Finite Group Databasebeta

← #30760821

Subgroups — #30760821

Order 17152.

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

normal subgroups only

This group has 320 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#22561148C211yesmaximal, characteristic
2#22560383C211yesmaximal, characteristic
3#22558979C211yesmaximal, characteristic
4#22558639C211yesmaximal, characteristic
5#22558637C211yesmaximal, characteristic
6#22558039C211yesmaximal, characteristic
7C67×#1570C211yesmaximal, characteristic
8C2×C4×Dic134C2211yescharacteristic
9#16863449C2211yescharacteristic
10C2×#14523978—22no—
11#16863420C2211yescharacteristic
12C2×#14523954C2211yescharacteristic
13#16863150—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×Dic134Q811yescharacteristic
22C2×#11761612—21no—
23C22×C67⋊C8D412yes—
24C4×Dic134—21no—
25C4×Dic134—21no—
26C4×Dic134—21no—
27C4×Dic134—21no—
28C2×C4⋊Dic67Q811yescharacteristic
29C2×Dic67⋊C4—21no—
30C2×C4×Dic67—21no—
31#14523978—44no—
32C4⋊(C67⋊C8)—21no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)—21no—
35C22⋊(C67⋊C8)—21no—
36#14523954—21no—
37#14523954—21no—
38C4×C67⋊C8D412yes—
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—
51C2×Dic134—42no—
52C22×Dic67—21no—
53C22×Dic67—21no—
54#11761612—42no—
55C2×C67⋊C8—22no—
56C2×C67⋊C8—22no—
57C2×C67⋊C8—22no—
58C2×C67⋊C8—41no—
59C2×C67⋊C8SD1614yes—
60C4⋊Dic67—21no—
61C4⋊Dic67—21no—
62Dic67⋊C4—41no—
63Dic67⋊C4—41no—
64C4×Dic67—41no—
65C4×Dic67—41no—
66C22×C4×C67C2×D411yescharacteristic
67C22×C4×C67C2×Q811yescharacteristic
68C22×C4×C67#4111yescharacteristic
69M16×C67—42no—
70C2×C8×C67—22no—
71C2×C8×C67—41no—
72C2×C8×C67—41no—
73C2×C8×C67—42no—
74C42×C67C2×D411yescharacteristic
75C42×C67C2×D411yescharacteristic
76C42×C67#4111yescharacteristic
77C42×C67#4111yescharacteristic
78C23×C67C22⋊Q811yescharacteristic
79C2×C4×C67—41no—
80C2×C4×C67—41no—
81C2×C4×C67C22⋊Q811yescharacteristic
82C2×C4×C67C22⋊Q811yescharacteristic
83C2×C4×C67#12311yescharacteristic
84C2×C4×C67#12311yescharacteristic
85C2×C4×C67#12311yescharacteristic
86C2×C4×C67#12711yescharacteristic
87C2×C4×C67C2×SD1612yes—
88C2×C4×C67#13712yes—
89C2×Dic67—22no—
90C2×Dic67—41no—
91C2×Dic67—41no—
92C2×Dic67—42no—
93C2×Dic67—41no—
94Dic134—22no—
95Dic134—22no—
96Dic134—41no—
97Dic134—41no—
98C8×C67—42no—
99C8×C67—81no—
100C8×C67—81no—
101C67⋊C8—24no—
102C67⋊C8—42no—
103C67⋊C8—81no—
104C22×C67—21no—
105C22×C67—21no—
106C22×C67#39911yescharacteristic
107C22×C67#47512yes—
108C4×C67—21no—
109C4×C67—21no—
110C4×C67—21no—
111C4×C67—21no—
112C4×C67#48312yes—
113C4×C67#49211yescharacteristic
114C4×C67#49711yescharacteristic
115Dic67—42no—
116Dic67—81no—
117#10663—671nomaximal, Sylow
118C2×C67—22no—
119C2×C67#209111yescharacteristic
120C2×C67#209611yescharacteristic
121C2×C67#209811yescharacteristic
122#1985—671no—
123#1849—671no—
124#1570—671no—
125#1570D6711yescharacteristic
126#1487—671no—
127#1487—671no—
128#1275—671no—
129C67#1066311yescharacteristic, complemented, Sylow
130C2×C4×Q8—671no—
131#431—671no—
132#431C2×D6711yescharacteristic
133C2×#108—1342no—
134#423—671no—
135#423C2×D6711yescharacteristic
136C2×C4⋊C8C2×D6711yescharacteristic
137C2×#97—671no—
138#336—1342no—
139#334—22no—
140#334—1342no—
141Q8⋊C8—1342no—
142Q8⋊C8—1342no—
143C22×Q8—671no—
144C2×M16—21no—
145C2×M16—1341no—
146C22×C8—21no—
147C22×C8—672no—
148C4×Q8—1341no—
149C4×Q8—1341no—
150C4×Q8—1341no—
151C4×Q8—1341no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—1341no—
154C2×C42—1341no—
155C2×C42C22×D6711yescharacteristic
156#108—2684no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—1341no—
163C4⋊C8—1341no—
164C4⋊C8—1341no—
165C22⋊C8—21no—
166C22⋊C8—1341no—
167#97—1341no—
168#97—1341no—
169C4×C8—672no—
170C4×C8C22⋊D6712yes—
171C2×Q8—2682no—
172C2×Q8—1341no—
173C2×Q8—1341no—
174C22×C4—1341no—
175C22×C4—1341no—
176C22×C4D4×D6711yescharacteristic
177C22×C4Q8×D6711yescharacteristic
178C22×C4#1176162811yescharacteristic
179M16—42no—
180M16—2682no—
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—2681no—
186C2×C8—674no—
187C2×C8—1342no—
188C2×C8—1342no—
189C2×C8—1342no—
190C4⋊C4—2681no—
191C4⋊C4—2681no—
192C4⋊C4—1341no—
193C4⋊C4—1341no—
194C42—2681no—
195C42—2681no—
196C42D4×D6711yescharacteristic
197C42C2×C22⋊D6711yescharacteristic
198C42#1176162811yescharacteristic
199C42#1176162911yescharacteristic
200C23#1452402111yescharacteristic
201Q8—2681no—
202Q8—2681no—
203Q8—1342no—
204Q8—1342no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—2681no—
208C2×C4—2681no—
209C2×C4—2682no—
210C2×C4—2681no—
211C2×C4—1342no—
212C2×C4#1452401911yescharacteristic
213C2×C4#1452402111yescharacteristic
214C2×C4#1452402611yescharacteristic
215C2×C4#1452402811yescharacteristic
216C2×C4#1452402811yescharacteristic
217C2×C4#1452403911yescharacteristic
218C2×C4SD16×D6712yes—
219C2×C4#1452407212yes—
220C8—42no—
221C8—81no—
222C8—81no—
223C8—2682no—
224C8—5361no—
225C8—1344no—
226C22—21no—
227C22—21no—
228C22#1686333111yescharacteristic
229C22#1686368012yes—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—2682no—
235C4—5361no—
236C4#1686372812yes—
237C4#1686376711yescharacteristic
238C4#1686378711yescharacteristic
239C2—22no—
240C2#2256172411yescharacteristic
241C2#2256174911yescharacteristic
242C2#2256176511yescharacteristic
subgroupG/H
centerC2×C4#14524039
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1570C2
socleC22×C67#399
prime radicalC23×C67C22⋊Q8
hypercenter#1570D67
hypocenterC67#10663
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
129. C67#10663semidirect

Maximal subgroups

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

subgroupindex
#225611482
#225603832
#225589792
#225586392
#225586372
#225580392
C67×#15702
#1066367