Finite Group Databasebeta

← #30766976

Subgroups — #30766976

Order 17152.

440 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560201C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558236C214yesmaximal, complemented
4C67×#2487C211yesmaximal, characteristic
5#16863420—21no—
6C2×#14523960—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863165C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11C2×C4×D4×C67C2211yescharacteristic
12C22×C4⋊C4×C67C411yescharacteristic
13#400×C67C2212yes—
14#399×C67C412yes—
15#389×C67—21no—
16#386×C67—21no—
17#385×C67—21no—
18#380×C67—21no—
19#380×C67C411yescharacteristic
20C2×#11761612—22no—
21C22×C67⋊C8—41no—
22C4⋊(C67⋊C8)—41no—
23C4⋊(C67⋊C8)—41no—
24#14523960—42no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22×D4×C67D411yescharacteristic
28C23×C4×C67—21no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4×D4×C67—41no—
32C4×D4×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67C2×C412yes—
39C2×C4⋊C4×C67C2311yescharacteristic
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48#11761612—42no—
49#11761612—42no—
50C2×C67⋊C8—42no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—82no—
53C24×C67C22⋊C411yescharacteristic
54C24×C67C22⋊C411yescharacteristic
55C2×D4×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67SD1612yes—
58C2×D4×C67Q1612yes—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—42no—
68C22×C4×C67—82no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22×C411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C4⋊C4×C67—41no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—82no—
80C22⋊C4×C67—41no—
81C22⋊C4×C67—41no—
82C22⋊C4×C67—42no—
83C22⋊C4×C67—82no—
84C42×C67—21no—
85C42×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67#10312yes—
95C23×C67#10312yes—
96C23×C67C2×C22⋊C411yescharacteristic
97D4×C67—42no—
98D4×C67—42no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—22no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—81no—
118C2×C4×C67—84no—
119C2×C4×C67#9911yescharacteristic
120C2×C4×C67#9911yescharacteristic
121C2×C4×C67C2×SD1611yescharacteristic
122C2×C4×C67C2×Q1611yescharacteristic
123C67⋊C8—82no—
124C67⋊C8—82no—
125C22×C67—21no—
126C22×C67—21no—
127C22×C67—22no—
128C22×C67—22no—
129C22×C67—42no—
130C22×C67—42no—
131C22×C67—42no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—41no—
136C22×C67—81no—
137C22×C67C2×#9911yescharacteristic
138C22×C67C2×#10311yescharacteristic
139C22×C67#42111yescharacteristic
140C4×C67—21no—
141C4×C67—21no—
142C4×C67—21no—
143C4×C67—41no—
144C4×C67—41no—
145C4×C67—41no—
146C4×C67—82no—
147C4×C67—82no—
148C4×C67#35212yes—
149#11903—671nomaximal, Sylow
150C2×C67—21no—
151C2×C67—21no—
152C2×C67—42no—
153C2×C67—42no—
154C2×C67#152011yescharacteristic
155C2×C67C2×#35211yescharacteristic
156C2×C67#213411yescharacteristic
157#2487D6711yescharacteristic
158#1800—671no—
159#1459—671no—
160#1353—674no—
161C67#1190311yescharacteristic, complemented, Sylow
162C2×C4×D4C2×D6711yescharacteristic
163C22×C4⋊C4Dic6711yescharacteristic
164#423—1341no—
165C2×#100—1341no—
166C2×C22⋊C8—1341no—
167#400C2×D6712yes—
168#399Dic6712yes—
169#389—21no—
170#386—21no—
171#385—21no—
172#380—21no—
173#380Dic6711yescharacteristic
174#343—672no—
175#324—672no—
176#323—1341no—
177C22×D4C22⋊D6711yescharacteristic
178C23×C4—21no—
179C23×C4C2×Dic6711yescharacteristic
180C23×C4C22⋊D6711yescharacteristic
181C2×M16—1342no—
182C22×C8—2681no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×Dic6712yes—
191C2×C4⋊C4C22×D6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C2×Dic6711yescharacteristic
195C2×C22⋊C4C22⋊D6711yescharacteristic
196C2×C42C22⋊D6711yescharacteristic
197C4⋊C8—2681no—
198C4⋊C8—2681no—
199#100—2682no—
200C22⋊C8—2681no—
201C22⋊C8—1342no—
202#95—22no—
203#95—22no—
204#95C2×Dic6712yes—
205C24C22⋊Dic6711yescharacteristic
206C24C22⋊Dic6711yescharacteristic
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#1176161512yes—
210C2×D4#1176161812yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4C22⋊Dic6711yescharacteristic
222C22×C4C22⋊Dic6711yescharacteristic
223C22×C4C22×Dic6711yescharacteristic
224C22×C4C2×C22⋊D6711yescharacteristic
225C22×C4C2×C22⋊D6711yescharacteristic
226M16—2682no—
227M16—2682no—
228C2×C8—2682no—
229C2×C8—2682no—
230C2×C8—5362no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—82no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C42—21no—
242C42—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1452396712yes—
252C23#1452396712yes—
253C23C2×C22⋊Dic6711yescharacteristic
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—84no—
276C2×C4#1452395811yescharacteristic
277C2×C4#1452395811yescharacteristic
278C2×C4C2×#1176161511yescharacteristic
279C2×C4C2×#1176161811yescharacteristic
280C8—5362no—
281C8—5362no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#1452395811yescharacteristic
295C22C2×#1452396711yescharacteristic
296C22#1686341311yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—82no—
304C4—82no—
305C4#1686318612yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#2255878111yescharacteristic
311C2C2×#1686318611yescharacteristic
312C2#2256190111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2487C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C4⋊C4×C67C23
hypercenter#2487D67
hypocenterC67#11903
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #22558236C2semidirect
161. C67#11903semidirect

Maximal subgroups

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

subgroupindex
#225602012
#225585272
#225582362
C67×#24872
#1190367