Finite Group Databasebeta

← #36355516

Subgroups — #36355516

Order 22144.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#26012291C211yesmaximal, characteristic
2#26011934C212yesmaximal, complemented
3#26011749C211yesmaximal, characteristic, complemented
4#26011302C212yesmaximal, complemented
5#349×C173C211yesmaximal, characteristic, complemented
6C2×#14583826—21no—
7#19149712—22no—
8#19149707—22no—
9#19149701—21no—
10#19149701C2212yes—
11C2×Dic692C2211yescharacteristic, complemented
12#19149658—21no—
13Dic1384—22no—
14#19149624—22no—
15#19149611C2211yescharacteristic, complemented
16#19149604—21no—
17C2×M16×C173C2211yescharacteristic
18M32×C173C2212yescomplemented
19#14583826—22no—
20#14583826—22no—
21#14583826—41no—
22#14583826—41no—
23C2×Dic346—21no—
24C2×Dic346—21no—
25C2×D692—21no—
26C2×C22⋊D173—41no—
27C2×C4×D173—41no—
28Dic692—21no—
29Dic692—22no—
30Dic692D412yes—
31#14583815—22no—
32#14583815—22no—
33D1384—21no—
34D1384D412yes—
35C4⋊Dic173—21no—
36#14583805—41no—
37C22⋊Dic173—41no—
38C22×C4×C173D411yescharacteristic
39M16×C173—22no—
40C2×C8×C173D411yescharacteristic
41C2×C8×C173C2311yescharacteristic
42C16×C173—22no—
43C22×D173—41no—
44Dic346—22no—
45Dic346—41no—
46Dic346—41no—
47Dic346—41no—
48D692—22no—
49D692—41no—
50C22⋊D173—42no—
51C22⋊D173—42no—
52C2×Dic173—41no—
53C2×Dic173—41no—
54C2×Dic173—41no—
55C4×D173—42no—
56C4×D173—42no—
57C23×C173D811yescharacteristic
58C2×C4×C173—21no—
59C2×C4×C173—21no—
60C2×C4×C173D811yescharacteristic
61C2×C4×C173C2×D411yescharacteristic
62C8×C173—21no—
63C8×C173C2×D412yes—
64C22×C173—21no—
65C22×C173—41no—
66C22×C173C2×D811yescharacteristic
67C2×D173—42no—
68C2×D173—82no—
69C4×C173—21no—
70C4×C173#12011yescharacteristic
71C4×C173AGL(1,ℤ/8)11yescharacteristic
72Dic173—42no—
73Dic173—81no—
74Dic173—81no—
75C2×C173—21no—
76C2×C173—41no—
77C2×C173#44711yescharacteristic
78D173—82no—
79C173#219111yescharacteristic, complemented, Sylow
80#2191—1731nomaximal, Sylow
81#575—1731no—
82#510—1732no—
83#470—1731no—
84#360—1732no—
85#349D17311yescharacteristic, complemented
86C2×#41—3461no—
87#137—3462no—
88AGL(1,ℤ/8)—3462no—
89#135—3461no—
90#135—1732no—
91C2×Q16—1731no—
92C2×M16C2×D17311yescharacteristic
93C22⋊Q8—3461no—
94Q32—3462no—
95SD32—3462no—
96M32C2×D17312yescomplemented
97#106—1731no—
98#103—3461no—
99#41—3462no—
100#41—3462no—
101#41—6921no—
102#41—6921no—
103C2×Q8—3461no—
104C2×Q8—3461no—
105C2×D4—3461no—
106C2×D4—6921no—
107C22×C4—6921no—
108C22×C4D69211yescharacteristic
109Q16—3461no—
110Q16—3462no—
111Q16—1732no—
112SD16—3462no—
113SD16—3462no—
114D8—3461no—
115D8—1732no—
116M16—22no—
117C2×C8D69211yescharacteristic
118C2×C8C22×D17311yescharacteristic
119C4⋊C4—3461no—
120C4⋊C4—6921no—
121C22⋊C4—6921no—
122C16—22no—
123C23—6921no—
124C23D138411yescharacteristic
125Q8—3462no—
126Q8—6921no—
127Q8—6921no—
128Q8—6921no—
129D4—3462no—
130D4—6921no—
131D4—6922no—
132D4—6922no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—6922no—
136C2×C4—6921no—
137C2×C4—6921no—
138C2×C4—6921no—
139C2×C4—6922no—
140C2×C4D138411yescharacteristic
141C2×C4C2×D69211yescharacteristic
142C8—21no—
143C8D4×D17312yes—
144C22—21no—
145C22—41no—
146C22—13842no—
147C22—6922no—
148C22C2×D138411yescharacteristic
149C4—21no—
150C4—13841no—
151C4—13841no—
152C4—6922no—
153C4#1914964911yescharacteristic
154C4#1914970711yescharacteristic
155C2—21no—
156C2—41no—
157C2—13842no—
158C2#2601162711yescharacteristic
subgroupG/H
centerC2#26011627
commutator, cosocleC2×C8×C173C23
FrattiniC2×C8C22×D173
Fitting#349×C173C2
socleC2×C173#447
prime radical#26012291C2
hypercenter#349D173
hypocenterC173#2191
LCSG ▷ C2×C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #26011934C2semidirect
3. #26011749C2semidirect
4. #26011302C2semidirect
5. #349×C173C2semidirect
11. C2×Dic692C22semidirect
15. #19149611C22semidirect
18. M32×C173C22semidirect
79. C173#2191semidirect
85. #349D173semidirect
96. M32C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122912
#260119342
#260117492
#260113022
#349×C1732
#2191173