Finite Group Databasebeta

← #30766583

Subgroups — #30766583

Order 17152.

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

normal subgroups only

This group has 368 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#22560292C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3#22558236C214yesmaximal, complemented
4C67×#2628C211yesmaximal, characteristic
5#16863453—21no—
6#16863360—21no—
7#16863165—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C2×C4⋊Q8×C67C411yescharacteristic
12C2×#117×C67C2211yescharacteristic
13#400×C67C2212yes—
14#399×C67C412yes—
15#395×C67—21no—
16#393×C67—21no—
17#389×C67—21no—
18#382×C67C411yescharacteristic
19#380×C67—21no—
20C2×#11761612—22no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—41no—
24C22⋊(C67⋊C8)—41no—
25#14523954—41no—
26C22×Q8×C67—21no—
27C23×C4×C67D411yescharacteristic
28C4⋊Q8×C67—41no—
29C4⋊Q8×C67—41no—
30#117×C67—41no—
31#117×C67—41no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—22no—
35C2×C4⋊C4×C67C2×C412yes—
36C2×C4⋊C4×C67D411yescharacteristic
37C2×C4⋊C4×C67C2311yescharacteristic
38C2×C22⋊C4×C67—22no—
39C2×C22⋊C4×C67C2×C411yescharacteristic
40C2×C22⋊C4×C67D411yescharacteristic
41C2×C42×C67C2×C411yescharacteristic
42C2×C42×C67D411yescharacteristic
43#95×C67—21no—
44#95×C67—22no—
45#95×C67—22no—
46#95×C67C2×C412yes—
47#11761612—42no—
48#11761612—42no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C24×C67C22⋊C411yescharacteristic
52C2×Q8×C67—42no—
53C2×Q8×C67—82no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—41no—
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22⋊C411yescharacteristic
62C22×C4×C67C22⋊C411yescharacteristic
63C22×C4×C67C22×C411yescharacteristic
64C22×C4×C67C2×D411yescharacteristic
65C22×C4×C67C2×D411yescharacteristic
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—82no—
73C22⋊C4×C67—41no—
74C22⋊C4×C67—41no—
75C22⋊C4×C67—82no—
76C42×C67—21no—
77C42×C67—21no—
78C42×C67—21no—
79C42×C67—21no—
80C23×C67—21no—
81C23×C67—21no—
82C23×C67—21no—
83C23×C67—41no—
84C23×C67—41no—
85C23×C67C2×C22⋊C411yescharacteristic
86Q8×C67—84no—
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—21no—
94C2×C4×C67—21no—
95C2×C4×C67—42no—
96C2×C4×C67—42no—
97C2×C4×C67—41no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—41no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67#9911yescharacteristic
108C2×C4×C67#9911yescharacteristic
109C2×C4×C67#13712yes—
110C67⋊C8—82no—
111C67⋊C8—82no—
112C22×C67—21no—
113C22×C67—21no—
114C22×C67—42no—
115C22×C67—41no—
116C22×C67—41no—
117C22×C67—41no—
118C22×C67—81no—
119C22×C67C2×#9911yescharacteristic
120C22×C67#41911yescharacteristic
121C22×C67#42111yescharacteristic
122C4×C67—22no—
123C4×C67—21no—
124C4×C67—21no—
125C4×C67—41no—
126C4×C67—41no—
127C4×C67—41no—
128C4×C67—41no—
129C4×C67—82no—
130C4×C67—82no—
131#11818—671nomaximal, Sylow
132C2×C67—21no—
133C2×C67—21no—
134C2×C67—81no—
135C2×C67#151111yescharacteristic
136C2×C67#212311yescharacteristic
137C2×C67#213411yescharacteristic
138#2628D6711yescharacteristic
139#1822—671no—
140#1464—671no—
141#1353—674no—
142C67#1181811yescharacteristic, complemented, Sylow
143C2×C4⋊Q8Dic6711yescharacteristic
144C2×#117C2×D6711yescharacteristic
145#432—1341no—
146#407—1341no—
147#400C2×D6712yes—
148#399Dic6712yes—
149#395—21no—
150#393—21no—
151#389—21no—
152#382Dic6711yescharacteristic
153#380—21no—
154#343—672no—
155#343—1341no—
156#324—672no—
157#323—1341no—
158C22×Q8—21no—
159C23×C4C22⋊D6711yescharacteristic
160C2×M16—1342no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C2×Dic6712yes—
169C2×C4⋊C4C22⋊D6711yescharacteristic
170C2×C4⋊C4C22×D6711yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4C2×Dic6711yescharacteristic
173C2×C22⋊C4C22⋊D6711yescharacteristic
174C2×C42C2×Dic6711yescharacteristic
175C2×C42C22⋊D6711yescharacteristic
176C4⋊C8—2681no—
177C22⋊C8—2681no—
178C22⋊C8—2681no—
179C22⋊C8—1342no—
180#97—2681no—
181#95—21no—
182#95—22no—
183#95—22no—
184#95C2×Dic6712yes—
185C24C22⋊Dic6711yescharacteristic
186C2×Q8—42no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C22⋊Dic6711yescharacteristic
195C22×C4C22⋊Dic6711yescharacteristic
196C22×C4C22⋊Dic6711yescharacteristic
197C22×C4C22×Dic6711yescharacteristic
198C22×C4C2×C22⋊D6711yescharacteristic
199C22×C4C2×C22⋊D6711yescharacteristic
200M16—2682no—
201M16—2682no—
202C2×C8—2682no—
203C2×C8—2682no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23C2×C22⋊Dic6711yescharacteristic
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4#1452395811yescharacteristic
246C2×C4#1452395811yescharacteristic
247C2×C4#1452407412yes—
248C8—5362no—
249C8—5362no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—81no—
257C22C2×#1452395811yescharacteristic
258C22#1686340511yescharacteristic
259C22#1686341311yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4—82no—
269C2—21no—
270C2—21no—
271C2—81no—
272C2#2255874211yescharacteristic
273C2#2256186311yescharacteristic
274C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2628C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2628D67
hypocenterC67#11818
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #22558236C2semidirect
142. C67#11818semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225585432
#225582362
C67×#26282
#1181867