Finite Group Databasebeta

← #14544127

Subgroups — #14544127

Order 2176.

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

normal subgroups only

This group has 252 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#11763831C211yesmaximal, characteristic
2C4×#63243C211yesmaximal, characteristic
3#11763659C212yesmaximal, complemented
4Q8⋊#3624C212yesmaximal, complemented
5C4×Q8×D17C211yesmaximal, characteristic
6C2×Q8×D17C411yescharacteristic
7C2×#63243—21no—
8C4×Q8×C17C411yescharacteristic
9Q8×Dic17C2211yescharacteristic
10C4⋊C4×D17—21no—
11C4⋊C4×D17C411yescharacteristic
12#10587573—21no—
13#10587573C2211yescharacteristic
14C42×D17—21no—
15C42×D17C2211yescharacteristic
16C4×Dic34—21no—
17C4×Dic34C411yescharacteristic
18#10587543—21no—
19#10587542—21no—
20#10587542—21no—
21C4⋊#3624C2212yes—
22C4×#3624C2212yes—
23C2×Q8×C17C2×C411yescharacteristic
24Q8×D17—22no—
25Q8×D17—22no—
26C2×C4×D17—21no—
27C2×C4×D17—21no—
28C2×C4×D17C2×C411yescharacteristic
29C2×C4×D17D411yescharacteristic
30C2×C4×D17D411yescharacteristic
31C2×Dic34—21no—
32C2×Dic34C2×C411yescharacteristic
33C2×#3624—22no—
34C2×#3624—22no—
35#63243—24no—
36C4⋊C4×C17—21no—
37C4⋊C4×C17C2×C411yescharacteristic
38C42×C17—21no—
39C42×C17C2×C411yescharacteristic
40C4⋊Dic17—21no—
41C4⋊Dic17C2×C411yescharacteristic
42#63225—21no—
43#63225—21no—
44#63225—21no—
45C4×Dic17—21no—
46C4×Dic17—21no—
47C4×Dic17—21no—
48C4×Dic17D411yescharacteristic
49C4×Dic17D411yescharacteristic
50C4×Dic17C2311yescharacteristic
51C22×D17C22⋊C411yescharacteristic
52Q8×C17—21no—
53Q8×C17M1612yescomplemented
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17C22⋊C411yescharacteristic
57C2×C4×C17C22⋊C411yescharacteristic
58C2×C4×C17C22×C411yescharacteristic
59C2×Dic17—21no—
60C2×Dic17—21no—
61C2×Dic17C22⋊C411yescharacteristic
62C2×Dic17C2×D411yescharacteristic
63C2×Dic17C2×D411yescharacteristic
64C4×D17—24no—
65C4×D17—22no—
66C4×D17—21no—
67C4×D17—21no—
68C4×D17—22no—
69C4×D17—42no—
70Dic34—24no—
71Dic34—21no—
72Dic34M1612yescomplemented
73#3624—24no—
74#3624—42no—
75#1491—171nomaximal, Sylow
76C22×C17C2×C22⋊C411yescharacteristic
77C2×D17—21no—
78C2×D17—21no—
79C2×D17#10412yes—
80C4×C17—22no—
81C4×C17—21no—
82C4×C17—21no—
83C4×C17—41no—
84C4×C17C2×M1611yescharacteristic
85C4×C17C2×M1611yescharacteristic
86Dic17—21no—
87Dic17—22no—
88Dic17—21no—
89Dic17—41no—
90Dic17#13712yes—
91C2×C4×Q8—171no—
92#432—171no—
93C4×M16—171no—
94Q8⋊C8—172no—
95Q8⋊C8—172no—
96C2×C17#40711yescharacteristic
97C2×C17#41911yescharacteristic
98C2×C17C2×#10411yescharacteristic
99D17—22no—
100C22×Q8—171no—
101C2×M16—341no—
102C4×Q8—171no—
103C4×Q8—171no—
104C4×Q8—171no—
105C4×Q8—341no—
106C4×Q8—341no—
107C4×Q8#59511yescharacteristic
108C2×C4⋊C4—171no—
109C2×C4⋊C4—341no—
110C2×C42—171no—
111C2×C42—341no—
112C4⋊C8—172no—
113C22⋊C8—341no—
114#97—341no—
115#97—341no—
116C4×C8—172no—
117C17#149111yescharacteristic, complemented, Sylow
118C2×Q8—171no—
119C2×Q8—341no—
120C2×Q8—342no—
121C2×Q8—342no—
122C2×Q8C2×#59511yescharacteristic
123C22×C4—171no—
124C22×C4—171no—
125C22×C4—171no—
126C22×C4—341no—
127C22×C4—341no—
128M16—344no—
129C2×C8—342no—
130C2×C8—342no—
131C4⋊C4—21no—
132C4⋊C4—171no—
133C4⋊C4—341no—
134C4⋊C4—341no—
135C4⋊C4—341no—
136C4⋊C4—341no—
137C4⋊C4C2×#59511yescharacteristic
138C42—21no—
139C42—171no—
140C42—171no—
141C42—171no—
142C42—341no—
143C42—341no—
144C42—341no—
145C42C2×#59511yescharacteristic
146C23—171no—
147Q8—21no—
148Q8—172no—
149Q8—344no—
150Q8—341no—
151Q8#6324312yescomplemented
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—171no—
155C2×C4—171no—
156C2×C4—171no—
157C2×C4—342no—
158C2×C4—342no—
159C2×C4—344no—
160C2×C4—341no—
161C2×C4—341no—
162C2×C4—341no—
163C2×C4—341no—
164C2×C4—682no—
165C2×C4C22⋊#59511yescharacteristic
166C2×C4C22⋊#59511yescharacteristic
167C2×C4C22×#59511yescharacteristic
168C8—344no—
169C8—682no—
170C22—172no—
171C22—341no—
172C22—341no—
173C22C2×C22⋊#59511yescharacteristic
174C4—22no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—172no—
179C4—341no—
180C4—342no—
181C4—341no—
182C4—681no—
183C4C2×#6324311yescharacteristic
184C4D17⋊M1611yescharacteristic
185C2—342no—
186C2#1176378011yescharacteristic
187C2#1176380211yescharacteristic
188C2D17⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC4×Q8×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×D17C22⋊C4
cosocleC4×Dic17C23
hypercenterC4×Q8#595
hypocenterC17#1491
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
3. #11763659C2semidirect
4. Q8⋊#3624C2semidirect
53. Q8×C17M16semidirect
72. Dic34M16semidirect
117. C17#1491semidirect
151. Q8#63243semidirect

Maximal subgroups

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

subgroupindex
#117638312
C4×#632432
#117636592
Q8⋊#36242
C4×Q8×D172
#149117