Finite Group Databasebeta

← #30766146

Subgroups — #30766146

Order 17152.

432 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 115 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#22559948C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558236C214yesmaximal, complemented
4C67×#2628C211yesmaximal, characteristic
5C22×#11761612—21no—
6#16863449—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—24no—
21C2×#11761612—22no—
22C2×#11761612—41no—
23C2×#11761612—42no—
24C22×C67⋊C8—41no—
25C4⋊(C67⋊C8)—41no—
26C22⋊(C67⋊C8)—22no—
27C4×C67⋊C8—41no—
28C22×Q8×C67—21no—
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32#117×C67—41no—
33#117×C67—41no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—22no—
37C2×C4⋊C4×C67C2×C412yes—
38C2×C4⋊C4×C67D411yescharacteristic
39C2×C4⋊C4×C67C2311yescharacteristic
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—21no—
46#95×C67—22no—
47#95×C67—22no—
48#95×C67C2×C412yes—
49#11761612—24no—
50#11761612—44no—
51#11761612—42no—
52C2×C67⋊C8—42no—
53C2×C67⋊C8—44no—
54C2×C67⋊C8—42no—
55C24×C67C22⋊C411yescharacteristic
56C2×Q8×C67—42no—
57C2×Q8×C67—82no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—22no—
63C22×C4×C67C22⋊C411yescharacteristic
64C22×C4×C67C22⋊C411yescharacteristic
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67SD1612yes—
67C22×C4×C67Q1612yes—
68C22×C4×C67C22×C411yescharacteristic
69C22×C4×C67C2×D411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C4⋊C4×C67—41no—
72C4⋊C4×C67—41no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—82no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—82no—
81C42×C67—21no—
82C42×C67—21no—
83C42×C67—21no—
84C42×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—41no—
88C23×C67—41no—
89C23×C67#10312yes—
90C23×C67C2×C22⋊C411yescharacteristic
91Q8×C67—84no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—22no—
95C2×C4×C67—22no—
96C2×C4×C67—22no—
97C2×C4×C67—22no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—41no—
105C2×C4×C67—41no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67#9911yescharacteristic
113C2×C4×C67#9911yescharacteristic
114C2×C4×C67#10312yes—
115C2×C4×C67#10412yes—
116C2×C4×C67C2×SD1611yescharacteristic
117C2×C4×C67C2×Q1611yescharacteristic
118C67⋊C8—44no—
119C67⋊C8—82no—
120C22×C67—21no—
121C22×C67—21no—
122C22×C67—22no—
123C22×C67—42no—
124C22×C67—41no—
125C22×C67—41no—
126C22×C67—81no—
127C22×C67C2×#9911yescharacteristic
128C22×C67C2×#10311yescharacteristic
129C22×C67C2×#10411yescharacteristic
130C4×C67—21no—
131C4×C67—21no—
132C4×C67—41no—
133C4×C67—41no—
134C4×C67—41no—
135C4×C67—41no—
136C4×C67—82no—
137C4×C67—82no—
138C4×C67#32812yes—
139C4×C67#32812yes—
140#11723—671nomaximal, Sylow
141C2×C67—21no—
142C2×C67—21no—
143C2×C67—42no—
144C2×C67C2×#32811yescharacteristic
145C2×C67#212311yescharacteristic
146C2×C67#213411yescharacteristic
147#2628D6711yescharacteristic
148#1742—671no—
149#1458—671no—
150#1353—674no—
151C67#1172311yescharacteristic, complemented, Sylow
152C22×M16—1341no—
153C2×C4⋊Q8Dic6711yescharacteristic
154C2×#117C2×D6711yescharacteristic
155#431—1341no—
156#400C2×D6712yes—
157#399Dic6712yes—
158#395—21no—
159#393—21no—
160#389—21no—
161#382Dic6711yescharacteristic
162#380—21no—
163#343—672no—
164#343—1341no—
165#324—672no—
166#323—1341no—
167C22×Q8—21no—
168C23×C4C22⋊D6711yescharacteristic
169C2×M16—2681no—
170C2×M16—2682no—
171C2×M16—1344no—
172C2×M16—1342no—
173C22×C8—2681no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×Dic6712yes—
182C2×C4⋊C4C22⋊D6711yescharacteristic
183C2×C4⋊C4C22×D6711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic6711yescharacteristic
186C2×C22⋊C4C22⋊D6711yescharacteristic
187C2×C42C2×Dic6711yescharacteristic
188C2×C42C22⋊D6711yescharacteristic
189C4⋊C8—2681no—
190C22⋊C8—1342no—
191C4×C8—2681no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95C2×Dic6712yes—
196C24C22⋊Dic6711yescharacteristic
197C2×Q8—42no—
198C2×Q8—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4C22⋊Dic6711yescharacteristic
205C22×C4C22⋊Dic6711yescharacteristic
206C22×C4C22⋊Dic6711yescharacteristic
207C22×C4#1176161512yes—
208C22×C4#1176161812yes—
209C22×C4C22×Dic6711yescharacteristic
210C22×C4C2×C22⋊D6711yescharacteristic
211C22×C4C2×C22⋊D6711yescharacteristic
212M16—2684no—
213M16—2682no—
214M16—1344no—
215C2×C8—2682no—
216C2×C8—2684no—
217C2×C8—2682no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1452396712yes—
237C23C2×C22⋊Dic6711yescharacteristic
238Q8—84no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#1452395811yescharacteristic
260C2×C4#1452395811yescharacteristic
261C2×C4#1452396712yes—
262C2×C4#1452397012yes—
263C2×C4C2×#1176161511yescharacteristic
264C2×C4C2×#1176161811yescharacteristic
265C8—2684no—
266C8—5362no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1452395811yescharacteristic
275C22C2×#1452396711yescharacteristic
276C22C2×#1452397011yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1686313312yes—
286C4#1686313312yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1686313311yescharacteristic
291C2#2256186311yescharacteristic
292C2#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#11723
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
151. C67#11723semidirect

Maximal subgroups

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

subgroupindex
#225599482
#225585232
#225582362
C67×#26282
#1172367