Finite Group Databasebeta

← C67×#10345 #30689596

Subgroups — C67×#10345 #30689596

Order 17152.

640 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2646C211yesmaximal, characteristic
2C67×#1787C212yesmaximal
3C67×#1446C211yesmaximal, characteristic
4C67×#1294C212yesmaximal
5C67×#1280C211yesmaximal, characteristic
6C23×Q8×C67C2211yescharacteristic
7C2×#103×C67—22no—
8C2×#103×C67—22no—
9C2×C22⋊C8×C67C2212yes—
10#400×C67—22no—
11#398×C67—22no—
12#385×C67—22no—
13#380×C67C2212yes—
14C2×#95×C67—22no—
15C2×#95×C67C2211yescharacteristic
16C2×#95×C67C2211yescharacteristic
17C22×Q8×C67—21no—
18C22×Q8×C67—22no—
19C22×Q8×C67—22no—
20C22×Q8×C67—44no—
21C22×Q8×C67Q812yes—
22C23×C4×C67—21no—
23C23×C4×C67—21no—
24C23×C4×C67D412yes—
25C23×C4×C67C2311yescharacteristic
26C22×C8×C67—22no—
27C2×C4⋊C4×C67—22no—
28C2×C4⋊C4×C67—22no—
29C2×C4⋊C4×C67D412yes—
30C2×C4⋊C4×C67D412yes—
31C2×C22⋊C4×C67—22no—
32#103×C67—44no—
33#103×C67—44no—
34C22⋊C8×C67—22no—
35C22⋊C8×C67—22no—
36#95×C67—22no—
37#95×C67—22no—
38#95×C67—22no—
39#95×C67—22no—
40#95×C67—22no—
41#95×C67—44no—
42C24×C67C2×D411yescharacteristic
43C2×Q8×C67—24no—
44C2×Q8×C67—24no—
45C2×Q8×C67—24no—
46C2×Q8×C67—44no—
47C2×Q8×C67—42no—
48C2×Q8×C67—42no—
49C2×Q8×C67—44no—
50C2×Q8×C67—48no—
51C2×Q8×C67—42no—
52C22×C4×C67—24no—
53C22×C4×C67—22no—
54C22×C4×C67—22no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—22no—
60C22×C4×C67—42no—
61C22×C4×C67—42no—
62C22×C4×C67—41no—
63C22×C4×C67—41no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—41no—
67C22×C4×C67—42no—
68C22×C4×C67—41no—
69C22×C4×C67—42no—
70C22×C4×C67C2×D412yes—
71C22×C4×C67C2×Q811yescharacteristic
72C22×C4×C67#4112yes—
73C22×C4×C67#4111yescharacteristic
74C2×C8×C67—42no—
75C2×C8×C67—42no—
76C2×C8×C67—42no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—82no—
80C4⋊C4×C67—82no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—42no—
83C23×C67—21no—
84C23×C67—21no—
85C23×C67—22no—
86C23×C67#12011yescharacteristic
87C23×C67C22⋊Q811yescharacteristic
88C23×C67#12411yescharacteristic
89C23×C67#13512yes—
90C23×C67#13712yes—
91Q8×C67—28no—
92Q8×C67—44no—
93Q8×C67—44no—
94Q8×C67—44no—
95C2×C4×C67—22no—
96C2×C4×C67—22no—
97C2×C4×C67—24no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—42no—
101C2×C4×C67—42no—
102C2×C4×C67—44no—
103C2×C4×C67—42no—
104C2×C4×C67—44no—
105C2×C4×C67—44no—
106C2×C4×C67—42no—
107C2×C4×C67—44no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
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—82no—
118C2×C4×C67—81no—
119C2×C4×C67—81no—
120C2×C4×C67—82no—
121C2×C4×C67C22⋊Q812yes—
122C2×C4×C67#12412yes—
123C8×C67—82no—
124C22×C67—22no—
125C22×C67—22no—
126C22×C67—22no—
127C22×C67—22no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67#39311yescharacteristic
133C22×C67#45212yes—
134C22×C67#45511yescharacteristic
135C22×C67#45611yescharacteristic
136C22×C67B(4,𝔽2)12yes—
137C22×C67#47912yes—
138C22×C67#48811yescharacteristic
139C22×C67#49111yescharacteristic
140C4×C67—22no—
141C4×C67—22no—
142C4×C67—44no—
143C4×C67—82no—
144C4×C67—82no—
145C4×C67—82no—
146C4×C67—82no—
147#10345C6711yesmaximal, characteristic, direct product factor, Sylow
148C2×C67—21no—
149C2×C67—22no—
150C2×C67—21no—
151C2×C67#160312yes—
152C2×C67#160612yes—
153C2×C67#202611yescharacteristic
154C2×C67#202811yescharacteristic
155C2×C67#202911yescharacteristic
156#2646C2×C6711yescharacteristic
157#1787C2×C6712yes—
158#1446C2×C6711yescharacteristic
159#1294C2×C6712yes—
160#1280C2×C6711yescharacteristic
161C67#1034511yescharacteristic, direct product factor, Sylow
162C23×Q8C22×C6711yescharacteristic
163C2×#103—22no—
164C2×#103—22no—
165C2×C22⋊C8C22×C6712yes—
166#400—22no—
167#398—22no—
168#385—22no—
169#380C22×C6712yes—
170C2×#95—22no—
171C2×#95C22×C6711yescharacteristic
172C2×#95C22×C6711yescharacteristic
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—44no—
177C22×Q8Q8×C6712yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4D4×C6712yes—
181C23×C4C23×C6711yescharacteristic
182C22×C8—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4D4×C6712yes—
186C2×C4⋊C4D4×C6712yes—
187C2×C22⋊C4—22no—
188#103—44no—
189#103—44no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—44no—
198C24C2×D4×C6711yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—44no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—48no—
207C2×Q8—42no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4C2×D4×C6712yes—
227C22×C4C2×Q8×C6711yescharacteristic
228C22×C4#41×C6712yes—
229C22×C4#41×C6711yescharacteristic
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#120×C6711yescharacteristic
243C23C22⋊Q8×C6711yescharacteristic
244C23#124×C6711yescharacteristic
245C23#135×C6712yes—
246C23#137×C6712yes—
247Q8—28no—
248Q8—44no—
249Q8—44no—
250Q8—44no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4C22⋊Q8×C6712yes—
278C2×C4#124×C6712yes—
279C8—82no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22#393×C6711yescharacteristic
289C22#452×C6712yes—
290C22#455×C6711yescharacteristic
291C22#456×C6711yescharacteristic
292C22B(4,𝔽2)×C6712yes—
293C22#479×C6712yes—
294C22#488×C6711yescharacteristic
295C22#491×C6711yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—82no—
301C4—82no—
302C4—82no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2C67×#160312yes—
307C2C67×#160612yes—
308C2C67×#202611yescharacteristic
309C2C67×#202811yescharacteristic
310C2C67×#202911yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC23×C4C23×C67
prime radicalC24×C67C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
147. #10345C67direct (N × H)
161. C67#10345direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#26462
C67×#17872
C67×#14462
C67×#12942
C67×#12802
#1034567