Finite Group Databasebeta

← #30766478

Subgroups — #30766478

Order 17152.

378 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 87 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#22558553C211yesmaximal, characteristic
3#22558235C214yesmaximal
4C4×#101×C67C211yesmaximal, characteristic
5C2×#14524043C411yescharacteristic
6#16863453—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
17C2×#101×C67—21no—
18C4×M16×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—
29#14523954—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
45#101×C67—24no—
46#97×C67—41no—
47C4×C8×C67—41no—
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—22no—
59C2×Q8×C67—22no—
60C2×Q8×C67—41no—
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22×C411yescharacteristic
66C22×C4×C67C2×D411yescharacteristic
67C22×C4×C67C2×D411yescharacteristic
68M16×C67—24no—
69M16×C67—42no—
70C2×C8×C67—42no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—41no—
74C42×C67—21no—
75C42×C67—21no—
76C42×C67—21no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C23×C67C2×C22⋊C411yescharacteristic
81Q8×C67—44no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—22no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—21no—
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—41no—
94C2×C4×C67—41no—
95C2×C4×C67—41no—
96C2×C4×C67#9911yescharacteristic
97C2×C4×C67#9911yescharacteristic
98C2×C4×C67#10412yes—
99C2×C4×C67#13712yes—
100C2×Dic67—42no—
101C2×Dic67—42no—
102C2×Dic67—44no—
103C2×Dic67—42no—
104C2×Dic67—42no—
105Dic134—82no—
106Dic134—82no—
107C8×C67—44no—
108C67⋊C8—82no—
109C22×C67—21no—
110C22×C67—21no—
111C22×C67C2×#9911yescharacteristic
112C22×C67#41911yescharacteristic
113C22×C67C2×#10411yescharacteristic
114C4×C67—21no—
115C4×C67—22no—
116C4×C67—22no—
117C4×C67—41no—
118C4×C67—41no—
119C4×C67—41no—
120C4×C67—41no—
121C4×C67#35512yes—
122Dic67—82no—
123Dic67—82no—
124#11794—671nomaximal, Sylow
125C2×C67—21no—
126C2×C67—21no—
127C2×C67#150811yescharacteristic
128C2×C67#212311yescharacteristic
129C2×C67C2×#35511yescharacteristic
130#2568—671no—
131C4×#101D6711yescharacteristic
132#1467—671no—
133#1353—674no—
134C67#1179411yescharacteristic, complemented, Sylow
135C2×C4⋊Q8—671no—
136C2×C4×Q8C2×D6711yescharacteristic
137#432—1341no—
138C2×#101—21no—
139C4×M16—21no—
140#400—672no—
141#400—672no—
142#391—1341no—
143#391—1341no—
144#391—1341no—
145#382—1341no—
146C4×C4⋊C4—671no—
147#343C2×D6712yes—
148#324—672no—
149#324—1341no—
150C22×Q8—1341no—
151C22×Q8C22⋊D6711yescharacteristic
152C2×M16—22no—
153C4⋊Q8—2681no—
154C4⋊Q8—2681no—
155C4×Q8—41no—
156C4×Q8—41no—
157C2×C4⋊C4—672no—
158C2×C4⋊C4—1341no—
159C2×C4⋊C4—1341no—
160C2×C4⋊C4C4×D6711yescharacteristic
161C2×C4⋊C4D26811yescharacteristic
162C2×C4⋊C4C22×D6711yescharacteristic
163C2×C42—1341no—
164C2×C42C4×D6711yescharacteristic
165C2×C42D26811yescharacteristic
166C2×C42C22⋊D6711yescharacteristic
167C4⋊C8—2681no—
168#101—24no—
169C22⋊C8—1342no—
170#97—41no—
171#97—2681no—
172C4×C8—41no—
173#95—672no—
174#95—1342no—
175#95—1342no—
176#95—1342no—
177#95—1342no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—2684no—
182C2×Q8—2681no—
183C2×Q8—2681no—
184C22×C4—1342no—
185C22×C4—1342no—
186C22×C4(C2×D67)⋊C411yescharacteristic
187C22×C4(C2×D67)⋊C411yescharacteristic
188C22×C4(C2×D67)⋊C411yescharacteristic
189C22×C4(C2×D67)⋊C411yescharacteristic
190C22×C4C2×C4×D6711yescharacteristic
191C22×C4C2×C22⋊D6711yescharacteristic
192C22×C4C2×D26811yescharacteristic
193M16—24no—
194M16—42no—
195C2×C8—42no—
196C2×C8—2682no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—2682no—
201C4⋊C4—2682no—
202C4⋊C4—2682no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—2682no—
210C23C2×(C2×D67)⋊C411yescharacteristic
211Q8—44no—
212Q8—5362no—
213Q8—5362no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—2682no—
229C2×C4—2684no—
230C2×C4—2682no—
231C2×C4—2682no—
232C2×C4—2682no—
233C2×C4#1452395711yescharacteristic
234C2×C4#1452395711yescharacteristic
235C2×C4#1452397112yes—
236C2×C4#1452407312yes—
237C8—44no—
238C8—5362no—
239C22—21no—
240C22—21no—
241C22C2×#1452395711yescharacteristic
242C22#1686340411yescharacteristic
243C22C2×#1452397111yescharacteristic
244C4—22no—
245C4—22no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—5362no—
252C4—5362no—
253C4#1686319212yes—
254C2—21no—
255C2—21no—
256C2#2255872811yescharacteristic
257C2#2256186411yescharacteristic
258C2C2×#1686319211yescharacteristic
subgroupG/H
centerC22C2×#14523957
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC4×#101×C67C2
socleC22×C67C2×#99
prime radicalC23×C67C2×C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC4×#101D67
hypocenterC67#11794
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
134. C67#11794semidirect

Maximal subgroups

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

subgroupindex
#225654212
#225585532
#225582352
C4×#101×C672
#1179467