Finite Group Databasebeta

← C67×#7398 #30686649

Subgroups — C67×#7398 #30686649

Order 17152.

258 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2018C211yesmaximal, characteristic
2C67×#1391C211yesmaximal, characteristic, complemented
3C67×#1321C211yesmaximal, characteristic, complemented
4C2×AGL(1,ℤ/8)×C67—21no—
5C22×D8×C67C411yescharacteristic, complemented
6C2×#100×C67—21no—
7#410×C67—21no—
8#408×C67C2211yescharacteristic
9#349×C67—21no—
10#341×C67C411yescharacteristic, complemented
11C2×#41×C67D411yescharacteristic
12C22×D4×C67—21no—
13AGL(1,ℤ/8)×C67—42no—
14AGL(1,ℤ/8)×C67—42no—
15C2×SD16×C67—41no—
16C2×D8×C67—41no—
17C2×D8×C67—41no—
18C2×D8×C67—42no—
19C2×M16×C67—21no—
20C2×M16×C67D411yescharacteristic
21C22×C8×C67C2×C411yescharacteristic
22#117×C67—41no—
23M32×C67—41no—
24#100×C67—42no—
25#99×C67—22no—
26#99×C67—41no—
27C22⋊C8×C67—41no—
28C24×C67—41no—
29#41×C67—41no—
30#41×C67—41no—
31C2×Q8×C67SD1611yescharacteristic
32C2×D4×C67—22no—
33C2×D4×C67—21no—
34C2×D4×C67—41no—
35C2×D4×C67—41no—
36C2×D4×C67—41no—
37C2×D4×C67—41no—
38C2×D4×C67—41no—
39C2×D4×C67D811yescharacteristic
40C22×C4×C67—21no—
41C22×C4×C67C22⋊C411yescharacteristic
42SD16×C67—44no—
43D8×C67—42no—
44D8×C67—42no—
45D8×C67—44no—
46M16×C67—22no—
47M16×C67—41no—
48M16×C67—41no—
49C2×C8×C67—21no—
50C2×C8×C67—21no—
51C2×C8×C67—41no—
52C2×C8×C67—41no—
53C4⋊C4×C67—81no—
54C22⋊C4×C67—42no—
55C42×C67—81no—
56C16×C67—81no—
57C23×C67—21no—
58C23×C67—42no—
59C23×C67—81no—
60C23×C67—81no—
61C23×C67—162no—
62C23×C67#10411yescharacteristic
63Q8×C67—41no—
64D4×C67—41no—
65D4×C67—42no—
66D4×C67—42no—
67D4×C67—81no—
68D4×C67—81no—
69D4×C67—81no—
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—41no—
74C2×C4×C67—41no—
75C2×C4×C67—82no—
76C2×C4×C67#9911yescharacteristic
77C2×C4×C67#10211yescharacteristic
78C8×C67—42no—
79C8×C67—41no—
80C8×C67—41no—
81C22×C67—21no—
82C22×C67—41no—
83C22×C67—41no—
84C22×C67—81no—
85C22×C67—82no—
86C22×C67—162no—
87C22×C67—162no—
88C22×C67—162no—
89C22×C67#32711yescharacteristic
90C4×C67—21no—
91C4×C67—41no—
92C4×C67—162no—
93C4×C67AΣL(1,𝔽16)11yescharacteristic
94C4×C67#35311yescharacteristic
95#7398C6711yesmaximal, characteristic, direct product factor, Sylow
96C2×C67—21no—
97C2×C67—41no—
98C2×C67—81no—
99C2×C67—162no—
100C2×C67#134311yescharacteristic
101#2018C2×C6711yescharacteristic
102#1391C2×C6711yescharacteristic, complemented
103#1321C2×C6711yescharacteristic, complemented
104C67#739811yescharacteristic, direct product factor, Sylow
105C2×AGL(1,ℤ/8)—21no—
106C22×D8C4×C6711yescharacteristic, complemented
107C2×#100—21no—
108#410—21no—
109#408C22×C6711yescharacteristic
110#349—21no—
111#341C4×C6711yescharacteristic, complemented
112C2×#41D4×C6711yescharacteristic
113C22×D4—21no—
114AGL(1,ℤ/8)—42no—
115AGL(1,ℤ/8)—42no—
116C2×SD16—41no—
117C2×D8—41no—
118C2×D8—41no—
119C2×D8—42no—
120C2×M16—21no—
121C2×M16D4×C6711yescharacteristic
122C22×C8C2×C4×C6711yescharacteristic
123#117—41no—
124M32—41no—
125#100—42no—
126#99—22no—
127#99—41no—
128C22⋊C8—41no—
129C24—41no—
130#41—41no—
131#41—41no—
132C2×Q8SD16×C6711yescharacteristic
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4D8×C6711yescharacteristic
141C22×C4—21no—
142C22×C4C22⋊C4×C6711yescharacteristic
143SD16—44no—
144D8—42no—
145D8—42no—
146D8—44no—
147M16—22no—
148M16—41no—
149M16—41no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—41no—
153C2×C8—41no—
154C4⋊C4—81no—
155C22⋊C4—42no—
156C42—81no—
157C16—81no—
158C23—21no—
159C23—42no—
160C23—81no—
161C23—81no—
162C23—162no—
163C23#104×C6711yescharacteristic
164Q8—41no—
165D4—41no—
166D4—42no—
167D4—42no—
168D4—81no—
169D4—81no—
170D4—81no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4#99×C6711yescharacteristic
178C2×C4#102×C6711yescharacteristic
179C8—42no—
180C8—41no—
181C8—41no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22—82no—
187C22—162no—
188C22—162no—
189C22—162no—
190C22#327×C6711yescharacteristic
191C4—21no—
192C4—41no—
193C4—162no—
194C4AΣL(1,𝔽16)×C6711yescharacteristic
195C4#353×C6711yescharacteristic
196C2—21no—
197C2—41no—
198C2—81no—
199C2—162no—
200C2C67×#134311yescharacteristic
subgroupG/H
center, socleC2×C67#1343
commutatorC22×C8C2×C4×C67
Frattini, cosocle#408C22×C67
prime radicalC67×#2018C2
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C2×C4×C67 ◁ C22×C4×C67 ◁ #408×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#1391C2semidirect
3. C67×#1321C2semidirect
5. C22×D8×C67C4semidirect
10. #341×C67C4semidirect
95. #7398C67direct (N × H)
102. #1391C2×C67semidirect
103. #1321C2×C67semidirect
104. C67#7398direct (N × H)
106. C22×D8C4×C67semidirect
111. #341C4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#20182
C67×#13912
C67×#13212
#739867