Finite Group Databasebeta

← #30760862

Subgroups — #30760862

Order 17152.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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#22561189C211yesmaximal, characteristic
2#22560362C211yesmaximal, characteristic
3#22558981C211yesmaximal, characteristic
4#22558613C211yesmaximal, characteristic
5#22558611C211yesmaximal, characteristic
6#22558037C211yesmaximal, characteristic
7C67×#1570C211yesmaximal, characteristic
8#16863953C2211yescharacteristic
9#16863449C2211yescharacteristic
10C2×C4⋊(C67⋊C8)C2211yescharacteristic
11C2×Dic67⋊C8—22no—
12#16863242C2211yescharacteristic
13#16863157—22no—
14#16863146—22no—
15#16863128—22no—
16#16863126—22no—
17#431×C67C2211yescharacteristic
18#423×C67C2211yescharacteristic
19C2×C4⋊C8×C67C2211yescharacteristic
20#334×C67—22no—
21C22×C67⋊C8—21no—
22#14524043Q811yescharacteristic
23#14524043Q811yescharacteristic
24#14524035—21no—
25#14524022—21no—
26#14524022—21no—
27C4×Dic134—21no—
28C4×Dic134—21no—
29#14524002—21no—
30C2×C4×Dic67D412yes—
31C4⋊(C67⋊C8)—21no—
32C4⋊(C67⋊C8)—21no—
33C4⋊(C67⋊C8)—21no—
34Dic67⋊C8—44no—
35C22⋊(C67⋊C8)—21no—
36C4×C67⋊C8D412yes—
37C67⋊#95—21no—
38C67⋊#95—21no—
39C2×M16×C67—21no—
40C22×C8×C67—21no—
41C2×C42×C67C2311yescharacteristic
42C4⋊C8×C67—21no—
43C4⋊C8×C67—21no—
44C4⋊C8×C67—21no—
45C4⋊C8×C67—21no—
46C4⋊C8×C67—21no—
47C22⋊C8×C67—21no—
48C4×C8×C67D412yes—
49C2×Dic134—21no—
50C2×Dic134—21no—
51C22×Dic67—22no—
52C2×C67⋊C8—22no—
53C2×C67⋊C8—41no—
54C2×C67⋊C8—42no—
55C4⋊Dic67—21no—
56C4⋊Dic67—21no—
57C4⋊Dic67—21no—
58C4⋊Dic67—21no—
59Dic67⋊C4—41no—
60Dic67⋊C4—41no—
61C22⋊Dic67—41no—
62C4×Dic67—22no—
63C4×Dic67—41no—
64C4×Dic67SD1614yes—
65C22×C4×C67C2×D411yescharacteristic
66C22×C4×C67#4111yescharacteristic
67C22×C4×C67#4111yescharacteristic
68M16×C67—42no—
69C2×C8×C67—22no—
70C2×C8×C67—41no—
71C2×C8×C67—41no—
72C2×C8×C67—42no—
73C42×C67C2×D411yescharacteristic
74C42×C67C2×D411yescharacteristic
75C42×C67C2×Q811yescharacteristic
76C42×C67#4111yescharacteristic
77C23×C67#12311yescharacteristic
78C2×C4×C67—41no—
79C2×C4×C67—41no—
80C2×C4×C67C22⋊Q811yescharacteristic
81C2×C4×C67C22⋊Q811yescharacteristic
82C2×C4×C67C22⋊Q811yescharacteristic
83C2×C4×C67#12311yescharacteristic
84C2×C4×C67#12311yescharacteristic
85C2×C4×C67#12711yescharacteristic
86C2×C4×C67C2×SD1612yes—
87C2×C4×C67#13712yes—
88C2×Dic67—24no—
89C2×Dic67—42no—
90C2×Dic67—42no—
91C2×Dic67—41no—
92C2×Dic67—41no—
93Dic134—42no—
94Dic134—42no—
95C8×C67—42no—
96C8×C67—81no—
97C8×C67—81no—
98C67⋊C8—42no—
99C67⋊C8—81no—
100C22×C67—21no—
101C22×C67—21no—
102C22×C67#39911yescharacteristic
103C22×C67#48312yes—
104C4×C67—21no—
105C4×C67—21no—
106C4×C67—21no—
107C4×C67—21no—
108C4×C67#47512yes—
109C4×C67#49211yescharacteristic
110C4×C67#49711yescharacteristic
111Dic67—44no—
112Dic67—81no—
113Dic67—81no—
114#10670—671nomaximal, Sylow
115C2×C67—22no—
116C2×C67#209611yescharacteristic
