Finite Group Databasebeta

← #14544214

Subgroups — #14544214

Order 2176.

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

normal subgroups only

This group has 264 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#11763828C211yesmaximal, characteristic, complemented
2#11763753C211yesmaximal, characteristic, complemented
3#11763661C212yesmaximal, complemented
4#11763660C212yesmaximal, complemented
5#11763356C211yesmaximal, characteristic
6C2×#63252—21no—
7C2×C4⋊#595—21no—
8C4×D4×C17C411yescharacteristic, complemented
9#10587626C411yescharacteristic, complemented
10#10587624C2211yescharacteristic
11C4×C22⋊D17—21no—
12#10587581C2211yescharacteristic
13#10587574C411yescharacteristic, complemented
14#10587571—21no—
15#10587570—21no—
16C4×D68C411yescharacteristic, complemented
17C42×D17C2211yescharacteristic
18(C2×#595)⋊C4—21no—
19(C2×#595)⋊C4C2212yes—
20#10587543C2212yes—
21C4⋊#3624—21no—
22C4×#3624—21no—
23C22×#595—22no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17—21no—
26C2×C22⋊D17—21no—
27C2×C22⋊D17—21no—
28C2×C22⋊D17—21no—
29#63252—42no—
30#63252—42no—
31C2×D68—21no—
32C2×D68C2×C411yescharacteristic
33C2×C4×D17—21no—
34C2×C4×D17D411yescharacteristic
35C2×C4×D17D411yescharacteristic
36C2×C4×D17C2311yescharacteristic
37C2×Dic34—21no—
38C2×#3624—22no—
39C4⋊#595—42no—
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×Dic17C2×C411yescharacteristic
49C4×Dic17D411yescharacteristic
50C4×Dic17D411yescharacteristic
51C23×C17—21no—
52C22×D17—21no—
53C22×D17C2×D411yescharacteristic
54C2×#595—42no—
55C2×#595—42no—
56C2×#595—42no—
57D4×C17—41no—
58C2×C4×C17—21no—
59C2×C4×C17—42no—
60C2×C4×C17C22⋊C411yescharacteristic
61C2×C4×C17C22⋊C411yescharacteristic
62C2×C4×C17C22×C411yescharacteristic
63C22⋊D17—42no—
64C22⋊D17—41no—
65C22⋊D17—41no—
66C22⋊D17—42no—
67C2×Dic17—21no—
68C2×Dic17C22⋊C411yescharacteristic
69C2×Dic17C22⋊C411yescharacteristic
70C2×Dic17C2×D411yescharacteristic
71D68—41no—
72D68—41no—
73D68—41no—
74C4×D17—41no—
75C4×D17—41no—
76C4×D17—42no—
77C4×D17D812yes—
78C4×D17SD1612yes—
79Dic34—42no—
80#3624—42no—
81#1517—171nomaximal, Sylow
82C22×C17—41no—
83C22×C17—41no—
84C22×C17—41no—
85C22×C17C2×C22⋊C411yescharacteristic
86C2×D17—21no—
87C2×D17—21no—
88C2×D17—41no—
89C2×D17—41no—
90C2×D17—41no—
91C2×D17C2×D811yescharacteristic
92C2×D17C2×SD1611yescharacteristic
93#595—82no—
94C4×C17—21no—
95C4×C17—21no—
96C4×C17—41no—
97C4×C17#10212yes—
98Dic17—21no—
99Dic17—21no—
100Dic17—41no—
101Dic17#10212yes—
102#532—171no—
103#431—171no—
104#382—171no—
105#327—172no—
106#327—172no—
107C2×C17—81no—
108C2×C17#41011yescharacteristic
109C2×C17C2×#10211yescharacteristic
110C2×C17#42111yescharacteristic
111D17—21no—
112D17—21no—
113D17—81no—
114C2×#41—341no—
115C4⋊Q8—171no—
116#127—171no—
117#124—341no—
118#121—171no—
119#121—171no—
120#121—341no—
121C4×D4—171no—
122C4×D4—341no—
123C4×D4#59511yescharacteristic, complemented
124C2×C4⋊C4—341no—
125C2×C42—171no—
126C4⋊C8—341no—
127C22⋊C8—172no—
128C4×C8—341no—
129#95—172no—
130#95—341no—
131C17#151711yescharacteristic, complemented, Sylow
132#41—682no—
133#41—682no—
134C2×Q8—341no—
135C2×D4—171no—
136C2×D4—341no—
137C2×D4—341no—
138C2×D4—341no—
139C2×D4—341no—
140C2×D4C2×#59511yescharacteristic
141C22×C4—21no—
142C22×C4—171no—
143C22×C4—171no—
144C22×C4—171no—
145C22×C4—341no—
146C22×C4—342no—
147C2×C8—342no—
148C4⋊C4—171no—
149C4⋊C4—341no—
150C4⋊C4—682no—
151C4⋊C4C2×#59511yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—341no—
154C22⋊C4—341no—
155C22⋊C4—341no—
156C42—171no—
157C42—171no—
158C42—171no—
159C42C2×#59511yescharacteristic
160C23—21no—
161C23—171no—
162C23—341no—
163Q8—682no—
164D4—41no—
165D4—681no—
166D4—681no—
167D4—681no—
168D4—682no—
169D4—682no—
170D4—681no—
171D4—681no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—171no—
175C2×C4—172no—
176C2×C4—172no—
177C2×C4—171no—
178C2×C4—171no—
179C2×C4—341no—
180C2×C4—681no—
181C2×C4—681no—
182C2×C4—682no—
183C2×C4—682no—
184C2×C4—682no—
185C2×C4—682no—
186C2×C4C22⋊#59511yescharacteristic
187C2×C4C22⋊#59511yescharacteristic
188C2×C4C22×#59511yescharacteristic
189C8—682no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—171no—
194C22—171no—
195C22—341no—
196C22—341no—
197C22—681no—
198C22—681no—
199C22—681no—
200C22C2×C22⋊#59511yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—172no—
205C4—341no—
206C4—341no—
207C4—681no—
208C4—1362no—
209C4#1058754712yes—
210C2—81no—
211C2—341no—
212C2—341no—
213C2—1361no—
214C2#1176378811yescharacteristic
215C2C2×#1058754711yescharacteristic
216C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radical#11763356C2
cosocleC2×C4×D17C23
hypercenterC4×D4#595
hypocenterC17#1517
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #11763828C2semidirect
2. #11763753C2semidirect
3. #11763661C2semidirect
4. #11763660C2semidirect
8. C4×D4×C17C4semidirect
9. #10587626C4semidirect
13. #10587574C4semidirect
16. C4×D68C4semidirect
123. C4×D4#595semidirect
131. C17#1517semidirect

Maximal subgroups

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

subgroupindex
#117638282
#117637532
#117636612
#117636602
#117633562
#151717