Finite Group Databasebeta

← #36352755

Subgroups — #36352755

Order 22144.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012162C211yesmaximal, characteristic
2#26011839C211yesmaximal, characteristic, complemented
3#26011698C211yesmaximal, characteristic, complemented
4#26011234C211yesmaximal, characteristic, complemented
5#26011221C211yesmaximal, characteristic, complemented
6D4⋊#13063436C211yesmaximal, characteristic, complemented
7#474×C173C211yesmaximal, characteristic, complemented
8C2×#14583826—21no—
9C2×#14583816—21no—
10#19149676C2211yescharacteristic
11#19149675—21no—
12#19149653—21no—
13#19149652—22no—
14C4×Dic346C2211yescharacteristic
15#19149641—21no—
16C4⋊#13063436C2211yescharacteristic
17#19149603—21no—
18#19149599—22no—
19C4×#13063436C2211yescharacteristic
20C4⋊Q8×C173C2211yescharacteristic
21C4×D4×C173C2211yescharacteristic
22#107×C173—21no—
23C4⋊C8×C173C2211yescharacteristic
24#102×C173—21no—
25#14583826—42no—
26#14583826—42no—
27C2×Dic346D411yescharacteristic
28C2×D692—21no—
29C2×D692—21no—
30C2×D692D412yes—
31C2×C22⋊D173—22no—
32C2×C22⋊D173—22no—
33C2×C4×D173—22no—
34#14583816—42no—
35C2×#13063436—21no—
36C2×#13063436—21no—
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583803—22no—
40C4×Dic173—21no—
41C2×Q8×C173—21no—
42C2×D4×C173D411yescharacteristic
43C22×C4×C173—21no—
44C2×C8×C173—21no—
45C4⋊C4×C173—21no—
46C4⋊C4×C173—21no—
47C4⋊C4×C173D411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—22no—
51Dic346—21no—
52Dic346D812yes—
53D692—42no—
54D692—41no—
55D692—41no—
56D692—41no—
57D692—41no—
58C22⋊D173—44no—
59C22⋊D173—82no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C4×D173—44no—
63#13063436—41no—
64#13063436—41no—
65C23×C173—21no—
66Q8×C173—41no—
67Q8×C173—41no—
68D4×C173—21no—
69D4×C173—21no—
70C2×C4×C173—21no—
71C2×C4×C173—41no—
72C2×C4×C173—41no—
73C2×C4×C173—41no—
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C2×C4×C173C2×D411yescharacteristic
77C8×C173—41no—
78C22×C173—41no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173#12011yescharacteristic
82C2×D173—42no—
83C2×D173—42no—
84C2×D173—42no—
85C4×C173—21no—
86C4×C173—41no—
87C4×C173—81no—
88C4×C173C2×D811yescharacteristic
89C4×C173#13511yescharacteristic
90C4×C173AGL(1,ℤ/8)11yescharacteristic
91C4×C173#13711yescharacteristic
92Dic173—22no—
93Dic173—41no—
94C2×C173—81no—
95C2×C173#44711yescharacteristic
96C2×C173#45211yescharacteristic
97C2×C173#45511yescharacteristic
98D173—82no—
99C173#164111yescharacteristic, complemented, Sylow
100#1641—1731nomaximal, Sylow
101#550—1731no—
102#486—1731no—
103#474D17311yescharacteristic, complemented
104#462—1731no—
105#331—1731no—
106Q8⋊C8—1731no—
107D4⋊C8—1731no—
108C2×#41—3461no—
109C2×Q16—3461no—
110C4⋊Q8C2×D17311yescharacteristic
111#127—3461no—
112#127—1731no—
113#121—3462no—
114#121—3461no—
115C4×Q8—1731no—
116C4×D4—3461no—
117C4×D4C2×D17311yescharacteristic
118#107—21no—
119C4⋊C8—1731no—
120C4⋊C8C2×D17311yescharacteristic
121#103—3461no—
122#102—21no—
123#102—3462no—
124C4×C8—1731no—
125#41—6922no—
126#41—6922no—
127C2×Q8—21no—
128C2×Q8—1731no—
129C2×D4—3462no—
130C2×D4—3462no—
131C2×D4—3461no—
132C2×D4—3461no—
133C2×D4—1732no—
134C2×D4C22⋊D17311yescharacteristic
135C22×C4—21no—
136C22×C4—3462no—
137Q16—6922no—
138C2×C8—21no—
139C2×C8—3461no—
140C2×C8—3461no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—3461no—
144C4⋊C4—1731no—
145C4⋊C4C22⋊D17311yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—3462no—
148C42—3461no—
149C42C22×D17311yescharacteristic
150C23—21no—
151C23—3462no—
152Q8—41no—
153Q8—41no—
154Q8—3461no—
155Q8—1732no—
156D4—21no—
157D4—21no—
158D4—13842no—
159D4—6922no—
160D4—6924no—
161D4—6921no—
162D4—6921no—
163D4—6921no—
164D4—6921no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3461no—
170C2×C4—3461no—
171C2×C4—6924no—
172C2×C4D4×D17311yescharacteristic
173C2×C4D4×D17311yescharacteristic
174C2×C4C2×C22⋊D17311yescharacteristic
175C8—41no—
176C8—6921no—
177C8—6921no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—6922no—
182C22—6922no—
183C22—6922no—
184C22#1914964711yescharacteristic
185C4—21no—
186C4—41no—
187C4—81no—
188C4—3462no—
189C4—6921no—
190C4D8×D17311yescharacteristic
191C4#1914969911yescharacteristic
192C4#1914970311yescharacteristic
193C4#1914971011yescharacteristic
194C2—81no—
195C2—13842no—
196C2#2601162311yescharacteristic
197C2#2601165111yescharacteristic
198C2#2601166711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#474×C173C2
socleC22×C173#120
prime radical#26012162C2
hypercenter#474D173
hypocenterC173#1641
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #26011839C2semidirect
3. #26011698C2semidirect
4. #26011234C2semidirect
5. #26011221C2semidirect
6. D4⋊#13063436C2semidirect
7. #474×C173C2semidirect
99. C173#1641semidirect
103. #474D173semidirect

Maximal subgroups

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

subgroupindex
#260121622
#260118392
#260116982
#260112342
#260112212
D4⋊#130634362
#474×C1732
#1641173