Finite Group Databasebeta

← #30753199

Subgroups — #30753199

Order 17152.

506 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 131 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#22569290C211yesmaximal, characteristic
2#22565194C211yesmaximal, characteristic
3#22558882C212yesmaximal
4#22558110C212yesmaximal
5C67×#1801C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7C2×#14524041—21no—
8Q8×C67⋊C8—24no—
9#16863449—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—42no—
24#14524041—42no—
25C2×C4⋊Dic67—21no—
26C2×C4⋊Dic67—21no—
27C2×C4⋊Dic67C2×C412yes—
28C2×Dic67⋊C4—24no—
29C2×C4×Dic67—21no—
30C4⋊(C67⋊C8)—24no—
31C4⋊(C67⋊C8)D414yes—
32C22⋊(C67⋊C8)—22no—
33C4×C67⋊C8—24no—
34C4×C67⋊C8—22no—
35C67⋊#95—24no—
36C67⋊#95C2×C412yes—
37C22×Q8×C67D411yescharacteristic
38C2×M16×C67—22no—
39C4×Q8×C67—22no—
40C4×Q8×C67—22no—
41C2×C4⋊C4×C67C2×C412yes—
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C42×C67D412yes—
44C2×C42×C67C2311yescharacteristic
45C4⋊C8×C67—22no—
46C4⋊C8×C67—22no—
47#101×C67—42no—
48C2×Dic134—42no—
49C2×Dic134—41no—
50C2×Dic134—41no—
51C2×Dic134—42no—
52C22×Dic67—22no—
53C22×Dic67—22no—
54C2×C67⋊C8—24no—
55C4⋊Dic67—24no—
56C4⋊Dic67—24no—
57C4⋊Dic67—41no—
58C4⋊Dic67—41no—
59Dic67⋊C4—48no—
60C4×Dic67—41no—
61C4×Dic67—41no—
62C2×Q8×C67—22no—
63C2×Q8×C67—22no—
64C2×Q8×C67D812yes—
65C2×Q8×C67SD1612yes—
66C22×C4×C67C22⋊C412yes—
67C22×C4×C67C22⋊C412yes—
68C22×C4×C67C22×C411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71M16×C67—42no—
72M16×C67—42no—
73C2×C8×C67—42no—
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—82no—
106C67⋊C8—44no—
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#2982—671no—
128#2546—671no—
129#1801D6711yescharacteristic
130#1546—672no—
131#1303—672no—
132C67#939411yescharacteristic, complemented, Sylow
133C2×C4⋊Q8—671no—
134C2×C4⋊Q8—1341no—
135C2×C4×Q8C2×D6711yescharacteristic
136C8×Q8—1344no—
137#431—672no—
138#431—1342no—
139#423C2×D6712yes—
140C2×#101—21no—
141#398—674no—
142#391—1341no—
143#389—1342no—
144C4×C4⋊C4—671no—
145#343—21no—
146#332—1344no—
147C22×Q8—1341no—
148C22×Q8C22⋊D6711yescharacteristic
149C2×M16—22no—
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⋊C4C22⋊D6711yescharacteristic
162C2×C42—1341no—
163C2×C42D26812yes—
164C2×C42C22×D6711yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—674no—
168C4⋊C8—1344no—
169#101—42no—
170C22⋊C8—1342no—
171C4×C8—1344no—
172C4×C8—1342no—
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×Q8#1176161312yes—
182C2×Q8#1176161512yes—
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—42no—
191M16—42no—
192C2×C8—42no—
193C2×C8—1344no—
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—
205C42D4×D6712yes—
206C42#1176162712yes—
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#1452396314yes—
225C2×C4(C22⋊D67)⋊C412yes—
226C2×C4#1452401311yescharacteristic
227C2×C4#1452401512yes—
228C2×C4#1452402711yescharacteristic
229C2×C4C2×#1176161311yescharacteristic
230C2×C4C2×#1176161511yescharacteristic
231C2×C4#1452406312yes—
232C8—82no—
233C8—2684no—
234C22—21no—
235C22—21no—
236C22#1686327311yescharacteristic
237C22C2×#1452396311yescharacteristic
238C22#1686339011yescharacteristic
239C4—21no—
240C4—24no—
241C4—22no—
242C4—42no—
243C4—2684no—
244C4—5362no—
245C4#1686358112yes—
246C4#1686362712yes—
247C4#1686363012yes—
248C2—21no—
249C2—21no—
250C2#2256060011yescharacteristic
251C2#2256063511yescharacteristic
252C2#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#9394
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
132. C67#9394semidirect

Maximal subgroups

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

subgroupindex
#225692902
#225651942
#225588822
#225581102
C67×#18012
#939467