Finite Group Databasebeta

← #30751880

Subgroups — #30751880

Order 17152.

560 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 135 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#22562827C211yesmaximal, characteristic
2#22558781C212yesmaximal
3C2×#16863122C211yesmaximal, characteristic
4#22558076C212yesmaximal
5C67×#2627C211yesmaximal, characteristic
6C2×#14524002—21no—
7C2×C22⋊(C67⋊C8)C2212yes—
8#16863295—24no—
9#16863256—21no—
10#16863256C2212yes—
11#16863133—24no—
12#16863122—22no—
13C2×C22⋊Q8×C67C2211yescharacteristic
14C22×C4⋊C4×C67C2211yescharacteristic
15#400×C67C412yescomplemented
16#398×C67—22no—
17#395×C67C412yescomplemented
18#385×C67—21no—
19#381×C67—21no—
20#380×C67C2211yescharacteristic
21C22×C67⋊C8—22no—
22#14524002—42no—
23#14524002—42no—
24C2×C4⋊Dic67—22no—
25C2×C4⋊Dic67—22no—
26C2×C4⋊Dic67—22no—
27C2×C22⋊Dic67—22no—
28C2×C4×Dic67—22no—
29C22⋊(C67⋊C8)—22no—
30C22⋊(C67⋊C8)—22no—
31C67⋊#95—22no—
32C67⋊#95D414yes—
33C22×Q8×C67C2×C411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C23×C4×C67C2311yescharacteristic
37C22⋊Q8×C67—22no—
38C22⋊Q8×C67—22no—
39C2×C4⋊C4×C67—22no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—22no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67C2×C411yescharacteristic
45C2×C4⋊C4×C67C2×C412yes—
46C2×C4⋊C4×C67D412yes—
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49#95×C67—22no—
50#95×C67—22no—
51#95×C67—22no—
52#95×C67C2×C412yes—
53C22×Dic67—24no—
54C2×C67⋊C8—42no—
55C2×C67⋊C8—82no—
56C4⋊Dic67—44no—
57C4⋊Dic67—82no—
58C4⋊Dic67—82no—
59C22⋊Dic67—42no—
60C22⋊Dic67—42no—
61C4×Dic67—44no—
62C24×C67C2×D411yescharacteristic
63C2×Q8×C67—22no—
64C2×Q8×C67—81no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—22no—
73C22×C4×C67—21no—
74C22×C4×C67—81no—
75C22×C4×C67—81no—
76C22×C4×C67C22⋊C412yes—
77C22×C4×C67C22⋊C412yes—
78C22×C4×C67SD1612yes—
79C22×C4×C67Q1612yes—
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C2×D412yes—
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67#4112yes—
84C4⋊C4×C67—24no—
85C4⋊C4×C67—24no—
86C4⋊C4×C67—22no—
87C4⋊C4×C67—42no—
88C4⋊C4×C67—81no—
89C4⋊C4×C67—82no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67—41no—
92C22⋊C4×C67—41no—
93C22⋊C4×C67—41no—
94C23×C67—41no—
95C23×C67—41no—
96C23×C67C2×C22⋊C411yescharacteristic
97C23×C67#12011yescharacteristic
98C23×C67#12311yescharacteristic
99C23×C67C2×SD1611yescharacteristic
100C23×C67C2×Q1611yescharacteristic
101C23×C67AGL(1,ℤ/8)11yescharacteristic
102C23×C67#13711yescharacteristic
103Q8×C67—82no—
104C2×C4×C67—22no—
105C2×C4×C67—22no—
106C2×C4×C67—22no—
107C2×C4×C67—22no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—22no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—42no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—42no—
124C2×C4×C67—81no—
125C2×C4×C67#10314yes—
126C2×C4×C67C4×D412yes—
127C2×C4×C67#12112yes—
128C2×Dic67—44no—
129C2×Dic67—44no—
130C2×Dic67—44no—
131C67⋊C8—82no—
132C22×C67—21no—
133C22×C67—21no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—21no—
137C22×C67—21no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67#38611yescharacteristic
143C22×C67C2×#10311yescharacteristic
144C22×C67#41711yescharacteristic
145C22×C67#45011yescharacteristic
146C22×C67#45111yescharacteristic
147C22×C67#48311yescharacteristic
148C22×C67#48411yescharacteristic
149C4×C67—24no—
150C4×C67—22no—
151C4×C67—22no—
152C4×C67—42no—
153C4×C67—82no—
154C4×C67—81no—
