Finite Group Databasebeta

← #30774917

Subgroups — #30774917

Order 17152.

370 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 101 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#22561928C212yesmaximal
2#22561080C211yesmaximal, characteristic
3#22558314C212yesmaximal
4#22558293C211yesmaximal, characteristic
5C67×#3162C211yesmaximal, characteristic
6C22×Dic268C2211yescharacteristic
7C2×#14523983C2212yes—
8C2×#14523975C2211yescharacteristic
9C2×#14523968—21no—
10#16863302—21no—
11#16863199—24no—
12#533×C67—21no—
13#501×C67—24no—
14#427×C67—21no—
15#427×C67C2212yes—
16C2×#97×C67C2211yescharacteristic
17C22×Dic134—21no—
18C2×Dic268—22no—
19C2×Dic268—22no—
20C2×Dic268—21no—
21C2×Dic268Q812yescomplemented
22C2×C4⋊Dic67—21no—
23#14523983—22no—
24#14523983—22no—
25C2×C67⋊C16—22no—
26#14523975—22no—
27#14523968—42no—
28C67⋊#95—42no—
29C22×C8×C67D411yescharacteristic
30C22×C8×C67C2311yescharacteristic
31C4⋊Q8×C67—22no—
32#125×C67—22no—
33C22⋊Q8×C67—42no—
34C4×Q8×C67—42no—
35#117×C67—22no—
36C2×C42×C67D411yescharacteristic
37#107×C67—22no—
38#107×C67D414yes—
39#106×C67—22no—
40#106×C67—22no—
41#97×C67—22no—
42C2×Dic134—22no—
43C2×Dic134—41no—
44C2×Dic134—41no—
45C2×Dic134—41no—
46C2×Dic134—41no—
47C2×Dic134—41no—
48C22×Dic67—41no—
49C22×Dic67—41no—
50Dic268—24no—
51Dic268—42no—
52C4⋊Dic67—41no—
53C4⋊Dic67—41no—
54C67⋊C16—42no—
55C2×Q8×C67—42no—
56C22×C4×C67D812yes—
57C22×C4×C67C2×D411yescharacteristic
58C2×C8×C67—21no—
59C2×C8×C67—21no—
60C2×C8×C67SD1612yes—
61C2×C8×C67C2×D412yes—
62C2×C8×C67C2×Q811yescharacteristic
63C2×C8×C67#4112yes—
64C2×C8×C67#4111yescharacteristic
65C4⋊C4×C67—24no—
66C4⋊C4×C67—44no—
67C4⋊C4×C67—42no—
68C22⋊C4×C67—42no—
69C42×C67—21no—
70C42×C67—21no—
71C42×C67—42no—
72C23×C67C2×D811yescharacteristic
73Q8×C67—84no—
74C2×C4×C67—22no—
75C2×C4×C67—42no—
76C2×C4×C67—44no—
77C2×C4×C67#12011yescharacteristic
78C2×C4×C67C22⋊Q811yescharacteristic
79C2×C4×C67#12411yescharacteristic
80C2×C4×C67C2×SD1611yescharacteristic
81C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C67#13711yescharacteristic
83C2×Dic67—42no—
84C2×Dic67—81no—
85C2×Dic67—81no—
86C2×Dic67—82no—
87C2×Dic67—81no—
88Dic134—42no—
89Dic134—42no—
90Dic134—81no—
91Dic134—81no—
92C8×C67—22no—
93C8×C67C22⋊Q812yes—
94C8×C67#12412yes—
95C22×C67—22no—
96C22×C67#44711yescharacteristic
97C22×C67#47411yescharacteristic
98C22×C67#48611yescharacteristic
99C4×C67—42no—
100C4×C67—84no—
101C4×C67#39311yescharacteristic
102C4×C67#44811yescharacteristic
103C4×C67#47611yescharacteristic
104C4×C67#48911yescharacteristic
105Dic67—82no—
106Dic67—161no—
107#13368—671nomaximal, Sylow
108C2×C67—21no—
109C2×C67—21no—
110C2×C67#202311yescharacteristic
111C2×C67#219112yes—
112#3162D6711yescharacteristic
113#2141—672no—
114#1971—671no—
115#1386—672no—
