Finite Group Databasebeta

← #30766997

Subgroups — #30766997

Order 17152.

388 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 99 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#22565414C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558238C214yesmaximal
4C67×#1801C211yesmaximal, characteristic
5C2×#14524043C411yescharacteristic
6#16863449—21no—
7#16863316C412yes—
8#16863316C2212yes—
9#16863303—21no—
10#16863294—21no—
11#16863294—21no—
12#16863265—21no—
13C4×C4⋊Dic67C411yescharacteristic
14#16863122—21no—
15#16863122C2212yes—
16C2×C4×Q8×C67C2211yescharacteristic
17#423×C67—21no—
18C2×#101×C67—21no—
19#343×C67C2212yes—
20C22×Dic134C2×C412yes—
21#14524043—41no—
22#14524043—41no—
23C2×C4⋊Dic67—21no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67—21no—
26C2×Dic67⋊C4—22no—
27C2×Dic67⋊C4—22no—
28C2×C4×Dic67—21no—
29C4⋊(C67⋊C8)—41no—
30C22⋊(C67⋊C8)—22no—
31C4×C67⋊C8—41no—
32C67⋊#95—22no—
33C67⋊#95—22no—
34C67⋊#95C2×C412yes—
35C22×Q8×C67D411yescharacteristic
36C2×M16×C67—22no—
37C4×Q8×C67—41no—
38C4×Q8×C67—41no—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C42×C67C2×C411yescharacteristic
43C2×C42×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C4⋊C8×C67—41no—
46C4⋊C8×C67—41no—
47#101×C67—42no—
48C2×Dic134—42no—
49C2×Dic134—82no—
50C22×Dic67—22no—
51C22×Dic67—22no—
52C2×C67⋊C8—42no—
53C4⋊Dic67—24no—
54C4⋊Dic67—42no—
55C4⋊Dic67—41no—
56C4⋊Dic67—41no—
57Dic67⋊C4—82no—
58Dic67⋊C4—82no—
59C4×Dic67—42no—
60C2×Q8×C67—41no—
61C2×Q8×C67—41no—
62C2×Q8×C67SD1612yes—
63C2×Q8×C67Q1612yes—
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C411yescharacteristic
67C22×C4×C67C22⋊C411yescharacteristic
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—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C42×C67—21no—
82C42×C67—21no—
83C23×C67C2×C22⋊C411yescharacteristic
84Q8×C67—42no—
85Q8×C67—42no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—22no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—41no—
97C2×C4×C67—41no—
98C2×C4×C67—41no—
99C2×C4×C67#9911yescharacteristic
100C2×C4×C67#9911yescharacteristic
101C2×C4×C67#10312yes—
102C2×C4×C67#10312yes—
103C2×C4×C67C2×SD1611yescharacteristic
104C2×C4×C67C2×Q1611yescharacteristic
105C2×Dic67—24no—
106C2×Dic67—42no—
107C2×Dic67—42no—
108C2×Dic67—42no—
109C2×Dic67—42no—
110C2×Dic67—42no—
111Dic134—84no—
112C8×C67—82no—
113C67⋊C8—82no—
114C22×C67—21no—
115C22×C67—21no—
116C22×C67C2×#9911yescharacteristic
117C22×C67C2×#10311yescharacteristic
118C22×C67#42111yescharacteristic
119C4×C67—21no—
120C4×C67—21no—
121C4×C67—22no—
122C4×C67—22no—
123C4×C67—21no—
124C4×C67—41no—
125C4×C67—41no—
126C4×C67—41no—
127C4×C67#35612yes—
128Dic67—44no—
129Dic67—82no—
130#11908—671nomaximal, Sylow
131C2×C67—21no—
132C2×C67—21no—
133C2×C67#152011yescharacteristic
134C2×C67#212211yescharacteristic
135C2×C67C2×#35611yescharacteristic
136#2567—671no—
137#1801D6711yescharacteristic
138#1461—671no—
139#1354—674no—
140C67#1190811yescharacteristic, complemented, Sylow
141C2×C4⋊Q8—671no—
142C2×C4×Q8C2×D6711yescharacteristic
143#431—1341no—
144#423—21no—
145C2×#101—21no—
146#395—672no—
147#395—672no—
148#391—1341no—
149#389—1341no—
150#389—1341no—
151#384—1341no—
152C4×C4⋊C4—671no—
153#343C2×D6712yes—
154#324—672no—
155#324—1341no—
156C22×Q8—672no—
157C22×Q8C22⋊D6711yescharacteristic
158C2×M16—22no—
159C4⋊Q8—2681no—
160C4⋊Q8—2681no—
161C4×Q8—41no—
162C4×Q8—41no—
163C2×C4⋊C4—1341no—
164C2×C4⋊C4—1341no—
165C2×C4⋊C4—1342no—
166C2×C4⋊C4—1342no—
167C2×C4⋊C4—1341no—
168C2×C4⋊C4C4×D6711yescharacteristic
169C2×C4⋊C4D26811yescharacteristic
170C2×C4⋊C4C22×D6711yescharacteristic
171C2×C42—1341no—
172C2×C42C4×D6711yescharacteristic
173C2×C42D26811yescharacteristic
174C2×C42C22⋊D6711yescharacteristic
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—2681no—
178#101—42no—
179C22⋊C8—1342no—
180C4×C8—2681no—
181#95—672no—
182#95—1342no—
183#95—1342no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—2682no—
187C2×Q8—5362no—
188C2×Q8#1176161612yes—
189C2×Q8#1176161812yes—
190C22×C4—1342no—
191C22×C4—1342no—
192C22×C4(C2×D67)⋊C411yescharacteristic
193C22×C4(C2×D67)⋊C411yescharacteristic
194C22×C4(C2×D67)⋊C411yescharacteristic
195C22×C4(C2×D67)⋊C411yescharacteristic
196C22×C4C2×C4×D6711yescharacteristic
197C22×C4C2×C22⋊D6711yescharacteristic
198C22×C4C2×D26811yescharacteristic
199M16—42no—
200M16—42no—
201C2×C8—42no—
202C2×C8—2682no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—2682no—
207C4⋊C4—2681no—
208C4⋊C4—2681no—
209C4⋊C4—5362no—
210C4⋊C4—5362no—
211C4⋊C4—1344no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—2682no—
219C23C2×(C2×D67)⋊C411yescharacteristic
220Q8—42no—
221Q8—42no—
222Q8—5364no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—2682no—
237C2×C4—2682no—
238C2×C4—2682no—
239C2×C4—2682no—
240C2×C4—2682no—
241C2×C4—1344no—
242C2×C4#1452395711yescharacteristic
243C2×C4#1452395711yescharacteristic
244C2×C4#1452396612yes—
245C2×C4#1452396612yes—
246C2×C4C2×#1176161611yescharacteristic
247C2×C4C2×#1176161811yescharacteristic
248C8—82no—
249C8—5362no—
250C22—21no—
251C22—21no—
252C22C2×#1452395711yescharacteristic
253C22C2×#1452396611yescharacteristic
254C22#1686341411yescharacteristic
255C4—21no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—2684no—
264C4—5362no—
265C4#1686319412yes—
266C2—21no—
267C2—21no—
268C2#2255878011yescharacteristic
269C2#2256186011yescharacteristic
270C2C2×#1686319411yescharacteristic
subgroupG/H
centerC22C2×#14523957
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#1801C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#1801D67
hypocenterC67#11908
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
140. C67#11908semidirect

Maximal subgroups

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

subgroupindex
#225654142
#225585332
#225582382
C67×#18012
#1190867