Finite Group Databasebeta

← C67×#8910 #30688161

Subgroups — C67×#8910 #30688161

Order 17152.

500 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2379C211yesmaximal, characteristic
2C67×#1959C211yesmaximal, characteristic
3C67×#1847C212yesmaximal, complemented
4C2×#329×C67C211yesmaximal, characteristic
5C67×#1295C212yesmaximal, complemented
6C2×#124×C67C2211yescharacteristic
7C2×#107×C67—22no—
8C2×#106×C67—22no—
9C2×#103×C67—21no—
10C2×#102×C67—21no—
11C2×#97×C67C2211yescharacteristic
12C2×#97×C67C2212yes—
13#391×C67—21no—
14#388×C67—22no—
15#386×C67—21no—
16#385×C67—22no—
17#384×C67C2211yescharacteristic
18#382×C67C2212yes—
19#329×C67C414yescomplemented
20C22×Q8×C67D411yescharacteristic
21C22×D4×C67D411yescharacteristic
22C23×C4×C67—21no—
23C22×C8×C67—22no—
24C22×C8×C67—21no—
25#124×C67—22no—
26#124×C67C2×C412yes—
27#124×C67Q812yes—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67—22no—
30C2×C4⋊C4×C67—21no—
31C2×C4⋊C4×C67D412yes—
32C2×C4⋊C4×C67D412yes—
33C2×C22⋊C4×C67—22no—
34C2×C22⋊C4×C67—22no—
35C2×C22⋊C4×C67—21no—
36C2×C42×C67—21no—
37C2×C42×C67C2311yescharacteristic
38#107×C67—44no—
39#106×C67—44no—
40#103×C67—42no—
41#102×C67—42no—
42#97×C67—22no—
43#97×C67—22no—
44#97×C67C2×C414yescomplemented
45#95×C67—22no—
46#95×C67—22no—
47#95×C67—22no—
48C24×C67—21no—
49C2×Q8×C67—21no—
50C2×Q8×C67—41no—
51C2×Q8×C67—41no—
52C2×Q8×C67C4⋊C412yes—
53C2×D4×C67—21no—
54C2×D4×C67—41no—
55C2×D4×C67—41no—
56C2×D4×C67C4⋊C412yes—
57C22×C4×C67—21no—
58C22×C4×C67—22no—
59C22×C4×C67—22no—
60C22×C4×C67—41no—
61C22×C4×C67—82no—
62C22×C4×C67—81no—
63C22×C4×C67C2×D411yescharacteristic
64C22×C4×C67C2×D412yes—
65C2×C8×C67—24no—
66C2×C8×C67—24no—
67C2×C8×C67—24no—
68C2×C8×C67—22no—
69C2×C8×C67—41no—
70C2×C8×C67—41no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—41no—
74C4⋊C4×C67—41no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—81no—
78C22⋊C4×C67—24no—
79C22⋊C4×C67—24no—
80C22⋊C4×C67—82no—
81C22⋊C4×C67—81no—
82C42×C67—81no—
83C42×C67C22×C411yescharacteristic
84C42×C67C2×Q811yescharacteristic
85C42×C67#4112yes—
86C23×C67—22no—
87C23×C67—42no—
88C23×C67—41no—
89C23×C67—41no—
90C23×C67—42no—
91C23×C67#12011yescharacteristic
92Q8×C67—42no—
93Q8×C67—81no—
94D4×C67—42no—
95D4×C67—81no—
96C2×C4×C67—22no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
109C2×C4×C67—84no—
110C2×C4×C67—81no—
111C2×C4×C67—82no—
112C2×C4×C67C2×C4⋊C411yescharacteristic
113C2×C4×C67C4×D412yes—
114C2×C4×C67C22⋊Q812yes—
115C2×C4×C67#12311yescharacteristic
116C8×C67—28no—
117C8×C67—42no—
118C22×C67—42no—
119C22×C67—44no—
120C22×C67—81no—
121C22×C67—81no—
122C22×C67—82no—
123C22×C67#38511yescharacteristic
124C22×C67#45311yescharacteristic
125C22×C67#45312yes—
126C22×C67#45511yescharacteristic
127C22×C67#45612yes—
128C4×C67—21no—
129C4×C67—21no—
130C4×C67—22no—
131C4×C67—22no—
132C4×C67—42no—
133C4×C67—82no—
134C4×C67—82no—
135C4×C67—161no—
136#8910C6711yesmaximal, characteristic, direct product factor, Sylow
137C2×C67—82no—
138C2×C67#165511yescharacteristic
