Finite Group Databasebeta

← #36355507

Subgroups — #36355507

Order 22144.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#26012288C211yesmaximal, characteristic, complemented
2#26011927C211yesmaximal, characteristic, complemented
3#26011741C211yesmaximal, characteristic, complemented
4#26011302C211yesmaximal, characteristic, complemented
5#26011301C211yesmaximal, characteristic, complemented
6#26011272C211yesmaximal, characteristic, complemented
7#509×C173C211yesmaximal, characteristic, complemented
8#41×D173—21no—
9#19149711—21no—
10#19149708—21no—
11#19149706—21no—
12#19149703—21no—
13#19149701C2211yescharacteristic
14#19149699—21no—
15C2×#14583809C2211yescharacteristic
16#19149650—21no—
17#19149623—21no—
18#19149620—21no—
19#19149619C2211yescharacteristic, complemented
20#19149611C2211yescharacteristic, complemented
21#19149600—21no—
22#135×C173C2211yescharacteristic
23C2×D8×C173C2211yescharacteristic, complemented
24SD32×C173—21no—
25D16×C173—21no—
26M32×C173C2211yescharacteristic, complemented
27#14583827—21no—
28#14583826—21no—
29#14583826—41no—
30#14583825—21no—
31#14583825—41no—
32Q8×D173—21no—
33C2×C22⋊D173—41no—
34D4×D173—21no—
35D4×D173—41no—
36C2×C4×D173—41no—
37C2×C4×D173D411yescharacteristic
38Dic692D411yescharacteristic, complemented
39#14583816—21no—
40#14583815—21no—
41#14583814—21no—
42#14583813—21no—
43D1384D411yescharacteristic, complemented
44#14583811—21no—
45#14583809—21no—
46#14583809—21no—
47C2×#13063436D411yescharacteristic
48C4⋊Dic173—21no—
49C22⋊Dic173—41no—
50#14583801—21no—
51#41×C173—21no—
52C2×D4×C173—21no—
53Q16×C173D411yescharacteristic, complemented
54SD16×C173—21no—
55D8×C173—21no—
56D8×C173—21no—
57D8×C173D411yescharacteristic, complemented
58C2×C8×C173C2311yescharacteristic
59C16×C173—21no—
60C22×D173—41no—
61C22×D173D811yescharacteristic
62Dic346—21no—
63Dic346—41no—
64D692—21no—
65D692—41no—
66C22⋊D173—41no—
67C22⋊D173—41no—
68C22⋊D173—41no—
69C22⋊D173—82no—
70C2×Dic173—41no—
71C2×Dic173—41no—
72C2×Dic173D811yescharacteristic
73C4×D173—21no—
74C4×D173—21no—
75C4×D173—41no—
76C4×D173—41no—
77C4×D173—41no—
78#13063436—21no—
79C23×C173—41no—
80Q8×C173—21no—
81D4×C173—21no—
82D4×C173—41no—
83D4×C173—41no—
84D4×C173—41no—
85C2×C4×C173—41no—
86C2×C4×C173C2×D411yescharacteristic
87C8×C173C2×D411yescharacteristic
88C8×C173C2×D411yescharacteristic
89C22×C173—41no—
90C22×C173—81no—
91C22×C173—81no—
92C22×C173—81no—
93C22×C173C2×D811yescharacteristic
94C2×D173—21no—
95C2×D173—41no—
96C2×D173—41no—
97C2×D173—82no—
98C4×C173—41no—
99C4×C173#12011yescharacteristic
100C4×C173AGL(1,ℤ/8)11yescharacteristic
101Dic173—21no—
102Dic173—41no—
103Dic173—81no—
104C2×C173—21no—
105C2×C173—81no—
106C2×C173—161no—
107C2×C173#44711yescharacteristic
108D173—41no—
109D173—81no—
110C173#219011yescharacteristic, complemented, Sylow
111#2190—1731nomaximal, Sylow
112#575—1731no—
113#509—1731no—
114#509D17311yescharacteristic, complemented
115#469—1731no—
116#360—1731no—
117#360—1731no—
118#349—1731no—
119C2×#41—3461no—
120#137—3461no—
121#137—3461no—
122AGL(1,ℤ/8)—3461no—
123AGL(1,ℤ/8)—3461no—
124#135—3461no—
125#135—1731no—
126#135C2×D17311yescharacteristic
127C2×D8C2×D17311yescharacteristic, complemented
128C2×M16—1731no—
129#121—3461no—
130SD32—21no—
131SD32—3461no—
132D16—21no—
133D16—3461no—
134M32—1731no—
135M32C2×D17311yescharacteristic, complemented
136#106—1731no—
137#102—3461no—
138#41—21no—
139#41—3461no—
140#41—3461no—
141#41—3461no—
142#41—6921no—
143#41—6921no—
144C2×Q8—3461no—
145C2×D4—21no—
146C2×D4—3461no—
147C2×D4—6921no—
148C2×D4—6921no—
149C22×C4—1731no—
150C22×C4—6921no—
151Q16—3461no—
152Q16—1731no—
153Q16C22⋊D17311yescharacteristic, complemented
154SD16—21no—
155SD16—3461no—
156SD16—3461no—
157SD16—3461no—
158D8—21no—
159D8—21no—
160D8—3461no—
161D8—1731no—
162D8C22⋊D17311yescharacteristic, complemented
163M16—3461no—
164M16—3461no—
165C2×C8—1731no—
166C2×C8C22×D17311yescharacteristic
167C4⋊C4—3461no—
168C22⋊C4—6921no—
169C16—21no—
170C16—3461no—
171C23—41no—
172C23—1731no—
173C23—6921no—
174Q8—21no—
175Q8—3461no—
176Q8—6921no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—3461no—
182D4—13842no—
183D4—6921no—
184D4—6921no—
185D4—6921no—
186D4—6921no—
187C2×C4—41no—
188C2×C4—3461no—
189C2×C4—3461no—
190C2×C4—1731no—
191C2×C4—6921no—
192C2×C4—6921no—
193C2×C4—6921no—
194C2×C4—6921no—
195C2×C4—6921no—
196C2×C4D4×D17311yescharacteristic
197C8—3461no—
198C8D4×D17311yescharacteristic
199C8C2×C22⋊D17311yescharacteristic
200C22—41no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—3461no—
205C22—13842no—
206C22—6921no—
207C22—6921no—
208C22D8×D17311yescharacteristic
209C4—41no—
210C4—3461no—
211C4—13841no—
212C4—6921no—
213C4#1914964711yescharacteristic
214C4#1914970311yescharacteristic
215C2—21no—
216C2—81no—
217C2—161no—
218C2—13841no—
219C2—6921no—
220C2#2601162311yescharacteristic
subgroupG/H
centerC2#26011623
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting#509×C173C2
socleC2×C173#447
hypercenter#509D173
hypocenterC173#2190
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #26012288C2semidirect
2. #26011927C2semidirect
3. #26011741C2semidirect
4. #26011302C2semidirect
5. #26011301C2semidirect
6. #26011272C2semidirect
7. #509×C173C2semidirect
19. #19149619C22semidirect
20. #19149611C22semidirect
23. C2×D8×C173C22semidirect
26. M32×C173C22semidirect
38. Dic692D4semidirect
43. D1384D4semidirect
53. Q16×C173D4semidirect
57. D8×C173D4semidirect
110. C173#2190semidirect
114. #509D173semidirect
127. C2×D8C2×D173semidirect
135. M32C2×D173semidirect
153. Q16C22⋊D173semidirect
162. D8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122882
#260119272
#260117412
#260113022
#260113012
#260112722
#509×C1732
#2190173