Finite Group Databasebeta

← #17191965

Subgroups — #17191965

Order 4352.

562 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 165 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
1Q8×D136C211yesmaximal, characteristic, complemented
2#14531007C213yesmaximal, complemented
3#14530921C213yesmaximal, complemented
4#14530816C213yesmaximal, complemented
5Q8×#63214C211yesmaximal, characteristic, complemented
6C4×#10587497C213yesmaximal, complemented
7C17×#3391C211yesmaximal, characteristic, complemented
8Q8×D68—21no—
9C2×#10587497C2213yes—
10#11763214C2213yes—
11#11763090—23no—
12C4×D136C2213yes—
13#11762634C2213yes—
14C4⋊#63214C2213yes—
15#11762610C2216yes—
16#11762610C2213yes—
17#11762609C2211yescharacteristic
18C4×#63214C2213yes—
19C17×#550—21no—
20C17×#493C2213yes—
21C17×#459—23no—
22C8×Q8×C17C2211yescharacteristic
23C4×D8×C17C2213yes—
24C2×#41×C17—41no—
25C2×Q8×D17—41no—
26C2×D8×C17—23no—
27C2×D8×C17C2313yes—
28C17×#127—23no—
29C17×#121—43no—
30C4×Q8×C17D411yescharacteristic
31C4×D4×C17—23no—
32C2×D136C2311yescharacteristic
33#10587583—43no—
34C4×D68—23no—
35#10587555—23no—
36C17×#107C2311yescharacteristic
37C17×C4⋊C8D413yes—
38C17×#102—23no—
39C4×C8×C17C2313yes—
40#10587497—26no—
41#10587492—23no—
42#10587489—23no—
43#10587489C2313yes—
44C2×#63214C2313yes—
45C2×#63214C2311yescharacteristic
46#41×C17—84no—
47C2×Q8×C17D811yescharacteristic
48C2×D4×C17—23no—
49C2×D4×C17—43no—
50C2×D4×C17—43no—
51C22×C4×C17—43no—
52Q8×D17—84no—
53C2×D68—21no—
54C2×C4×D17—43no—
55C2×Dic34—43no—
56D8×C17—23no—
57D8×C17—23no—
58D8×C17—83no—
59C2×C8×C17C2×D413yes—
60C2×C8×C17C2411yescharacteristic
61C4⋊C4×C17—21no—
62C4⋊C4×C17D813yes—
63C22⋊C4×C17—43no—
64C42×C17C2×D413yes—
65#63227—43no—
66C4⋊Dic17—23no—
67C4⋊Dic17—43no—
68D136—21no—
69D136#4112yescomplemented
70#63214—23no—
71#63214—21no—
72#33925—171nomaximal, Sylow
73C23×C17—43no—
74C22×D17—41no—
75Q8×C17D1614yescomplemented
76D4×C17—43no—
77D4×C17—43no—
78D4×C17—86no—
79D4×C17—163no—
80C2×C4×C17—41no—
81C2×C4×C17—86no—
82C2×C4×C17C2×D813yes—
83C2×C4×C17C2×D813yes—
84C2×C4×C17C22×D411yescharacteristic
85C2×Dic17—43no—
86D68—22no—
87D68—41no—
88C4×D17—86no—
89Dic34—86no—
90C8×C17—23no—
91C8×C17C2×#4111yescharacteristic
92C8×C172−1+411yescharacteristic
93#3391D1711yescharacteristic, complemented
94Q8×D8—171no—
95#2240—173no—
96#2224—173no—
97#2206—173no—
98Q8×C16—171no—
99C4×D16—173no—
100C22×C17—83no—
101C22×C17—83no—
102C22×C17—83no—
103C22×C17C22×D811yescharacteristic
104C2×D17—42no—
105C2×D17—81no—
106C2×D17—81no—
107C4×C17—23no—
108C4×C17—81no—
109C4×C17C2×D1616yes—
110C4×C17#54811yescharacteristic
111C4×C17#57711yescharacteristic
112Dic17—83no—
113#550—21no—
114D4×Q8—341no—
115C2×D16—173no—
116#500—173no—
117#493C2×D1713yes—
