Finite Group Databasebeta

← #30760731

Subgroups — #30760731

Order 17152.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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#22558937C211yesmaximal, characteristic
2#22558907C211yesmaximal, characteristic
3#22558683C211yesmaximal, characteristic
4#22558120C211yesmaximal, characteristic
5#22558079C211yesmaximal, characteristic
6#22558071C211yesmaximal, characteristic
7C67×#1570C211yesmaximal, characteristic
8#16863953C2211yescharacteristic
9#16863453C2211yescharacteristic
10#16863453C2211yescharacteristic
11#16863453C2211yescharacteristic
12#16863144—21no—
13#16863143—21no—
14#16863143—21no—
15#16863141—21no—
16#16863128—22no—
17#431×C67C2211yescharacteristic
18#423×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20#334×C67—22no—
21#14524043D411yescharacteristic
22#14524043D411yescharacteristic
23#14524035—21no—
24#14524022—21no—
25#14524022—21no—
26C4×Dic134—21no—
27C4×Dic134—21no—
28#14524002—21no—
29C4⋊(C67⋊C8)—21no—
30C4⋊(C67⋊C8)—21no—
31C4⋊(C67⋊C8)—21no—
32C22⋊(C67⋊C8)—21no—
33C22⋊(C67⋊C8)—21no—
34C22⋊(C67⋊C8)—21no—
35#14523954—21no—
36#14523954—21no—
37#14523954—21no—
38C2×M16×C67—21no—
39C22×C8×C67—21no—
40C2×C42×C67C2311yescharacteristic
41C4⋊C8×C67—21no—
42C4⋊C8×C67—21no—
43C4⋊C8×C67—21no—
44C4⋊C8×C67—21no—
45C4⋊C8×C67—21no—
46C22⋊C8×C67—21no—
47C4×C8×C67D412yes—
48C2×Dic134—21no—
49C2×Dic134—21no—
50C2×C67⋊C8—41no—
51C2×C67⋊C8—41no—
52C2×C67⋊C8—41no—
53C4⋊Dic67—21no—
54C4⋊Dic67—21no—
55C4⋊Dic67—21no—
56C4⋊Dic67—21no—
57Dic67⋊C4—41no—
58Dic67⋊C4—41no—
59C22⋊Dic67—41no—
60C4×Dic67—41no—
61C22×C4×C67#4111yescharacteristic
62C22×C4×C67#4111yescharacteristic
63C22×C4×C67#4111yescharacteristic
64M16×C67—81no—
65C2×C8×C67—22no—
66C2×C8×C67—41no—
67C2×C8×C67—41no—
68C2×C8×C67—81no—
69C42×C67C2×D411yescharacteristic
70C42×C67C2×D411yescharacteristic
71C42×C67#4111yescharacteristic
72C42×C67#4111yescharacteristic
73C23×C67#12611yescharacteristic
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—41no—
77C2×C4×C67—41no—
78C2×C4×C67#12111yescharacteristic
79C2×C4×C67#12311yescharacteristic
80C2×C4×C67#12311yescharacteristic
81C2×C4×C67#12311yescharacteristic
82C2×C4×C67#12411yescharacteristic
83C2×C4×C67#12611yescharacteristic
84C2×Dic67—41no—
85C2×Dic67—41no—
86Dic134—41no—
87Dic134—41no—
88Dic134—81no—
89C8×C67—42no—
90C8×C67—81no—
91C8×C67—81no—
92C67⋊C8—81no—
93C67⋊C8—81no—
94C67⋊C8—81no—
95C22×C67—21no—
96C22×C67—21no—
97C22×C67—21no—
98C22×C67#39711yescharacteristic
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—21no—
102C4×C67—21no—
103C4×C67—21no—
104C4×C67#47111yescharacteristic
105C4×C67#47111yescharacteristic
106Dic67—81no—
107Dic67—81no—
108#10648—671nomaximal, Sylow
109C2×C67—41no—
110C2×C67#207911yescharacteristic
111C2×C67#208011yescharacteristic
112C2×C67#208211yescharacteristic
113#1570D6711yescharacteristic
114#1559—671no—
115#1552—671no—
116#1497—671no—
117#1307—671no—
118#1290—671no—
119#1287—671no—
120C67#1064811yescharacteristic, complemented, Sylow
121#533—671no—
122#432—671no—
123#432—671no—
124#432—671no—
125#431C2×D6711yescharacteristic
126#423C2×D6711yescharacteristic
127C2×C4⋊C8C2×D6711yescharacteristic
128#334—22no—
129#333—1341no—
130#332—1341no—
131#332—1341no—
132#332—1341no—
133Q8⋊C8—1342no—
134C2×M16—21no—
135C22×C8—21no—
136C4⋊Q8—671no—
137C4⋊Q8—671no—
138#125—1341no—
139C22⋊Q8—1341no—
140C22⋊Q8—1341no—
141C4×Q8—1341no—
142C4×Q8—1341no—
143#117—1341no—
144C2×C42C22×D6711yescharacteristic
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—1341no—
151C4⋊C8—1341no—
152C4⋊C8—1341no—
153C22⋊C8—21no—
154C22⋊C8—1341no—
155C22⋊C8—1341no—
156C22⋊C8—1341no—
157#97—1341no—
158#97—1341no—
159#97—1341no—
160C4×C8C22⋊D6712yes—
161C2×Q8—1341no—
162C2×Q8—1341no—
163C22×C4#1176162711yescharacteristic
164C22×C4#1176162811yescharacteristic
165C22×C4#1176162911yescharacteristic
166M16—81no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—81no—
171C2×C8—2681no—
172C2×C8—2681no—
173C2×C8—2681no—
174C4⋊C4—2681no—
175C4⋊C4—2681no—
176C4⋊C4—1341no—
177C4⋊C4—1341no—
178C4⋊C4—1341no—
179C4⋊C4—1341no—
180C22⋊C4—2681no—
181C42—2681no—
182C42D4×D6711yescharacteristic
183C42C2×C22⋊D6711yescharacteristic
184C42#1176162811yescharacteristic
185C42#1176162911yescharacteristic
186C23#1452403811yescharacteristic
187Q8—2681no—
188Q8—2681no—
189Q8—5361no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—2681no—
195C2×C4—2681no—
196C2×C4#1452401411yescharacteristic
197C2×C4#1452402611yescharacteristic
198C2×C4#1452402811yescharacteristic
199C2×C4#1452402811yescharacteristic
200C2×C4#1452403211yescharacteristic
201C2×C4#1452403811yescharacteristic
202C8—42no—
203C8—81no—
204C8—81no—
205C8—5361no—
206C8—5361no—
207C8—5361no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22#1686332211yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—5361no—
218C4—5361no—
219C4#1686365611yescharacteristic
220C4#1686365611yescharacteristic
221C2—41no—
222C2#2256165511yescharacteristic
223C2#2256166011yescharacteristic
224C2#2256167411yescharacteristic
subgroupG/H
centerC22#16863322
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1570C2
socleC22×C67#397
prime radicalC23×C67#126
hypercenter#1570D67
hypocenterC67#10648
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
120. C67#10648semidirect

Maximal subgroups

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

subgroupindex
#225589372
#225589072
#225586832
#225581202
#225580792
#225580712
C67×#15702
#1064867