Finite Group Databasebeta

← C67×#7502 #30686753

Subgroups — C67×#7502 #30686753

Order 17152.

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×#355×C67C211yesmaximal, characteristic
2C2×#354×C67C211yesmaximal, characteristic
3C67×#1958C211yesmaximal, characteristic
4C2×#124×C67C2211yescharacteristic
5C2×#103×C67—21no—
6C2×#102×C67—21no—
7C2×#101×C67—21no—
8C2×#99×C67—21no—
9C2×C4×C8×C67C411yescharacteristic, complemented
10#382×C67C411yescharacteristic, complemented
11#355×C67—21no—
12#355×C67C412yescomplemented
13#354×C67—21no—
14#354×C67C412yescomplemented
15C22×Q8×C67D411yescharacteristic
16C22×D4×C67D411yescharacteristic
17C2×M16×C67—41no—
18C22×C8×C67—21no—
19#124×C67—41no—
20#124×C67C2×C411yescharacteristic
21#124×C67C2×C411yescharacteristic
22#124×C67D411yescharacteristic
23#124×C67Q811yescharacteristic
24C2×C4⋊C4×C67—21no—
25C2×C22⋊C4×C67—41no—
26C2×C22⋊C4×C67—41no—
27C2×C42×C67C2×C411yescharacteristic
28#103×C67—42no—
29#102×C67—42no—
30#101×C67—22no—
31#101×C67—41no—
32#99×C67—22no—
33#99×C67—41no—
34C4×C8×C67—41no—
35#95×C67—41no—
36C24×C67—21no—
37C2×Q8×C67—21no—
38C2×Q8×C67—41no—
39C2×Q8×C67—41no—
40C2×Q8×C67C22⋊C411yescharacteristic
41C2×Q8×C67C4⋊C411yescharacteristic
42C2×D4×C67—21no—
43C2×D4×C67—41no—
44C2×D4×C67—41no—
45C2×D4×C67C22⋊C411yescharacteristic
46C2×D4×C67C4⋊C411yescharacteristic
47C22×C4×C67—21no—
48C22×C4×C67—21no—
49C22×C4×C67—41no—
50C22×C4×C67—81no—
51C22×C4×C67C22⋊C411yescharacteristic
52M16×C67—42no—
53M16×C67—81no—
54C2×C8×C67—22no—
55C2×C8×C67—41no—
56C2×C8×C67—41no—
57C2×C8×C67—81no—
58C4⋊C4×C67—41no—
59C4⋊C4×C67—41no—
60C22⋊C4×C67—41no—
61C22⋊C4×C67—41no—
62C22⋊C4×C67—41no—
63C22⋊C4×C67—41no—
64C22⋊C4×C67—42no—
65C22⋊C4×C67—81no—
66C42×C67—21no—
67C42×C67C4211yescharacteristic
68C42×C67C4⋊C411yescharacteristic
69C23×C67—21no—
70C23×C67—21no—
71C23×C67—41no—
72C23×C67—41no—
73C23×C67—82no—
74C23×C67#9911yescharacteristic
75Q8×C67—41no—
76Q8×C67—41no—
77Q8×C67—81no—
78D4×C67—41no—
79D4×C67—41no—
80D4×C67—81no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—42no—
88C2×C4×C67—41no—
89C2×C4×C67—41no—
90C2×C4×C67—82no—
91C2×C4×C67—82no—
92C2×C4×C67—81no—
93C2×C4×C67—162no—
94C2×C4×C67#9511yescharacteristic
95C2×C4×C67#9911yescharacteristic
96C8×C67—42no—
97C8×C67—82no—
98C22×C67—22no—
99C22×C67—41no—
100C22×C67—41no—
101C22×C67—81no—
102C22×C67—81no—
103C22×C67—82no—
104C22×C67—161no—
105C22×C67#34211yescharacteristic
106C22×C67#35311yescharacteristic
107C22×C67#35411yescharacteristic
108C4×C67—21no—
109C4×C67—21no—
110C4×C67—41no—
111C4×C67—41no—
112C4×C67—41no—
113C4×C67—41no—
114C4×C67—161no—
115C4×C67—162no—
116#7502C6711yesmaximal, characteristic, direct product factor, Sylow
117C2×C67—21no—
118C2×C67—21no—
119C2×C67—81no—
120C2×C67—81no—
121C2×C67#139311yescharacteristic
122C2×C67#141412yes—
123C2×#355C2×C6711yescharacteristic
124C2×#354C2×C6711yescharacteristic
125#1958C2×C6711yescharacteristic
126C67#750211yescharacteristic, direct product factor, Sylow
127C2×#124C22×C6711yescharacteristic
128C2×#103—21no—
129C2×#102—21no—
130C2×#101—21no—
131C2×#99—21no—
132C2×C4×C8C4×C6711yescharacteristic, complemented
133#382C4×C6711yescharacteristic, complemented
134#355—21no—
135#355C4×C6712yescomplemented
136#354—21no—
137#354C4×C6712yescomplemented
138C22×Q8D4×C6711yescharacteristic
139C22×D4D4×C6711yescharacteristic
140C2×M16—41no—
141C22×C8—21no—
142#124—41no—
143#124C2×C4×C6711yescharacteristic
144#124C2×C4×C6711yescharacteristic
145#124D4×C6711yescharacteristic
146#124Q8×C6711yescharacteristic
147C2×C4⋊C4—21no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42C2×C4×C6711yescharacteristic
151#103—42no—
152#102—42no—
153#101—22no—
154#101—41no—
155#99—22no—
156#99—41no—
157C4×C8—41no—
158#95—41no—
159C24—21no—
160C2×Q8—21no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8C22⋊C4×C6711yescharacteristic
164C2×Q8C4⋊C4×C6711yescharacteristic
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4C22⋊C4×C6711yescharacteristic
169C2×D4C4⋊C4×C6711yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—81no—
174C22×C4C22⋊C4×C6711yescharacteristic
175M16—42no—
176M16—81no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—81no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—42no—
188C22⋊C4—81no—
189C42—21no—
190C42C42×C6711yescharacteristic
191C42C4⋊C4×C6711yescharacteristic
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—82no—
197C23#99×C6711yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—81no—
201D4—41no—
202D4—41no—
203D4—81no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—162no—
217C2×C4#95×C6711yescharacteristic
218C2×C4#99×C6711yescharacteristic
219C8—42no—
220C8—82no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—82no—
227C22—161no—
228C22#342×C6711yescharacteristic
229C22#353×C6711yescharacteristic
230C22#354×C6711yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—161no—
238C4—162no—
239C2—21no—
240C2—21no—
241C2—81no—
242C2—81no—
243C2C67×#139311yescharacteristic
244C2C67×#141412yes—
subgroupG/H
center, socleC22×C67#354
commutatorC42C42×C67
Frattini, cosocleC2×#124C22×C67
prime radicalC22×D4×C67D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C67 ◁ C22×C4×C67 ◁ C2×#124×C67 ◁ G

Complements

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

NH ≅ G/Ntype
9. C2×C4×C8×C67C4semidirect
10. #382×C67C4semidirect
12. #355×C67C4semidirect
14. #354×C67C4semidirect
116. #7502C67direct (N × H)
126. C67#7502direct (N × H)
132. C2×C4×C8C4×C67semidirect
133. #382C4×C67semidirect
135. #355C4×C67semidirect
137. #354C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#355×C672
C2×#354×C672
C67×#19582
#750267