Finite Group Databasebeta

← #30758935

Subgroups — #30758935

Order 17152.

400 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#22568572C211yesmaximal, characteristic
2#22561186C211yesmaximal, characteristic
3#22558774C212yesmaximal
4#22558068C212yesmaximal
5C67×#1276C211yesmaximal, characteristic
6#16863953C2211yescharacteristic
7C2×#14524035—21no—
8#16863420C2211yescharacteristic
9#16863294—24no—
10#16863266—21no—
11#16863262C2212yes—
12#16863156—21no—
13#16863156—21no—
14#16863134—24no—
15#431×C67C2212yes—
16C43×C67C2211yescharacteristic
17C2×#11761612—21no—
18#14524043Q812yes—
19#14524035—21no—
20#14524035—42no—
21#14524035—42no—
22#14524022—22no—
23C4×Dic134—22no—
24#14524002—21no—
25C2×C4⋊Dic67—24no—
26C2×C4⋊Dic67—22no—
27C2×Dic67⋊C4—24no—
28C4⋊(C67⋊C8)—22no—
29C67⋊#95—22no—
30C67⋊#95D414yes—
31C2×C42×C67—22no—
32C2×C42×C67D411yescharacteristic
33C2×C42×C67D411yescharacteristic
34C2×C42×C67C2311yescharacteristic
35C4⋊C8×C67—22no—
36C22⋊C8×C67—22no—
37C4×C8×C67—22no—
38C2×Dic134—22no—
39C22×Dic67—24no—
40#11761612—41no—
41#11761612—41no—
42C2×C67⋊C8—41no—
43C4⋊Dic67—22no—
44C4⋊Dic67—22no—
45C4⋊Dic67—48no—
46C4⋊Dic67—42no—
47C4⋊Dic67—42no—
48Dic67⋊C4—48no—
49Dic67⋊C4—42no—
50C22⋊Dic67—41no—
51C4×Dic67—41no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67C2×D412yes—
55C22×C4×C67C2×D411yescharacteristic
56C2×C8×C67—42no—
57C42×C67—21no—
58C42×C67—21no—
59C42×C67—44no—
60C42×C67SD1612yes—
61C42×C67Q1612yes—
62C42×C67C2×Q811yescharacteristic
63C42×C67#4111yescharacteristic
64C42×C67#4112yes—
65C23×C67#12011yescharacteristic
66C2×C4×C67—24no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67C22⋊Q812yes—
72C2×C4×C67C22⋊Q811yescharacteristic
73C2×C4×C67#12412yes—
74C2×C4×C67#12411yescharacteristic
75C2×C4×C67C2×SD1611yescharacteristic
76C2×C4×C67C2×Q1611yescharacteristic
77C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
78C2×C4×C67#13711yescharacteristic
79C2×Dic67—48no—
80C2×Dic67—42no—
81C2×Dic67—44no—
82Dic134—82no—
83C8×C67—82no—
84C67⋊C8—81no—
85C22×C67—22no—
86C22×C67#39311yescharacteristic
87C22×C67#45011yescharacteristic
88C22×C67#45111yescharacteristic
89C4×C67—24no—
90C4×C67—22no—
91C4×C67—22no—
92C4×C67—42no—
93C4×C67#47511yescharacteristic
94C4×C67#47711yescharacteristic
95C4×C67#48711yescharacteristic
96C4×C67#48911yescharacteristic
97Dic67—84no—
98Dic67—82no—
99#10365—671nomaximal, Sylow
100C2×C67—21no—
101C2×C67—21no—
102C2×C67#202411yescharacteristic
103C2×C67#203112yes—
104#2859—671no—
105#1994—671no—
106#1518—672no—
107#1286—672no—
108#1276D6711yescharacteristic
109C67#1036511yescharacteristic, complemented, Sylow
110#533—671no—
111C2×#125—1341no—
112#431C2×D6712yes—
113#423—671no—
114#389—1344no—
115#384—1341no—
116#382—672no—
117C43C2×D6711yescharacteristic
118#339—1341no—
119#339—1341no—
120#328—1344no—
121C2×M16—1341no—
122C4⋊Q8—672no—
123#125—2682no—
124#125—2682no—
125#125—1341no—
126C22⋊Q8—1342no—
127C4×Q8—1342no—
128#117—1341no—
129C2×C4⋊C4—1344no—
130C2×C4⋊C4—1344no—
131C2×C4⋊C4—1342no—
132C2×C42—22no—
133C2×C42D26811yescharacteristic
134C2×C42D26811yescharacteristic
135C2×C42C22×D6711yescharacteristic
136C4⋊C8—22no—
137C4⋊C8—1342no—
138C22⋊C8—22no—
139C4×C8—22no—
140#95—674no—
141#95—1342no—
142C2×Q8—1342no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1344no—
146C22×C4D4×D6712yes—
147C22×C4C2×D26811yescharacteristic
148M16—2681no—
149M16—2681no—
150C2×C8—42no—
151C2×C8—2681no—
152C4⋊C4—2688no—
153C4⋊C4—2688no—
154C4⋊C4—2682no—
155C4⋊C4—2682no—
156C4⋊C4—2682no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C22⋊C4—2681no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—2681no—
164C42#1176161712yes—
165C42Dic26812yes—
166C42Q8×D6711yescharacteristic
167C42#1176162711yescharacteristic
168C42#1176162812yes—
169C23#1452401311yescharacteristic
170Q8—5362no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—2688no—
177C2×C4—2684no—
178C2×C4—2682no—
179C2×C4#1452402112yes—
180C2×C4#1452402311yescharacteristic
181C2×C4#1452402912yes—
182C2×C4#1452403111yescharacteristic
183C2×C4C2×#1176161711yescharacteristic
184C2×C4C2×Dic26811yescharacteristic
185C2×C4#1452407111yescharacteristic
186C2×C4#1452407611yescharacteristic
187C8—82no—
188C8—5361no—
189C22—22no—
190C22#1686330911yescharacteristic
191C22#1686354911yescharacteristic
192C22#1686355411yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—5364no—
198C4—5362no—
199C4#1686368211yescharacteristic
200C4#1686369211yescharacteristic
201C4#1686374711yescharacteristic
202C4#1686375611yescharacteristic
203C2—21no—
204C2—21no—
205C2#2256135111yescharacteristic
206C2#2256138212yes—
subgroupG/H
centerC22#16863309
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1276C2
socleC22×C67#393
prime radicalC23×C67#120
hypercenter#1276D67
hypocenterC67#10365
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
109. C67#10365semidirect

Maximal subgroups

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

subgroupindex
#225685722
#225611862
#225587742
#225580682
C67×#12762
#1036567