Finite Group Databasebeta

← #14544099

Subgroups — #14544099

Order 2176.

234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×C4⋊#3624C211yesmaximal, characteristic, complemented
2C4×#63247C211yesmaximal, characteristic, complemented
3#11763659C212yesmaximal, complemented
4D4⋊#3624C212yesmaximal, complemented
5#11763340C211yesmaximal, characteristic
6C2×#63247—21no—
7C22×#3624—21no—
8C17×#121C411yescharacteristic
9#10587622C411yescharacteristic
10#10587621—21no—
11D4×Dic17—21no—
12D4×Dic17C2211yescharacteristic
13#10587610C411yescharacteristic
14C2×C4×Dic17C2211yescharacteristic
15#10587576C411yescharacteristic
16#10587573C2211yescharacteristic
17#10587565—21no—
18#10587563—21no—
19#10587542—21no—
20C4⋊#3624—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23C2×D4×C17—21no—
24C2×D4×C17C2×C411yescharacteristic
25C22×C4×C17C2×C411yescharacteristic
26C22×Dic17—21no—
27C22×Dic17D411yescharacteristic
28C22×Dic17D411yescharacteristic
29C2×Dic34C2×C411yescharacteristic
30#63247—24no—
31C2×#3624—22no—
32C2×#3624—22no—
33C2×#3624—42no—
34C4⋊C4×C17C2×C411yescharacteristic
35C22⋊C4×C17—21no—
36C22⋊Dic17—21no—
37C22⋊Dic17—21no—
38C22⋊Dic17—21no—
39C22⋊Dic17—21no—
40C4⋊Dic17—21no—
41C4⋊Dic17C2×C411yescharacteristic
42#63225—21no—
43#63225—21no—
44#63225—21no—
45C4×Dic17—21no—
46C4×Dic17C2×C411yescharacteristic
47C4×Dic17D411yescharacteristic
48C4×Dic17D411yescharacteristic
49C4×Dic17C2311yescharacteristic
50C23×C17—21no—
51C23×C17C22⋊C411yescharacteristic
52D4×C17—22no—
53D4×C17—42no—
54C2×C4×C17—22no—
55C2×C4×C17—21no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22×C411yescharacteristic
58C2×Dic17—22no—
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17—21no—
62C2×Dic17—21no—
63C2×Dic17—41no—
64C2×Dic17—41no—
65C2×Dic17C22⋊C411yescharacteristic
66C2×Dic17C22⋊C411yescharacteristic
67C2×Dic17C2×D411yescharacteristic
68C2×Dic17C2×D411yescharacteristic
69Dic34—22no—
70#3624—24no—
71#3624—42no—
72#1483—171nomaximal, Sylow
73C22×C17—21no—
74C22×C17—21no—
75C22×C17—41no—
76C22×C17—41no—
77C22×C17—41no—
78C22×C17#10412yes—
79C22×C17C2×C22⋊C411yescharacteristic
80C4×C17—21no—
81C4×C17—41no—
82C4×C17#13112yes—
83Dic17—21no—
84Dic17—21no—
85Dic17—21no—
86Dic17—41no—
87Dic17—41no—
88Dic17AGL(1,ℤ/8)11yescharacteristic
89Dic17#13711yescharacteristic
90#529—171no—
91C2×C4⋊C8—171no—
92C4×M16—171no—
93Q8⋊C8—172no—
94D4⋊C8—172no—
95C2×C17—22no—
96C2×C17—81no—
97C2×C17#40811yescharacteristic
98C2×C17#41711yescharacteristic
99C2×C17C2×#10411yescharacteristic
100C2×M16—341no—
101C22×C8—341no—
102#126—341no—
103#125—171no—
104#124—171no—
105#123—341no—
106C22⋊Q8—171no—
107#121#59511yescharacteristic
108C4×Q8—171no—
109C4×D4—171no—
110C4×D4—341no—
111#117—341no—
112C2×C42—171no—
113C4⋊C8—172no—
114C4⋊C8—341no—
115#97—341no—
116C4×C8—172no—
117C17#148311yescharacteristic, complemented, Sylow
118C2×Q8—171no—
119C2×D4—21no—
120C2×D4C2×#59511yescharacteristic
121C22×C4—171no—
122C22×C4—171no—
123C22×C4—341no—
124C22×C4C2×#59511yescharacteristic
125M16—344no—
126C2×C8—342no—
127C2×C8—342no—
128C2×C8—682no—
129C4⋊C4—171no—
130C4⋊C4—341no—
131C4⋊C4—341no—
132C4⋊C4—341no—
133C4⋊C4—341no—
134C4⋊C4C2×#59511yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—341no—
137C22⋊C4—341no—
138C22⋊C4—341no—
139C22⋊C4—341no—
140C42—171no—
141C42—171no—
142C42—171no—
143C42—171no—
144C42—341no—
145C23—21no—
146C23C22⋊#59511yescharacteristic
147Q8—342no—
148D4—22no—
149D4—42no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—171no—
153C2×C4—171no—
154C2×C4—171no—
155C2×C4—171no—
156C2×C4—341no—
157C2×C4—341no—
158C2×C4—342no—
159C2×C4—341no—
160C2×C4—341no—
161C2×C4—681no—
162C2×C4—681no—
163C2×C4C22⋊#59511yescharacteristic
164C2×C4C22×#59511yescharacteristic
165C8—344no—
166C8—682no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1058754812yes—
173C22C2×C22⋊#59511yescharacteristic
174C4—21no—
175C4—41no—
176C4—171no—
177C4—171no—
178C4—341no—
179C4—341no—
180C4—341no—
181C4—681no—
182C4—681no—
183C4#1058767812yes—
184C2—22no—
185C2—81no—
186C2#1176378211yescharacteristic
187C2#1176380011yescharacteristic
188C2C2×#1058754811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
Fitting, prime radicalC17×#121C4
socleC22×C17C2×C22⋊C4
cosocleC4×Dic17C23
hypercenter#121#595
hypocenterC17#1483
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#3624C2semidirect
2. C4×#63247C2semidirect
3. #11763659C2semidirect
4. D4⋊#3624C2semidirect
117. C17#1483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#36242
C4×#632472
#117636592
D4⋊#36242
#117633402
#148317