155C4×C67—81no—
156Dic67—84no—
157#9200—671nomaximal, Sylow
158C2×C67—21no—
159C2×C67—21no—
160C2×C67—21no—
161C2×C67—21no—
162C2×C67—21no—
163C2×C67—21no—
164C2×C67#187211yescharacteristic
165C2×C67#191211yescharacteristic
166C2×C67#191311yescharacteristic
167#2627D6711yescharacteristic
168#2316—671no—
169#1520—672no—
170C2×#324—671no—
171#1289—672no—
172C67#920011yescharacteristic, complemented, Sylow
173C2×C22⋊Q8C2×D6711yescharacteristic
174C2×#117—1341no—
175C22×C4⋊C4C2×D6711yescharacteristic
176C2×C22⋊C8—672no—
177#400Dic6712yescomplemented
178#398—22no—
179#395Dic6712yescomplemented
180#389—1344no—
181#385—21no—
182#381—21no—
183#380—672no—
184#380—1341no—
185#380C2×D6711yescharacteristic
186#328—1344no—
187#324—1342no—
188C22×Q8C2×Dic6711yescharacteristic
189C23×C4C22⋊D6711yescharacteristic
190C23×C4C22⋊D6711yescharacteristic
191C23×C4C22×D6711yescharacteristic
192C22×C8—1342no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195#117—2682no—
196#117—2682no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1342no—
203C2×C4⋊C4—1342no—
204C2×C4⋊C4—1342no—
205C2×C4⋊C4C2×Dic6711yescharacteristic
206C2×C4⋊C4C2×Dic6712yes—
207C2×C4⋊C4C22⋊D6712yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—1342no—
211C2×C42—1342no—
212C22⋊C8—1342no—
213C22⋊C8—1342no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—674no—
218#95—1342no—
219#95C2×Dic6712yes—
220C24C2×C22⋊D6711yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—1344no—
235C22×C4C22⋊Dic6712yes—
236C22×C4C22⋊Dic6712yes—
237C22×C4#1176161512yes—
238C22×C4#1176161812yes—
239C22×C4C22×Dic6711yescharacteristic
240C22×C4D4×D6712yes—
241C22×C4C2×C22⋊D6711yescharacteristic
242C22×C4#1176162712yes—
243C2×C8—2682no—
244C2×C8—5362no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—2684no—
252C4⋊C4—5362no—
253C4⋊C4—5362no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—2682no—
259C22⋊C4—2682no—
260C42—2684no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊Dic6711yescharacteristic
264C23#1452401111yescharacteristic
265C23#1452402411yescharacteristic
266C23C2×#1176161511yescharacteristic
267C23C2×#1176161811yescharacteristic
268C23#1452406811yescharacteristic
269C23#1452407311yescharacteristic
270Q8—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—2684no—
293C2×C4—2684no—
294C2×C4—2684no—
295C2×C4#1452396714yes—
296C2×C4D4×Dic6712yes—
297C2×C4#1452401512yes—
298C8—5362no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1686327611yescharacteristic
310C22C2×#1452396711yescharacteristic
311C22#1686339711yescharacteristic
312C22#1686354611yescharacteristic
313C22#1686355111yescharacteristic
314C22#1686372411yescharacteristic
315C22#1686372911yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—5364no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2256051711yescharacteristic
331C2#2256074711yescharacteristic
332C2#2256075211yescharacteristic
subgroupG/H
centerC22#16863276
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC67×#2627C2
socleC22×C67#386
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenter#2627D67
hypocenterC67#9200
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
15. #400×C67C4semidirect
17. #395×C67C4semidirect
172. C67#9200semidirect
177. #400Dic67semidirect
179. #395Dic67semidirect

Maximal subgroups

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

subgroupindex
#225628272
#225587812
C2×#168631222
#225580762
C67×#26272
#920067