescharacteristic
128#1546—672no—
129#1303—672no—
130C67#940211yescharacteristic, complemented, Sylow
131C2×C4⋊Q8—671no—
132C2×C4⋊Q8—1341no—
133C2×C4×Q8C2×D6711yescharacteristic
134#446—1344no—
135#432—1342no—
136#431—672no—
137#423C2×D6712yes—
138C2×#101—21no—
139#398—674no—
140#391—1341no—
141#389—1342no—
142C4×C4⋊C4—671no—
143#343—21no—
144#332—1344no—
145C22×Q8—1341no—
146C22×Q8C22⋊D6711yescharacteristic
147C2×M16—22no—
148C4⋊Q8—2684no—
149C4⋊Q8—1342no—
150C4⋊Q8—1342no—
151C4×Q8—22no—
152C4×Q8—22no—
153C2×C4⋊C4—672no—
154C2×C4⋊C4—1344no—
155C2×C4⋊C4—1341no—
156C2×C4⋊C4—1341no—
157C2×C4⋊C4C4×D6712yes—
158C2×C4⋊C4C22⋊D6711yescharacteristic
159C2×C42—1341no—
160C2×C42D26812yes—
161C2×C42C22×D6711yescharacteristic
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—674no—
165C4⋊C8—1344no—
166#101—42no—
167C22⋊C8—1342no—
168#97—1344no—
169C4×C8—1342no—
170#95—672no—
171#95—1344no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—2682no—
177C2×Q8—2682no—
178C2×Q8—2681no—
179C2×Q8—2681no—
180C22×C4—1342no—
181C22×C4—1342no—
182C22×C4(C2×D67)⋊C412yes—
183C22×C4(C2×D67)⋊C412yes—
184C22×C4C2×C4×D6711yescharacteristic
185C22×C4C2×C22⋊D6711yescharacteristic
186C22×C4C2×D26811yescharacteristic
187M16—42no—
188M16—42no—
189C2×C8—42no—
190C2×C8—1344no—
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—2688no—
194C4⋊C4—2681no—
195C4⋊C4—2681no—
196C4⋊C4—1344no—
197C4⋊C4—1344no—
198C42—22no—
199C42—22no—
200C42—2682no—
201C42D4×D6712yes—
202C42#1176162712yes—
203C23C2×(C2×D67)⋊C411yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—5362no—
207Q8—5362no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—2682no—
216C2×C4—2682no—
217C2×C4—2682no—
218C2×C4—2682no—
219C2×C4—2682no—
220C2×C4—1344no—
221C2×C4(C22⋊D67)⋊C412yes—
222C2×C4#1452401311yescharacteristic
223C2×C4#1452401512yes—
224C2×C4#1452402711yescharacteristic
225C2×C4#1452406812yes—
226C2×C4#1452406911yescharacteristic
227C2×C4#1452407411yescharacteristic
228C8—82no—
229C8—2684no—
230C22—21no—
231C22—21no—
232C22#1686327311yescharacteristic
233C22#1686339511yescharacteristic
234C22#1686340011yescharacteristic
235C4—21no—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—2684no—
240C4—5362no—
241C4#1686358112yes—
242C4#1686364412yes—
243C4#1686364712yes—
244C2—21no—
245C2—21no—
246C2#2256060011yescharacteristic
247C2#2256064511yescharacteristic
248C2#2256075911yescharacteristic
subgroupG/H
centerC22#16863273
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1801C2
socleC22×C67#386
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C42×C67C23
hypercenter#1801D67
hypocenterC67#9402
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
130. C67#9402semidirect

Maximal subgroups

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

subgroupindex
#225692942
#225651942
#225588822
#225581102
C67×#18012
#940267