Finite Group Databasebeta

← #36352027

Subgroups — #36352027

Order 22144.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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#26012028C211yesmaximal, characteristic
2#26011542C211yesmaximal, characteristic, complemented
3#26011513C211yesmaximal, characteristic, complemented
4Q8⋊#13063436C211yesmaximal, characteristic, complemented
5#26011220C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7Q8⋊C8×C173C211yesmaximal, characteristic, complemented
8C2×#14583809—21no—
9#19149674—21no—
10#19149670C411yescharacteristic
11#19149668C411yescharacteristic
12#19149664—21no—
13#19149655C411yescharacteristic
14#19149654C411yescharacteristic
15C4×D692—21no—
16C4×D692C2211yescharacteristic
17#19149636—21no—
18C42×D173C2211yescharacteristic
19C4⋊#13063436C2211yescharacteristic
20#19149608—21no—
21#19149608—21no—
22#19149591—21no—
23C4×#13063436C2211yescharacteristic
24C4×Q8×C173C2211yescharacteristic
25C4⋊C8×C173C2211yescharacteristic
26C4×C8×C173C2211yescharacteristic
27C2×D692—21no—
28C2×D692C2×C411yescharacteristic
29C2×C4×D173—21no—
30C2×C4×D173C2×C411yescharacteristic
31C2×C4×D173D411yescharacteristic
32C2×C4×D173D411yescharacteristic
33#14583809—42no—
34C2×#13063436—21no—
35C2×#13063436—21no—
36C4⋊Dic173—21no—
37C4⋊Dic173C2×C411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583803—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45C4×Dic173C2×C411yescharacteristic
46C4×Dic173D411yescharacteristic
47C4×Dic173D411yescharacteristic
48C2×Q8×C173C2×C411yescharacteristic
49C2×C8×C173—21no—
50C2×C8×C173—21no—
51C4⋊C4×C173—21no—
52C4⋊C4×C173C2×C411yescharacteristic
53C42×C173—21no—
54C42×C173C2311yescharacteristic
55C22×D173—21no—
56C22×D173C22⋊C411yescharacteristic
57D692—21no—
58D692—21no—
59D692—42no—
60C2×Dic173—21no—
61C2×Dic173C22⋊C411yescharacteristic
62C2×Dic173C22⋊C411yescharacteristic
63C2×Dic173C22⋊C411yescharacteristic
64C4×D173—21no—
65C4×D173—21no—
66C4×D173—21no—
67C4×D173—21no—
68C4×D173—41no—
69C4×D173—41no—
70C4×D173—41no—
71#13063436—41no—
72#13063436—41no—
73Q8×C173—21no—
74Q8×C173—21no—
75C2×C4×C173—21no—
76C2×C4×C173—21no—
77C2×C4×C173C22×C411yescharacteristic
78C2×C4×C173C2×D411yescharacteristic
79C2×C4×C173C2×D411yescharacteristic
80C8×C173—41no—
81C8×C173—41no—
82C22×C173C2×C22⋊C411yescharacteristic
83C2×D173—21no—
84C2×D173—21no—
85C2×D173—21no—
86C2×D173—41no—
87C2×D173—41no—
88C2×D173—41no—
89C4×C173—21no—
90C4×C173—41no—
91C4×C173—41no—
92C4×C173#13111yescharacteristic
93C4×C173#13111yescharacteristic
94C4×C173AGL(1,ℤ/8)11yescharacteristic
95C4×C173#13711yescharacteristic
96Dic173—21no—
97Dic173—21no—
98Dic173—21no—
99Dic173—41no—
100C2×C173#40811yescharacteristic
101C2×C173#41711yescharacteristic
102C2×C173#42111yescharacteristic
103D173—41no—
104D173—81no—
105C173#149511yescharacteristic, complemented, Sylow
106#1495—1731nomaximal, Sylow
107#529—1731no—
108#431—1731no—
109#423—1731no—
110Q8⋊C8—1731no—
111Q8⋊C8D17311yescharacteristic, complemented
112D4⋊C8—1731no—
113D4⋊C8—1731no—
114C2×M16—3461no—
115#126—3461no—
116#125—1731no—
117#124—1731no—
118#123—3461no—
119C22⋊Q8—1731no—
120#121—1731no—
121C4×Q8C2×D17311yescharacteristic
122C4×D4—3461no—
123C4×D4—1731no—
124#117—3461no—
125C2×C42—1731no—
126C4⋊C8—3461no—
127C4⋊C8—3461no—
128C4⋊C8—1731no—
129C4⋊C8C2×D17311yescharacteristic
130C22⋊C8—3461no—
131C4×C8—1731no—
132C4×C8C2×D17311yescharacteristic
133C2×Q8C4×D17311yescharacteristic
134C2×D4—3461no—
135C2×D4—1731no—
136C22×C4—3461no—
137C22×C4—1731no—
138C22×C4—1731no—
139C22×C4—1731no—
140M16—6922no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—3461no—
144C2×C8—3461no—
145C4⋊C4—21no—
146C4⋊C4—3461no—
147C4⋊C4—3461no—
148C4⋊C4—3461no—
149C4⋊C4—1731no—
150C4⋊C4C4×D17311yescharacteristic
151C22⋊C4—3461no—
152C22⋊C4—3461no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C22⋊C4—3461no—
156C42—21no—
157C42—1731no—
158C42—1731no—
159C42—1731no—
160C42C22×D17311yescharacteristic
161C23—3461no—
162C23—1731no—
163Q8—21no—
164Q8—21no—
165D4—3461no—
166D4—3461no—
167D4—6922no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—3461no—
171C2×C4—3461no—
172C2×C4—3461no—
173C2×C4—3461no—
174C2×C4—3461no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6921no—
181C2×C4C2×C4×D17311yescharacteristic
182C2×C4D4×D17311yescharacteristic
183C2×C4D4×D17311yescharacteristic
184C8—41no—
185C8—41no—
186C8—6921no—
187C8—6921no—
188C22—3461no—
189C22—3461no—
190C22—3461no—
191C22—6921no—
192C22—6921no—
193C22—6921no—
194C22C22⋊C4×D17311yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—3461no—
199C4—3461no—
200C4—3461no—
201C4—6921no—
202C4#1914968511yescharacteristic
203C4#1914968711yescharacteristic
204C4#1914970611yescharacteristic
205C4#1914971111yescharacteristic
206C2—13841no—
207C2—6921no—
208C2#2601145611yescharacteristic
209C2#2601149111yescharacteristic
210C2#2601150611yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
FittingQ8⋊C8×C173C2
socleC22×C173C2×C22⋊C4
prime radical#19149654C4
cosocleC42×C173C23
hypercenterQ8⋊C8D173
hypocenterC173#1495
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #26011542C2semidirect
3. #26011513C2semidirect
4. Q8⋊#13063436C2semidirect
5. #26011220C2semidirect
6. #26011218C2semidirect
7. Q8⋊C8×C173C2semidirect
105. C173#1495semidirect
111. Q8⋊C8D173semidirect

Maximal subgroups

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

subgroupindex
#260120282
#260115422
#260115132
Q8⋊#130634362
#260112202
#260112182
Q8⋊C8×C1732
#1495173