Finite Group Databasebeta

← #14544118

Subgroups — #14544118

Order 2176.

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

normal subgroups only

This group has 246 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#11763831C211yesmaximal, characteristic, complemented
2#11763828C211yesmaximal, characteristic, complemented
3Q8⋊#3624C212yesmaximal, complemented
4#11763657C212yesmaximal, complemented
5C4×#63256C211yesmaximal, characteristic
6C2×#63256C411yescharacteristic
7C4×Q8×C17C411yescharacteristic
8Q8×Dic17C2211yescharacteristic
9#10587579—21no—
10#10587579C2211yescharacteristic
11#10587578—21no—
12#10587578C411yescharacteristic
13C4×D68—21no—
14C4×D68C411yescharacteristic
15C42×D17—21no—
16C42×D17C2211yescharacteristic
17#10587543—21no—
18#10587543—21no—
19#10587542—21no—
20C4⋊#3624—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23C2×Q8×C17C2×C411yescharacteristic
24#63256—22no—
25#63256—22no—
26C2×D68—21no—
27C2×D68C2×C411yescharacteristic
28C2×C4×D17—21no—
29C2×C4×D17—21no—
30C2×C4×D17—21no—
31C2×C4×D17C2×C411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×C4×D17D411yescharacteristic
34C2×#3624—22no—
35C2×#3624—22no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17C2×C411yescharacteristic
38C42×C17—21no—
39C42×C17C2×C411yescharacteristic
40#63227—21no—
41#63227—21no—
42#63227—21no—
43C4⋊Dic17—21no—
44C4⋊Dic17C2×C411yescharacteristic
45C4×Dic17—21no—
46C4×Dic17—21no—
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17C2311yescharacteristic
50C22×D17—21no—
51C22×D17C22⋊C411yescharacteristic
52Q8×C17—21no—
53Q8×C17M1612yes—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C2×Dic17—21no—
60C2×Dic17C22⋊C411yescharacteristic
61C2×Dic17C2×D411yescharacteristic
62C2×Dic17C2×D411yescharacteristic
63D68—24no—
64D68—21no—
65D68M1612yes—
66C4×D17—24no—
67C4×D17—21no—
68C4×D17—21no—
69C4×D17—22no—
70C4×D17—41no—
71C4×D17—42no—
72C4×D17—42no—
73#3624—42no—
74#3624—42no—
75#1488—171nomaximal, Sylow
76C22×C17C2×C22⋊C411yescharacteristic
77C2×D17—21no—
78C2×D17—22no—
79C2×D17—21no—
80C2×D17—21no—
81C2×D17—41no—
82C2×D17—41no—
83C4×C17—22no—
84C4×C17—21no—
85C4×C17—21no—
86C4×C17—41no—
87C4×C17C2×M1611yescharacteristic
88C4×C17C2×M1611yescharacteristic
89Dic17—21no—
90Dic17—21no—
91Dic17—41no—
92Dic17#13512yes—
93C4×#41—171no—
94#432—171no—
95#431—171no—
96Q8⋊C8—172no—
97D4⋊C8—172no—
98C2×C17#40711yescharacteristic
99C2×C17#41611yescharacteristic
100C2×C17#42111yescharacteristic
101D17—41no—
102D17—42no—
103C2×#41—171no—
104C4×Q8—171no—
105C4×Q8#59511yescharacteristic
106C4×D4—171no—
107C4×D4—171no—
108C4×D4—341no—
109C4×D4—341no—
110#117—171no—
111#117—341no—
112C2×C42—171no—
113C2×C42—341no—
114C4⋊C8—172no—
115C4⋊C8—341no—
116C22⋊C8—341no—
117C22⋊C8—341no—
118#97—341no—
119C4×C8—172no—
120C17#148811yescharacteristic, complemented, Sylow
121#41—342no—
122#41—342no—
123C2×Q8C2×#59511yescharacteristic
124C2×D4—171no—
125C2×D4—341no—
126C22×C4—171no—
127C22×C4—171no—
128C22×C4—171no—
129C22×C4—341no—
130C22×C4—341no—
131C22×C4—341no—
132C2×C8—342no—
133C2×C8—342no—
134C4⋊C4—21no—
135C4⋊C4—171no—
136C4⋊C4—341no—
137C4⋊C4C2×#59511yescharacteristic
138C22⋊C4—341no—
139C22⋊C4—341no—
140C22⋊C4—341no—
141C42—21no—
142C42—171no—
143C42—171no—
144C42—171no—
145C42—341no—
146C42—341no—
147C42C2×#59511yescharacteristic
148C23—171no—
149C23—341no—
150Q8—21no—
151Q8#6324312yes—
152D4—172no—
153D4—344no—
154D4—341no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—171no—
158C2×C4—171no—
159C2×C4—171no—
160C2×C4—342no—
161C2×C4—344no—
162C2×C4—341no—
163C2×C4—341no—
164C2×C4—341no—
165C2×C4—681no—
166C2×C4—682no—
167C2×C4—682no—
168C2×C4C22⋊#59511yescharacteristic
169C2×C4C22⋊#59511yescharacteristic
170C2×C4C22×#59511yescharacteristic
171C8—682no—
172C8—682no—
173C22—342no—
174C22—341no—
175C22—341no—
176C22—341no—
177C22—681no—
178C22—681no—
179C22C2×C22⋊#59511yescharacteristic
180C4—22no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—172no—
185C4—341no—
186C4—341no—
187C4—681no—
188C4C2×#6324311yescharacteristic
189C4D17⋊M1611yescharacteristic
190C2—681no—
191C2—682no—
192C2#1176378011yescharacteristic
193C2#1176379911yescharacteristic
194C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#63256C4
cosocleC4×Dic17C23
hypercenterC4×Q8#595
hypocenterC17#1488
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #11763831C2semidirect
2. #11763828C2semidirect
3. Q8⋊#3624C2semidirect
4. #11763657C2semidirect
120. C17#1488semidirect

Maximal subgroups

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

subgroupindex
#117638312
#117638282
Q8⋊#36242
#117636572
C4×#632562
#148817