Finite Group Databasebeta

← C67×#11546 #30690797

Subgroups — C67×#11546 #30690797

Order 17152.

460 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C67×#2482C211yesmaximal, characteristic
2C67×#1464C211yesmaximal, characteristic, complemented
3C67×#1458C211yesmaximal, characteristic, complemented
4C67×#1326C212yesmaximal, complemented
5C67×#1316C212yesmaximal
6C2×#117×C67C2211yescharacteristic
7C22×C22⋊C4×C67C411yescharacteristic
8#432×C67—21no—
9#431×C67—21no—
10#399×C67—21no—
11#397×C67—21no—
12#388×C67—21no—
13#388×C67—21no—
14#385×C67C411yescharacteristic
15#385×C67C2211yescharacteristic
16#381×C67C411yescharacteristic
17#379×C67C411yescharacteristic
18#379×C67C2211yescharacteristic
19#324×C67—21no—
20#324×C67—21no—
21#323×C67C2212yes—
22#323×C67C2212yes—
23C25×C67C2×C411yescharacteristic
24C23×C4×C67C2×C411yescharacteristic
25C23×C4×C67C2×C411yescharacteristic
26#117×C67—41no—
27#117×C67—41no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67D411yescharacteristic
31C2×C4⋊C4×C67D411yescharacteristic
32C2×C22⋊C4×C67—21no—
33C2×C22⋊C4×C67—21no—
34C2×C22⋊C4×C67—21no—
35C2×C22⋊C4×C67—21no—
36C2×C22⋊C4×C67—21no—
37C2×C22⋊C4×C67—21no—
38C2×C22⋊C4×C67—41no—
39C2×C22⋊C4×C67—41no—
40C2×C22⋊C4×C67C2×C411yescharacteristic
41C2×C22⋊C4×C67C2×C411yescharacteristic
42C2×C22⋊C4×C67C2×C411yescharacteristic
43C2×C22⋊C4×C67C2311yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C4⋊C8×C67—41no—
47C4⋊C8×C67—41no—
48C22⋊C8×C67—22no—
49C22⋊C8×C67—22no—
50#97×C67—41no—
51C4×C8×C67—41no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67—21no—
55#95×C67—21no—
56C24×C67—21no—
57C24×C67—22no—
58C24×C67—41no—
59C24×C67—42no—
60C24×C67—41no—
61C24×C67—81no—
62C24×C67C22×C411yescharacteristic
63C22×C4×C67—22no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—41no—
70C22×C4×C67—41no—
71C22×C4×C67—81no—
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C2×C8×C67—42no—
79C2×C8×C67—42no—
80C4⋊C4×C67—41no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—81no—
83C4⋊C4×C67—81no—
84C22⋊C4×C67—22no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—41no—
87C22⋊C4×C67—42no—
88C22⋊C4×C67—42no—
89C22⋊C4×C67—42no—
90C22⋊C4×C67—41no—
91C22⋊C4×C67—41no—
92C22⋊C4×C67—41no—
93C22⋊C4×C67—81no—
94C22⋊C4×C67—81no—
95C42×C67—21no—
96C42×C67—21no—
97C42×C67—21no—
98C42×C67—21no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—22no—
102C23×C67—41no—
103C23×C67—41no—
104C23×C67—41no—
105C23×C67—41no—
106C23×C67—41no—
107C23×C67—42no—
108C23×C67—42no—
109C23×C67—41no—
110C23×C67—41no—
111C23×C67—41no—
112C23×C67—82no—
113C23×C67—84no—
114C23×C67—84no—
115C23×C67—82no—
116C23×C67C2×C22⋊C411yescharacteristic
117C23×C67#13112yes—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—42no—
128C2×C4×C67—42no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—82no—
143C2×C4×C67#9911yescharacteristic
144C2×C4×C67#9911yescharacteristic
145C8×C67—82no—
146C8×C67—82no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—22no—
150C22×C67—42no—
151C22×C67—42no—
152C22×C67—42no—
153C22×C67—42no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—42no—
157C22×C67—81no—
158C22×C67—81no—
159C22×C67—84no—
160C22×C67—84no—
161C22×C67—81no—
162C22×C67—81no—
163C22×C67#40811yescharacteristic
164C22×C67C2×#9911yescharacteristic
165C22×C67#41311yescharacteristic
166C4×C67—21no—
167C4×C67—21no—
168C4×C67—41no—
