Finite Group Databasebeta

← #14545047

Subgroups — #14545047

Order 2176.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#11763796C211yesmaximal, characteristic
2#11763789C211yesmaximal, characteristic
3#11763712C212yesmaximal, complemented
4#11763711C212yesmaximal, complemented
5#11763381C211yesmaximal, characteristic
6C2×#63254C2211yescharacteristic
7C22×Dic34C411yescharacteristic, complemented
8C2×#63247—21no—
9#10587677—21no—
10C17×#123C411yescharacteristic, complemented
11#10587621C411yescharacteristic, complemented
12#10587615C2211yescharacteristic
13#10587582—21no—
14#10587574—21no—
15#10587572C2211yescharacteristic
16#10587571—21no—
17#10587564—21no—
18#10587563C411yescharacteristic, complemented
19#10587553C2212yes—
20#10587551—21no—
21#10587551C2212yes—
22#10587545—21no—
23C2×D4×C17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17D411yescharacteristic
26C2×C22⋊D17C2311yescharacteristic
27C22×Dic17C2×C411yescharacteristic
28C22×Dic17C2×C411yescharacteristic
29C22×Dic17D411yescharacteristic
30#63254—41no—
31#63254—41no—
32C2×C4×D17D411yescharacteristic
33C2×Dic34—21no—
34C2×Dic34—21no—
35C2×Dic34—41no—
36C2×Dic34—41no—
37C2×Dic34D411yescharacteristic
38C22⋊#595—22no—
39#63247—22no—
40#63247—41no—
41C2×#3624—41no—
42C4⋊#595—41no—
43C4×#595—41no—
44C4⋊C4×C17—21no—
45C22⋊C4×C17—21no—
46C22⋊C4×C17C2×C411yescharacteristic
47C22⋊Dic17—21no—
48C22⋊Dic17C2×C411yescharacteristic
49#63227—21no—
50#63227—21no—
51C4⋊Dic17—21no—
52#63225—21no—
53#63225—21no—
54#63225—21no—
55C4×Dic17—21no—
56C23×C17C22⋊C411yescharacteristic
57C23×C17C22×C411yescharacteristic
58C22×D17C2×D411yescharacteristic
59C2×#595—42no—
60D4×C17—41no—
61C2×C4×C17—21no—
62C2×C4×C17—21no—
63C2×C4×C17—41no—
64C2×C4×C17C22⋊C411yescharacteristic
65C22⋊D17—41no—
66C22⋊D17—41no—
67C2×Dic17—21no—
68C2×Dic17—21no—
69C2×Dic17—21no—
70C2×Dic17—21no—
71C2×Dic17—21no—
72C2×Dic17—21no—
73C2×Dic17—41no—
74C2×Dic17C22⋊C411yescharacteristic
75C2×Dic17C22⋊C411yescharacteristic
76C2×Dic17C2×D411yescharacteristic
77C4×D17—21no—
78C4×D17—21no—
79Dic34—41no—
80Dic34—42no—
81Dic34—41no—
82Dic34—41no—
83#3624—42no—
84#2123—171nomaximal, Sylow
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—41no—
88C22×C17—41no—
89C22×C17C2×C22⋊C411yescharacteristic
90C2×D17—21no—
91C2×D17—41no—
92#595—82no—
93C4×C17—21no—
94C4×C17—41no—
95C4×C17—41no—
96Dic17—21no—
97Dic17—21no—
98Dic17—41no—
99Dic17—41no—
100Dic17#9911yescharacteristic
101Dic17#9911yescharacteristic
102#536—171no—
103#413—171no—
104#410—171no—
105#352—172no—
106#352—172no—
107C2×C17—21no—
108C2×C17—41no—
109C2×C17—41no—
110C2×C17C2×#9911yescharacteristic
111D17—41no—
112C2×#41—171no—
113C22×Q8—171no—
114C2×M16—341no—
115C4⋊Q8—341no—
116#124—341no—
117#123—171no—
118#123—171no—
119#123—171no—
120#123#59511yescharacteristic, complemented
121C22⋊Q8—341no—
122C22⋊Q8—341no—
123#117—171no—
124#117—341no—
125#101—341no—
126#100—172no—
127#99—172no—
128#99—341no—
129C17#212311yescharacteristic, complemented, Sylow
130#41—681no—
131#41—681no—
132C2×Q8—171no—
133C2×Q8—341no—
134C2×Q8—341no—
135C2×Q8—681no—
136C2×Q8—681no—
137C2×D4—171no—
138C2×D4—171no—
139C2×D4C2×#59511yescharacteristic
140C22×C4—171no—
141C22×C4—171no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4C2×#59511yescharacteristic
145M16—342no—
146M16—681no—
147C2×C8—681no—
148C4⋊C4—21no—
149C4⋊C4—341no—
150C4⋊C4—341no—
151C4⋊C4—341no—
152C4⋊C4—341no—
153C4⋊C4—681no—
154C22⋊C4—21no—
155C22⋊C4—171no—
156C22⋊C4—341no—
157C22⋊C4—341no—
158C22⋊C4—341no—
159C22⋊C4—342no—
160C22⋊C4C2×#59511yescharacteristic
161C42—341no—
162C42—681no—
163C23—171no—
164C23C22⋊#59511yescharacteristic
165C23C22×#59511yescharacteristic
166Q8—681no—
167Q8—681no—
168Q8—681no—
169Q8—682no—
170D4—41no—
171D4—681no—
172D4—681no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—171no—
177C2×C4—171no—
178C2×C4—171no—
179C2×C4—341no—
180C2×C4—341no—
181C2×C4—341no—
182C2×C4—341no—
183C2×C4—341no—
184C2×C4—341no—
185C2×C4—341no—
186C2×C4—341no—
187C2×C4—681no—
188C2×C4—682no—
189C2×C4C22⋊#59511yescharacteristic
190C8—682no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—341no—
196C22—681no—
197C22C2×C22⋊#59511yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—171no—
202C4—171no—
203C4—341no—
204C4—341no—
205C4—681no—
206C4—681no—
207C4—1362no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—681no—
212C2D17⋊#9911yescharacteristic
subgroupG/H
centerC2D17⋊#99
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#123C4
socleC2×C17C2×#99
prime radicalC2×#63254C22
cosocleC2×C22⋊D17C23
hypercenter#123#595
hypocenterC17#2123
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #11763712C2semidirect
4. #11763711C2semidirect
7. C22×Dic34C4semidirect
10. C17×#123C4semidirect
11. #10587621C4semidirect
18. #10587563C4semidirect
120. #123#595semidirect
129. C17#2123semidirect

Maximal subgroups

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

subgroupindex
#117637962
#117637892
#117637122
#117637112
#117633812
#212317