Finite Group Databasebeta

← #30774921

Subgroups — #30774921

Order 17152.

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

normal subgroups only

This group has 370 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#22570733C211yesmaximal, characteristic
2#22561930C212yesmaximal
3#22561080C211yesmaximal, characteristic
4#22558315C212yesmaximal
5C67×#1377C211yesmaximal, characteristic
6C22×Dic268C2211yescharacteristic
7#16863953—21no—
8#16863805—24no—
9#16863434—21no—
10#16863434C2212yes—
11C2×#14523975C2211yescharacteristic
12C2×#14523968—21no—
13#16863302—21no—
14#16863201—24no—
15C2×M32×C67C2212yes—
16C2×#97×C67C2211yescharacteristic
17C22×Dic134—21no—
18C2×Dic268—22no—
19C2×Dic268—22no—
20C2×Dic268—21no—
21C2×Dic268Q812yescomplemented
22#14524043—22no—
23#14524035—22no—
24#14524022—42no—
25C4×Dic134—42no—
26#14524002—22no—
27C2×C4⋊Dic67—21no—
28#14523977—22no—
29#14523977D414yes—
30#14523975—22no—
31#14523975—22no—
32#14523975—22no—
33#14523968—42no—
34C67⋊#95—42no—
35C22×C8×C67D411yescharacteristic
36C22×C8×C67C2311yescharacteristic
37C2×C42×C67D411yescharacteristic
38M32×C67—22no—
39M32×C67—22no—
40C2×C16×C67—22no—
41#97×C67—22no—
42C2×Dic134—22no—
43C2×Dic134—41no—
44C2×Dic134—41no—
45C2×Dic134—41no—
46C2×Dic134—41no—
47C2×Dic134—42no—
48C2×Dic134—41no—
49C22×Dic67—41no—
50C22×Dic67—41no—
51Dic268—24no—
52Dic268—42no—
53C4⋊Dic67—24no—
54C4⋊Dic67—42no—
55C4⋊Dic67—41no—
56C4⋊Dic67—41no—
57Dic67⋊C4—44no—
58C22⋊Dic67—42no—
59C4×Dic67—42no—
60C22×C4×C67D812yes—
61C22×C4×C67C2×D411yescharacteristic
62C2×C8×C67—21no—
63C2×C8×C67—21no—
64C2×C8×C67SD1612yes—
65C2×C8×C67C2×D412yes—
66C2×C8×C67C2×Q811yescharacteristic
67C2×C8×C67#4111yescharacteristic
68C2×C8×C67#4112yes—
69C42×C67—21no—
70C42×C67—21no—
71C16×C67—42no—
72C23×C67C2×D811yescharacteristic
73C2×C4×C67—22no—
74C2×C4×C67—42no—
75C2×C4×C67#12011yescharacteristic
76C2×C4×C67C22⋊Q811yescharacteristic
77C2×C4×C67#12411yescharacteristic
78C2×C4×C67C2×SD1611yescharacteristic
79C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C67#13711yescharacteristic
81C2×Dic67—44no—
82C2×Dic67—42no—
83C2×Dic67—81no—
84C2×Dic67—81no—
85C2×Dic67—82no—
86C2×Dic67—81no—
87Dic134—42no—
88Dic134—42no—
89Dic134—84no—
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#39311yescharacteristic
101C4×C67#44811yescharacteristic
102C4×C67#47611yescharacteristic
103C4×C67#48911yescharacteristic
104Dic67—84no—
105Dic67—82no—
106Dic67—161no—
107#13368—671nomaximal, Sylow
108C2×C67—21no—
109C2×C67—21no—
110C2×C67#202311yescharacteristic
111C2×C67#219112yes—
112#3162—671no—
113#2141—672no—
114#1971—671no—
115#1386—672no—
116#1377D6711yescharacteristic
117C67#1336811yescharacteristic, complemented, Sylow
118C22×Q16—671no—
119#533—1341no—
120C2×M32C2×D6712yes—
121#501—1344no—
122#427—672no—
123#427—1341no—
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×C8D26811yescharacteristic
135C22×C8C22×D6711yescharacteristic
136C4⋊Q8—1342no—
137#125—1342no—
138C22⋊Q8—2682no—
139C4×Q8—2682no—
140#117—1342no—
141C2×C4⋊C4—1341no—
142C2×C42D26811yescharacteristic
143M32—22no—
144M32—22no—
145C2×C16—22no—
146#107—674no—
147#107—1342no—
148#106—1342no—
149#106—1342no—
150#106—1342no—
151#103—2682no—
152#97—22no—
153#95—2682no—
154C2×Q8—2682no—
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×C4D53612yes—
164C22×C4C2×D26811yescharacteristic
165Q16—2682no—
166Q16—1344no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1176161712yes—
170C2×C8D4×D6712yes—
171C2×C8Q8×D6711yescharacteristic
172C2×C8#1176162711yescharacteristic
173C2×C8#1176162812yes—
174C4⋊C4—2684no—
175C4⋊C4—2682no—
176C4⋊C4—2681no—
177C4⋊C4—2681no—
178C4⋊C4—1344no—
179C22⋊C4—2682no—
180C42—21no—
181C42—21no—
182C42—2682no—
183C16—42no—
184C23C2×D53611yescharacteristic
185Q8—2682no—
186Q8—2682no—
187Q8—5364no—
188Q8—5361no—
189Q8—5361no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—2684no—
193C2×C4—2682no—
194C2×C4—5361no—
195C2×C4—5361no—
196C2×C4—5362no—
197C2×C4—5361no—
198C2×C4#1452401311yescharacteristic
199C2×C4#1452402311yescharacteristic
200C2×C4#1452403111yescharacteristic
201C2×C4C2×#1176161711yescharacteristic
202C2×C4#1452407111yescharacteristic
203C2×C4#1452407611yescharacteristic
204C8—22no—
205C8#1452402112yes—
206C8#1452402912yes—
207C22—22no—
208C22#1686353211yescharacteristic
209C22#1686367711yescharacteristic
210C22#1686374311yescharacteristic
211C4—42no—
212C4—5362no—
213C4—5364no—
214C4—10721no—
215C4#1686330911yescharacteristic
216C4#1686353711yescharacteristic
217C4#1686368711yescharacteristic
218C4#1686375611yescharacteristic
219C2—21no—
220C2—21no—
221C2#2256134611yescharacteristic
222C2#2256216212yes—
subgroupG/H
centerC22#16863532
commutator, cosocleC22×C8×C67C23
FrattiniC22×C8C22×D67
FittingC67×#1377C2
socleC22×C67#447
prime radicalC23×C67C2×D8
hypercenter#1377D67
hypocenterC67#13368
LCSG ▷ C22×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

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
#225707332
#225619302
#225610802
#225583152
C67×#13772
#1336867