Finite Group Databasebeta

← #14544115

Subgroups — #14544115

Order 2176.

246 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763830C211yesmaximal, characteristic, complemented
2#11763826C211yesmaximal, characteristic, complemented
3Q8⋊#3624C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5#41×Dic17C211yesmaximal, characteristic
6C2×#41×C17C411yescharacteristic
7Q8×Dic17C411yescharacteristic
8Q8×Dic17C2211yescharacteristic
9D4×Dic17—21no—
10D4×Dic17—21no—
11D4×Dic17C411yescharacteristic
12D4×Dic17C2211yescharacteristic
13#10587612—21no—
14#10587612C411yescharacteristic
15C2×C4×Dic17—21no—
16C2×C4×Dic17C2211yescharacteristic
17C22⋊#3624—21no—
18C22⋊#3624—21no—
19#10587544—21no—
20#10587542—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23#41×C17—22no—
24#41×C17—22no—
25C2×Q8×C17C2×C411yescharacteristic
26C2×D4×C17—21no—
27C2×D4×C17C2×C411yescharacteristic
28C22×C4×C17—21no—
29C22×C4×C17C2×C411yescharacteristic
30C22×Dic17—21no—
31C22×Dic17—21no—
32C22×Dic17D411yescharacteristic
33C22×Dic17D411yescharacteristic
34C2×#3624—22no—
35C2×#3624—22no—
36C22⋊Dic17—21no—
37C22⋊Dic17—21no—
38C22⋊Dic17—21no—
39C4⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17C2×C411yescharacteristic
42C4⋊Dic17C2×C411yescharacteristic
43C4×Dic17—21no—
44C4×Dic17—21no—
45C4×Dic17—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17C2311yescharacteristic
50C23×C17—21no—
51C23×C17C22⋊C411yescharacteristic
52Q8×C17—21no—
53Q8×C17M1612yes—
54D4×C17—24no—
55D4×C17—21no—
56D4×C17M1612yes—
57C2×C4×C17—24no—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17C22⋊C411yescharacteristic
62C2×C4×C17C22×C411yescharacteristic
63C2×Dic17—22no—
64C2×Dic17—21no—
65C2×Dic17—21no—
66C2×Dic17—41no—
67C2×Dic17—42no—
68C2×Dic17—42no—
69C2×Dic17C22⋊C411yescharacteristic
70C2×Dic17C22⋊C411yescharacteristic
71C2×Dic17C2×D411yescharacteristic
72C2×Dic17C2×D411yescharacteristic
73#3624—42no—
74#3624—42no—
75#1488—171nomaximal, Sylow
76C22×C17—21no—
77C22×C17—22no—
78C22×C17—21no—
79C22×C17—21no—
80C22×C17—41no—
81C22×C17—41no—
82C22×C17C2×C22⋊C411yescharacteristic
83C4×C17—21no—
84C4×C17—22no—
85C4×C17C2×M1611yescharacteristic
86C4×C17C2×M1611yescharacteristic
87Dic17—21no—
88Dic17—21no—
89Dic17—21no—
90Dic17—41no—
91Dic17—41no—
92Dic17#13512yes—
93C4×#41—171no—
94#432—171no—
95#431—171no—
96Q8⋊C8—172no—
97D4⋊C8—172no—
98C2×C17—41no—
99C2×C17—42no—
100C2×C17#40711yescharacteristic
101C2×C17#41611yescharacteristic
102C2×C17#42111yescharacteristic
103C2×#41#59511yescharacteristic
104C4×Q8—171no—
105C4×Q8—171no—
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—22no—
122#41—22no—
123C2×Q8C2×#59511yescharacteristic
124C2×D4—21no—
125C2×D4C2×#59511yescharacteristic
126C22×C4—21no—
127C22×C4—171no—
128C22×C4—171no—
129C22×C4—341no—
130C22×C4—341no—
131C22×C4C2×#59511yescharacteristic
132C2×C8—342no—
133C2×C8—342no—
134C4⋊C4—171no—
135C4⋊C4—171no—
136C4⋊C4—341no—
137C4⋊C4—341no—
138C22⋊C4—341no—
139C22⋊C4—341no—
140C22⋊C4—341no—
141C42—171no—
142C42—171no—
143C42—171no—
144C42—171no—
145C42—341no—
146C42—341no—
147C42—341no—
148C23—21no—
149C23C22⋊#59511yescharacteristic
150Q8—21no—
151Q8#6324712yes—
152D4—24no—
153D4—21no—
154D4#6324712yes—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—171no—
160C2×C4—171no—
161C2×C4—171no—
162C2×C4—171no—
163C2×C4—342no—
164C2×C4—341no—
165C2×C4—341no—
166C2×C4—682no—
167C2×C4—681no—
168C2×C4—682no—
169C2×C4C22⋊#59511yescharacteristic
170C2×C4C22×#59511yescharacteristic
171C8—682no—
172C8—682no—
173C22—21no—
174C22—22no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22C2×C22⋊#59511yescharacteristic
180C4—21no—
181C4—22no—
182C4—172no—
183C4—341no—
184C4—341no—
185C4—341no—
186C4—681no—
187C4—681no—
188C4C2×#6324711yescharacteristic
189C4C2×#6324711yescharacteristic
190C2—41no—
191C2—42no—
192C2#1176377911yescharacteristic
193C2#1176379911yescharacteristic
194C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
Fitting, prime radicalC2×#41×C17C4
socleC22×C17C2×C22⋊C4
cosocleC4×Dic17C23
hypercenterC2×#41#595
hypocenterC17#1488
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #11763830C2semidirect
2. #11763826C2semidirect
3. Q8⋊#3624C2semidirect
4. D4⋊#3624C2semidirect
120. C17#1488semidirect

Maximal subgroups

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

subgroupindex
#117638302
#117638262
Q8⋊#36242
D4⋊#36242
#41×Dic172
#148817