118#474—343no—
119#459—23no—
120C8×Q8C2×D1711yescharacteristic
121C4×D8—173no—
122C4×D8C2×D1713yes—
123#366—173no—
124C4⋊C16—173no—
125#357—173no—
126#357—176no—
127#357—171no—
128C4×C16—173no—
129C2×C17—163no—
130C2×C17#335811yescharacteristic
131C2×C17C22×D1611yescharacteristic
132C2×C17#341611yescharacteristic
133D17—82no—
134C2×#41—41no—
135C22×Q8—681no—
136C2×D8—23no—
137C2×D8—171no—
138C2×D8C22×D1713yes—
139C4⋊Q8—343no—
140#127—23no—
141C22⋊Q8—683no—
142#121—43no—
143C4×Q8C22⋊D1711yescharacteristic
144C4×D4—23no—
145C4×D4—343no—
146D16—346no—
147C2×C16—171no—
148C2×C16—173no—
149#107—173no—
150#107—343no—
151#107C22×D1711yescharacteristic
152C4⋊C8C22⋊D1713yes—
153#102—23no—
154#102—343no—
155C4×C8C22×D1713yes—
156C17#3392511yescharacteristic, complemented, Sylow
157#41—84no—
158C2×Q8—683no—
159C2×Q8—1364no—
160C2×Q8#6322811yescharacteristic
161C2×D4—23no—
162C2×D4—43no—
163C2×D4—43no—
164C2×D4—341no—
165C22×C4—43no—
166C22×C4—683no—
167D8—23no—
168D8—23no—
169D8—83no—
170D8—172no—
171D8—341no—
172C2×C8C2×C22⋊D1713yes—
173C2×C8C23×D1711yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—343no—
176C4⋊C4—683no—
177C4⋊C4#6322813yes—
178C22⋊C4—43no—
179C22⋊C4—683no—
180C42C2×C22⋊D1713yes—
181C16—343no—
182C16—341no—
183C23—43no—
184C23—681no—
185Q8—1366no—
186Q8#1058749714yescomplemented
187D4—43no—
188D4—43no—
189D4—86no—
190D4—163no—
191D4—342no—
192D4—681no—
193C2×C4—41no—
194C2×C4—86no—
195C2×C4—683no—
196C2×C4—1366no—
197C2×C4C2×#6322813yes—
198C2×C4C2×#6322813yes—
199C2×C4C22×C22⋊D1711yescharacteristic
200C8—23no—
201C8#41×D1711yescharacteristic
202C8#1058769711yescharacteristic
203C22—83no—
204C22—83no—
205C22—83no—
206C22—682no—
207C22—1361no—
208C22—1361no—
209C22C22×#6322811yescharacteristic
210C4—23no—
211C4—81no—
212C4—1363no—
213C4C2×#1058749716yes—
214C4#1176346611yescharacteristic
215C4#1176362011yescharacteristic
216C2—163no—
217C2—1362no—
218C2#1454141511yescharacteristic
219C2C22×#1058749711yescharacteristic
220C2#1454193111yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#3391C2
socleC22×C17C22×D8
hypercenter#3391D17
hypocenterC17#33925
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3391

Complements

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

NH ≅ G/Ntype
1. Q8×D136C2semidirect
2. #14531007C2semidirect
3. #14530921C2semidirect
4. #14530816C2semidirect
5. Q8×#63214C2semidirect
6. C4×#10587497C2semidirect
7. C17×#3391C2semidirect
69. D136#41semidirect
75. Q8×C17D16semidirect
93. #3391D17semidirect
156. C17#33925semidirect
186. Q8#10587497semidirect

Maximal subgroups

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

subgroupindex
Q8×D1362
#145310072
#145309212
#145308162
Q8×#632142
C4×#105874972
C17×#33912
#3392517