Finite Group Databasebeta

← #14540239

Subgroups — #14540239

Order 2176.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#11763426C212yesmaximal, complemented
2#11763103C212yesmaximal, complemented
3#11763087C212yesmaximal, complemented
4#11763053C212yesmaximal, complemented
5#11763050C211yesmaximal, characteristic, complemented
6#11763030C212yesmaximal, complemented
7#11763022C212yesmaximal, complemented
8#11762835C211yesmaximal, characteristic, complemented
9C17×#466C211yesmaximal, characteristic
10C2×D8×C17C2211yescharacteristic
11C22×C8×C17C2211yescharacteristic
12C17×#121C2212yes—
13#10587622—22no—
14#10587622C2212yes—
15#10587621—22no—
16D4×Dic17C2212yes—
17C2×#63229C2212yes—
18#10587612C2211yescharacteristic
19#10587610C2212yes—
20C2×#63223C2211yescharacteristic
21C2×Dic68C2211yescharacteristic
22#10587574C2212yes—
23#10587573C2212yes—
24#10587565—22no—
25#10587564—22no—
26C17×#107C2211yescharacteristic
27C17×#102C2212yes—
28#10587503C2212yes—
29#10587489C2211yescharacteristic
30#10587487C2212yes—
31#10587485C2212yes—
32#10587485C2212yes—
33#10587481C2212yes—
34#10587478C2212yes—
35C2×D4×C17—22no—
36C2×D4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—22no—
39C2×Dic34—22no—
40C2×Dic34C2312yes—
41D8×C17—41no—
42C2×C8×C17—41no—
43C2×C8×C17C2311yescharacteristic
44C2×C8×C17C2311yescharacteristic
45C4⋊C4×C17C2312yes—
46C22⋊C4×C17—22no—
47C22⋊Dic17—22no—
48C22⋊Dic17—22no—
49C22⋊Dic17—22no—
50C22⋊Dic17—22no—
51C22⋊Dic17D412yes—
52#63229—42no—
53C4⋊Dic17D411yescharacteristic
54C4⋊Dic17D411yescharacteristic
55C4⋊Dic17C2312yes—
56#63225—22no—
57#63225—22no—
58#63225—22no—
59C4×Dic17—22no—
60C4×Dic17C2312yes—
61#63223—41no—
62C2×#3622C2312yes—
63Dic68—41no—
64C23×C17—22no—
65C23×C17C2×D411yescharacteristic
66D4×C17—22no—
67D4×C17—22no—
68D4×C17—82no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—22no—
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2411yescharacteristic
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—82no—
77C2×Dic17C2×D412yes—
78C2×Dic17C2×D412yes—
79Dic34—22no—
80Dic34—22no—
81Dic34—82no—
82C8×C17—21no—
83C8×C17—21no—
84#3622—22no—
85#3262—171nomaximal, Sylow
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17—42no—
92C22×C17C22×D411yescharacteristic
93C4×C17—21no—
94C4×C17—42no—
95C4×C17H(5,𝔽2)11yescharacteristic
96C4×C172−1+411yescharacteristic
97Dic17—22no—
98Dic17—22no—
99Dic17—42no—
100Dic17—42no—
101#542—172no—
102#477—172no—
103#474—172no—
104#468—172no—
105#467—171no—
106#466D1711yescharacteristic
107#464—172no—
108#463—172no—
109#424—171no—
110C2×C17—41no—
111C2×C17—82no—
112C2×C17#53711yescharacteristic
113C2×C17#57611yescharacteristic
114C2×C17#57811yescharacteristic
115C2×Q16—171no—
116C2×SD16—172no—
117C2×D8C2×D1711yescharacteristic
118C2×M16—171no—
119C22×C8C2×D1711yescharacteristic
120C4⋊Q8—172no—
121#126—342no—
122#124—172no—
123#124—342no—
124#123—342no—
125C22⋊Q8—172no—
126C22⋊Q8—342no—
127#121C2×D1712yes—
128C4×Q8—172no—
129C4×D4—172no—
130#117—171no—
131#107—171no—
132#107—172no—
133#107C2×D1711yescharacteristic
134C4⋊C8—172no—
135C4⋊C8—172no—
136#103—172no—
137#103—172no—
138#102—172no—
139#102C2×D1712yes—
140C17#326211yescharacteristic, complemented, Sylow
141C2×Q8—172no—
142C2×Q8—342no—
143C2×D4—22no—
144C2×D4C22×D1712yes—
145C22×C4—342no—
146C22×C4C22×D1711yescharacteristic
147Q16—681no—
148SD16—682no—
149D8—41no—
150M16—681no—
151C2×C8—41no—
152C2×C8—172no—
153C2×C8C22×D1711yescharacteristic
154C2×C8C22×D1711yescharacteristic
155C4⋊C4—171no—
156C4⋊C4—171no—
157C4⋊C4—172no—
158C4⋊C4—342no—
159C4⋊C4—342no—
160C4⋊C4—342no—
161C4⋊C4C22×D1712yes—
162C22⋊C4—22no—
163C22⋊C4—172no—
164C22⋊C4—342no—
165C22⋊C4—342no—
166C22⋊C4—342no—
167C22⋊C4—342no—
168C42—172no—
169C42—342no—
170C23—22no—
171C23D4×D1711yescharacteristic
172Q8—342no—
173Q8—342no—
174Q8—1362no—
175D4—22no—
176D4—22no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—172no—
182C2×C4—172no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—1362no—
186C2×C4D4×D1711yescharacteristic
187C2×C4C23×D1711yescharacteristic
188C8—21no—
189C8—21no—
190C8—342no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22C2×D4×D1711yescharacteristic
198C4—21no—
199C4—42no—
200C4—342no—
201C4—342no—
202C4—682no—
203C4—682no—
204C4#1058769111yescharacteristic
205C4#1058769711yescharacteristic
206C2—41no—
207C2—82no—
208C2#1176338611yescharacteristic
209C2#1176361011yescharacteristic
210C2#1176362311yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#466C2
socleC22×C17C22×D4
hypercenter#466D17
hypocenterC17#3262
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #11763426C2semidirect
2. #11763103C2semidirect
3. #11763087C2semidirect
4. #11763053C2semidirect
5. #11763050C2semidirect
6. #11763030C2semidirect
7. #11763022C2semidirect
8. #11762835C2semidirect
140. C17#3262semidirect

Maximal subgroups

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

subgroupindex
#117634262
#117631032
#117630872
#117630532
#117630502
#117630302
#117630222
#117628352
C17×#4662
#326217