Finite Group Databasebeta

← #30766993

Subgroups — #30766993

Order 17152.

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

normal subgroups only

This group has 372 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#22565421C211yesmaximal, characteristic
2#22558533C211yesmaximal, characteristic
3#22558235C214yesmaximal
4C67×#1801C211yesmaximal, characteristic
5C2×#14524043C411yescharacteristic
6#16863449—21no—
7#16863335C412yes—
8#16863335C2212yes—
9#16863303—21no—
10#16863302—21no—
11#16863302—21no—
12#16863261—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×Dic134—21no—
21#14524043—41no—
22#14524043—41no—
23C2×C4⋊Dic67—21no—
24C2×C4⋊Dic67—21no—
25C2×C4⋊Dic67C2×C412yes—
26C2×C4×Dic67—21no—
27C4⋊(C67⋊C8)—41no—
28C22⋊(C67⋊C8)—22no—
29C4×C67⋊C8—41no—
30C67⋊#95—22no—
31C67⋊#95—22no—
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—44no—
49C2×Dic134—41no—
50C2×Dic134—41no—
51C22×Dic67—22no—
52C22×Dic67—22no—
53C2×C67⋊C8—42no—
54C4⋊Dic67—42no—
55C4⋊Dic67—42no—
56C4⋊Dic67—42no—
57C4×Dic67—42no—
58C2×Q8×C67—41no—
59C2×Q8×C67—41no—
60C2×Q8×C67SD1612yes—
61C2×Q8×C67Q1612yes—
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
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—41no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—41no—
75C42×C67—21no—
76C42×C67—21no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C42×C67—21no—
81C23×C67C2×C22⋊C411yescharacteristic
82Q8×C67—42no—
83Q8×C67—42no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—22no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—41no—
95C2×C4×C67—41no—
96C2×C4×C67—41no—
97C2×C4×C67#9911yescharacteristic
98C2×C4×C67#9911yescharacteristic
99C2×C4×C67#10312yes—
100C2×C4×C67#10312yes—
101C2×C4×C67C2×SD1611yescharacteristic
102C2×C4×C67C2×Q1611yescharacteristic
103C2×Dic67—42no—
104C2×Dic67—42no—
105C2×Dic67—44no—
106C2×Dic67—42no—
107C2×Dic67—42no—
108Dic134—82no—
109Dic134—82no—
110C8×C67—82no—
111C67⋊C8—82no—
112C22×C67—21no—
113C22×C67—21no—
114C22×C67C2×#9911yescharacteristic
115C22×C67C2×#10311yescharacteristic
116C22×C67#42111yescharacteristic
117C4×C67—21no—
118C4×C67—21no—
119C4×C67—22no—
120C4×C67—22no—
121C4×C67—21no—
122C4×C67—41no—
123C4×C67—41no—
124C4×C67—41no—
125C4×C67#35512yes—
126Dic67—82no—
127Dic67—82no—
128#11907—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67#152011yescharacteristic
132C2×C67#212311yescharacteristic
133C2×C67C2×#35511yescharacteristic
134#2568—671no—
135#1801D6711yescharacteristic
136#1461—671no—
137#1353—674no—
138C67#1190711yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—671no—
140C2×C4×Q8C2×D6711yescharacteristic
141#431—1341no—
142#423—21no—
143C2×#101—21no—
144#400—672no—
145#400—672no—
146#391—1341no—
147#391—1341no—
148#391—1341no—
149#382—1341no—
150C4×C4⋊C4—671no—
151#343C2×D6712yes—
152#324—672no—
153#324—1341no—
154C22×Q8—1341no—
155C22×Q8C22⋊D6711yescharacteristic
156C2×M16—22no—
157C4⋊Q8—2681no—
158C4⋊Q8—2681no—
159C4×Q8—41no—
160C4×Q8—41no—
161C2×C4⋊C4—672no—
162C2×C4⋊C4—1341no—
163C2×C4⋊C4—1341no—
164C2×C4⋊C4C4×D6711yescharacteristic
165C2×C4⋊C4D26811yescharacteristic
166C2×C4⋊C4C22×D6711yescharacteristic
167C2×C42—1341no—
168C2×C42C4×D6711yescharacteristic
169C2×C42D26811yescharacteristic
170C2×C42C22⋊D6711yescharacteristic
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—2681no—
174#101—42no—
175C22⋊C8—1342no—
176C4×C8—2681no—
177#95—672no—
178#95—1342no—
179#95—1342no—
180#95—1342no—
181#95—1342no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—2684no—
185C2×Q8—2681no—
186C2×Q8—2681no—
187C2×Q8#1176161612yes—
188C2×Q8#1176161812yes—
189C22×C4—1342no—
190C22×C4—1342no—
191C22×C4(C2×D67)⋊C411yescharacteristic
192C22×C4(C2×D67)⋊C411yescharacteristic
193C22×C4(C2×D67)⋊C411yescharacteristic
194C22×C4(C2×D67)⋊C411yescharacteristic
195C22×C4C2×C4×D6711yescharacteristic
196C22×C4C2×C22⋊D6711yescharacteristic
197C22×C4C2×D26811yescharacteristic
198M16—42no—
199M16—42no—
200C2×C8—42no—
201C2×C8—2682no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—2682no—
206C4⋊C4—2682no—
207C4⋊C4—2682no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—2682no—
215C23C2×(C2×D67)⋊C411yescharacteristic
216Q8—42no—
217Q8—42no—
218Q8—5362no—
219Q8—5362no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—2682no—
234C2×C4—2684no—
235C2×C4—2682no—
236C2×C4—2682no—
237C2×C4—2682no—
238C2×C4#1452395711yescharacteristic
239C2×C4#1452395711yescharacteristic
240C2×C4#1452396612yes—
241C2×C4#1452396612yes—
242C2×C4C2×#1176161611yescharacteristic
243C2×C4C2×#1176161811yescharacteristic
244C8—82no—
245C8—5362no—
246C22—21no—
247C22—21no—
248C22C2×#1452395711yescharacteristic
249C22C2×#1452396611yescharacteristic
250C22#1686341411yescharacteristic
251C4—21no—
252C4—22no—
253C4—22no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—5362no—
260C4—5362no—
261C4#1686319212yes—
262C2—21no—
263C2—21no—
264C2#2255878011yescharacteristic
265C2#2256186411yescharacteristic
266C2C2×#1686319211yescharacteristic
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#11907
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
138. C67#11907semidirect

Maximal subgroups

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

subgroupindex
#225654212
#225585332
#225582352
C67×#18012
#1190767