Finite Group Databasebeta

← #14545046

Subgroups — #14545046

Order 2176.

234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 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#11763795C211yesmaximal, characteristic
2#11763788C211yesmaximal, characteristic
3#11763711C214yesmaximal, complemented
4#11763383C211yesmaximal, characteristic
5C22×Q8×C17C411yescharacteristic, complemented
6C2×#63252C2211yescharacteristic
7#10587677—21no—
8D17⋊M16—21no—
9#10587633—21no—
10#10587632—21no—
11#10587632—21no—
12#10587630—21no—
13#10587615C412yescomplemented
14#10587615C2212yes—
15#10587612C411yescharacteristic, complemented
16#10587553C2212yes—
17#10587551C2212yes—
18#10587545—21no—
19#10587539—21no—
20C2×Q8×C17—21no—
21C2×Q8×C17—21no—
22C2×Q8×C17—41no—
23C2×Q8×C17—41no—
24C22×C4×C17C2×C412yes—
25C22×C4×C17C2×C411yescharacteristic
26C2×C22⋊D17C2×C411yescharacteristic
27C2×C22⋊D17C2311yescharacteristic
28#63252—41no—
29#63252—41no—
30C2×D68D411yescharacteristic
31C2×C4×D17D411yescharacteristic
32C2×C4×D17D411yescharacteristic
33C2×Dic34D411yescharacteristic
34C22⋊#595—22no—
35#63247—22no—
36C4⋊#595—41no—
37C4×#595—41no—
38#63243—41no—
39#63242—41no—
40C22⋊Dic17C2×C412yes—
41#63227—22no—
42#63227—22no—
43C4⋊Dic17—21no—
44#63225—22no—
45#63225—22no—
46C4×Dic17—21no—
47C23×C17C22×C411yescharacteristic
48C22×D17C22⋊C411yescharacteristic
49C22×D17C2×D411yescharacteristic
50C2×#595—42no—
51Q8×C17—42no—
52Q8×C17—41no—
53Q8×C17—41no—
54C2×C4×C17—21no—
55C2×C4×C17—21no—
56C2×C4×C17—22no—
57C2×C4×C17—42no—
58C2×C4×C17C22⋊C411yescharacteristic
59C2×C4×C17C22⋊C411yescharacteristic
60C22⋊D17—41no—
61C22⋊D17—41no—
62C2×Dic17—22no—
63C2×Dic17C22⋊C411yescharacteristic
64C2×Dic17C2×D411yescharacteristic
65D68—41no—
66C4×D17—21no—
67C4×D17—21no—
68C4×D17—21no—
69C4×D17—21no—
70Dic34—41no—
71#3624—42no—
72#2123—171nomaximal, Sylow
73C22×C17—21no—
74C22×C17—41no—
75C22×C17C2×C22⋊C411yescharacteristic
76C2×D17—21no—
77C2×D17—21no—
78C2×D17—41no—
79C2×D17—41no—
80#595—82no—
81C4×C17—21no—
82C4×C17—42no—
83C4×C17#9911yescharacteristic
84C4×C17#9911yescharacteristic
85Dic17—21no—
86Dic17—21no—
87Dic17—42no—
88#536—171no—
89#413—171no—
90#410—171no—
91#352—174no—
92C2×C17—21no—
93C2×C17—41no—
94C2×C17C2×#9911yescharacteristic
95D17—41no—
96D17—41no—
97C2×#41—171no—
98C22×Q8#59511yescharacteristic, complemented
99C2×M16—341no—
100C4⋊Q8—341no—
101#124—341no—
102#123—172no—
103#123—172no—
104C22⋊Q8—341no—
105C22⋊Q8—341no—
106#117—171no—
107#117—341no—
108#101—341no—
109#100—172no—
110#99—172no—
111#99—341no—
112C17#212311yescharacteristic, complemented, Sylow
113#41—681no—
114#41—681no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—171no—
120C2×D4—171no—
121C2×D4—171no—
122C2×D4—171no—
123C22×C4—171no—
124C22×C4—171no—
125C22×C4C2×#59512yes—
126C22×C4C2×#59511yescharacteristic
127M16—342no—
128M16—681no—
129C2×C8—681no—
130C4⋊C4—342no—
131C4⋊C4—342no—
132C4⋊C4—341no—
133C4⋊C4—681no—
134C22⋊C4—172no—
135C22⋊C4—342no—
136C22⋊C4—342no—
137C22⋊C4—342no—
138C42—341no—
139C42—681no—
140C23—171no—
141C23—171no—
142C23C22×#59511yescharacteristic
143Q8—42no—
144Q8—41no—
145Q8—41no—
146Q8—681no—
147D4—681no—
148D4—681no—
149D4—681no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—171no—
155C2×C4—171no—
156C2×C4—341no—
157C2×C4—341no—
158C2×C4—341no—
159C2×C4—341no—
160C2×C4—342no—
161C2×C4—682no—
162C2×C4C22⋊#59511yescharacteristic
163C2×C4C22⋊#59511yescharacteristic
164C8—682no—
165C22—21no—
166C22—41no—
167C22—341no—
168C22—341no—
169C22—681no—
170C22—681no—
171C22C2×C22⋊#59511yescharacteristic
172C4—21no—
173C4—42no—
174C4—341no—
175C4—341no—
176C4—682no—
177C4—1362no—
178C4#1058755111yescharacteristic
179C4#1058755111yescharacteristic
180C2—21no—
181C2—41no—
182C2—681no—
183C2—681no—
184C2C2×#1058755111yescharacteristic
subgroupG/H
centerC2C2×#10587551
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC22×Q8×C17C4
socleC2×C17C2×#99
prime radicalC2×#63252C22
cosocleC2×C22⋊D17C23
hypercenterC22×Q8#595
hypocenterC17#2123
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #11763711C2semidirect
5. C22×Q8×C17C4semidirect
13. #10587615C4semidirect
15. #10587612C4semidirect
98. C22×Q8#595semidirect
112. C17#2123semidirect

Maximal subgroups

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

subgroupindex
#117637952
#117637882
#117637112
#117633832
#212317