Finite Group Databasebeta

← #17191976

Subgroups — #17191976

Order 4352.

506 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 167 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#14541790C211yesmaximal, characteristic, complemented
2#14531013C213yesmaximal, complemented
3#14530923C213yesmaximal, complemented
4#14530829C213yesmaximal
5Q8×#63214C211yesmaximal, characteristic, complemented
6C4×#10587498C213yesmaximal
7Q8×D8×C17C211yesmaximal, characteristic
8#11763538—21no—
9C2×#10587498C2213yes—
10#11763189C2213yes—
11#11763020—23no—
12C4×Dic68C2213yes—
13#11762636C2213yes—
14C4⋊#63214C2213yes—
15#11762612C2216yes—
16#11762612C2213yes—
17#11762610C2211yescharacteristic
18C4×#63214C2213yes—
19D4×Q8×C17—21no—
20C17×#500C2213yes—
21C17×#474—23no—
22C8×Q8×C17C2211yescharacteristic
23C4×D8×C17C2213yes—
24C22×Q8×C17—41no—
25C2×D8×C17C2311yescharacteristic
26C17×C4⋊Q8—23no—
27C17×C22⋊Q8—43no—
28C4×Q8×C17D411yescharacteristic
29C4×D4×C17—23no—
30Q8×Dic17—41no—
31C2×Dic68—23no—
32C2×Dic68C2313yes—
33#10587576—43no—
34#10587555—23no—
35C4×Dic34—23no—
36C17×#107—23no—
37C17×#107C2313yes—
38C17×C4⋊C8D413yes—
39C17×#102—23no—
40C4×C8×C17C2313yes—
41#10587498—26no—
42#10587489C2311yescharacteristic
43#10587487—23no—
44C2×#63214C2313yes—
45C2×#63214C2311yescharacteristic
46C2×Q8×C17—43no—
47C2×Q8×C17—84no—
48C2×Q8×C17D811yescharacteristic
49C2×D4×C17—21no—
50C22×C4×C17—43no—
51C2×Dic34—23no—
52D8×C17—21no—
53D8×C17#4112yes—
54C2×C8×C17C2×D413yes—
55C2×C8×C17C2411yescharacteristic
56C4⋊C4×C17—23no—
57C4⋊C4×C17—43no—
58C4⋊C4×C17D813yes—
59C22⋊C4×C17—43no—
60C42×C17C2×D413yes—
61C4⋊Dic17—21no—
62C4⋊Dic17—43no—
63C4⋊Dic17—43no—
64#63225—43no—
65C4×Dic17—43no—
66Dic68—23no—
67Dic68—23no—
68Dic68—83no—
69#63214—23no—
70#63214—21no—
71#33926—171nomaximal, Sylow
72C23×C17—41no—
73Q8×C17—86no—
74Q8×C17SD3214yescomplemented
75D4×C17—22no—
76D4×C17—41no—
77C2×C4×C17—43no—
78C2×C4×C17—86no—
79C2×C4×C17C2×D813yes—
80C2×C4×C17C2×D813yes—
81C2×C4×C17C22×D411yescharacteristic
82C2×Dic17—43no—
83C2×Dic17—41no—
84Dic34—43no—
85Dic34—43no—
86C8×C17—23no—
87C8×C17C2×#4111yescharacteristic
88C8×C172−1+411yescharacteristic
89#3395—171no—
90Q8×D8D1711yescharacteristic
91#2241—173no—
92#2225—173no—
93#2208—173no—
94Q8×C16—171no—
95C4×SD32—173no—
96C22×C17—42no—
97C22×C17—81no—
98C22×C17—81no—
99C22×C17C22×D811yescharacteristic
100C4×C17—23no—
101C4×C17—83no—
102C4×C17C2×SD3216yes—
103C4×C17#54811yescharacteristic
104C4×C17#57711yescharacteristic
105Dic17—83no—
106Dic17—81no—
107#557—341no—
108D4×Q8—21no—
109C2×SD32—173no—
110#500C2×D1713yes—
111#494—173no—
112#474—23no—
113#462—343no—
114C8×Q8C2×D1711yescharacteristic
115C4×Q16—173no—
116C4×D8C2×D1713yes—
117#367—173no—
118C4⋊C16—173no—
119#358—173no—
120#358—176no—
121#357—171no—
122C4×C16—173no—
123C2×C17—82no—
124C2×C17#335811yescharacteristic
125C2×C17C22×SD3211yescharacteristic
126C2×C17#341711yescharacteristic
127C22×Q8—41no—
128C2×Q16—173no—
129C2×Q16—343no—
130C2×D8C22×D1711yescharacteristic
131C4⋊Q8—23no—
132C4⋊Q8—343no—
133#125—683no—
134C22⋊Q8—43no—
135C4×Q8—343no—
136C4×Q8—681no—
137C4×Q8C22⋊D1711yescharacteristic
138C4×D4—23no—
139SD32—346no—
140C2×C16—171no—
141C2×C16—173no—
142#107—23no—
143#107—171no—
144#107C22×D1713yes—
145C4⋊C8C22⋊D1713yes—
146#103—343no—
147#102—23no—
148C4×C8C22×D1713yes—
149C17#3392611yescharacteristic, complemented, Sylow
150C2×Q8—43no—
151C2×Q8—84no—
152C2×Q8—343no—
153C2×Q8#6322811yescharacteristic
154C2×D4—21no—
155C22×C4—43no—
156Q16—343no—
157Q16—343no—
158Q16—1363no—
159D8—21no—
160D8#6325612yes—
161C2×C8C2×C22⋊D1713yes—
162C2×C8C23×D1711yescharacteristic
163C4⋊C4—23no—
164C4⋊C4—43no—
165C4⋊C4—341no—
166C4⋊C4—683no—
167C4⋊C4—683no—
168C4⋊C4—683no—
169C4⋊C4#6322813yes—
170C22⋊C4—43no—
171C42—683no—
172C42C2×C22⋊D1713yes—
173C16—343no—
174C16—341no—
175C23—41no—
176Q8—86no—
177Q8—683no—
178Q8—683no—
179Q8#1058749814yescomplemented
180D4—22no—
181D4—41no—
182C2×C4—43no—
183C2×C4—86no—
184C2×C4—681no—
185C2×C4—683no—
186C2×C4C2×#6322813yes—
187C2×C4C2×#6322813yes—
188C2×C4C22×C22⋊D1711yescharacteristic
189C8—23no—
190C8C2×#6325611yescharacteristic
191C8#1058769411yescharacteristic
192C22—42no—
193C22—81no—
194C22—81no—
195C22C22×#6322811yescharacteristic
196C4—23no—
197C4—83no—
198C4—1363no—
199C4—1361no—
200C4C2×#1058749816yes—
201C4#1176347011yescharacteristic
202C4#1176361711yescharacteristic
203C2—82no—
204C2#1454141911yescharacteristic
205C2C22×#1058749811yescharacteristic
206C2#1454193511yescharacteristic
subgroupG/H
centerC22C22×#63228
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingQ8×D8×C17C2
socleC22×C17C22×D8
prime radicalC2×D8×C17C23
hypercenterQ8×D8D17
hypocenterC17#33926
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
1. #14541790C2semidirect
2. #14531013C2semidirect
3. #14530923C2semidirect
5. Q8×#63214C2semidirect
74. Q8×C17SD32semidirect
149. C17#33926semidirect
179. Q8#10587498semidirect

Maximal subgroups

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

subgroupindex
#145417902
#145310132
#145309232
#145308292
Q8×#632142
C4×#105874982
Q8×D8×C172
#3392617