116#1377—671no—
117C67#1336811yescharacteristic, complemented, Sylow
118C22×Q16—671no—
119#533—21no—
120C2×M32—672no—
121#501—24no—
122#427—21no—
123#427C2×D6712yes—
124C2×#106—671no—
125C2×#103—1341no—
126C2×#97C2×D6711yescharacteristic
127#391—1341no—
128#358—1344no—
129C22×Q8—1341no—
130C2×Q16—672no—
131C2×Q16—1342no—
132C2×Q16—1341no—
133C2×Q16—1342no—
134C22×C8C22⋊D6711yescharacteristic
135C22×C8C22×D6711yescharacteristic
136C4⋊Q8—22no—
137#125—22no—
138C22⋊Q8—42no—
139C4×Q8—42no—
140#117—22no—
141C2×C4⋊C4—1341no—
142C2×C42C22⋊D6711yescharacteristic
143M32—1342no—
144M32—1342no—
145C2×C16—1342no—
146#107—22no—
147#107C22⋊D6714yes—
148#106—22no—
149#106—22no—
150#106—1342no—
151#103—2682no—
152#97—22no—
153#95—2682no—
154C2×Q8—42no—
155C2×Q8—2681no—
156C2×Q8—2681no—
157C2×Q8—2681no—
158C2×Q8—2681no—
159C2×Q8—2681no—
160C2×Q8—1342no—
161C22×C4—2681no—
162C22×C4—2681no—
163C22×C4#1176161312yes—
164C22×C4C2×C22⋊D6711yescharacteristic
165Q16—2682no—
166Q16—1344no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1176161512yes—
170C2×C8C2×C22⋊D6712yes—
171C2×C8Q8×D6711yescharacteristic
172C2×C8#1176162912yes—
173C2×C8#1176162911yescharacteristic
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—2681no—
178C4⋊C4—2681no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—42no—
183C16—2682no—
184C23C2×#1176161311yescharacteristic
185Q8—84no—
186Q8—2682no—
187Q8—2682no—
188Q8—5361no—
189Q8—5361no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—2682no—
194C2×C4—5361no—
195C2×C4—5361no—
196C2×C4—5362no—
197C2×C4—5361no—
198C2×C4#1452401211yescharacteristic
199C2×C4#1452401911yescharacteristic
200C2×C4#1452403211yescharacteristic
201C2×C4C2×#1176161511yescharacteristic
202C2×C4#1452406911yescharacteristic
203C2×C4#1452407411yescharacteristic
204C8—22no—
205C8#1452401912yes—
206C8#1452403212yes—
207C22—22no—
208C22#1686353011yescharacteristic
209C22#1686367311yescharacteristic
210C22#1686374411yescharacteristic
211C4—42no—
212C4—84no—
213C4—5362no—
214C4—10721no—
215C4#1686330711yescharacteristic
216C4#1686353511yescharacteristic
217C4#1686368311yescharacteristic
218C4#1686375811yescharacteristic
219C2—21no—
220C2—21no—
221C2#2256134211yescharacteristic
222C2#2256216012yes—
subgroupG/H
centerC22#16863530
commutator, cosocleC22×C8×C67C23
FrattiniC22×C8C22×D67
FittingC67×#3162C2
socleC22×C67#447
prime radicalC23×C67C2×D8
hypercenter#3162D67
hypocenterC67#13368
LCSG ▷ C22×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #3162

Complements

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

NH ≅ G/Ntype
21. C2×Dic268Q8semidirect
117. C67#13368semidirect

Maximal subgroups

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

subgroupindex
#225619282
#225610802
#225583142
#225582932
C67×#31622
#1336867