Finite Group Databasebeta

← #14544643

Subgroups — #14544643

Order 2176.

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

normal subgroups only

This group has 312 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×#10587544C211yesmaximal, characteristic, complemented
2C2×C4⋊#3624C211yesmaximal, characteristic, complemented
3C2×#10587543C212yesmaximal
4#11763675C212yesmaximal, complemented
5C2×#10587578C211yesmaximal, characteristic
6C22×C4×D17C2211yescharacteristic
7C22×#3624C2212yes—
8C22×#3624C2212yes—
9C2×C4⋊C4×C17C411yescharacteristic
10C2×#63227C411yescharacteristic
11C2×#63227C411yescharacteristic
12C2×C4⋊Dic17C411yescharacteristic
13C2×C4×Dic17C2211yescharacteristic
14C2×C4×Dic17C2211yescharacteristic
15#10587578—22no—
16#10587578—22no—
17#10587544—22no—
18#10587543—24no—
19C4⋊#3624—22no—
20C23×D17C2×C411yescharacteristic
21C22×C4×C17C2×C411yescharacteristic
22C22×C4×C17C2×C411yescharacteristic
23C22×C4×C17C2×C411yescharacteristic
24C22×Dic17C2×C411yescharacteristic
25C22×Dic17C2×C411yescharacteristic
26C22×Dic17C2311yescharacteristic
27C2×C4×D17—21no—
28C2×C4×D17—21no—
29C2×C4×D17—21no—
30C2×C4×D17—21no—
31C2×C4×D17D412yes—
32C2×C4×D17Q812yes—
33C2×#3624—22no—
34C2×#3624—22no—
35C2×#3624—22no—
36C2×#3624—22no—
37C2×#3624—22no—
38C2×#3624—22no—
39C4⋊C4×C17—22no—
40#63227—22no—
41#63227—22no—
42C4⋊Dic17—22no—
43C4×Dic17—21no—
44C4×Dic17—21no—
45C4×Dic17D412yes—
46C4×Dic17D412yes—
47C23×C17C22×C411yescharacteristic
48C22×D17—21no—
49C22×D17—21no—
50C22×D17—22no—
51C22×D17—22no—
52C22×D17C4⋊C412yes—
53C2×C4×C17—21no—
54C2×C4×C17—21no—
55C2×C4×C17—22no—
56C2×C4×C17—21no—
57C2×C4×C17—22no—
58C2×C4×C17C22⋊C412yes—
59C2×C4×C17C4⋊C412yes—
60C2×Dic17—22no—
61C2×Dic17—21no—
62C2×Dic17—22no—
63C2×Dic17C22⋊C412yes—
64C2×Dic17C2×D411yescharacteristic
65C2×Dic17C2×D411yescharacteristic
66C2×Dic17C2×D411yescharacteristic
67C2×Dic17C2×Q811yescharacteristic
68C2×Dic17#4111yescharacteristic
69C2×Dic17#4111yescharacteristic
70C4×D17—22no—
71C4×D17—22no—
72C4×D17—22no—
73C4×D17—22no—
74#3624—42no—
75#3624—42no—
76#1798—171nomaximal, Sylow
77C22×C17C2×C22⋊C411yescharacteristic
78C22×C17C2×C4⋊C411yescharacteristic
79C22×C17#11711yescharacteristic
80C22×C17#13112yes—
81C22×C17#13112yes—
82C2×D17—22no—
83C2×D17—24no—
84C2×D17—41no—
85C2×D17—41no—
86C2×D17—42no—
87C4×C17—22no—
88C4×C17—22no—
89C4×C17—41no—
90Dic17—22no—
91Dic17—41no—
92Dic17#12111yescharacteristic
93Dic17#12111yescharacteristic
94Dic17C22⋊Q811yescharacteristic
95Dic17C22⋊Q811yescharacteristic
96C2×#117—171no—
97C2×C4⋊C8—171no—
98C2×C4⋊C8—171no—
99C2×C22⋊C8—172no—
100#336—172no—
101C2×C17#38011yescharacteristic
102C2×C17#40811yescharacteristic
103C2×C17#40811yescharacteristic
104C2×C17#42412yes—
105C2×C17#43312yes—
106D17—42no—
107C23×C4—171no—
108C22×C8—172no—
109C22×C8—172no—
110#117—342no—
111#117—342no—
112C2×C4⋊C4—171no—
113C2×C4⋊C4#59511yescharacteristic
114C2×C22⋊C4—171no—
115C2×C22⋊C4—171no—
116C2×C42—171no—
117C2×C42—171no—
118C4⋊C8—342no—
119C4⋊C8—342no—
120C22⋊C8—344no—
121C17#179811yescharacteristic, complemented, Sylow
122C24—171no—
123C22×C4—171no—
124C22×C4—171no—
125C22×C4—172no—
126C22×C4—172no—
127C22×C4—171no—
128C22×C4—341no—
129C22×C4—341no—
130C22×C4—341no—
131C22×C4—341no—
132C22×C4C2×#59511yescharacteristic
133C22×C4C2×#59511yescharacteristic
134C22×C4C2×#59511yescharacteristic
135C2×C8—342no—
136C2×C8—342no—
137C2×C8—342no—
138C2×C8—342no—
139C2×C8—342no—
140C2×C8—342no—
141C4⋊C4—22no—
142C4⋊C4—342no—
143C22⋊C4—342no—
144C22⋊C4—342no—
145C42—172no—
146C42—172no—
147C42—341no—
148C42—341no—
149C23—172no—
150C23—341no—
151C23—342no—
152C23—342no—
153C23—341no—
154C23C22×#59511yescharacteristic
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—171no—
161C2×C4—171no—
162C2×C4—171no—
163C2×C4—171no—
164C2×C4—171no—
165C2×C4—171no—
166C2×C4—172no—
167C2×C4—342no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—341no—
174C2×C4C4⋊#59512yes—
175C2×C4C22⋊#59512yes—
176C8—682no—
177C8—682no—
178C22—342no—
179C22—344no—
180C22—681no—
181C22—681no—
182C22—682no—
183C22C2×C4⋊#59511yescharacteristic
184C22#1058767711yescharacteristic
185C22#1058767812yes—
186C22#1058768012yes—
187C22C2×C22⋊#59511yescharacteristic
188C4—22no—
189C4—22no—
190C4—41no—
191C4—171no—
192C4—171no—
193C4—171no—
194C4—171no—
195C4—342no—
196C4—681no—
197C2—682no—
198C2#1176375111yescharacteristic
199C2#1176378211yescharacteristic
200C2#1176378411yescharacteristic
201C2#1176381612yes—
202C2#1176383412yes—
subgroupG/H
center, FrattiniC23C22×#595
commutator, socleC23×C17C22×C4
FittingC2×C4⋊C4×C17C4
prime radicalC23×D17C2×C4
cosocleC22×Dic17C23
hypercenterC2×C4⋊C4#595
hypocenterC17#1798
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587544C2semidirect
2. C2×C4⋊#3624C2semidirect
4. #11763675C2semidirect
121. C17#1798semidirect

Maximal subgroups

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

subgroupindex
C2×#105875442
C2×C4⋊#36242
C2×#105875432
#117636752
C2×#105875782
#179817