Finite Group Databasebeta

← #36352626

Subgroups — #36352626

Order 22144.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#26012163C211yesmaximal, characteristic, complemented
2#26011867C211yesmaximal, characteristic, complemented
3#26011683C211yesmaximal, characteristic, complemented
4#26011234C211yesmaximal, characteristic, complemented
5Q8⋊#13063436C211yesmaximal, characteristic, complemented
6#26011218C211yesmaximal, characteristic, complemented
7#460×C173C211yesmaximal, characteristic, complemented
8C2×#14583827—21no—
9C2×#14583811—21no—
10C2×#14583811—22no—
11#19149676—21no—
12#19149676C2211yescharacteristic
13#19149654—21no—
14#19149654—22no—
15C4×D692—21no—
16C4×D692C2211yescharacteristic
17C4⋊#13063436C2211yescharacteristic
18#19149600—21no—
19C4×#13063436C2211yescharacteristic
20C2×SD16×C173—21no—
21#127×C173C2211yescharacteristic
22C4×Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25#14583827—42no—
26#14583827—42no—
27C2×D692—22no—
28C2×D692—22no—
29C2×D692—21no—
30C2×D692—21no—
31C2×D692D411yescharacteristic
32C2×D692D412yes—
33C2×C4×D173—22no—
34C2×C4×D173—21no—
35#14583811—44no—
36#14583811—42no—
37C2×#13063436—21no—
38C2×#13063436—21no—
39C4⋊Dic173D411yescharacteristic
40#14583803—22no—
41#14583803—21no—
42C2×Q8×C173D411yescharacteristic
43C2×D4×C173—21no—
44C2×D4×C173—21no—
45C2×D4×C173—21no—
46SD16×C173—42no—
47C2×C8×C173—21no—
48C4⋊C4×C173—21no—
49C4⋊C4×C173D411yescharacteristic
50C42×C173—21no—
51C42×C173C2311yescharacteristic
52C22×D173—21no—
53C22×D173—22no—
54D692—21no—
55D692—21no—
56D692—42no—
57D692—44no—
58D692—41no—
59D692—41no—
60D692—41no—
61D692—41no—
62D692—82no—
63C2×Dic173—21no—
64C4×D173—44no—
65C4×D173—41no—
66C4×D173—41no—
67#13063436—22no—
68#13063436—41no—
69C23×C173—41no—
70Q8×C173—21no—
71Q8×C173D812yescomplemented
72D4×C173—42no—
73D4×C173—41no—
74D4×C173—41no—
75D4×C173—41no—
76D4×C173—41no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173C2×D411yescharacteristic
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D411yescharacteristic
82C8×C173—41no—
83C22×C173—81no—
84C22×C173—81no—
85C22×C173—81no—
86C22×C173#12011yescharacteristic
87C2×D173—42no—
88C2×D173—41no—
89C2×D173—41no—
90C2×D173—42no—
91C2×D173—42no—
92C2×D173—41no—
93C4×C173—22no—
94C4×C173—21no—
95C4×C173—41no—
96C4×C173C2×D811yescharacteristic
97C4×C173#13511yescharacteristic
98C4×C173AGL(1,ℤ/8)11yescharacteristic
99C4×C173AGL(1,ℤ/8)11yescharacteristic
100Dic173—41no—
101C2×C173—161no—
102C2×C173#44711yescharacteristic
103C2×C173#44911yescharacteristic
104C2×C173#45311yescharacteristic
105D173—81no—
106D173—82no—
107C173#162111yescharacteristic, complemented, Sylow
108#1621—1731nomaximal, Sylow
109#550—1731no—
110#493—1731no—
111#460D17311yescharacteristic, complemented
112#459—1731no—
113#331—1731no—
114Q8⋊C8—1731no—
115D4⋊C8—1731no—
116C2×#41—3461no—
117C2×SD16—21no—
118C2×D8—3461no—
119C2×D8—3462no—
120#127—3461no—
121#127—1731no—
122#127C2×D17311yescharacteristic
123#121—3462no—
124#121—3461no—
125C4×Q8C2×D17311yescharacteristic
126C4×D4—3461no—
127C4×D4—1731no—
128C4⋊C8—1731no—
129C4⋊C8C2×D17311yescharacteristic
130#102—21no—
131#102—3461no—
132C4×C8—1731no—
133#41—6922no—
134#41—6922no—
135C2×Q8C22⋊D17311yescharacteristic
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—3462no—
140C2×D4—3462no—
141C2×D4—3461no—
142C2×D4—3461no—
143C2×D4—1731no—
144C2×D4—1732no—
145C22×C4—3462no—
146C22×C4—3461no—
147SD16—42no—
148D8—6924no—
149D8—6922no—
150C2×C8—21no—
151C2×C8—3461no—
152C2×C8—3461no—
153C4⋊C4—21no—
154C4⋊C4—1731no—
155C4⋊C4C22⋊D17311yescharacteristic
156C22⋊C4—3462no—
157C22⋊C4—3461no—
158C42—21no—
159C42C22×D17311yescharacteristic
160C23—41no—
161C23—3461no—
162C23—3462no—
163Q8—21no—
164Q8#1458381112yescomplemented
165D4—42no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—3461no—
171D4—3461no—
172D4—13842no—
173D4—6922no—
174D4—6924no—
175D4—6921no—
176D4—6921no—
177D4—6921no—
178D4—6921no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—3461no—
182C2×C4—6924no—
183C2×C4—6921no—
184C2×C4—6921no—
185C2×C4D4×D17311yescharacteristic
186C2×C4D4×D17311yescharacteristic
187C2×C4C2×C22⋊D17311yescharacteristic
188C8—41no—
189C8—3462no—
190C8—6921no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—6921no—
195C22—6922no—
196C22—6922no—
197C22—6921no—
198C22—6921no—
199C22—6922no—
200C22#1914964711yescharacteristic
201C4—21no—
202C4—22no—
203C4—41no—
204C4—6921no—
205C4C2×#1458381111yescharacteristic
206C4#1914970011yescharacteristic
207C4#1914970411yescharacteristic
208C4#1914970611yescharacteristic
209C2—161no—
210C2—13841no—
211C2—13842no—
212C2#2601162411yescharacteristic
213C2#2601163511yescharacteristic
214C2#2601165711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#460×C173C2
socleC22×C173#120
hypercenter#460D173
hypocenterC173#1621
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #26012163C2semidirect
2. #26011867C2semidirect
3. #26011683C2semidirect
4. #26011234C2semidirect
5. Q8⋊#13063436C2semidirect
6. #26011218C2semidirect
7. #460×C173C2semidirect
71. Q8×C173D8semidirect
107. C173#1621semidirect
111. #460D173semidirect
164. Q8#14583811semidirect

Maximal subgroups

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

subgroupindex
#260121632
#260118672
#260116832
#260112342
Q8⋊#130634362
#260112182
#460×C1732
#1621173