Finite Group Databasebeta

← #14544830

Subgroups — #14544830

Order 2176.

362 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 66 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#11763876C211yesmaximal, characteristic, complemented
2#11763807C212yesmaximal, complemented
3D17⋊#100C211yesmaximal, characteristic, complemented
4#11763676C212yesmaximal, complemented
5#11763372C211yesmaximal, characteristic
6C2×#63254C2211yescharacteristic
7C2×D4×D17C2211yescharacteristic
8C2×C22⋊#595—22no—
9D4×#595—24no—
10#10587677C2212yes—
11D17⋊M16C2212yes—
12C17×#127C411yescharacteristic, complemented
13#10587626C411yescharacteristic, complemented
14#10587625—22no—
15#10587623—22no—
16#10587622C411yescharacteristic, complemented
17#10587561C411yescharacteristic, complemented
18#10587558C2211yescharacteristic
19#10587553—22no—
20#10587548—24no—
21C23×D17D412yes—
22C22×#595—24no—
23C2×D4×C17—22no—
24C2×D4×C17C2×C411yescharacteristic
25C2×D4×C17C2×C411yescharacteristic, complemented
26C2×C22⋊D17—21no—
27C2×C22⋊D17—22no—
28C2×C22⋊D17D412yes—
29C22×Dic17—21no—
30#63254—22no—
31#63254—22no—
32D4×D17—22no—
33D4×D17—22no—
34C2×D68C2×C411yescharacteristic
35C2×C4×D17C2311yescharacteristic
36C2×Dic34C2×C411yescharacteristic, complemented
37C22⋊#595—24no—
38C22⋊#595—22no—
39#63247—22no—
40C4⋊#595—22no—
41C4×#595D414yescomplemented
42#63243—22no—
43#63242—22no—
44C42×C17C2×C411yescharacteristic, complemented
45C22⋊Dic17—22no—
46#63227—22no—
47#63227—21no—
48#63225—21no—
49C4×Dic17C2×C411yescharacteristic, complemented
50C23×C17—21no—
51C23×C17C22⋊C412yes—
52C22×D17—22no—
53C22×D17—22no—
54C22×D17—82no—
55C22×D17C22⋊C412yes—
56C22×D17C2×D411yescharacteristic
57C2×#595—24no—
58C2×#595—84no—
59D4×C17—22no—
60D4×C17—22no—
61D4×C17—82no—
62C2×C4×C17—21no—
63C2×C4×C17C22×C411yescharacteristic
64C22⋊D17—24no—
65C22⋊D17—41no—
66C22⋊D17—41no—
67C22⋊D17—82no—
68C2×Dic17—21no—
69C2×Dic17—21no—
70C2×Dic17—42no—
71C2×Dic17C2×D411yescharacteristic
72D68—22no—
73C4×D17C2×D412yes—
74C4×D17#4112yes—
75Dic34—22no—
76#3624—42no—
77#1911—171nomaximal, Sylow
78C22×C17—22no—
79C22×C17—42no—
80C22×C17—42no—
81C22×C17—41no—
82C22×C17C2×C22⋊C411yescharacteristic
83C2×D17—22no—
84C2×D17—41no—
85C2×D17—44no—
86C2×D17—42no—
87C2×D17—82no—
88C2×D17#12011yescharacteristic
89C2×D17#12311yescharacteristic
90#595—44no—
91C4×C17—41no—
92C4×C17C4×D412yes—
93Dic17—41no—
94Dic17—41no—
95Dic17#12112yes—
96#535—171no—
97#518—171no—
98#421—172no—
99C2×#100—171no—
100H(3,ℤ/4)—172no—
101C2×C17—21no—
102C2×C17—42no—
103C2×C17—81no—
104C2×C17#38611yescharacteristic
105D17—21no—
106D17—21no—
107D17—42no—
108C2×#41—171no—
109C22×D4—171no—
110C2×M16—172no—
111#127—171no—
112#127#59511yescharacteristic, complemented
113#124—171no—
114#124—171no—
115#121—342no—
116#120—342no—
117C4×D4—344no—
118#117—172no—
119#117—171no—
120C2×C22⋊C4—342no—
121#104—344no—
122#100—342no—
123C17#191111yescharacteristic, complemented, Sylow
124C24—172no—
125#41—342no—
126#41—342no—
127C2×Q8—171no—
128C2×D4—22no—
129C2×D4—172no—
130C2×D4—171no—
131C2×D4—342no—
132C2×D4—342no—
133C2×D4—341no—
134C2×D4—342no—
135C2×D4C2×#59511yescharacteristic
136C2×D4C2×#59511yescharacteristic, complemented
137C22×C4—171no—
138C22×C4—341no—
139C22×C4—344no—
140M16—342no—
141M16—342no—
142C2×C8—342no—
143C4⋊C4—341no—
144C4⋊C4—342no—
145C22⋊C4—341no—
146C22⋊C4—342no—
147C22⋊C4—342no—
148C22⋊C4—344no—
149C22⋊C4—342no—
150C42—174no—
151C42—171no—
152C42C2×#59511yescharacteristic, complemented
153C23—21no—
154C23—171no—
155C23—172no—
156C23—342no—
157C23—342no—
158C23—1362no—
159C23C22⋊#59512yes—
160Q8—342no—
161D4—22no—
162D4—22no—
163D4—82no—
164D4—344no—
165D4—342no—
166D4—681no—
167D4—681no—
168D4—1362no—
169C2×C4—21no—
170C2×C4—171no—
171C2×C4—172no—
172C2×C4—172no—
173C2×C4—344no—
174C2×C4—341no—
175C2×C4—341no—
176C2×C4—682no—
177C2×C4—1364no—
178C2×C4C22×#59511yescharacteristic
179C8—682no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—171no—
185C22—171no—
186C22—342no—
187C22—681no—
188C22—682no—
189C22—684no—
190C22—1362no—
191C22C2×C22⋊#59511yescharacteristic
192C4—41no—
193C4—172no—
194C4—681no—
195C4—684no—
196C4—681no—
197C4D4×#59512yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—341no—
202C2—341no—
203C2—682no—
204C2#1176375711yescharacteristic
subgroupG/H
centerC2#11763757
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#127C4
socleC2×C17#386
prime radical#11763372C2
cosocleC2×C4×D17C23
hypercenter#127#595
hypocenterC17#1911
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #11763876C2semidirect
2. #11763807C2semidirect
3. D17⋊#100C2semidirect
4. #11763676C2semidirect
12. C17×#127C4semidirect
13. #10587626C4semidirect
16. #10587622C4semidirect
17. #10587561C4semidirect
25. C2×D4×C17C2×C4semidirect
36. C2×Dic34C2×C4semidirect
41. C4×#595D4semidirect
44. C42×C17C2×C4semidirect
49. C4×Dic17C2×C4semidirect
112. #127#595semidirect
123. C17#1911semidirect
136. C2×D4C2×#595semidirect
152. C42C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117638762
#117638072
D17⋊#1002
#117636762
#117633722
#191117