Finite Group Databasebeta

← #14534451

Subgroups — #14534451

Order 2176.

428 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 145 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#11762732C212yesmaximal
2#11762732C212yesmaximal
3#11762707C212yesmaximal, complemented
4#11762706C212yesmaximal, complemented
5#11762696C212yesmaximal, complemented
6#11762693C212yesmaximal, complemented
7#11762672C211yesmaximal, characteristic
8C4×C22⋊Dic17C211yesmaximal, characteristic
9C4×C22⋊C4×C17C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C2×C22⋊C4×C17C2212yes—
12C2×C42×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2211yescharacteristic
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24C2×C4×Dic17C2211yescharacteristic
25C17×#95C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C22×C4×C17—41no—
33C22×C4×C17—41no—
34C22×C4×C17Q814yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22⋊C4×C17—24no—
44C42×C17—41no—
45C42×C17—41no—
46C22⋊Dic17—24no—
47C22⋊Dic17—41no—
48C22⋊Dic17—41no—
49C4⋊Dic17—24no—
50C4⋊Dic17—42no—
51#63225—24no—
52#63225—24no—
53#63225—24no—
54#63225—42no—
55C4×Dic17—41no—
56C4×Dic17—41no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—21no—
60C23×C17—21no—
61C23×C17C2×Q812yes—
62C23×C17C2411yescharacteristic
63C2×C4×C17—22no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—44no—
73C2×C4×C17C2×Q812yes—
74C2×C4×C17C2×Q812yes—
75C2×C4×C17#4114yes—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17#4114yes—
88#2608—171nomaximal, Sylow
89C22×C17—22no—
90C22×C17—22no—
91C22×C17—22no—
92C22×C17—42no—
93C22×C17—42no—
94C22×C17C22×Q811yescharacteristic
95C22×C17C2×#4112yes—
96C22×C17C2×#4112yes—
97C22×C17H(5,𝔽2)11yescharacteristic
98C22×C17H(5,𝔽2)11yescharacteristic
99C4×C17—22no—
100C4×C17—22no—
101C4×C17—24no—
102C4×C17—42no—
103Dic17—24no—
104Dic17—42no—
105Dic17—42no—
106Dic17—42no—
107#396—172no—
108#396—172no—
109#389—172no—
110#389—172no—
111#387—172no—
112#387—172no—
113#381—171no—
114C4×C22⋊C4—171no—
115C4×C22⋊C4D1711yescharacteristic
116C2×C17—42no—
117C2×C17#53311yescharacteristic
118C2×C17#53311yescharacteristic
119C2×C17#54311yescharacteristic
120C2×C17#55312yes—
121C2×C17#55312yes—
122C23×C4C2×D1711yescharacteristic
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4—172no—
129C2×C22⋊C4—172no—
130C2×C22⋊C4—171no—
131C2×C22⋊C4—171no—
132C2×C22⋊C4C2×D1712yes—
133C2×C42—171no—
134C2×C42—171no—
135C2×C42C2×D1711yescharacteristic
136C2×C42C2×D1711yescharacteristic
137#95—172no—
138#95—172no—
139#95—172no—
140#95—172no—
141#95—172no—
142#95C2×D1712yes—
143C17#260811yescharacteristic, complemented, Sylow
144C24C22×D1711yescharacteristic
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—172no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4Dic3414yes—
152C22×C4C22×D1712yes—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1712yes—
156C4⋊C4—344no—
157C4⋊C4—344no—
158C4⋊C4—344no—
159C4⋊C4—344no—
160C4⋊C4—682no—
161C4⋊C4—682no—
162C22⋊C4—24no—
163C22⋊C4—344no—
164C22⋊C4—681no—
165C22⋊C4—681no—
166C42—41no—
167C42—41no—
168C42—681no—
169C42—681no—
170C23—22no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23C2×Dic3412yes—
175C23C23×D1711yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—44no—
186C2×C4—174no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4C2×Dic3412yes—
199C2×C4C2×Dic3412yes—
200C2×C4#6325214yes—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—42no—
205C22—42no—
206C22C22×Dic3411yescharacteristic
207C22C2×#6325212yes—
208C22#1058769111yescharacteristic
209C22#1058769111yescharacteristic
210C22#41×D1712yes—
211C4—24no—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—344no—
216C4—682no—
217C4—682no—
218C4—682no—
219C2—42no—
220C2#1176336211yescharacteristic
221C2#1176336611yescharacteristic
222C2#1176343511yescharacteristic
223C2#1176350812yes—
224C2#1176350912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC4×C22⋊C4×C17C2
prime radicalC24×C17C23
hypercenterC4×C22⋊C4D17
hypocenterC17#2608
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #11762707C2semidirect
4. #11762706C2semidirect
5. #11762696C2semidirect
6. #11762693C2semidirect
143. C17#2608semidirect

Maximal subgroups

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

subgroupindex
#117627322
#117627072
#117627062
#117626962
#117626932
#117626722
C4×C22⋊Dic172
C4×C22⋊C4×C172
#260817