Finite Group Databasebeta

← #14544909

Subgroups — #14544909

Order 2176.

384 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 92 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×C4⋊#3624C211yesmaximal, characteristic, complemented
2C2×#10587543C212yesmaximal, complemented
3C2×C22⋊#3624C212yesmaximal, complemented
4C2×#10587542C211yesmaximal, characteristic, complemented
5C2×#10587567C211yesmaximal, characteristic
6C22×C22⋊D17C411yescharacteristic
7C23×Dic17C2211yescharacteristic
8C22×C4×D17C2211yescharacteristic
9C22×#3624C2212yes—
10C22×#3624C2212yes—
11C2×C22⋊C4×C17C411yescharacteristic
12C2×#63227C411yescharacteristic
13C2×#63225C411yescharacteristic
14C2×C4×Dic17C2211yescharacteristic
15#10587567—22no—
16#10587567—22no—
17C22⋊#3624—24no—
18#10587543—24no—
19#10587542—22no—
20C4⋊#3624—22no—
21C24×C17C2×C411yescharacteristic
22C23×D17C2×C411yescharacteristic
23C22×C4×C17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17—22no—
26C2×C22⋊D17—22no—
27C2×C22⋊D17—22no—
28C22×Dic17—22no—
29C22×Dic17—21no—
30C22×Dic17—21no—
31C22×Dic17—22no—
32C22×Dic17C2×C411yescharacteristic
33C22×Dic17C2×C411yescharacteristic
34C22×Dic17C2311yescharacteristic
35C2×C4×D17—22no—
36C2×C4×D17—21no—
37C2×C4×D17—21no—
38C2×C4×D17—22no—
39C2×#3624—22no—
40C2×#3624—22no—
41C2×#3624—22no—
42C2×#3624—22no—
43C2×#3624—22no—
44C2×#3624D414yescomplemented
45C22⋊C4×C17—22no—
46#63227—22no—
47#63225—22no—
48C4×Dic17D412yes—
49C4×Dic17D412yes—
50C23×C17—21no—
51C23×C17—22no—
52C23×C17—21no—
53C23×C17—22no—
54C23×C17M1612yes—
55C23×C17C22×C411yescharacteristic
56C22×D17—21no—
57C22×D17—22no—
58C22×D17—21no—
59C22×D17—22no—
60C22×D17M1612yes—
61C2×C4×C17—22no—
62C2×C4×C17—21no—
63C2×C4×C17—22no—
64C2×C4×C17C22⋊C412yes—
65C22⋊D17—28no—
66C2×Dic17—21no—
67C2×Dic17—21no—
68C2×Dic17—21no—
69C2×Dic17—22no—
70C2×Dic17—42no—
71C2×Dic17—41no—
72C2×Dic17—41no—
73C2×Dic17C22⋊C412yes—
74C2×Dic17C2×D411yescharacteristic
75C2×Dic17C2×D412yes—
76C2×Dic17C2×D411yescharacteristic
77C2×Dic17#4112yes—
78C4×D17—42no—
79C4×D17—41no—
80C4×D17—41no—
81#3624—24no—
82#3624—42no—
83#1955—171nomaximal, Sylow
84C22×C17—22no—
85C22×C17—24no—
86C22×C17—41no—
87C22×C17—41no—
88C22×C17—42no—
89C22×C17C2×C22⋊C411yescharacteristic
90C22×C17C4×D412yes—
91C22×C17C2×M1611yescharacteristic
92C22×C17C2×M1611yescharacteristic
93C22×C17#13112yes—
94C2×D17—22no—
95C2×D17—24no—
96C2×D17—41no—
97C2×D17—41no—
98C2×D17—42no—
99C4×C17—22no—
100C4×C17—41no—
101Dic17—22no—
102Dic17—41no—
103Dic17#12112yes—
104Dic17#12411yescharacteristic
105Dic17#12711yescharacteristic
106C2×C4×D4—171no—
107C2×C4⋊C8—171no—
108C2×C22⋊C8—172no—
109C2×C22⋊C8—172no—
110C2×#97—171no—
111C2×C17—42no—
112C2×C17#39011yescharacteristic
113C2×C17#40711yescharacteristic
114C2×C17#40811yescharacteristic
115C2×C17#43512yes—
116C2×C17#43512yes—
117D17—42no—
118C22×D4—171no—
119C23×C4—171no—
120C23×C4—171no—
121C22×C8—172no—
122C22×C8—172no—
123C4×D4—342no—
124C4×D4—342no—
125C2×C4⋊C4—171no—
126C2×C22⋊C4—171no—
127C2×C22⋊C4#59511yescharacteristic
128C2×C42—171no—
129C4⋊C8—342no—
130C22⋊C8—344no—
131C22⋊C8—344no—
132#97—342no—
133C17#195511yescharacteristic, complemented, Sylow
134C24—171no—
135C24C2×#59511yescharacteristic
136C2×D4—342no—
137C2×D4—342no—
138C2×D4—342no—
139C22×C4—171no—
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—342no—
143C22×C4—341no—
144C22×C4—341no—
145C22×C4—342no—
146C22×C4—342no—
147C22×C4—341no—
148C22×C4—341no—
149C22×C4—342no—
150C22×C4C2×#59511yescharacteristic
151C22×C4C2×#59511yescharacteristic
152C2×C8—174no—
153C2×C8—342no—
154C2×C8—342no—
155C2×C8—342no—
156C2×C8—342no—
157C2×C8—342no—
158C4⋊C4—342no—
159C22⋊C4—22no—
160C22⋊C4—342no—
161C42—172no—
162C42—172no—
163C23—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—172no—
168C23—341no—
169C23—342no—
170C23—342no—
171C23—341no—
172C23#6324312yes—
173C23C22×#59511yescharacteristic
174D4—348no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—172no—
179C2×C4—172no—
180C2×C4—171no—
181C2×C4—171no—
182C2×C4—172no—
183C2×C4—341no—
184C2×C4—341no—
185C2×C4—342no—
186C2×C4—341no—
187C2×C4—682no—
188C2×C4—681no—
189C2×C4—681no—
190C2×C4—682no—
191C2×C4—681no—
192C2×C4—681no—
193C2×C4C22⋊#59512yes—
194C8—344no—
195C8—682no—
196C22—22no—
197C22—24no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—342no—
202C22—344no—
203C22—681no—
204C22—681no—
205C22—682no—
206C22C2×#6324311yescharacteristic
207C22D17⋊M1611yescharacteristic
208C22#1058767812yes—
209C22D4×#59512yes—
210C22C2×C22⋊#59511yescharacteristic
211C4—22no—
212C4—41no—
213C4—172no—
214C4—171no—
215C4—171no—
216C4—342no—
217C4—681no—
218C2—42no—
219C2—682no—
220C2#1176376111yescharacteristic
221C2#1176378011yescharacteristic
222C2#1176378211yescharacteristic
223C2#1176383812yes—
224C2#1176384112yes—
subgroupG/H
center, FrattiniC23C22×#595
commutator, socleC23×C17C22×C4
FittingC2×C22⋊C4×C17C4
prime radicalC22×C22⋊D17C4
cosocleC22×Dic17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1955
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#3624C2semidirect
2. C2×#10587543C2semidirect
3. C2×C22⋊#3624C2semidirect
4. C2×#10587542C2semidirect
44. C2×#3624D4semidirect
133. C17#1955semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#36242
C2×#105875432
C2×C22⋊#36242
C2×#105875422
C2×#105875672
#195517