139C2×C67#166011yescharacteristic
140C2×C67#166112yes—
141C2×C67#185611yescharacteristic
142C2×C67#185912yes—
143#2379C2×C6711yescharacteristic
144#1959C2×C6711yescharacteristic
145#1847C2×C6712yescomplemented
146C2×#329C2×C6711yescharacteristic
147#1295C2×C6712yescomplemented
148C67#891011yescharacteristic, direct product factor, Sylow
149C2×#124C22×C6711yescharacteristic
150C2×#107—22no—
151C2×#106—22no—
152C2×#103—21no—
153C2×#102—21no—
154C2×#97C22×C6711yescharacteristic
155C2×#97C22×C6712yes—
156#391—21no—
157#388—22no—
158#386—21no—
159#385—22no—
160#384C22×C6711yescharacteristic
161#382C22×C6712yes—
162#329C4×C6714yescomplemented
163C22×Q8D4×C6711yescharacteristic
164C22×D4D4×C6711yescharacteristic
165C23×C4—21no—
166C22×C8—22no—
167C22×C8—21no—
168#124—22no—
169#124C2×C4×C6712yes—
170#124Q8×C6712yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4D4×C6712yes—
175C2×C4⋊C4D4×C6712yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42C23×C6711yescharacteristic
181#107—44no—
182#106—44no—
183#103—42no—
184#102—42no—
185#97—22no—
186#97—22no—
187#97C2×C4×C6714yescomplemented
188#95—22no—
189#95—22no—
190#95—22no—
191C24—21no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C4⋊C4×C6712yes—
196C2×D4—21no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4C4⋊C4×C6712yes—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—41no—
204C22×C4—82no—
205C22×C4—81no—
206C22×C4C2×D4×C6711yescharacteristic
207C22×C4C2×D4×C6712yes—
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—81no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—82no—
224C22⋊C4—81no—
225C42—81no—
226C42C22×C4×C6711yescharacteristic
227C42C2×Q8×C6711yescharacteristic
228C42#41×C6712yes—
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#120×C6711yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—42no—
238D4—81no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4C2×C4⋊C4×C6711yescharacteristic
256C2×C4C4×D4×C6712yes—
257C2×C4C22⋊Q8×C6712yes—
258C2×C4#123×C6711yescharacteristic
259C8—28no—
260C8—42no—
261C22—42no—
262C22—44no—
263C22—81no—
264C22—81no—
265C22—82no—
266C22#385×C6711yescharacteristic
267C22#453×C6711yescharacteristic
268C22#453×C6712yes—
269C22#455×C6711yescharacteristic
270C22#456×C6712yes—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—82no—
277C4—82no—
278C4—161no—
279C2—82no—
280C2C67×#165511yescharacteristic
281C2C67×#166011yescharacteristic
282C2C67×#166112yes—
283C2C67×#185611yescharacteristic
284C2C67×#185912yes—
subgroupG/H
center, socleC23×C67#120
commutatorC42C22×C4×C67
Frattini, cosocleC2×C42C23×C67
prime radicalC22×D4×C67D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#1847C2semidirect
5. C67×#1295C2semidirect
19. #329×C67C4semidirect
44. #97×C67C2×C4semidirect
136. #8910C67direct (N × H)
145. #1847C2×C67semidirect
147. #1295C2×C67semidirect
148. C67#8910direct (N × H)
162. #329C4×C67semidirect
187. #97C2×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#23792
C67×#19592
C67×#18472
C2×#329×C672
C67×#12952
#891067