Finite Group Databasebeta

← #36352497

Subgroups — #36352497

Order 22144.

310 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 53 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#26012072C211yesmaximal, characteristic, complemented
2#26011625C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011217C211yesmaximal, characteristic, complemented
5#394×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7C2×#14583811—22no—
8#19149659—22no—
9#19149654—22no—
10#19149653—21no—
11#19149653C2212yes—
12#19149600—22no—
13C22⋊#13063436C2212yes—
14C22×D4×C173C2211yescharacteristic
15C2×C4⋊C4×C173C2211yescharacteristic
16C2×C22⋊C4×C173—21no—
17C2×C22⋊C4×C173—21no—
18#95×C173C2211yescharacteristic
19#14583826—42no—
20#14583826—42no—
21C2×Dic346—21no—
22C2×D692—21no—
23C2×D692—21no—
24C2×D692D412yes—
25C2×C22⋊D173—22no—
26C2×C4×D173—22no—
27#14583811—44no—
28C2×#13063436—22no—
29C4⋊Dic173—21no—
30C4⋊Dic173D412yes—
31#14583803—22no—
32#14583803—22no—
33C24×C173—21no—
34C2×D4×C173—22no—
35C2×D4×C173—42no—
36C22×C4×C173—21no—
37C22×C4×C173D411yescharacteristic
38C22×C4×C173D411yescharacteristic
39C22×C4×C173C2311yescharacteristic
40C4⋊C4×C173—22no—
41C22⋊C4×C173—42no—
42C22⋊C4×C173—42no—
43C22×D173—22no—
44Dic346—42no—
45D692—42no—
46D692—42no—
47D692—81no—
48C22⋊D173—42no—
49C22⋊D173—42no—
50C2×Dic173—22no—
51C4×D173—44no—
52#13063436—42no—
53C23×C173—22no—
54C23×C173—41no—
55C23×C173—41no—
56C23×C173—41no—
57C23×C173—41no—
58C23×C173—41no—
59C23×C173—41no—
60C23×C173C2×D411yescharacteristic
61D4×C173—24no—
62D4×C173—42no—
63C2×C4×C173—21no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—41no—
67C2×C4×C173—41no—
68C2×C4×C173—41no—
69C2×C4×C173—41no—
70C2×C4×C173—41no—
71C2×C4×C173D812yescomplemented
72C2×C4×C173C2×D412yes—
73C22×C173—21no—
74C22×C173—21no—
75C22×C173—42no—
76C22×C173—42no—
77C22×C173—42no—
78C22×C173—81no—
79C22×C173—81no—
80C22×C173—81no—
81C22×C173—81no—
82C22×C173#12011yescharacteristic
83C22×C173C2×D811yescharacteristic
84C22×C173AGL(1,ℤ/8)11yescharacteristic
85C2×D173—42no—
86C2×D173—42no—
87C2×D173—42no—
88C4×C173—22no—
89C4×C173—22no—
90C4×C173—41no—
91C4×C173—81no—
92Dic173—42no—
93C2×C173—21no—
94C2×C173—21no—
95C2×C173—82no—
96C2×C173#44711yescharacteristic
97C2×C173#45411yescharacteristic
98C2×C173B(4,𝔽2)11yescharacteristic
99D173—82no—
100C173#159811yescharacteristic, complemented, Sylow
101#1598—1731nomaximal, Sylow
102#537—1731no—
103#447—1732no—
104#394D17311yescharacteristic, complemented
105#327—1732no—
106#324—1731no—
107C2×#41—3461no—
108C22×D4C2×D17311yescharacteristic
109C2×D8—3462no—
110C22⋊Q8—3462no—
111#121—3462no—
112#121—3461no—
113#121—1732no—
114C2×C4⋊C4C2×D17311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—3462no—
118C22⋊C8—1732no—
119#95C2×D17311yescharacteristic
120C24—21no—
121#41—6922no—
122#41—6922no—
123C2×Q8—3461no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—3461no—
127C2×D4—3461no—
128C2×D4—3462no—
129C2×D4—1732no—
130C22×C4—21no—
131C22×C4—3462no—
132C22×C4C22⋊D17311yescharacteristic
133C22×C4C22⋊D17311yescharacteristic
134C22×C4C22×D17311yescharacteristic
135D8—6924no—
136C2×C8—3462no—
137C4⋊C4—22no—
138C4⋊C4—3461no—
139C4⋊C4—1732no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—3462no—
143C22⋊C4—3462no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—3462no—
152C23C2×C22⋊D17311yescharacteristic
153Q8—6922no—
154D4—24no—
155D4—42no—
156D4—13841no—
157D4—6922no—
158D4—6922no—
159D4—6922no—
160D4—6922no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3462no—
170C2×C4—6924no—
171C2×C4#1458381112yescomplemented
172C2×C4D4×D17312yes—
173C8—6922no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—6922no—
184C22—6922no—
185C22—6922no—
186C22#1914964711yescharacteristic
187C22C2×#1458381111yescharacteristic
188C22#1914970411yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—6922no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—13842no—
198C2#2601162411yescharacteristic
199C2#2601166311yescharacteristic
200C2#2601167711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#394×C173C2
socleC22×C173#120
hypercenter#394D173
hypocenterC173#1598
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #26012072C2semidirect
2. #26011625C2semidirect
3. #26011224C2semidirect
4. #26011217C2semidirect
5. #394×C173C2semidirect
71. C2×C4×C173D8semidirect
100. C173#1598semidirect
104. #394D173semidirect
171. C2×C4#14583811semidirect

Maximal subgroups

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

subgroupindex
#260120722
#260116252
#260112242
#260112172
#394×C1732
#1598173