169C4×C67—41no—
170C4×C67—41no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—81no—
174C4×C67—81no—
175C4×C67—81no—
176#11546C6711yesmaximal, characteristic, direct product factor, Sylow
177C2×C67—21no—
178C2×C67—21no—
179C2×C67—42no—
180C2×C67—82no—
181C2×C67#146811yescharacteristic
182C2×C67#212011yescharacteristic
183C2×C67#212211yescharacteristic
184#2482C2×C6711yescharacteristic
185#1464C2×C6711yescharacteristic, complemented
186#1458C2×C6711yescharacteristic, complemented
187#1326C2×C6712yescomplemented
188#1316C2×C6712yes—
189C67#1154611yescharacteristic, direct product factor, Sylow
190C2×#117C22×C6711yescharacteristic
191C22×C22⋊C4C4×C6711yescharacteristic
192#432—21no—
193#431—21no—
194#399—21no—
195#397—21no—
196#388—21no—
197#388—21no—
198#385C4×C6711yescharacteristic
199#385C22×C6711yescharacteristic
200#381C4×C6711yescharacteristic
201#379C4×C6711yescharacteristic
202#379C22×C6711yescharacteristic
203#324—21no—
204#324—21no—
205#323C22×C6712yes—
206#323C22×C6712yes—
207C25C2×C4×C6711yescharacteristic
208C23×C4C2×C4×C6711yescharacteristic
209C23×C4C2×C4×C6711yescharacteristic
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4D4×C6711yescharacteristic
215C2×C4⋊C4D4×C6711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4C2×C4×C6711yescharacteristic
225C2×C22⋊C4C2×C4×C6711yescharacteristic
226C2×C22⋊C4C2×C4×C6711yescharacteristic
227C2×C22⋊C4C23×C6711yescharacteristic
228C2×C42D4×C6711yescharacteristic
229C2×C42D4×C6711yescharacteristic
230C4⋊C8—41no—
231C4⋊C8—41no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234#97—41no—
235C4×C8—41no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240C24—21no—
241C24—22no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—81no—
246C24C22×C4×C6711yescharacteristic
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—81no—
256C22×C4C22⋊C4×C6711yescharacteristic
257C22×C4C22⋊C4×C6711yescharacteristic
258C22×C4C22⋊C4×C6711yescharacteristic
259C22×C4C22⋊C4×C6711yescharacteristic
260C22×C4C2×D4×C6711yescharacteristic
261C22×C4C2×D4×C6711yescharacteristic
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C22⋊C4—22no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—82no—
297C23—84no—
298C23—84no—
299C23—82no—
300C23C2×C22⋊C4×C6711yescharacteristic
301C23#131×C6712yes—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—82no—
327C2×C4#99×C6711yescharacteristic
328C2×C4#99×C6711yescharacteristic
329C8—82no—
330C8—82no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—84no—
344C22—84no—
345C22—81no—
346C22—81no—
347C22#408×C6711yescharacteristic
348C22C2×#99×C6711yescharacteristic
349C22#413×C6711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—82no—
364C2C67×#146811yescharacteristic
365C2C67×#212011yescharacteristic
366C2C67×#212211yescharacteristic
subgroupG/H
center, socleC22×C67C2×#99
commutatorC24C22×C4×C67
Frattini, cosocleC2×C22⋊C4C23×C67
prime radicalC25×C67C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4×C67 ◁ C2×#117×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#1464C2semidirect
3. C67×#1458C2semidirect
4. C67×#1326C2semidirect
176. #11546C67direct (N × H)
185. #1464C2×C67semidirect
186. #1458C2×C67semidirect
187. #1326C2×C67semidirect
189. C67#11546direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#24822
C67×#14642
C67×#14582
C67×#13262
C67×#13162
#1154667