Finite Group Databasebeta

← C67×#7503 #30686754

Subgroups — C67×#7503 #30686754

Order 17152.

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2132C211yesmaximal, characteristic
2C67×#2126C211yesmaximal, characteristic
3C67×#1963C211yesmaximal, characteristic
4#532×C67C2211yescharacteristic
5#423×C67C411yescharacteristic, complemented
6C2×#104×C67—21no—
7C2×#104×C67—21no—
8#413×C67—21no—
9#410×C67—21no—
10#376×C67C411yescharacteristic, complemented
11#356×C67C412yescomplemented
12#355×C67—21no—
13#354×C67—21no—
14#353×C67C412yescomplemented
15C2×#41×C67D411yescharacteristic
16C2×#41×C67D411yescharacteristic
17C2×M16×C67—21no—
18C2×M16×C67—41no—
19C4⋊Q8×C67C2×C411yescharacteristic
20#127×C67C2×C411yescharacteristic
21#124×C67D411yescharacteristic
22#124×C67Q811yescharacteristic
23#121×C67—41no—
24C4×D4×C67—41no—
25#117×C67—41no—
26C2×C42×C67—21no—
27C2×C42×C67C2×C411yescharacteristic
28C4⋊C8×C67—41no—
29#104×C67—42no—
30#104×C67—42no—
31#101×C67—22no—
32#100×C67—41no—
33#99×C67—22no—
34#99×C67—41no—
35#95×C67—41no—
36#41×C67—41no—
37#41×C67—41no—
38#41×C67—41no—
39#41×C67—41no—
40C2×Q8×C67C22⋊C411yescharacteristic
41C2×Q8×C67C4⋊C411yescharacteristic
42C2×D4×C67—21no—
43C2×D4×C67—21no—
44C2×D4×C67—41no—
45C2×D4×C67C22⋊C411yescharacteristic
46C2×D4×C67C4⋊C411yescharacteristic
47C22×C4×C67—21no—
48C22×C4×C67—21no—
49C22×C4×C67—21no—
50C22×C4×C67—41no—
51C22×C4×C67C22⋊C411yescharacteristic
52M16×C67—42no—
53M16×C67—41no—
54M16×C67—41no—
55M16×C67—81no—
56C2×C8×C67—41no—
57C2×C8×C67—81no—
58C4⋊C4×C67—41no—
59C4⋊C4×C67—81no—
60C22⋊C4×C67—41no—
61C22⋊C4×C67—41no—
62C22⋊C4×C67—42no—
63C42×C67—21no—
64C42×C67—22no—
65C42×C67—41no—
66C42×C67—81no—
67C42×C67C4211yescharacteristic
68C42×C67C4⋊C411yescharacteristic
69C23×C67—21no—
70C23×C67—21no—
71C23×C67#9911yescharacteristic
72Q8×C67—41no—
73Q8×C67—41no—
74D4×C67—41no—
75D4×C67—41no—
76D4×C67—81no—
77D4×C67—81no—
78D4×C67—82no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—21no—
85C2×C4×C67—41no—
86C2×C4×C67—41no—
87C2×C4×C67—41no—
88C2×C4×C67—41no—
89C2×C4×C67—42no—
90C2×C4×C67—81no—
91C2×C4×C67—82no—
92C2×C4×C67—82no—
93C2×C4×C67#9511yescharacteristic
94C2×C4×C67#9911yescharacteristic
95C8×C67—82no—
96C8×C67—81no—
97C22×C67—21no—
98C22×C67—41no—
99C22×C67—41no—
100C22×C67—41no—
101C22×C67—81no—
102C22×C67—81no—
103C22×C67#34211yescharacteristic
104C4×C67—21no—
105C4×C67—41no—
106C4×C67—41no—
107C4×C67—41no—
108C4×C67—41no—
109C4×C67—82no—
110C4×C67—162no—
111C4×C67#35311yescharacteristic
112C4×C67#35411yescharacteristic
113#7503C6711yesmaximal, characteristic, direct product factor, Sylow
114C2×C67—21no—
115C2×C67—41no—
116C2×C67—81no—
117C2×C67—81no—
118C2×C67#139311yescharacteristic
119#2132C2×C6711yescharacteristic
120#2126C2×C6711yescharacteristic
121#1963C2×C6711yescharacteristic
122C67#750311yescharacteristic, direct product factor, Sylow
123#532C22×C6711yescharacteristic
124#423C4×C6711yescharacteristic, complemented
125C2×#104—21no—
126C2×#104—21no—
127#413—21no—
128#410—21no—
129#376C4×C6711yescharacteristic, complemented
130#356C4×C6712yescomplemented
131#355—21no—
132#354—21no—
133#353C4×C6712yescomplemented
134C2×#41D4×C6711yescharacteristic
135C2×#41D4×C6711yescharacteristic
136C2×M16—21no—
137C2×M16—41no—
138C4⋊Q8C2×C4×C6711yescharacteristic
139#127C2×C4×C6711yescharacteristic
140#124D4×C6711yescharacteristic
141#124Q8×C6711yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—41no—
145C2×C42—21no—
146C2×C42C2×C4×C6711yescharacteristic
147C4⋊C8—41no—
148#104—42no—
149#104—42no—
150#101—22no—
151#100—41no—
152#99—22no—
153#99—41no—
154#95—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8C22⋊C4×C6711yescharacteristic
160C2×Q8C4⋊C4×C6711yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C22⋊C4×C6711yescharacteristic
165C2×D4C4⋊C4×C6711yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4C22⋊C4×C6711yescharacteristic
171M16—42no—
172M16—41no—
173M16—41no—
174M16—81no—
175C2×C8—41no—
176C2×C8—81no—
177C4⋊C4—41no—
178C4⋊C4—81no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—42no—
182C42—21no—
183C42—22no—
184C42—41no—
185C42—81no—
186C42C42×C6711yescharacteristic
187C42C4⋊C4×C6711yescharacteristic
188C23—21no—
189C23—21no—
190C23#99×C6711yescharacteristic
191Q8—41no—
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—81no—
196D4—81no—
197D4—82no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#95×C6711yescharacteristic
213C2×C4#99×C6711yescharacteristic
214C8—82no—
215C8—81no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22#342×C6711yescharacteristic
223C4—21no—
224C4—41no—
225C4—41no—
226C4—41no—
227C4—41no—
228C4—82no—
229C4—162no—
230C4#353×C6711yescharacteristic
231C4#354×C6711yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2C67×#139311yescharacteristic
subgroupG/H
centerC4×C67#354
commutatorC42C42×C67
Frattini, cosocle#532C22×C67
socleC2×C67#1393
prime radical#532×C67C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C67 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ #532×C67 ◁ G

Complements

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

NH ≅ G/Ntype
5. #423×C67C4semidirect
10. #376×C67C4semidirect
11. #356×C67C4semidirect
14. #353×C67C4semidirect
113. #7503C67direct (N × H)
122. C67#7503direct (N × H)
124. #423C4×C67semidirect
129. #376C4×C67semidirect
130. #356C4×C67semidirect
133. #353C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#21322
C67×#21262
C67×#19632
#750367