Finite Group Databasebeta

← #36352548

Subgroups — #36352548

Order 22144.

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

normal subgroups only

This group has 280 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#26012053C211yesmaximal, characteristic
2#26011804C212yesmaximal, complemented
3#26011224C212yesmaximal, complemented
4#26011215C211yesmaximal, characteristic, complemented
5#397×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173—21no—
7#19149663—22no—
8#19149653—21no—
9#19149653C2212yes—
10#19149649—22no—
11#19149612—22no—
12#19149599—22no—
13C22⋊#13063436C2212yescomplemented
14C2×C4⋊C4×C173C2211yescharacteristic, complemented
15C2×C22⋊C4×C173—21no—
16C2×C22⋊C4×C173C2211yescharacteristic
17#95×C173—21no—
18#95×C173C2211yescharacteristic, complemented
19C23×D173—21no—
20C2×D692—21no—
21C2×D692D412yes—
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—42no—
25C2×C22⋊D173—42no—
26C22×Dic173—21no—
27C2×#13063436—22no—
28C4⋊Dic173—21no—
29C4⋊Dic173D412yes—
30#14583803—22no—
31#14583803—22no—
32C24×C173D411yescharacteristic
33C22×C4×C173—21no—
34C22×C4×C173—21no—
35C22×C4×C173D411yescharacteristic
36C22×C4×C173C2311yescharacteristic
37C4⋊C4×C173—22no—
38C22⋊C4×C173—22no—
39C22⋊C4×C173—42no—
40C22×D173—22no—
41C22×D173—42no—
42C22×D173—81no—
43C22×D173—81no—
44D692—42no—
45D692—81no—
46C22⋊D173—44no—
47C22⋊D173—42no—
48C22⋊D173—42no—
49C2×Dic173—22no—
50C2×Dic173—42no—
51#13063436—42no—
52C23×C173—21no—
53C23×C173—21no—
54C23×C173—41no—
55C23×C173—41no—
56C23×C173D812yes—
57C23×C173C2×D411yescharacteristic
58C2×C4×C173—21no—
59C2×C4×C173—41no—
60C2×C4×C173—41no—
61C2×C4×C173—41no—
62C2×C4×C173—41no—
63C2×C4×C173—41no—
64C2×C4×C173—41no—
65C2×C4×C173—41no—
66C2×C4×C173—41no—
67C2×C4×C173C2×D412yes—
68C22×C173—21no—
69C22×C173—21no—
70C22×C173—22no—
71C22×C173—42no—
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—81no—
75C22×C173#12011yescharacteristic
76C22×C173C2×D811yescharacteristic
77C22×C173AGL(1,ℤ/8)11yescharacteristic
78C2×D173—42no—
79C2×D173—42no—
80C2×D173—42no—
81C2×D173—84no—
82C4×C173—22no—
83C4×C173—41no—
84C4×C173—81no—
85C4×C173—81no—
86Dic173—42no—
87C2×C173—21no—
88C2×C173—21no—
89C2×C173—42no—
90C2×C173#44711yescharacteristic
91C2×C173#45511yescharacteristic
92C2×C173#45811yescharacteristic
93D173—82no—
94C173#160711yescharacteristic, complemented, Sylow
95#1607—1731nomaximal, Sylow
96#534—1731no—
97#480—1732no—
98#397D17311yescharacteristic, complemented
99#327—1732no—
100#323—1731no—
101C22×D4—3461no—
102#123—3462no—
103#121—3461no—
104#121—1732no—
105#120—3462no—
106C2×C4⋊C4C2×D17311yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D17311yescharacteristic
109#107—3462no—
110#102—3462no—
111C22⋊C8—1732no—
112#95—21no—
113#95C2×D17311yescharacteristic, complemented
114C24—3461no—
115C24C22⋊D17311yescharacteristic
116C2×D4—3461no—
117C2×D4—3462no—
118C2×D4—3462no—
119C2×D4—1732no—
120C2×D4—6922no—
121C2×D4—6922no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—3461no—
125C22×C4C22⋊D17311yescharacteristic
126C22×C4C22×D17311yescharacteristic
127C2×C8—3462no—
128C4⋊C4—22no—
129C4⋊C4—3461no—
130C4⋊C4—1732no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—3462no—
134C22⋊C4—3462no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—3462no—
140C23—13841no—
141C23—13841no—
142C23—6922no—
143C23#1458381112yes—
144C23C2×C22⋊D17311yescharacteristic
145D4—13841no—
146D4—6922no—
147D4—6924no—
148D4—6922no—
149D4—6922no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3462no—
160C2×C4—6922no—
161C2×C4D4×D17312yes—
162C8—6922no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—13844no—
171C22—6922no—
172C22—6922no—
173C22—6922no—
174C22#1914964711yescharacteristic
175C22C2×#1458381111yescharacteristic
176C22#1914970411yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—6922no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—13842no—
186C2#2601162411yescharacteristic
187C2#2601166811yescharacteristic
188C2#2601168011yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#397×C173C2
socleC22×C173#120
prime radical#26012053C2
hypercenter#397D173
hypocenterC173#1607
LCSG ▷ C22×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #26011804C2semidirect
3. #26011224C2semidirect
4. #26011215C2semidirect
5. #397×C173C2semidirect
13. C22⋊#13063436C22semidirect
14. C2×C4⋊C4×C173C22semidirect
18. #95×C173C22semidirect
94. C173#1607semidirect
98. #397D173semidirect
106. C2×C4⋊C4C2×D173semidirect
113. #95C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120532
#260118042
#260112242
#260112152
#397×C1732
#1607173