Finite Group Databasebeta

← #36368879

Subgroups — #36368879

Order 22144.

234 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 62 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
1D173⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
2C4×#14583836C211yesmaximal, characteristic, complemented
3Q8⋊#13063443C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5#26012020C211yesmaximal, characteristic
6C2×#14583836—21no—
7C2×D173⋊C8—21no—
8#19149746—21no—
9C4⋊#13063443C2212yes—
10#19149734—21no—
11C4×#13063443C2212yes—
12#19149672—21no—
13#19149671C411yescharacteristic
14#19149661—21no—
15#19149655C411yescharacteristic
16#19149650C411yescharacteristic
17Q8×Dic173C2211yescharacteristic
18D4×Dic173C2211yescharacteristic
19#19149639—21no—
20#19149635—21no—
21C42×D173C2211yescharacteristic
22#124×C173C411yescharacteristic
23#14583836—24no—
24D173⋊C8—42no—
25C2×#13063443—22no—
26C2×#13063443—22no—
27C2×C22⋊D173—21no—
28C22×Dic173—21no—
29C2×C4×D173C2×C411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C2×C4×D173D411yescharacteristic
32C4⋊Dic173—21no—
33C4⋊Dic173C2×C411yescharacteristic
34C4⋊Dic173C2×C411yescharacteristic
35#14583805—21no—
36#14583805—21no—
37#14583805—21no—
38C22⋊Dic173—21no—
39C22⋊Dic173—21no—
40#14583803—21no—
41C4×Dic173—21no—
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C4×Dic173C2311yescharacteristic
45C2×Q8×C173C2×C411yescharacteristic
46C2×D4×C173C2×C411yescharacteristic
47C22⋊C4×C173—21no—
48C22⋊C4×C173—21no—
49C42×C173C2×C411yescharacteristic
50#13063443—24no—
51#13063443—42no—
52C22×D173C22⋊C411yescharacteristic
53C22⋊D173—42no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—41no—
57C2×Dic173—41no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C4×D173—22no—
62C4×D173—22no—
63C4×D173—21no—
64C4×D173—21no—
65C23×C173—21no—
66Q8×C173—22no—
67D4×C173—22no—
68C2×C4×C173—21no—
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22⋊C411yescharacteristic
71C2×C4×C173C22×C411yescharacteristic
72C22×C173—41no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173C2×C22⋊C411yescharacteristic
76C2×D173—21no—
77C2×D173—21no—
78C2×D173#10412yes—
79C4×C173—21no—
80C4×C173—21no—
81C4×C173—41no—
82C4×C173#13112yes—
83Dic173—21no—
84Dic173—21no—
85Dic173—41no—
86Dic173—41no—
87Dic173AGL(1,ℤ/8)11yescharacteristic
88Dic173#13711yescharacteristic
89C2×C173—81no—
90C2×C173#40811yescharacteristic
91C2×C173#41711yescharacteristic
92C2×C173C2×#10411yescharacteristic
93D173—22no—
94C173#148311yescharacteristic, complemented, Sylow
95#1483—1731nomaximal, Sylow
96#529—1731no—
97C2×C4⋊C8—1731no—
98C4×M16—1731no—
99Q8⋊C8—1732no—
100D4⋊C8—1732no—
101C2×M16—3461no—
102C22×C8—3461no—
103#126—3461no—
104#125—1731no—
105#124#1062192411yescharacteristic
106#123—3461no—
107C22⋊Q8—1731no—
108#121—1731no—
109C4×Q8—1731no—
110C4×D4—3461no—
111C4×D4—1731no—
112#117—3461no—
113C2×C42—1731no—
114C4⋊C8—3461no—
115C4⋊C8—1732no—
116#97—3461no—
117C4×C8—1732no—
118C2×Q8C2×#1062192411yescharacteristic
119C2×D4—3461no—
120C2×D4C2×#1062192411yescharacteristic
121C22×C4—3461no—
122C22×C4—1731no—
123C22×C4—1731no—
124C22×C4—1731no—
125M16—3464no—
126C2×C8—3462no—
127C2×C8—3462no—
128C2×C8—6922no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4—1731no—
134C4⋊C4—1731no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—3461no—
138C22⋊C4—3461no—
139C22⋊C4—3461no—
140C42—3461no—
141C42—1731no—
142C42—1731no—
143C42—1731no—
144C42C2×#1062192411yescharacteristic
145C23—21no—
146C23—1731no—
147Q8—22no—
148D4—22no—
149D4—6922no—
150C2×C4—21no—
151C2×C4—3462no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—3462no—
155C2×C4—3461no—
156C2×C4—3461no—
157C2×C4—1731no—
158C2×C4—1731no—
159C2×C4—1731no—
160C2×C4—6921no—
161C2×C4—6921no—
162C2×C4C22⋊#1062192411yescharacteristic
163C2×C4C22⋊#1062192411yescharacteristic
164C2×C4C22×#1062192411yescharacteristic
165C8—3464no—
166C8—6922no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—3461no—
171C22—3461no—
172C22—1732no—
173C22C2×C22⋊#1062192411yescharacteristic
174C4—21no—
175C4—21no—
176C4—41no—
177C4—3461no—
178C4—3461no—
179C4—1731no—
180C4—1731no—
181C4—6921no—
182C4—6921no—
183C4#1914976612yes—
184C2—81no—
185C2—3462no—
186C2#2601246511yescharacteristic
187C2#2601248211yescharacteristic
188C2D173⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#124×C173C4
socleC22×C173C2×C22⋊C4
prime radical#19149650C4
cosocleC4×Dic173C23
hypercenter#124#10621924
hypocenterC173#1483
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D173⋊(C4⋊C8)C2semidirect
2. C4×#14583836C2semidirect
3. Q8⋊#13063443C2semidirect
4. D4⋊#13063443C2semidirect
94. C173#1483semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C4⋊C8)2
C4×#145838362
Q8⋊#130634432
D4⋊#130634432
#260120202
#1483173