117C2×C67#209811yescharacteristic
118C2×C67#209811yescharacteristic
119#1995—671no—
120#1843—671no—
121#1570—671no—
122#1570D6711yescharacteristic
123#1481—671no—
124#1481—671no—
125#1274—671no—
126C67#1067011yescharacteristic, complemented, Sylow
127#533—671no—
128#431—671no—
129#431C2×D6711yescharacteristic
130#423C2×D6711yescharacteristic
131C2×C4⋊C8—671no—
132C2×C4⋊C8—1342no—
133C2×C4⋊C8C2×D6711yescharacteristic
134#376—671no—
135#339—1342no—
136#334—22no—
137#334—1342no—
138Q8⋊C8—1342no—
139Q8⋊C8—1342no—
140C2×M16—21no—
141C22×C8—21no—
142C22×C8—1341no—
143C4⋊Q8—671no—
144C4⋊Q8—671no—
145#125—1341no—
146C22⋊Q8—1341no—
147C22⋊Q8—1341no—
148C4×Q8—1341no—
149C4×Q8—1341no—
150#117—1341no—
151C2×C42—672no—
152C2×C42C22×D6711yescharacteristic
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—2684no—
159C4⋊C8—1341no—
160C4⋊C8—1341no—
161C4⋊C8—1341no—
162C22⋊C8—21no—
163C22⋊C8—1341no—
164C4×C8—672no—
165C4×C8C22⋊D6712yes—
166#95—1341no—
167#95—1341no—
168C2×Q8—1341no—
169C2×Q8—1341no—
170C22×C4—1342no—
171C22×C4D4×D6711yescharacteristic
172C22×C4#1176162811yescharacteristic
173C22×C4#1176162911yescharacteristic
174M16—42no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—2681no—
180C2×C8—2682no—
181C2×C8—1342no—
182C4⋊C4—2681no—
183C4⋊C4—2681no—
184C4⋊C4—1341no—
185C4⋊C4—1341no—
186C4⋊C4—1341no—
187C4⋊C4—1341no—
188C22⋊C4—2681no—
189C42—2681no—
190C42—674no—
191C42—1342no—
192C42D4×D6711yescharacteristic
193C42C2×C22⋊D6711yescharacteristic
194C42Q8×D6711yescharacteristic
195C42#1176162811yescharacteristic
196C23#1452402811yescharacteristic
197Q8—2682no—
198Q8—2682no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—2682no—
202C2×C4—2682no—
203C2×C4—2681no—
204C2×C4—2681no—
205C2×C4—1344no—
206C2×C4#1452401911yescharacteristic
207C2×C4#1452402111yescharacteristic
208C2×C4#1452402111yescharacteristic
209C2×C4#1452402611yescharacteristic
210C2×C4#1452402811yescharacteristic
211C2×C4#1452403911yescharacteristic
212C2×C4SD16×D6712yes—
213C2×C4#1452407212yes—
214C8—42no—
215C8—81no—
216C8—81no—
217C8—2682no—
218C8—5361no—
219C22—21no—
220C22—21no—
221C22#1686333111yescharacteristic
222C22#1686372812yes—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—2684no—
228C4—5361no—
229C4—5361no—
230C4#1686368012yes—
231C4#1686376711yescharacteristic
232C4#1686378711yescharacteristic
233C2—22no—
234C2#2256174911yescharacteristic
235C2#2256176111yescharacteristic
236C2#2256176511yescharacteristic
subgroupG/H
centerC2×C4#14524039
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1570C2
socleC22×C67#399
prime radicalC23×C67#123
hypercenter#1570D67
hypocenterC67#10670
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
126. C67#10670semidirect

Maximal subgroups

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

subgroupindex
#225611892
#225603622
#225589812
#225586132
#225586112
#225580372
C67×#15702
#1067067