Finite Group Databasebeta

← #14540130

Subgroups — #14540130

Order 2176.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763340C212yesmaximal, complemented
2#11763115C212yesmaximal, complemented
3#11763107C212yesmaximal, complemented
4#11763050C211yesmaximal, characteristic, complemented
5#11763036C212yesmaximal, complemented
6#11763030C212yesmaximal, complemented
7#11763022C212yesmaximal, complemented
8C2×#10587485C211yesmaximal, characteristic, complemented
9C17×#466C211yesmaximal, characteristic
10C2×D8×C17C2211yescharacteristic
11C22×C8×C17C2211yescharacteristic
12C17×#121C2212yes—
13#10587622C2212yes—
14#10587621—22no—
15D4×Dic17—22no—
16D4×Dic17C2212yes—
17C2×#63229C2212yes—
18#10587610C2212yes—
19C2×C4×Dic17C2211yescharacteristic
20C22×#3622C2211yescharacteristic
21C2×Dic68C2211yescharacteristic
22#10587576C2212yes—
23#10587573C2212yes—
24#10587565—22no—
25#10587563—22no—
26C17×#107C2211yescharacteristic
27C17×#102C2212yes—
28#10587503C2212yes—
29#10587489C2211yescharacteristic
30#10587487C2212yes—
31#10587485C2212yes—
32#10587485C2212yes—
33#10587481C2212yes—
34#10587479C2212yes—
35C2×D4×C17—22no—
36C2×D4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—22no—
39C22×Dic17D411yescharacteristic
40C22×Dic17D411yescharacteristic
41C2×Dic34C2312yes—
42D8×C17—41no—
43C2×C8×C17—41no—
44C2×C8×C17C2311yescharacteristic
45C2×C8×C17C2311yescharacteristic
46C4⋊C4×C17C2312yes—
47C22⋊C4×C17—22no—
48C22⋊Dic17—22no—
49C22⋊Dic17—22no—
50C22⋊Dic17—22no—
51C22⋊Dic17—22no—
52#63229—42no—
53C4⋊Dic17—22no—
54C4⋊Dic17C2312yes—
55#63225—22no—
56#63225—22no—
57#63225—22no—
58C4×Dic17—22no—
59C4×Dic17D412yes—
60C4×Dic17C2312yes—
61C2×#3622—41no—
62C2×#3622C2312yes—
63Dic68—41no—
64C23×C17—22no—
65C23×C17C2×D411yescharacteristic
66D4×C17—22no—
67D4×C17—22no—
68D4×C17—44no—
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—22no—
77C2×Dic17—41no—
78C2×Dic17—44no—
79C2×Dic17C2×D412yes—
80C2×Dic17C2×D412yes—
81Dic34—22no—
82Dic34—22no—
83C8×C17—21no—
84C8×C17—21no—
85#3622—22no—
86#3250—171nomaximal, Sylow
87C22×C17—21no—
88C22×C17—21no—
89C22×C17—42no—
90C22×C17—42no—
91C22×C17—42no—
92C22×C17#13512yes—
93C22×C17C22×D411yescharacteristic
94C4×C17—21no—
95C4×C17—42no—
96C4×C17H(5,𝔽2)11yescharacteristic
97C4×C172−1+411yescharacteristic
98Dic17—22no—
99Dic17—22no—
100Dic17—42no—
101Dic17—42no—
102#529—172no—
103#479—172no—
104#478—172no—
105#467—171no—
106#466D1711yescharacteristic
107#465—172no—
108#464—172no—
109#463—172no—
110C2×C4⋊C8—171no—
111C2×C17—22no—
112C2×C17—82no—
113C2×C17#53711yescharacteristic
114C2×C17C2×#13511yescharacteristic
115C2×C17#57511yescharacteristic
116C2×Q16—171no—
117C2×SD16—172no—
118C2×D8C2×D1711yescharacteristic
119C22×C8—171no—
120C22×C8C2×D1711yescharacteristic
121#126—342no—
122#125—172no—
123#124—172no—
124#123—342no—
125C22⋊Q8—172no—
126#121C2×D1712yes—
127C4×Q8—172no—
128C4×D4—172no—
129C4×D4—342no—
130#117—342no—
131C2×C42—171no—
132#107—171no—
133#107C2×D1711yescharacteristic
134#106—172no—
135C4⋊C8—172no—
136C4⋊C8—172no—
137#103—172no—
138#103—172no—
139#102—172no—
140#102C2×D1712yes—
141C17#325011yescharacteristic, complemented, Sylow
142C2×Q8—172no—
143C2×D4—22no—
144C2×D4C22×D1712yes—
145C22×C4—171no—
146C22×C4—171no—
147C22×C4—342no—
148C22×C4C22×D1711yescharacteristic
149Q16—681no—
150SD16—682no—
151D8—41no—
152C2×C8—41no—
153C2×C8—172no—
154C2×C8—681no—
155C2×C8C22×D1711yescharacteristic
156C2×C8C22×D1711yescharacteristic
157C4⋊C4—172no—
158C4⋊C4—342no—
159C4⋊C4—342no—
160C4⋊C4—342no—
161C4⋊C4—342no—
162C4⋊C4C22×D1712yes—
163C22⋊C4—22no—
164C22⋊C4—342no—
165C22⋊C4—342no—
166C22⋊C4—342no—
167C22⋊C4—342no—
168C42—172no—
169C42—172no—
170C42—342no—
171C23—22no—
172C23D4×D1711yescharacteristic
173Q8—342no—
174Q8—342no—
175D4—22no—
176D4—22no—
177D4—44no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—172no—
182C2×C4—172no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—681no—
187C2×C4—684no—
188C2×C4D4×D1711yescharacteristic
189C2×C4C23×D1711yescharacteristic
190C8—21no—
191C8—21no—
192C8—342no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22#1058759812yes—
199C22C2×D4×D1711yescharacteristic
200C4—21no—
201C4—42no—
202C4—342no—
203C4—342no—
204C4—682no—
205C4—682no—
206C4#1058769111yescharacteristic
207C4#1058769711yescharacteristic
208C2—22no—
209C2—82no—
210C2#1176338611yescharacteristic
211C2C2×#1058759811yescharacteristic
212C2#1176360611yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#466C2
socleC22×C17C22×D4
hypercenter#466D17
hypocenterC17#3250
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. #11763340C2semidirect
2. #11763115C2semidirect
3. #11763107C2semidirect
4. #11763050C2semidirect
5. #11763036C2semidirect
6. #11763030C2semidirect
7. #11763022C2semidirect
8. C2×#10587485C2semidirect
141. C17#3250semidirect

Maximal subgroups

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

subgroupindex
#117633402
#117631152
#117631072
#117630502
#117630362
#117630302
#117630222
C2×#105874852
C17×#4662
#325017