Finite Group Databasebeta

← #14535157

Subgroups — #14535157

Order 2176.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#10587610C212yesmaximal, complemented
2C2×#10587564C212yesmaximal, complemented
3#11762732C212yesmaximal, complemented
4#11762732C212yesmaximal, complemented
5#11762730C212yesmaximal, complemented
6#11762712C211yesmaximal, characteristic
7#11762706C212yesmaximal, complemented
8#11762689C211yesmaximal, characteristic
9C17×#386C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C23×Dic17C2211yescharacteristic
13C22×Dic34C2212yes—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23#10587610—28no—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587564—28no—
30C17×#95C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33C24×C17C2311yescharacteristic
34C24×C17C2311yescharacteristic
35C2×D4×C17—41no—
36C2×D4×C17—41no—
37C2×D4×C17Q814yescomplemented
38C22×C4×C17—24no—
39C22×C4×C17—41no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×Dic17—24no—
44C22×Dic17—41no—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C22×Dic17C2312yes—
49C2×Dic34—24no—
50C2×Dic34—24no—
51C2×Dic34—42no—
52C22⋊C4×C17—24no—
53C22⋊C4×C17—41no—
54C22⋊Dic17—24no—
55C22⋊Dic17—24no—
56C22⋊Dic17—24no—
57C22⋊Dic17—41no—
58C4⋊Dic17—24no—
59C4⋊Dic17—24no—
60C4⋊Dic17—41no—
61#63225—24no—
62#63225—24no—
63#63225—42no—
64#63225D418yescomplemented
65C4×Dic17—41no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17C2×Q812yes—
75C23×C17C2×Q812yes—
76C23×C17C2411yescharacteristic
77D4×C17—44no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—44no—
86C2×C4×C17C2×D414yes—
87C2×C4×C17C2×Q812yes—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—44no—
99C2×Dic17C2×D414yes—
100C2×Dic17C2×D414yes—
101Dic34—48no—
102#2683—171nomaximal, Sylow
103C22×C17—22no—
104C22×C17—22no—
105C22×C17—24no—
106C22×C17—24no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—41no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—41no—
113C22×C17C22×D412yes—
114C22×C17C22×Q811yescharacteristic
115C22×C17H(5,𝔽2)11yescharacteristic
116C22×C17H(5,𝔽2)11yescharacteristic
117C22×C172−1+412yes—
118C4×C17—22no—
119C4×C17—24no—
120C4×C17—42no—
121Dic17—24no—
122Dic17—24no—
123Dic17—42no—
124Dic17—42no—
125C2×C22⋊Q8—172no—
126C2×C22⋊Q8—172no—
127#396—172no—
128#396—172no—
129#395—172no—
130#390—171no—
131#389—172no—
132#386—171no—
133#386D1711yescharacteristic
134C2×C17—42no—
135C2×C17—42no—
136C2×C17#53712yes—
137C2×C17#54311yescharacteristic
138C2×C17D4211yescharacteristic
139C2×C17D4×Q812yes—
140C2×C17#56311yescharacteristic
141C22×Q8—172no—
142C22×D4C2×D1711yescharacteristic
143C23×C4—171no—
144C23×C4C2×D1711yescharacteristic
145C22⋊Q8—348no—
146C22⋊Q8—348no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—172no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—171no—
154C2×C22⋊C4—172no—
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C22⋊C4—171no—
158C2×C22⋊C4C2×D1712yes—
159C2×C22⋊C4C2×D1711yescharacteristic
160C2×C42—171no—
161#95—172no—
162#95—172no—
163#95C2×D1712yes—
164C17#268311yescharacteristic, complemented, Sylow
165C24C22×D1711yescharacteristic
166C24C22×D1711yescharacteristic
167C2×Q8—344no—
168C2×Q8—344no—
169C2×Q8—682no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4Dic3414yescomplemented
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—172no—
176C22×C4—172no—
177C22×C4—172no—
178C22×C4—172no—
179C22×C4—344no—
180C22×C4—681no—
181C22×C4C22×D1712yes—
182C22×C4C22×D1711yescharacteristic
183C22×C4C22×D1712yes—
184C4⋊C4—178no—
185C4⋊C4—344no—
186C4⋊C4—344no—
187C4⋊C4—344no—
188C4⋊C4—344no—
189C4⋊C4—682no—
190C4⋊C4—681no—
191C22⋊C4—24no—
192C22⋊C4—41no—
193C22⋊C4—344no—
194C22⋊C4—344no—
195C22⋊C4—344no—
196C22⋊C4—681no—
197C42—681no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23—22no—
206C23C2×Dic3412yes—
207C23C2×Dic3412yes—
208C23C23×D1711yescharacteristic
209Q8—688no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—174no—
220C2×C4—174no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—684no—
232C2×C4C2×Dic3412yes—
233C2×C4D4×D1714yes—
234C22—24no—
235C22—24no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22C22×Dic3411yescharacteristic
245C22C2×D4×D1712yes—
246C22#1058769111yescharacteristic
247C22#1058769111yescharacteristic
248C22#1058769712yes—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—344no—
253C4—344no—
254C4—682no—
255C4—682no—
256C2—42no—
257C2—42no—
258C2#1176338812yes—
259C2#1176343511yescharacteristic
260C2#1176344111yescharacteristic
261C2D4×Dic3412yes—
262C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2683
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#10587610C2semidirect
2. C2×#10587564C2semidirect
3. #11762732C2semidirect
4. #11762732C2semidirect
5. #11762730C2semidirect
7. #11762706C2semidirect
37. C2×D4×C17Q8semidirect
64. #63225D4semidirect
164. C17#2683semidirect
172. C2×D4Dic34semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
C2×#105875642
#117627322
#117627302
#117627122
#117627062
#117626892
C17×#3862
#268317