Finite Group Databasebeta

← #30753363

Subgroups — #30753363

Order 17152.

436 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 99 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#22570585C211yesmaximal, characteristic
2#22558922C212yesmaximal
3#22558847C211yesmaximal, characteristic, complemented
4#22558080C212yesmaximal, complemented
5C67×#2987C211yesmaximal, characteristic, complemented
6C2×#14524040C2211yescharacteristic
7#16863762—22no—
8#16863744C414yes—
9#16863453C2212yes—
10#16863400—22no—
11C4×#11761612C2211yescharacteristic
12#16863139—24no—
13#16863138—22no—
14#533×C67C2211yescharacteristic
15#446×C67—24no—
16#433×C67—22no—
17#432×C67C2212yes—
18C22×D268—21no—
19C22×D268—22no—
20C2×#11761612—21no—
21#14524040—21no—
22#14524040—22no—
23#14524040C2×C412yes—
24C4⋊(C67⋊C8)—22no—
25#14523963—28no—
26C22⋊(C67⋊C8)—22no—
27#14523954—21no—
28#14523954—22no—
29C4×C67⋊C8C2×C412yes—
30C4⋊Q8×C67C2×C412yes—
31#125×C67D412yes—
32C22⋊Q8×C67—22no—
33C4×Q8×C67—22no—
34#117×C67D412yes—
35C2×C42×C67C2311yescharacteristic
36C4⋊C8×C67—24no—
37C4⋊C8×C67D414yes—
38C22⋊C8×C67—22no—
39#97×C67—22no—
40#97×C67—22no—
41C4×C8×C67—22no—
42C23×D67—41no—
43C2×D268—22no—
44C2×D268—24no—
45C2×D268—44no—
46C2×D268—41no—
47C2×D268—42no—
48C2×D268—84no—
49#11761612—42no—
50C2×C67⋊C8—22no—
51C2×C67⋊C8—42no—
52C2×Q8×C67—22no—
53C22×C4×C67C2×D411yescharacteristic
54C22×C4×C67#4112yes—
55C2×C8×C67—24no—
56C4⋊C4×C67—22no—
57C4⋊C4×C67—24no—
58C4⋊C4×C67C22⋊C414yes—
59C22⋊C4×C67—22no—
60C42×C67—22no—
61C42×C67C22×C411yescharacteristic
62C42×C67C2×D412yes—
63C42×C67C2×D411yescharacteristic
64C23×C67#12311yescharacteristic
65C22×D67—42no—
66C22×D67—82no—
67C22×D67—84no—
68Q8×C67—82no—
69C2×C4×C67—22no—
70C2×C4×C67—24no—
71C2×C4×C67—42no—
72C2×C4×C67C2×C22⋊C411yescharacteristic
73C2×C4×C67C4×D412yes—
74C2×C4×C67#12011yescharacteristic
75C2×C4×C67#12112yes—
76D268—44no—
77D268—84no—
78D268—84no—
79D268—82no—
80C8×C67—44no—
81C67⋊C8—42no—
82C67⋊C8—82no—
83C22×C67—21no—
84C22×C67—22no—
85C22×C67#38611yescharacteristic
86C2×D67—82no—
87C2×D67—84no—
88C2×D67—164no—
89C4×C67—21no—
90C4×C67—22no—
91C4×C67—22no—
92C4×C67—44no—
93C4×C67#45412yes—
94#9424—671nomaximal, Sylow
95C2×C67—41no—
96C2×C67#188811yescharacteristic
97C2×C67#190312yes—
98D67—162no—
99#3138—671no—
100#2987D6711yescharacteristic, complemented
101#1555—672no—
102#1537—671no—
103#1291—672no—
104C67#942411yescharacteristic, complemented, Sylow
105#533C2×D6711yescharacteristic
106C2×#127—671no—
107#490—1342no—
108#486—674no—
109#446—24no—
110#433—22no—
111#432—672no—
112#432C2×D6712yes—
113#418—1342no—
114C4×M16—671no—
115#331—1344no—
116#330—1342no—
117C22×D4—1341no—
118C22×D4—1342no—
119C2×M16—1341no—
120C4⋊Q8C4×D6712yes—
121#127—672no—
122#127—1342no—
123#127—1341no—
124#125D26812yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117C22⋊D6712yes—
128C2×C42C22×D6711yescharacteristic
129C4⋊C8—24no—
130C4⋊C8—1342no—
131C4⋊C8C22⋊D6714yes—
132#102—1348no—
133C22⋊C8—22no—
134C22⋊C8—1342no—
135#97—22no—
136#97—22no—
137#97—1342no—
138#97—1341no—
139C4×C8—22no—
140C4×C8—672no—
141C24—2681no—
142C2×Q8—22no—
143C2×D4—2684no—
144C2×D4—2681no—
145C2×D4—2682no—
146C2×D4—5364no—
147C2×D4—1342no—
148C2×D4—1344no—
149C22×C4C2×C22⋊D6711yescharacteristic
150C22×C4#1176162812yes—
151M16—2682no—
152C2×C8—24no—
153C2×C8—2682no—
154C2×C8—1342no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4(C2×D67)⋊C414yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D6711yescharacteristic
161C42C2×C22⋊D6712yes—
162C42C2×D26811yescharacteristic
163C23—2682no—
164C23—5362no—
165C23—5364no—
166C23#1452402611yescharacteristic
167Q8—82no—
168D4—2684no—
169D4—5364no—
170D4—5364no—
171D4—5362no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×(C2×D67)⋊C411yescharacteristic
176C2×C4C4×C22⋊D6712yes—
177C2×C4#1452401211yescharacteristic
178C2×C4#1452401712yes—
179C8—44no—
180C8—2682no—
181C8—5362no—
182C22—21no—
183C22—22no—
184C22—5362no—
185C22—5364no—
186C22—10724no—
187C22#1686327511yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1686356912yes—
193C2—41no—
194C2—10722no—
195C2#2256061211yescharacteristic
196C2#2256068912yes—
subgroupG/H
centerC22#16863275
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2987C2
socleC22×C67#386
prime radicalC2×#14524040C22
cosocleC2×C42×C67C23
hypercenter#2987D67
hypocenterC67#9424
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
3. #22558847C2semidirect
4. #22558080C2semidirect
5. C67×#2987C2semidirect
100. #2987D67semidirect
104. C67#9424semidirect

Maximal subgroups

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

subgroupindex
#225705852
#225589222
#225588472
#225580802
C67×#29872
#942467