Finite Group Databasebeta

← #14544084

Subgroups — #14544084

Order 2176.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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#11763828C211yesmaximal, characteristic, complemented
2#11763827C211yesmaximal, characteristic, complemented
3#11763657C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5C4×D4×D17C211yesmaximal, characteristic
6C2×D4×D17C411yescharacteristic
7C22×C4×D17—21no—
8C4×D4×C17C411yescharacteristic
9D4×Dic17C2211yescharacteristic
10C4×C22⋊D17—21no—
11#10587579C2211yescharacteristic
12C4⋊C4×D17C411yescharacteristic
13#10587567—21no—
14C22⋊C4×D17—21no—
15C4×D68C411yescharacteristic
16C42×D17C2211yescharacteristic
17#10587544—21no—
18#10587543—21no—
19#10587543—21no—
20C4⋊#3624C2212yescomplemented
21C4×#3624—21no—
22C4×#3624C2212yescomplemented
23C23×D17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C22×Dic17—21no—
28D4×D17—22no—
29D4×D17C814yescomplemented
30C2×D68C2×C411yescharacteristic
31C2×C4×D17—21no—
32C2×C4×D17—21no—
33C2×C4×D17—21no—
34C2×C4×D17—44no—
35C2×C4×D17C2×C411yescharacteristic
36C2×C4×D17D411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×#3624—22no—
39C2×#3624—22no—
40C4⋊C4×C17C2×C411yescharacteristic
41C22⋊C4×C17—21no—
42C42×C17C2×C411yescharacteristic
43C22⋊Dic17—21no—
44#63227—21no—
45#63227—21no—
46C4⋊Dic17C2×C411yescharacteristic
47#63225—21no—
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C4×Dic17C2311yescharacteristic
51C23×C17—21no—
52C22×D17—21no—
53C22×D17—24no—
54C22×D17—42no—
55C22×D17—42no—
56C22×D17C22⋊C411yescharacteristic
57D4×C17—21no—
58D4×C17C2×C812yes—
59C2×C4×C17—21no—
60C2×C4×C17—42no—
61C2×C4×C17C22⋊C411yescharacteristic
62C2×C4×C17C22⋊C411yescharacteristic
63C2×C4×C17C22×C411yescharacteristic
64C22⋊D17—24no—
65C2×Dic17—21no—
66C2×Dic17—42no—
67C2×Dic17C22⋊C411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69C2×Dic17C2×D411yescharacteristic
70D68—21no—
71D68C2×C812yes—
72C4×D17—22no—
73C4×D17—41no—
74C4×D17—42no—
75C4×D17—42no—
76C4×D17—42no—
77C4×D17C2×C812yes—
78C4×D17M1612yes—
79#3624—42no—
80#3624—42no—
81#1478—171nomaximal, Sylow
82C22×C17—22no—
83C22×C17—41no—
84C22×C17—41no—
85C22×C17C2×C22⋊C411yescharacteristic
86C2×D17—21no—
87C2×D17—21no—
88C2×D17—22no—
89C2×D17—48no—
90C2×D17—41no—
91C2×D17—41no—
92C2×D17C22⋊C812yes—
93C4×C17—21no—
94C4×C17—21no—
95C4×C17—41no—
96C4×C17C22×C811yescharacteristic
97C4×C17C2×M1611yescharacteristic
98Dic17—21no—
99Dic17—41no—
100Dic17C22⋊C812yes—
101Dic17AGL(1,ℤ/8)12yes—
102C2×C4×D4—171no—
103#431—171no—
104#431—171no—
105D4⋊C8—172no—
106D4⋊C8—172no—
107C2×C17—42no—
108C2×C17C2×C22⋊C811yescharacteristic
109C2×C17#41811yescharacteristic
110C2×C17#42111yescharacteristic
111D17—22no—
112D17—42no—
113C22×D4—171no—
114C23×C4—341no—
115C4×D4—171no—
116C4×D4—171no—
117C4×D4—171no—
118C4×D4—341no—
119C4×D4—341no—
120C4×D4#59511yescharacteristic
121C2×C4⋊C4—171no—
122C2×C22⋊C4—341no—
123C2×C42—171no—
124C4⋊C8—172no—
125C4⋊C8—341no—
126C22⋊C8—341no—
127C22⋊C8—341no—
128C4×C8—172no—
129C4×C8—341no—
130C17#147811yescharacteristic, complemented, Sylow
131C24—341no—
132C2×D4—174no—
133C2×D4—171no—
134C2×D4—342no—
135C2×D4—341no—
136C2×D4C2×#59511yescharacteristic
137C22×C4—21no—
138C22×C4—171no—
139C22×C4—171no—
140C22×C4—171no—
141C22×C4—341no—
142C22×C4—341no—
143C22×C4—341no—
144C22×C4—341no—
145C22×C4—684no—
146C2×C8—342no—
147C2×C8—342no—
148C4⋊C4—171no—
149C4⋊C4—341no—
150C4⋊C4C2×#59511yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—341no—
153C22⋊C4—341no—
154C22⋊C4—341no—
155C42—171no—
156C42—171no—
157C42—171no—
158C42C2×#59511yescharacteristic
159C23—21no—
160C23—171no—
161C23—344no—
162C23—341no—
163C23—682no—
164C23—682no—
165D4—21no—
166D4—172no—
167D4—344no—
168D4—341no—
169D4#6324212yes—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—171no—
173C2×C4—171no—
174C2×C4—172no—
175C2×C4—172no—
176C2×C4—171no—
177C2×C4—342no—
178C2×C4—341no—
179C2×C4—681no—
180C2×C4—682no—
181C2×C4—682no—
182C2×C4—682no—
183C2×C4—682no—
184C2×C4C22⋊#59511yescharacteristic
185C2×C4C22⋊#59511yescharacteristic
186C2×C4C22×#59511yescharacteristic
187C8—682no—
188C8—682no—
189C22—22no—
190C22—41no—
191C22—41no—
192C22—172no—
193C22—341no—
194C22—341no—
195C22—342no—
196C22—688no—
197C22—681no—
198C22—681no—
199C22C2×C22⋊#59511yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—172no—
204C4—172no—
205C4—341no—
206C4—681no—
207C4C2×#6324211yescharacteristic
208C4D17⋊M1611yescharacteristic
209C2—42no—
210C2—342no—
211C2—682no—
212C2#1176377711yescharacteristic
213C2#1176380111yescharacteristic
214C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC4×C17C22×C8
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×D4×D17C4
cosocleC4×Dic17C23
hypercenterC4×D4#595
hypocenterC17#1478
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763828C2semidirect
2. #11763827C2semidirect
3. #11763657C2semidirect
4. D4⋊#3624C2semidirect
20. C4⋊#3624C22semidirect
22. C4×#3624C22semidirect
29. D4×D17C8semidirect
130. C17#1478semidirect

Maximal subgroups

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

subgroupindex
#117638282
#117638272
#117636572
D4⋊#36242
C4×D4×D172
#147817