Finite Group Databasebeta

← #36369518

Subgroups — #36369518

Order 22144.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012558C211yesmaximal, characteristic
2#26012502C212yesmaximal, complemented
3#26012443C211yesmaximal, characteristic
4#26012363C212yesmaximal, complemented
5#100×D173C211yesmaximal, characteristic
6D4×#10621924—22no—
7D4×#10621924—22no—
8#19149758C2212yes—
9C2×C4⋊#10621924C2211yescharacteristic
10C2×C22⋊#10621924—22no—
11C2×C22⋊#10621924—22no—
12C2×C4×#10621924C2211yescharacteristic
13#19149748—22no—
14#19149747—22no—
15C2×D4×D173C2211yescharacteristic
16M16×D173C2212yes—
17#19149596C411yescharacteristic, complemented
18#19149595C412yescomplemented
19#100×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21C22×#10621924—22no—
22C22×#10621924—21no—
23C22×#10621924—21no—
24C4⋊#10621924—22no—
25C4⋊#10621924D412yes—
26C4⋊#10621924D412yes—
27C22⋊#10621924—24no—
28C22⋊#10621924—22no—
29C22⋊#10621924—44no—
30C4×#10621924—22no—
31C4×#10621924—22no—
32C23×D173D412yes—
33C2×D692C2×C411yescharacteristic
34C2×C22⋊D173Q812yes—
35D4×D173—22no—
36D4×D173—22no—
37C2×C4×D173C2311yescharacteristic
38#14583810C2×C412yescomplemented
39#14583809—22no—
40C8×D173—22no—
41C2×D4×C173C2×C411yescharacteristic
42M16×C173C2×C412yescomplemented
43C2×#10621924—22no—
44C2×#10621924—22no—
45C2×#10621924—22no—
46C2×#10621924—41no—
47C2×#10621924—41no—
48C2×#10621924—42no—
49C2×#10621924—41no—
50C2×#10621924—82no—
51C2×#10621924—82no—
52C22×D173—22no—
53C22×D173—22no—
54C22×D173—42no—
55C22×D173—42no—
56C22×D173C4⋊C412yes—
57C22×D173C2×D411yescharacteristic
58D692—22no—
59C22⋊D173—24no—
60C2×Dic173C2×Q811yescharacteristic
61C4×D173C2×D412yes—
62C4×D173#4112yes—
63#13063436—22no—
64C23×C173C4⋊C412yes—
65D4×C173—22no—
66C2×C4×C173C22×C411yescharacteristic
67C8×C173—22no—
68C22×C173—22no—
69C22×C173—42no—
70C22×C173C2×C4⋊C411yescharacteristic
71C2×D173—21no—
72C2×D173—21no—
73C2×D173—22no—
74C2×D173—42no—
75C2×D173—42no—
76C2×D173—42no—
77C2×D173—82no—
78C2×D173#12011yescharacteristic
79C2×D173#12311yescharacteristic
80#10621924—42no—
81#10621924—42no—
82#10621924—42no—
83#10621924—81no—
84C4×C173C4×D412yes—
85Dic173C22⋊Q812yes—
86C2×C173—21no—
87C2×C173—42no—
88C2×C173#38511yescharacteristic
89D173—21no—
90D173—42no—
91D173#45311yescharacteristic
92D173#45511yescharacteristic
93C173#185611yescharacteristic, complemented, Sylow
94#1856—1731nomaximal, Sylow
95#518—1731no—
96#428—1732no—
97C2×#100—1731no—
98#390—1731no—
99#339—1732no—
100C22×D4—1731no—
101C2×M16—1732no—
102C4×D4—3462no—
103C4×D4—3462no—
104#117—1732no—
105C2×C4⋊C4—1731no—
106C2×C22⋊C4—3462no—
107C2×C22⋊C4—3462no—
108C2×C42—1731no—
109#107—3462no—
110#106—3462no—
111#100—1731no—
112#100—1732no—
113#100#1062192411yescharacteristic, complemented
114C24—1732no—
115C2×D4—3462no—
116C2×D4—3462no—
117C2×D4—1732no—
118C2×D4—1731no—
119C2×D4C2×#1062192411yescharacteristic
120C22×C4—3462no—
121C22×C4—3462no—
122C22×C4—3461no—
123C22×C4—3461no—
124C22×C4—1731no—
125M16—3462no—
126M16—1732no—
127M16C2×#1062192412yescomplemented
128C2×C8—3462no—
129C4⋊C4—3462no—
130C4⋊C4—1732no—
131C4⋊C4—1732no—
132C22⋊C4—3464no—
133C22⋊C4—3462no—
134C22⋊C4—6924no—
135C42—3462no—
136C42—3462no—
137C23—3462no—
138C23—3462no—
139C23—1731no—
140C23—1732no—
141C23—6922no—
142C23—6922no—
143C23C4⋊#1062192412yes—
144D4—22no—
145D4—3464no—
146D4—3462no—
147C2×C4—3462no—
148C2×C4—3462no—
149C2×C4—3462no—
150C2×C4—13842no—
151C2×C4—13842no—
152C2×C4—1731no—
153C2×C4—1732no—
154C2×C4—1732no—
155C2×C4—6922no—
156C2×C4—6921no—
157C2×C4—6921no—
158C2×C4—6921no—
159C2×C4C22×#1062192411yescharacteristic
160C8—22no—
161C8—3462no—
162C22—22no—
163C22—42no—
164C22—3461no—
165C22—3461no—
166C22—3462no—
167C22—13842no—
168C22—1731no—
169C22—1731no—
170C22—6922no—
171C22—6922no—
172C22—6922no—
173C22C2×C4⋊#1062192411yescharacteristic
174C4—13841no—
175C4—1732no—
176C4—6922no—
177C4—6922no—
178C4—6922no—
179C4D4×#1062192412yes—
180C2—21no—
181C2—42no—
182C2—3461no—
183C2—1731no—
184C2—1731no—
185C2—6922no—
186C2#2601243811yescharacteristic
subgroupG/H
centerC2#26012438
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#100×C173C4
socleC2×C173#385
prime radicalC2×D4×D173C22
cosocleC2×C4×D173C23
hypercenter#100#10621924
hypocenterC173#1856
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #26012502C2semidirect
4. #26012363C2semidirect
17. #19149596C4semidirect
18. #19149595C4semidirect
19. #100×C173C4semidirect
38. #14583810C2×C4semidirect
42. M16×C173C2×C4semidirect
93. C173#1856semidirect
113. #100#10621924semidirect
127. M16C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
#260125582
#260125022
#260124432
#260123632
#100×D1732
#1856173